MACHINE DESIGN BY CHARLES H. BENJAMIN, DEAN OP THE SCHOOLS OF ENGINEERING, PUBDUE UNIVERSITY AND JAMES D. HOFFMAN PKOFESSOB OF MECHANICAL ENGINEEBING AND PBACTICAL MECHANICS, UNIVEBSITY OF NEBBASKA NEW YORK HENRY HOLT AND COMPANY 1913 -^ COPYBIGHT, 1909 BY JAMES D. HOFFMAN COPYRIGHT, 1906, 1913 BY HENRY HOLT AND COMPANY THE. MAPLE. PRESS. YORK. PA PREFACE The present book represents the consolidation of two texts on this subject, Benjamin's Machine Design and Hoffman's Elementary Machine Design. As now arranged, the book serves two purposes: That of a text for the classroom, embodying the theory and practice of design, and that of a reference book for the drafting room, illustrating the design of complete machines. The authors recognize the fact that there are two methods of teaching this subject, one by details separately treated as ele- ments, one by a consideration of the complete machine, i.e., one method is synthetic and one analytic. It is believed that this book will afford a means of using either method or both combined. Some important additions to the text are worthy of mention. Chapter II, on Materials, has been rewritten. Much additional matter on the subjec^ of cast-iron frames has been introduced, involving the results of numerous experiments. The theoretical and experimental strength of steel tubes under collapsing pres- sures is quite fully discussed and additional data are given on the failure of pipe fittings. Other subjects which receive in this volume fuller treatment than heretofore are Flat plates, Crane hooks, Leaf springs, Bear- ings, both plain and rolling, Clutches, Gear teeth and Belting. in TABLE OF CONTENTS CHAPTER PAGE INTRODUCTION. UNITS AND FORMULAS 1 1. Units. 2. Abbreviations. 3. Notation. 4. Formulas. 5. Profiles of uniform strength. I. MATERIALS 9 6. Primary classification. 7. Iron. 8. Steel. 9. Steel alloys. 10. Copper alloys. 11. Strength and Elasticity. II. FRAME DESIGN 21 12. General principles of design. 13. Machine supports. 14. Machine frames. 15. Tests on simple beams. 16. Shapes of frames. 17. Stresses in frames. 18. Professor Jenkin's ex- periments. 19. Purdue tests. 20. Principles of design. III. CYLINDERS AND PIPES 50 21. Thin shells. 22. Thick shells. 23. Steel and wrought iron pipe. 24. Strength of boiler tubes. 25. Theory. 26. Tube joints. 27. Tubes under concentrated loads. 28. Pipe fit- tings. 29. Flanged pittings. 30. Steam cylinders. 31. Thick- ness of flat plates. 32. Steel plates. IV. FASTENINGS 91 33. Bolts and nuts. 34. Crane hooks. 35. Riveted joints. 36. Lap joints. 37. Butt joint with two straps. 38. Effi- ciency of joints. 39. Butt joints with unequal straps. 40. Practical rules. 41. Riveted joints for narrow plates. 42. Joint pins. 43. Cotters. V. SPRINGS 107 44. Helical springs. 45. Square wire. 46. Experiments. 47. Springs in torsion. 48. Flat springs. 49. Elliptic and semi- elliptic springs. VI. SLIDING BEARINGS 120 50. Slides in general. 51. Angular slides. 52. Gibbed slides. 53. Flat slides. 54. Circular guides. 55. Stuffing boxes. VII. JOURNALS, PIVOTS AND BEARINGS 128 56. Journals. 57. Adjustment. 58. Lubrication. 59. Fric- tion of journals. 60. Limits of pressure. 61. Heating of jour- nals. 62. Experiments. 63. Strength and stiffness of jour- nals. 64. Caps and bolts. 65. Step bearings. 66. Friction of pivots. 67. Flat collar. 68. Conical pivot. 69. Schiele's pivot. 70. Multiple bearing. vi TABLE OF CONTENTS CHAPTER PAGE VIII. BALL AND ROLLER BEARINGS 153 71. General principles. 72. Journal bearings. 73. Step bear- ings. 74. Materials and wear. 75. Design of bearings. 76. Endurance of ball bearings. 77. Roller bearings. 78. Grant roller bearings. 79. Hyatt rollers. 80. Roller step bearings. 81. Design of roller bearings. IX. SHAFTING, COUPLINGS AND HANGERS 167 82. Strength of shafting. 83. Combined tension and bending. 84. Couplings. 85. Clutches. 86. Automobile clutches. 87. Coupling bolts. 88. Shafting keys. 89. Strength of keyed shafts. 90. Hangers and boxes. X. GEARS, PULLEYS AND CRANKS 186 91. Gear teeth. 92. Strength of gear teeth. 93. Lewis' for- mula. 94. Experimental data. 95. Modern practice. 96. Teeth of bevel gears. 97. Rim and arms. 98. Sprocket wheels and chains. 99. Silent chains. 100. Cranks and levers. XL FLY-WHEELS 204 101. In general. 102. Safe speed for wheels. 103. Experi- ments on fly-wheels. 104. Wooden pulleys. 105. Rims of cast- iron gears. 106. Rotating discs. 107. Plain discs. 108. Con- ical discs. 109. Discs with logarithmic profile. 110. Bursting speeds. 111. Tests of discs. XII. TRANSMISSION BY BELTS AND ROPES 221 112. Friction of belting. 113. Slip of belt. 114. Coefficient of friction. 115. Strength of belting. 116. Taylor's experi- ments. 117. Rules for width of belts. 118. Speed of belting. 119. Manila rope transmission. 120. Strength of Manila ropes. 121. Cotton rope transmission. 122. Wire rope transmission. XIII. DESIGN OF TOGGLE-JOINT PRESS 235 123. Introductory. 124. Drawings. 125. Calculations. 126. Analysis of forces. 127. Design of lever. 128. Shapes of levers. 129. Hole in lever. 130. Fastening for standard. 131. Design of standard. 132. Toggle joint. 133. Shapes of toggle members. 134. Die heads. 135. Frame or bed. 136. Stability of frame. 137. Toggle press. 138. Vertical hand- power press. 139. Foot-power press. 140. Hand-power punch. 141. Punch and shear. XIV. DESIGN OF BELT-DRIVEN PUNCH OR SHEAR 265 142. General statement. 143. Requirements of design. 144. Design of frame. 145. Outline of frame. 146. Shearing force. 147. Depth of cut. 148. Sizes of pulleys. 149. Weight of fly-wheel. 150. Driving shaft. 151. Gears. 152. Main shaft. 153. Sliding head. 154. Clutches and transmission devices. TABLE OF CONTENTS vii CHAPTER PAGE 155. Punch, die and holders. 156. The. bevel shear. 157. Horizontal power punch. 158. The bull-dozer. 159. Power press. 160. Rotary shear. 161. Sheet metal flanger. 162. Flanging machine. XV. DESIGN OF Am HOIST AND RIVETER 299 163. Air hoist. 164, 165. Portable riveter. 166. Alligator riveter. 167. Mud-ring riveter. 168. Lever riveter. 169. Hydraulic riveter. 170. Triple pressure hydraulic riveter. XVI. STUDIES IN THE KINEMATICS OF MACHINES 309 171. Gear tooth outlines. 172. Planer cam. 173. Sewing machine cam. 174. Sewing machine bobbin winder. 175. Constant diameter cam. 176, 177, 178, 179. Quick return motions. 180. Cam and oscillating arm. 181. Two-motion cam. 182. Conical cam. 183, 184. Motion problems. 185. Forming machine for wire clips. 186-202. Motion problems. 203. Mechanism of inertia governor. 204. Mechanism of centrifugal governors. 205. Straight line governor. 206. Stephenson link motion. 207. Walschaert valve gear. INDEX . 336 MACHINE DESIGN TABLES TABLE PAGE I. VALUES OP Q IN COLUMN FORMULA 4 I a. VALUES OF S AND K IN COLUMN FORMULA ...... 5 II. CONSTANTS OF CROSS-SECTION 6 III. FORMULAS FOR LOADED BEAMS 7 IV. CLASSIFICATION OF METALS 9 V. COMPOSITION OF BRONZES 16 VI. STRENGTH OF WROUGHT METALS . . 18 VII. STRENGTH OF CAST METALS 19 VIII. STRENGTH OF CAST IRON BEAMS 30 IX. STRENGTH OF CAST IRON BEAMS 32 X. STRENGTH OF CAST IRON BEAMS 33 XI. STRENGTH OF CAST IRON BEAMS 35 XII. STRESSES IN MACHINE FRAMES 38 XIII. STRENGTH OF RIVETER FRAMES 41 XIV. STRENGTH OF RIVETER FRAMES 44 XV. SIZES OF IRON AND STEEL PIPE 56 XVI. SIZES OF EXTRA STRONG PIPE 58 XVII. SIZES OF DOUBLE EXTRA STRONG PIPE ........ 59 XVIII. SIZES OF IRON AND STEEL BOILER TUBES 60 XIX. COLLAPSING PRESSURE OF TUBES 64 XX. STIFFNESS OF STEEL HOOPS 70 XXI. STRENGTH OF STANDARD SCREWED PIPE FITTINGS ... 72 XXII. STRENGTH OF FLANGED FITTINGS 74 XXIII. BURSTING STRENGTH OF CAST IRON CYLINDERS .... 80 XXIV. STRENGTH OF REINFORCED CYLINDERS 82 XXV. STRENGTH OF CAST IRON PLATES 86 XXVI. STRENGTH OF CAST IRON PLATES 87 XXVII. STRESSES IN FLAT PLATES 89 XXVIII. STRENGTH OF IRON OR STEEL BOLTS 91 XXIX. DIMENSIONS OF MACHINE SCREWS 94 XXX. ELASTIC LIMIT OF CRANE HOOKS 94 XXXI. DIMENSIONS OF RIVETED LAP JOINTS 101 XXXII. DIMENSIONS OF RIVETED BUTT JOINTS 101 XXXIII. STRENGTH AND STIFFNESS OF HELICAL SPRINGS . . . .110 XXXIV. FRICTION OF PISTON ROD PACKINGS 126 XXXV. FRICTION OF PISTON ROD PACKINGS 126 XXXVI. FRICTION OF PISTON ROD PACKINGS 127 XXXVII. FRICTION OF JOURNAL BEARINGS . . 139 XXXVIII. TESTS OF LARGE JOURNALS 140 ix PLATES TABLE PAGE XXXIX. MARINE THRUST BEARINGS 151 XL. FRICTION OF ROLLER AND PLAIN BEARINGS 161 XLI. COEFFICIENTS OF FRICTION 161 XLII. COEFFICIENTS OF FRICTION 162 XLIII. SAFE LOADS FOR ROLLER BEARINGS 164 XLIV. SAFE LOADS FOR ROLLER STEP BEARINGS 165 XLV. VALUES OF K FOR ROLLER THRUST BEARINGS . . . .166 XL VI. DIAMETERS OF SHAFTING 168 XL VII. POWER OF CLUTCHES 177 XLVIII. EFFICIENCY OF KEYED SHAFTS 181 XLIX. PROPORTIONS OF GEAR TEETH 187 L. SIZES OF TEST FLY-WHEELS 208 LI. SIZES OF TEST FLY-WHEELS 209 LII. FLANGES AND BOLTS OF TEST FLY-WHEELS 209 LIII. FAILURE OF FLANGED JOINTS 210 LIV. SIZES OF LINKED JOINTS 210 LV. FAILURE OF LINKED JOINTS 210 LVI. BURSTING SPEEDS OF ROTATING Discs 218 LVII. BURSTING SPEEDS OF ROTATING Discs 219 LVIII. HORSE-POWER OF MANILA ROPE 231 LIX. HORSE-POWER OF COTTON ROPE 232 LX. HCRSE-POWER OF WIRE ROPE . 233 PLATES PLATES PAGE C 1. TOGGLE JOINT PRESS ASSEMBLY Facing 236 C 2. TOGGLE JOINT PRESS DETAILS . Facing 238 C 3. TOGGLE JOINT PRESS DETAILS Facing 240 C 4. SINGLE POWER PUNCH ASSEMBLY 287 C 5. SINGLE POWER PUNCH DETAILS 288 C 6. SINGLE POWER PUNCH DETAILS 289 C 7. SINGLE POWER PUNCH DETAILS Facing 290 C 8. SINGLE POWER PUNCH DETAILS . . 290 MACHINE DESIGN INTRODUCTION UNITS AND FORMULAS 1. Units. In this book the following units will be used unless otherwise stated. Dimensions in inches. Forces in pounds. Stresses in pounds per square inch. Velocities in feet per second. Work and energy in foot pounds. Moments in pounds inches. Speeds of lotation in revolutions per minute. The word stress will be used to denote the resistance of material to distortion per unit of sectional area. The word deformation will be used to denote the distortion of a piece per unit of length. The word set will be used to denote total permanent distortion. In making calculations the use of the slide-rule and of four- place logarithms is recommended; accuracy is expected only to three significant figures. 2. Abbreviations. The following abbreviations are among those recommended by a committee of the American Society of Mechanical Engineers in December, 1904, and will be used throughout the book. 1 NAME ABBREVIATION Inches in. Feet ft. Yards yd. 1 Tr. A. S. M. E., Vol. XXVI, p. 60. 1 2' MACHINE DESIGN NAME ABBREVIATION Miles spell out. Pounds lb. Tons spell out. Gallons gal. Seconds sec. Minutes min. Hours hr. Linear . lin. Square sq. Cubic cu. Per spell out. Fahrenheit fahr. Percentage % or per cent. Revolutions per minute r.p.m. Brake horse power b.h.p. Electric horse power e.h.p. Indicated horse power i.h.p. British thermal units B.t.u. Diameter Diam. 3. Notation. Arc of contact = 6 radians. Area of section =A sq. in. Breadth of section =b in. Coefficient of friction =/ Deflection of beam A in. Degrees deg. Depth of section =h in. Diameter of circular section =d in. Distance of neutral axis from outer fiber =y in. Elasticity, modulus of, in tension and compression E in shearing and torsion =G Heaviness, weight per cu. ft. =w Length of any member =1 in. Load or dead weight = W lb. Moment, in bending = M Ib.-in. in twisting = T Ib.-in. FORMULAS 3 t Moment of inertia rectangular =7 polar =/ Pitch of teeth, rivets, etc. =p in. Radius of gyration =r in. Section modulus, bending =-=Z twisting =-=Z p y Stress per unit of area = S Velocity in feet per second =v ft. per sec. 4. Formulas. Simple Stress W Tension, compression or shear, S=r (1) A. Bending under Transverse Load SI General equation, M = (2) y Rectangular section, M = - (3) Rectangular section, bh 2 = ~ (4) Circular section, M = ^-^> (5) Circular section, Torsion or Twisting Q T General equation, T= (7) \s Circular section, 7 T = -. (8) o.l 3 /^ ^ rp Circular section, d = \ ^ (9) o Hollow circular section, T = ~ - - 1 . (10) o.l d Other values of - and - may be taken from Table II. y y MACHINE DESIGN Combined Bending and Twisting Calculate shaft for a bending moment, Column subject to Bending W S Use Rankine's formula, - = (11) (12) The values of r 2 may be taken from Table II. The subjoined table gives the average values of g, while S is the compressive strength of the material. TABLE I VALUES OP q IN FORMULA 12 Material Both ends fixed Fixed and round Both ends round Fixed and free TimVpr 1 1.78 4 16 3000 1 3000 1.78 3000 4 3000 16 Cast iron 5000 1 5000 1.78 5000 4 5000 16 qtpol 36000 1 36000 1.78 36000 4 36000 16 25000 25000 25000 25000 Carnegie's hand-book gives S = 50,000 for medium steel columns and q= 35000; 2^017 an d TSOOO f r the three first columns in above table. In this formula, as in all such, the values of the constant should be determined for the material used by direct experiment if possible. W I Or use straight line formula, - r = Sk (12a) COLUMNS TABLE la VALUES OF S AND k IN FORMULA (12a) (Merriman's Mechanics of Materials) Kind of column S k Limit - Wrought Iron: Flat ends . . 42,000 128 218 Hinged ends 42,000 157 178 Round ends 42,000 203 138 Mild Steel: Flat ends 52,500 179 195 Hinged ends 52,500 220 159 Round ends 52,500 284 123 Cast Iron: Flat ends 80,000 438 122 Hinged ends Round ends Oak: Flat ends 80,000 80,000 5,400 537 693 28 99 77 128 Carnegie's hand-book gives allowable stress for structural columns of mild steel as 12,000 for lengths less than 90 radii, and as 17,100 57 - for longer columns. This allows a factor of safety of about four. MACHINE DESIGN TABLE II CONSTANTS OF CROSS-SECTION Form of section and area A Square of radius of gyration Moment of inertia Section modulus y Polar moment of inertia J Torsion modulus J y Rectangle V bh 3 bh* W+Vk bh 3 + b 3 h bh 12 12 6 12 6^6'+ h* Square d 2 d 4 d 3 d 4 d 3 d 2 12 12 6 6 4.24 Hollow bh 3 61 h 3 i bh 3 bih 3 i bh 3 b l h 3 i /-beam 12(bh-bihi) 12 6h bh-bihi Circle d 2 ;rd 4 d 3 ;rd 4 d 16 64 10.2 32 5.1 If Hollow d 4 -d*i d 2 +d 2 i 3i(d 4 d 4 i) ^4_ ( f4 1 re(d 4 d 4 i) 5.1d 16 64 10.2d 32 Ellipse it a 2 xba 3 6 2 x(ba 3 + ab 3 ) ba 3 +ab 3 16 64 10.2 64 10.2a Values of 7 and J for more complicated sections can be worked out from those in table. LOADED BEAMS TABLE III FORMULAS FOR LOADED BEAMS Beams of uniform cross-section Maximum moment M Maximum deflection A Wl Wl 3 Cantilever uniform load Wl 3EI Wl 3 Simple beam load at middle 2 Wl 8EI Wl 3 Simple beam uniform load 4 Wl 8EI 5W1 3 Beam fixed at one end, supported at other, 8 3WI 384EI .Q182W1 3 load at middle. Beam fixed at one end, suppported at other, 16 Wl El .0054 Wl 3 uniform load. Beam fixed at both ends load at middle 8 Wl El Wl 3 Beam fixed at both ends, uniform load Beam fixed at both ends, load at one end, 8 Wl ^2 Wl I92EI Wl 3 384EI Wl 3 (pulley arm). 2 12EI 5. Profiles of Uniform Strength. In a bracket or beam of uniform cross-section the stress on the outer row of fibers in- creases as the bending moment increases and the piece is most liable to break at the point where the moment is a maximum. This difficulty can be remedied by varying the cross-section in such a way as to keep the fiber stress constant along the top or bottom of the piece. The following table shows the shapes to be used under different conditions. MACHINE DESIGN Type Load Plan Elevation Cantilever Cantilever. . . Center. . . . Uniform . Rectangle . . . Rectangle . . . Parabola, axis horizontal. Triangle. Simp. Beam Center Rectangle Two parabolas intersecting Simp, beam Uniform. . Rectangle . . . under load. Ellipse, major axis hori- zontal. The material is best economized by maintaining a constant breadth and varying the depth as indicated. This method of design is applicable to cast pieces rather than to those that are forged or cut. The maximum deflection of cantilevers and beams having a profile of uniform strength is greater than when the cross-section is uniform, 50 per cent greater if the breadth varies, and 100 per cent greater if the depth varies. CHAPTER I MATERIALS 6. Primary Classification. The materials used in machine construction are practically all metals. They may be classified in two ways : (a) According to the principal metallic constituents such as iron, copper, tin, etc.; (6) as cast or wrought metals according to the methods employed in preparing them for use. The following table combines the two methods of classification. TABLE IV Principal metal Cast Wrought Cast iron Wrought iron. Malleable iron Soft steel Copper < Steel castings Bronze Brass Tool steel. Alloy steel. Brass wire. Sheet brass. Tin Aluminum Babbitt metal Bronze Rolled or drawn. 7. Iron. Commercial iron is produced from iron ore by reduction in a blast furnace. Most iron ores are oxides and also contain earthy impurities such as silica and alumina. The oxygen is removed by the burning of the coke used as fuel, while the limestone used as a flux unites with the silica and alumina forming a glassy slag which floats on the molten iron. Pig Iron. The coarse-grained impure iron thus formed is the pig iron of commerce and from it is made ordinary cast iron by remelting in the cupola of the foundry. Pig iron contains besides iron various quantities of carbon, silicon, manganese, phosphorus and sulphur. The last two are impurities and if 10 MACHINE DESIGN present in any considerable quantity render the pig unsuitable for the manufacture of high-grade irons or steels. The phos- phorus comes from the ore and the sulphur from the fuel used. The use of high-grade ore and of coke made from a non-sulphur coal is necessary to the production of pure iron. Pig iron may be used in the foundry for the manufacture of iron castings, in the puddling mill for producing wrought iron, or in the steel mill for the manufacture of Bessemer or of open-hearth steel. Cast Iron. Iron castings are made in the foundry by melting pig iron in a cupola using coke for a fuel. The quality of the cast iron depends largely upon the character of the pig iron used, as there is little chemical change affected in the cupola. A certain amount of scrap cast iron may be melted with the charge; remelting of iron makes it finer grained and harder. Wrought iron or steel shavings mixed with the molten cast iron produces a tough fine-grained iron, sometimes called semi-steel. The addition of about 25 per cent of steel scrap makes a fine- grained soft iron having a tensile strength about 50 per cent greater than that of the cast iron without the steel. Carbon exists in cast iron in two forms: (a) chemically com- bined with the iron; (6) as free carbon or graphite. The larger the proportion of free carbon, the softer and weaker is the iron. Remelting and cooling increases the amount of combined carbon and makes the iron harder as before noticed. The total amount of carbon present varies from 2 to 5 per cent in different irons. Silicon is an important element in iron and influences the rate of cooling. The more slowly iron cools after melting, the more graphite forms, the less the shrinkage and the softer the iron. Two per cent of silicon gives a soft gray iron with a high tensile strength. Machinery iron contains usually from 1^ to 2 per cent of silicon. Chilled iron is cast iron which has been cooled suddenly in the mold by contact with metal or some other good conductor of heat. Chilling increases the amount of combined carbon and makes the iron white and hard. It is used on surfaces which need to be extremely hard and durable, as the treads of car wheels and the outside of the rolls used on steel mills. The depth of the chill depends on the amount of metal used in the cooling surface of the mold. CAST IRON 11 All castings are chilled slightly on the surface. An'examina- tion of a freshly fractured casting shows whiter and finer-grained metal around the edges than at the center. For this reason, castings having considerable surface or "skin" in proportion to their weight are relatively stronger (see Art. 15). In selecting cast iron for various machine members, soft gray irons should be chosen where workability rather than strength is desired. Medium gray irons having a fine grain should be used where moderate strength and hardness are necessary as in the cylinders of steam engines and pumps. Hard gray iron is only suitable for heavy castings which require little or no machining, as it is brittle and not easily worked. An examination of the fracture of a sample of iron is a guide in determining its desira- bility for any particular case. Cast iron is the cheapest and best material for pieces of irregu- lar and complicated shape; it has a high compressive and a low tensile strength; it is brittle and cannot be welded or forged; but it resists corrosion much better than wrought iron. For its use in machine construction, see Art. 14. Malleable Iron. Malleable iron is cast iron annealed and partially decarbonized by being heated in an annealing oven in contact with some oxidizing material such as hematite ore, and then being allowed to cool slowly. A white cast iron is best for this process as the presence of graphitic carbon interferes with its success. An iron containing a small amount of silicon and considerable manganese promotes the formation of combined carbon just as silicon promotes the formation of free carbon. The castings before being annealed are hard and brittle, the fracture showing a silvery appearance. They are packed in air-tight cast-iron boxes with the oxidizing material and are kept at a red heat for several days. They are allowed to cool slowly and when removed are tough and ductile with a dull gray fracture. The oxidation removes some of the total carbon from the surface of the material and the heating and slow cooling changes the most of that remaining to graphite. An iron which originally contains 2.8 per cent combined and 0.20 per cent free carbon, after annealing may show 0.20 per cent combined and 1.8 per cent free carbon. 12 MACHINE DESIGN Malleable castings are particularly suitable for small parts having irregular shapes. The metal does not possess as much ductility or tensile strength as wrought iron but occupies a place intermediate between that and cast iron. As the process of malleablizing is to a certain extent a super- ficial one, it is best adapted to thin metal, although castings an inch or more in thickness have been successfully treated. Wrought Iron. Wrought iron is commercially pure iron which is made from pig iron by decarbonizing it in the puddling furnace. This furnace is a reverberatory one in which the molten pig is subjected to the action of the hot gases from the fuel. The silicon, manganese and carbon are oxidized or burned out, either by the action of the gas or by oxide of iron introduced with the charge. A part of the phosphorus and sulphur is also oxidized in the puddling. The molten mass is continually stirred during the process and finally assumes a pasty consis- tency. It is then squeezed to remove the slag and rolled into bars. These are cut, piled and welded into either bar or plate iron. The particles of iron in the puddling process are more or less enveloped in the slag and as the mass is squeezed and rolled, these particles become fibers separated from each other by a thin sheath or covering of slag, and it is this which gives wrought iron its characteristic structure. The presence of either sulphur or phosphorus in the iron renders it less reliable. Wrought iron possesses moderate tensile strength and high ductility. It can be forged and welded readily. Hammering or rolling it cold increases its strength and stiffness to a certain degree and raises artificially its elastic limit. For most purposes, it has been replaced of late years by soft steel. Either of these metals may be rendered superficially hard by the process known as case hardening. The pieces to be treated are packed in air- tight boxes together with pulverized carbon in some form, usually bone-black. The boxes are brought to a red heat and kept so for several hours. The pieces are then removed and quenched suddenly in water. The surface of the iron has com- bined with the carbon in which it was packed and changed to a high-carbon or hardening steel. Such pieces have a soft, ductile STEEL 13 center and a hard surface. Case hardening can be done after finishing but is liable to warp the metal. 8. Steel. Steel is made from molten pig iron by burning out the silicon and carbon with a hot blast, either passing through the liquid as in the Bessemer converter, or over its surface as in the open-hearth furnace. A suitable quantity of carbon and manganese is then added and the metal poured into ingot molds. If the ingots are reheated and rolled, structural steel and rods or rails are the result. Manganese has the effect of preventing blow holes and giving the steel a more uniform texture. Open-hearth steel differs but little from Bessemer in its chemical composition but is more uniform in quality on account of the more deliberate nature of the process of manufacture. Boiler p'late, structural steel, and in general material which is respon- sible for the safety of life and limb should be of open-hearth rather than Bessemer steel. Steel containing not more than 0.6 per cent of carbon is known as soft steel. It has a higher elastic limit and greater tensile strength than wrought iron, which metal it has practically sup- planted in the manufacture of machine parts. It is very ductile and malleable and may be welded if not too high in carbon. Crucible steel is made by melting steel or a mixture of iron and carbon in a crucible and pouring the melted metal into molds, and hence is commonly known as cast steel. This method is used for producing the harder steels suitable for cutting tools. The amount of carbon will vary from 0.5 to 1.5 per cent according to the use to be made of the steel. Such steel contains small amounts of silicon and manganese but must be practically free from sulphur and phosphorus. It is relatively high priced and is not used for ordinary machine parts. It cannot be readily welded but possesses the very useful characteristic of hardening when heated to a red heat and cooled suddenly. The degree of hardness can be controlled by accu- rately measuring the temperature of heating and by using various cooling agents such as water, brine and different kinds of oil. The steel can be tempered or softened after hardening by reheat- ing to a slight degree. 14 MACHINE DESIGN In machine construction crucible steel is only used for screws, spindles, ratchets, etc., which need to be extremely hard. It has a high tensile and compressive strength but is brittle and liable to contain hardening cracks. Steel castings are made by pouring fluid open-hearth steel directly into molds. They possess somewhat the same charac- teristics as malleable castings, being relatively tough and ductile. It has been somewhat difficult in the past to obtain reliable castings of this material as the great shrinkage about double that of cast iron has tended to make them porous and spongy in spots. Furthermore, steel which was sufficiently low in carbon to make soft castings was not fluid enough to run sharply in the mold. These difficulties have been to a large extent overcome and it is now possible to obtain steel castings which are reasonably clean and sound. They have about the same chemical composi- tion as mild rolled steel, the carbon varying from 0.2 to 0.6 per cent, the silicon about the same and manganese from 0.5 to 1 per cent. Steel castings when first poured are coarse-grained and should be annealed to make them tough and ductile. 9. Steel Alloys. Steel alloys are compounds of steel with chromium, vanadium, manganese, etc.; strictly speaking, all steels are alloys of iron with other substances, but when the term steel is used without qualification, it is understood to mean carbon steel. Nickel steel is both stronger and tougher than carbon steel. A high carbon steel is strong but brittle; the same or greater strength can be obtained by the addition of nickel without materially diminishing the ductility. This metal is suitable for pieces which are subject to severe shocks. Manganese steel is an alloy containing about 1 per cent of carbon and from 10 to 20 per cent of manganese; 14 per cent of manganese gives the maximum of strength and ductility com- bined. This metal is strong, tough and extremely hard, so that it cannot be readily finished except by grinding. It can be used for cutting tools, and like nickel steel is valuable for pieces ALLOY STEELS 15 subjected to great stress and wear. Its strength is increased by heating and sudden cooling. Chromium is sometimes added to nickel steel in the manu- facture of safes and armor plate. Mushet steel is an alloy of high carbon steel with tungsten and manganese and was the first of the air-hardening steels used for cutting tools. Like all of this class of tool steels, it must be worked at a yellow heat and hardens when cooled slowly in the air. The so-called air-hardening or high-speed steels are of various chemical compositions, containing carbon, manganese, tungsten, chromium, molybdenum or titanium, but the exact ingredients and proportions are for the most part trade secrets. Such steels are usually purchased in small sections and are used in special tool holders. They are forged with great difficulty and are generally heated in special furnaces with pyrometers for determining the exact temperature, and cooled in an air blast or by dipping in oil baths. The difference of a few degrees in the temperature of the metal will make or mar the cutting efficiency. They are of no use in machine construction, but affect it indirectly by requiring much greater strength, rigidity and power in machine tools. It is not an uncommon thing for the power consumption of a lathe or planer to be increased six or eight times by the use of the newer tools. Vanadium steel is one of the latest claimants for favor among the steel alloys. The addition of a small amount of this metal, 0.1 or 0.2 per cent, increases the strength and stiffness of mild steel in a marked degree with comparatively little increase in its cost. It is already used extensively in machine construction, particularly in marine work. 10. Copper Alloys. These metals are alloys of copper and tin, copper and zinc or of all three. Copper is not used alone in machine construction except for electric conductors. Phos- phorus, aluminum and manganese are also used in combination with copper. The copper-tin alloys are commonly known as bronzes and are 16 MACHINE DESIGN expensive on account of the large proportion of copper, from 85 to 90 per cent. Copper-zinc alloys, on the other hand, are called brass, and for maximum strength and ductility should contain from 60 to 70 per cent of copper. Bronzes high in tin and low in copper are weak, but have considerable ductility and make good metals for bearings. Tin 80, copper 10 and antimony 10 is Babbitt metal, so much used to line journal bearings, the antimony increasing the hardness. The late Dr. Thurston's experiments on the copper-tin-zinc alloys showed a maximum strength for copper 55, zinc 43 and tin 2 per cent. The tensile strength of this mixture was nearly 70,000 Ib. per square inch. Phosphor bronze is a copper alloy with a small amount of phosphorus added to prevent oxidation of the copper and thereby strengthen the alloy. Manganese bronze is an alloy of copper and manganese, usually containing iron and sometimes tin. A bronze containing about 84 per cent copper, 14 per cent manganese and a little iron, has much the same physical characteristics as soft steel and resists corrosion better. There is practically no limit to the varieties of color, hardness, ductility and" durability among the copper alloys. Some of the more common mixtures are here given. TABLE V COMPOSITION OP BRONZES ' Name Composition Gun metal Bell metal Yellow brass Muntz metal Aluminum bronze Phosphor bronze Manganese bronze (1) . Manganese bronze (2) . Copper .90, tin .10 Copper .77, tin .23 Copper .65, zinc .35 Copper .60, zinc .40 Copper .90, aluminum .10 Copper .89, tin .09, phosphorus .01 Copper .84, manganese .14, iron .02 [Copper .675, manganese .18 \Zinc .13, aluminum .01, silicon .005 FACTORS OF SAFETY 17 11. Strength and Elasticity. The constants for strength and elasticity given in the tables are only fair average values and should be determined for any special material by direct experi- ment when it is practicable. Many of the constants are not given in the table on account of the lack of reliable data for their determination. The strength of steel, either rolled or cast, depends so much upon the percentages of carbon, phosphorus and manganese, that any general figures are liable to be misleading. Structural steel usually has a tensile strength of about 65,000 Ib. per square inch, while boiler plate usually has less carbon, a low tensile strength and good ductility. Factors of Safety. A factor of safety is the ratio of the ultimate strength of any member to the ordinary working load which will come upon it. This factor is intended to allow for: (a) Overloading either intentional or accidental, (b) Sudden blows or shocks, (c) Gradual fatigue or deterioration of material. (d) Flaws or imperfections in the material. To a certain extent the term "factor of ignorance" is justifiable since allowance is made for the unknown. Certain fixed laws may guide one, however, in making the selection of a factor. It is a well-known fact that loads in excess of the elastic limit are liable to cause failure in time. Therefore, when the elastic limit of the material can be determined, it should be used as a basis rather than to use the ultimate strength. Furthermore, suddenly applied loads will cause about double the stress due to dead loads. These considerations indicate four as the least factor that should be used when the ultimate strength is taken as a basis. Pieces subject to stress alternately in oppo- site directions should have large factors of safety. The following table shows the factors used in good practice under various conditions: Structural steel in buildings 4 Structural steel in bridges 5 Steel in machine construction 6 Steel in engine construction 10 Steel plate in boilers 5 Cast iron in machines 6 to 15 Castings of bronze or steel should have larger factors than rolled or forged metal on account of the possibility of flaws. 18 MACHINE DESIGN us 1*11 a J us ^ M > - & a W I 8 ^ S oo flj O -g So . 3 1 '3 i ^ll 1 !- rston ha 60,000 + Prof. rbon STRENGTH OF METALS 19 3 -M c! l-slf g ll r=H 7s t^ S M -+3 V 73 O ft J O S S bO I x T-l t ^ 20 MACHINE DESIGN Cast iron should not be used in pieces subject to tension or bending if there is a liability of shocks or blows coming on the piece. NOTE. In giving references to transactions and periodicals, the follow- ing abbreviations will be used: Transactions of American Society of Mechanical 1 m ~ ,_ -n . > ir. A. fe. M. H/. Engineers. J American Machinist Am. Mach. Cassier's Magazine Cass. Engineering Magazine Eng. Mag. Engineering News Eng. News Machinery Mchy. REFERENCES Materials of Machines. A. W. Smith. Mechanics of Materials. Merriman. Materials of Engineering. Thurston. CHAPTER II FRAME DESIGN 12. General Principles of Design. The working or moving parts should be designed first and the frame adapted to them. The moving parts can be first arranged to give the motions and velocities desired, special attention being paid to compact- ness and to the convenience of the operator. Novel and complicated mechanisms should be avoided and the more simple and well-tried devices used. Any device which is new should be first tried in a working model before being introduced in the design. The dimensions of the working parts for strength and stiffness must next be determined and the design for the frame completed. This may involve some modification of the moving parts. In designing any part of the machine, the metal must be put in the line of stress and bending avoided as far as possible. . Straight lines should be used for the outlines of pieces exposed to tension or compression, circular cross-sections for all parts in torsion, and profile of uniform fiber stress for pieces subjected to bending action. Superfluous metal must be avoided and this excludes all ornamentation as such. There should be a good practical reason for every pound of metal in the machine. An excess of metal is sometimes needed to give inertia and solidity and prevent vibration, as in the frames of machines having reciprocating parts, like engines, planers, slotting ma- chines, etc. Mr. Oberlin Smith has characterized this as the " anvil" style of design in contradistinction to the " fiddle" style. In one the designer relies on the mass of the metal, in the other on the distribution of the metal, to resist the applied forces. A comparison of the massive Tangye bed of some large high- speed engines with the comparatively slight girder frame used in most Corliss engines, will emphasize this difference. 21 22 MACHINE DESIGN It may be sometimes necessary to waste metal in order to save labor in finishing, and in general the aim should be to economize labor rather than stock. The designers should be familiar with all the shop processes as well as the principles of strength and stability. The usual tendency in design, especially of cast-iron work, is toward unnecessary weight. All corners should be rounded for the comfort and convenience of the operator, no cracks or sharp internal angles left where dirt and grease may accumulate, and in general special attention should be paid to so designing the machine that it may be safely and conveniently operated, that it may be easily kept clean, and that oil holes are readily accessible. The appearance of a machine in use is a key to its working condition. Polished metal should be avoided on account of its tendency to rust, and neither varnish nor bright colors tolerated. The paint should be of some neutral tint and have a dead finish so as not to show scratches or dirt. Beauty is an element of machine design, but it can only be attained by legitimate means which are appropriate to the material and the surroundings. Beauty is a natural result of correct mechanical construction but should never be made the object of design. Harmony of design may be secured by adopting one type of cross-section and adhering to it throughout, never combining cored or box sections with ribbed sections. In cast pieces the thickness of metal should be uniform to avoid cooling strains, and for the same reason sharp corners should be absent. The lines of crystallization in castings are normal to the cooled surface and where two flat pieces come together at right angles, the interference of the two sets of crystals forms a plane of weakness at the corner. This is best obviated by joining the two planes with a bend or sweep. Rounding the external corner and filleting the internal one is usually sufficient. Where two parts come together in such a way as to cause an increase of thickness of the metal there are apt to be "blow holes" or "hot spots" at the junction due to the uneven cooling. "Strengthening" flanges when of improper proportions or in FRAMES 23 * the wrong location are frequently a source of weakness rather than strength. A cast rib or flange on the tension side of a plate exposed to bending, will sometimes cause rupture by crack- FIG. 1. OLD PLANING MACHINE. AN EXAMPLE OF ELABORATE ORNAMENTATION. ing on the outer edge. When a crack is once started rupture follows almost immediately. When apertures are cut in a 24 MACHINE DESIGN frame either for core-prints or for lightness, the hole or aperture hsould be the symmetrical figure, and not the metal that sur- rounds it, to make the design pleasing to the eye. The design should be in harmony with the material used and not imitation. For example, to imitate structural work either of wood or iron in a cast-iron frame is silly and meaningless. Machine design has been a process of evolution. The earlier types of machines were built before the general introduction of cast-iron frames and had frames made of wood or stone, paneled, carved and decorated as in cabinet or architectural designs. FRAMES 25 t When cast-iron frames and supports were first introduced they were made to imitate wood and stone construction, so that in the earlier forms we find panels, moldings, gothic traceries and elaborate decorations of vines, fruit and flowers, the whole covered with contrasting colors of paint and varnished as carefully as a piece of furniture for the drawing-room. Relics of this transition period in machine architecture may be seen in almost every shop. One man has gone down to posterity as actually advertising an upright drill designed in pure Tuscan. 13. Machine Supports. The fewer the number of supports the better. Heavy frames, as of large engines, lathes, planers, etc., are best made so as to rest directly on a masonry foundation. Short frames as those of shapers, screw machines and milling machines, should have one support of the cabinet form. The use of a cabinet at one end and legs at the other is offensive to the eye, being inharmonious. If two cabinets are used provision should be made for a cradle or pivot at one end to prevent twisting of the frame by an uneven foundation. The use of intermediate supports is always to be condemned, as it tends to make the frame conform to the inequalities of the floor or foundation on what has been aptly termed the " caterpillar principle." A distinction must be made between cabinets or supports which are broad at the base and intended to be fastened to the founda- tion, and legs similar to those of a table or chair. The latter are intended to simply rest on the floor, should be firmly fastened to the machine and should be larger at the upper end where the greatest bending moment will come. The use of legs instead of cabinets is an assumption that the frame is stiff enough to withstand all stresses that come upon it, unaided by the foundation, and if that is the case intermediate supports are unnecessary. Whether legs or cabinets are best adapted to a certain machine the designer must determine for himself. Where two supports or pairs of legs are necessary under a frame, it is best to have them set a certain distance from the ends, and make the overhanging part of the frame of a parabolic form, as this divides up the bending moment and allows less deflection at the center. Trussing a long cast-iron frame with 26 MACHINE DESIGN iron or steel rods is objectionable on account of the difference in expansion of the two metals and the liability of the tension nuts being tampered with by workmen. The sprawling double curved leg which originated in the time of Louis XIV and which has served in turn for chairs, pianos, stoves and finally for engine lathes is wrong both from a practical and esthetic standpoint. It is incorrect in principle and is therefore ugly. EXERCISE 1. Apply the foregoing principles in making a written criticism of some engine or machine frame and its supports. (a) Girder frame of engine. (b) Tangye bed of air compressor. (c) Bed, uprights and supports of iron planing machine. (d) Bed and supports of engine lathe. (e) Cabinet of shaping or milling machine. (f) Frame of upright drill. 14. Machine Frames. Cast iron is the material most used but steel castings are now becoming common in situations where the stresses are unusually great, as in the frames of presses, shears and rolls for shaping steel. Cored vs. Rib Sections. Formerly the flanged or rib section was used almost exclusively, as but a few castings were made from each pattern and the cost of the latter was a considerable item. Of late years the use of hollow sections has become more common; the patterns are more durable and more easily molded than those having many projections and the frames when finished are more pleasing in appearance. The first cost of a pattern for hollow work, including the cost of the core-box, is sometimes considerably more but the pattern is less likely to change its shape and in these days of many castings from one pattern, this latter point is of more importance. Finally, it may be said that hollow sections are usually stronger for the same weight of metal than any that can be shaped from webs and flanges. Resistance to Bending. Most machine frames are exposed to bending in one or two directions. If the section is to be ribbed it should be of the form shown in Fig. 3. The metal being of FRAMES 27 nearly uniform thickness and the flange which is in tension having an area three or four times that of the compression flange. In a steel casting these may be more nearly equal. The hollow section may be of the shape shown in Fig. 4, a hollow rectangle .with the tension side re-enforced and slightly thicker than the other three sides. The re-enforcing flanges at A and B may often be utilized for the attaching of other members to the frame as in shapers or drill presses. The box section has one great advantage over the I section in that its moment of resistance to side bending FIG. 3. FIG. 4. or to twisting is usually much greater. The double I or the U section is common where it is necessary to have two parallel ways for sliding pieces as in lathes and planers. As is shown in Fig. 5 the two Fs are usually connected at intervals by cross girts. Besides making the cross-section of the most economical form, it is often desirable to have such a longitudinal profile as shall give a uniform fiber stress from end to end. This necessitates a parabolic or elliptic outline of which the best instance is the housing or upright of a modern iron planer. Resistance to Twisting. 'The hollow cir- cular section is the ideal form for all frames or machine mem- bers which are subjected to torsion. If subjected also to bend- ing the section may be made elliptical or, as is more common, thickened on two sides by making the core oval. See Fig. 6. As has already been pointed out the box sections are in general better adapted to resist twisting than the ribbed or I sections. FIG. 5. 28 MACHINE DESIGN FIG. 6. Frames of Machine Tools. The beds of lathes are subjected to bending on account of their own weight and that of the saddle and on account of the downward pressure on the tool when work is being turned. They are usually subjected to torsion on ac- count of the uneven pressure of the supports. The box section is then the best; the double I commonly used is very weak against twisting. The same principle would apply in designing the beds of planers but the usual method of driving the table by means of a gear and rack prevents the use of the box sec- tion. The uprights of planers and the cross rail are subjected to severe bending moments and should have profiles of uni- form strength. The uprights are also sub- ject to side bending when the tool is taking a heavy side cut near the top. To provide for this the uprights may be of a box sec- tion or may be reinforced by outside ribs. The upright of a drill press or vertical shaper is exposed to a constant bending moment equal to the upward pressure on the cutter multiplied by the distance from center of cutter to center of upright. It should then be of constant cross-section from the bottom to the top of the straight part. The curved or goose-necked portion should then taper gradually. The frame of a shear press or punch is usually of the G shape in profile with the inner fibers in tension and the outer in compression. The cross-section should be as in Fig. 3 or Fig. 4, preferably the latter, and should be graduated to the magnitude of the bending moment at each point. (See Fig. 7.) 15. Tests on Simple Beams. In 1902, a series of experiments was made on cast-iron beams of various sections at the Case School of Applied Science. The work was done by Messrs. FIG. 7. CAST IRON BEAMS 29 A. F. Kwis and R. H. West 1 under the direction of the author and the results were reported by him in 1906. The patterns were all 20 in. long and had the same cross-section of 4.15 sq. in. As may be seen from the tables, the areas of the cast beams varied slightly. The castings of each set were all made from the same ladle of iron and were cast on end. A soft gray iron was used and a. large flush basin distributed the molten metal to the mold, giving a uniform temperature and quality. The castings were prepared by Mr. Thomas D. West and proved to be remarkably uniform in quality and free from imperfections. The specimens were all tested by loading transversely at the center, the supports being 18 in. apart. Object. The investigation had two distinct objects in view and two classes of test pieces were used. The first class com- prised Nos. 1 to 11 and Nos. 22 to 32, and these specimens had sections such as are used in parts of machines. The second class comprised Nos. 12 to 21 and 33 to 42, all having sections similar to those used in the rims of fly-wheels. The sections tested were such as shown by the diagrams in the tables. The areas given in the table are those of the specimens at the point of rupture. There are two specimens of each shape cast from the same pattern. The section modulus - was calculated from the dimensions of the casting at the breaking point, y being the distance from neutral axis to extreme fiber in tension. In testing each specimen the load was applied gradually and readings of the deflection were taken at regular intervals. When the "set" load was reached, the pressure was removed and a reading of the perma- nent set was taken. The load was again applied and observations made on the deflection up to near the time of rupture. The load-deflection curves plotted from these observations are nearly all smooth and uniform in character, as may be seen by reference to Fig. 8 which shows the curves for No. 33. The initial line curves gradually from the start showing an imperfect elasticity, while the set line is nearly straight and approximately parallel to the tangent of the curve at the vertex. 1 Mchy., May, 1906. 30 MACHINE DESIGN >> Q o 8^ o_ o_ o^ o q^ o^ o^ q^ o a -M < | o*~o*'o''o~"o~'o~o~o~o~'o* p^tn OOOi^Oi-OOiCOCOT^cocOOi g QJ 00*^ 00^ 00** Ol" CO" CO" CO* ^ ^OOO(N P-, og * 00 h^ ^ b 8i888888 g fl PQ fl ^ ^ ^ T^TfiiOCO OOO -^O ^3^^'^ CO COiO^OOTHOOOO O ^ * O IO IO CO CO CAST IRON BEAMS 31 The so-called moduli of elasticity were calculated 'from the set lines using the formula E 48 A/ In each test a reading of the load was taken at the instant when the deflection measured 0.03 in., and these loads may be taken as a fair measure of the "stiffness" of the section. The modulus of rupture was calculated from the breaking load and the section modulus, using the formula: My Wly ~T'- = ^T The modulus of rupture, as S is generally called, is supposed to represent the tensile stress on the outer fibers at the point of rupture and to measure in a way the transverse strength of the material. In the absence of a better measure we will use this, and take the circular and square sections as our standards. The average value of S for the four is 24,360 Ib. per square inch. This is a low value even for soft gray iron. The remarkable fluctuations in the value of $ for specimens of different cross- section, from a minimum of 18,700 to a maximum of 36,000, show that the ordinary method of calculation would not be of much value in predicting the breaking load of such beams. Comparison of Strength. An investigation of the values in Table VIII shows that the hollow circular and elliptic sections are much stronger than the solid sections, the increase in strength being greater than that of the section modulus. The average value of S for the last six numbers in Table VIII is 31,600 as against 24,000 for the six solid sections, an apparent increase in the strength of the material itself of over 25 per cent. This is partly due to the thinner metal, the greater surface of hard "skin" and the freedom from shrinkage strains. The absence of corners and the consequent uniformity of metal make this an ideal form of section. The hollow rectangles and the I-sections given in Table IX have an average value of S = 22,450. No. 8 is lower than the average and Nos. 28 and 32 considerably higher. These discrepancies are due to some accidental condi- 32 MACHINE DESIGN 1*1 c 5 > 00000 O O j 2 .12 3 o 88888 8 8 8 8 8 t < 11 o o o o o 05 CO tf CO CM O l> CO l> 1 CO o 1 o S o : CO >O CO *O IO IO o * "* no t 5 o o o o o o o o o 3 g O 00 O O CM IO rH t^ rH CM 8 CM CM 1 ^ ' rH 00 I s - rH rH CM CM (N rH CM rH CM 1 2 8 O T-H 00 O 1C CM CO to O5 l1 O O O O O O o O O O 1 CM CM CM CM CM o 8 o" o 8 o" CM o" 1 1 CO 1 CO 88888 8 8 8 8 8 .a g II oT CM" oo" CM" r-T co" CM~ 05" oT CO* 00 < 3 Js iO O O iO iO 8 o o S CM rH CM CM O5 rH rH CM ^ (^| ^J (JQ ^J CO CM rH rH 00 M CM CO CM CM CM CO CO CO- CO CO -|. a 3 3 *N I >> 00 rt< Tj< i I CO O ft O5 CO CO =0 05 g CM 8 "S ^ o i ' XO iO O iO CO CO CO CO CO co OQ 5 1 co o o oo r^> CO -^ CM 00 00 rH oo CO CO CO CM 1 1 Jl ^ ^ ^ ^ ^ Th ^ * TH IO (3 O +3 08 D OH H H H H H H H * I s * 00 00 Oi O^ CM CM . o rH rH CO rH rH CM CO WHEEL RIMS 33 * >. 0000000000 % i oooooooooo 1! rfl>-COT li>-cO>OOiOThi t>.iC^COi-HOOI>OCOt^ OOOOOOOOrHOOOsO r-H T-H i-H T-H i-H rH *o 3 3 ^^ os co t^* ^o co co t^* T H oo 11 O tH OOOOSOOO^OTtiTj- CO *O CM co co t^* CO T-H 00 co oo - !>. CO t^ 1 S 1! & ** o o o o TJH" of 10" oo" (M O 00 co co co T i d T-H CN 73 d 9 o ^ *g 2 ^ ,f o *o o >o iO iO O O iO >O O5 CM CO T* CO CO b- co o oo 1>- Os j5, 0000 0000 o3 kC CO O^ Ol O 5 T^H ^J ^^ CO C^ ^D .9 l> O *O CM TH (M O iO PQ T-H T 1 T-H (N T-H CM d ; (^ ,-|H r- ^ QQ CO O l> CO OO *O rt 3 *H I ^ r^l r J Q^ QQ 00 CO t^ CO t^ OS o ^3 || ^ 1 T-H (M J *fi 3 1 1 a> S S g o3 rH CP TH o .S ' i o H 1 ri^ O g 1 O = It, o o 8 a R R c^ CO CO (M W op 10 TjT CO 00 S H to for the stress at the outer surface. Fig. 16 illustrates the variation in S from inner to outer surface. Solving for d 2 in (19) we have ^S/fzfj- (21) A discussion of Lamp's formula may be found in most works on strength of materials. PROBLEMS 1. A hydraulic cylinder has an inner diameter of 12 in., a thickness of 4 in. and an internal pressure of 1500 Ib. per square inch. Determine the maximum stress on the metal by Barlow's and Lame's formulas. 2. Design a cast-iron cylinder 8 in. internal diameter to carry a working pressure of 1200 Ib. per square inch with a factor of safety of 10. 3. A cast-iron water pipe is 1 in. thick and 18 in. internal diameter. Required head of water which it will carry with a factor of safety of 6. 23. Steel and Wrought-iron Pipe. Pipe for the transmission of steam, gas or water may be made of wrought iron or steel. Cast iron is used for water mains to a certain extent, but its use for either steam or gas has been mostly abandoned. The weight of cast-iron pipe and its unreliability forbid its use for high pressure work. Wrought-iron pipe up to and including 1 in. in diameter is usually butt-welded, and above that is lap-welded. Steel pipes may be either welded or may be drawn without any seam. Electric welding has been successfully applied to all kinds of steel tubing, both for transmitting fluids and for boiler tubes. The tables on pp. 56 to 61 are taken by permission from the catalogue of the Crane Company and show the standard dimen- sions for steam pipe and for boiler tubes. Ordinary standard pipe is used for pressures not exceeding 100 Ib. per square inch, extra strong pipe for the pressures pre- vailing in steam plants where compound and triple expansion engines are used, while the double extra is employed in hydrau- lic work under the heavy pressures peculiar to that sort of transmission. BOILER TUBES 55 Tests made by the Crane Company on ordinary commercial pipe such as is listed in Table XV showed the following pressures: 8 in. diam ........ 2,000 Ib. per square inch. 10 in. diam ........ 2,300 Ib. per square inch. 12 in. diam ........ 1,500 Ib. per square inch. The pipe was not ruptured at these pressures. 24. Strength of Boiler Tubes. When tubes are used in a so- called fire-tube boiler with the gas inside and the water outside, they are exposed to a collapsing pressure. The same is true of the furnace flues of internally fired boilers. Such a member is in unstable equilibrium and it is difficult to predict just when failure will occur. Experiments on small wrought-iron tubes have shown the collapsing pressure to be about 80 per cent of the bursting- pressure. With short tubes set in tube sheets the length would have considerable influence on the strength, but ordinary boiler tubes collapsing at the middle of the length would not be in- fluenced by the setting. The strength of such tubes is proportional to some function of -, where t is the thickness and d is the diameter. The formulas a heretofore in use are very limited in their application, being founded on experiments covering but a few diameters and thicknesses. Fairbairn's formula is the oldest and best known of these and was established by him as a result of experiments on wrought- iron flues not over 5 ft. in length and having relatively thin walls. p = 9,672,000 - all dimensions being in inches and p being the collapsing pressure. D. K. Clark gives for large iron flues the following formula: 200,000*' ~~d^~ where P is the collapsing pressure in pounds per square inch. These flues had diameters varying from 30 in. to 50 in. and thick- ness of metal from f in. to y 7 ^- in. 56 MACHINE DESIGN P GO r? 1=1 ^ O o.l *f! s^i 3 a tS g^ O2 *H n < rQ ^ ti aH"? ^ z> 00 00 TjH HIM HN HM HN 00 oo i-r Oi & *i Pounds. 1 1 ^ c? ^ os o 1C I> s 1C TH 00 rH ^* T> CO co' 00 t- co 0> co CO 1 10 1C J> ' . Pipe Containing One Cubic Foot. I CO T-> 1C C3 CO o? ^H CO OS 1C o? s CO o TH OS 9 i 1C os Pipe per Foot of II MtM i 1C t 1 ^* TH OS ^ CO !> i> CO TH 1C CO 1 ci {> CO oi 9 oo J^ 3 1C "^ o? 1 2 !* External Surface. 1 fc ^ ^ O5 1 J> i 1C ^ 1C TI? co" CO TH co' TH c? i 00 o OS 3 II CO 1 CO OS ^H CO c^ CO CO ^t 1 tC OS ^ o t^ os t^ ^H J> o 1 CO -* o> O7 5 "3 s 5 J3 CO 1 n i> TH OS TH 00 1C CO 1 1 GO s CO I 00 00 cc I rH I T- Ci s 3 2 1 CO" 1 t> r. ^r JO I T-^ O5 i CO JO O5 ^f ^ CO 00 I- M H o - ill! 00 2% 00 eS'SlSl^lS'i CQ ?|^ ^3 |S 1 10 CO ^ ^ I JO CO I COCOrJ|00 I ! 05 ,CO |tjj __ CO e^ si 1 i M w I g SSlgii- O T-l I^H CO.U 5? BOILER TUBES 61 1 1 ~ a II M ,Q 1^3 o H Q ij H ^ i <1 J a* i L -is ifa CO ,00 OO I- lt> co o co 00 co os I CD CO i CO 1 I . fill it- OS OO CO ^ Ico os t- loo I l< 1O CO 'CO OO lOO I OS CO ~* 1^ 199 S 2 SB'S ^ ^^ 1 10 *o O5 o? i "^ TT ^o t^ ill! s 1 OS ICO ^* Tj< CO OS SIS CO 1^ s\ CO lei s OS i |^| CD T-H |CO ' 3 2 1 !1 CO I'* CO ICO T-l LO OS -^ \"& OO OS |O T-H -I-H SI i> o i ^ t- o H TH CO CO CO CO U eol- 11 S CO CO CO CO CO I CO j^ 00 CO |00 I I I ,00 .s-g 63 o a a " M TH OS CD CD 10 10 lOCDC-COOSOr-iC ! ^ 'S Ico co 11 :i* 111 111 * "o M H r 3 1 5 '-S *8 62 MACHINE DESIGN In 1906, Professor R. T. Stewart reported to the American Society of Mechanical Engineers some very comprehensive and interesting experiments on lap-welded boiler tubes of Bessemer steel. 1 The tests were conducted at the works of the National Tube Company on tubes manufactured by that firm and were in progress for four years. Two series of experiments were made one on tubes 8f in. outside diameter of different thicknesses and of different lengths, for the purpose of testing the applicability of existing formulas to tubes of this character; one on tubes 20 ft. long and of different diameters and thicknesses for the purpose of establishing empirical formulas for the strength of such tubes. The formulas of Fairbairn, Clark, Unwin, Grashof, etc., were tested by comparison with the results of the first series of experi- ments and were all found inapplicable, sometimes giving less than one-third the actual collapsing pressure. The general conclusions reached by Professor Stewart are thus stated by him: "1. The length of tube, between transverse joints tending to hold it to a circular form, has no practical influence upon the collapsing pressure of a commercial lap-welded steel tube, so long as this length is not less than about six diameters of tube. 2. The formulas, as based upon the present research, for the collapsing pressure of modern lap-welded Bessemer steel tubes, are as follows: P = 1000(1 - ^l - 1600 -) (A) P = 86,670-^-1386. (B) a Where P = collapsing pressure, pounds per square inch d = outside diameter of tube in inches t = thickness of wall in inches. Formula (A) is for values of P less than 581 lb., or for values of j less than 0.023, while formula (B) is for values greater than these. 1 Trans. A. S. M. E., Vol. XXVII. STEEL TUBES 63 These formulas, while strictly correct for tubes thajt are 20 ft. in length between transverse joints tending to hold them to a circular form, are, at the same time, substantially correct for all lengths greater than about six diameters. They have been tested for seven diameters, ranging from 3 to 10 in., in all obtainable thicknesses of wall, and are known to be correct for this range. 3. The apparent' fiber stress under which the different tubes failed varied from about 7000 Ib. for the relatively thinnest to 35,000 Ib. per square inch for the relatively thickest walls. Since the average yield-point of the material was 37,000 and the tensile strength 58,000 Ib. per square inch, it would appear that the strength of a tube subjected to a collapsing fluid pressure is not dependent alone upon either the elastic limit or ultimate strength of the material constituting it." The following tables are condensed from those published by Professor Stewart and give average dimensions and pressures for each size tested, each result being the average of five tubes: The reader is referred to the published paper for further details of this most valuable contribution to a hitherto neglected subject. 25. Theory. In January, 1911, Professor Stewart presented a discussion of the theory of collapsed tubes based on the experi- ments above described. 1 Considering a ring or annulus of the tube 1 in. long near the middle of its length, he treats each half of the ring as a column fixed at both ends and compressed uniformly along its center line, abc, Fig. 17. The ring is subjected to a uniform radial external pressure of p pounds per square inch and is therefore in the same condition as the thin shell in Art. 21 except that the resultant stress is now. compression instead of tension. By equation (15), ~ pr_pd and 2tS , . (a) 1 Trans. A. S. M. E., Vol. XXXIII. 64 MACHINE DESIGN TABLE XIX COLLAPSING PRESSURE OF TUBES Test number Average outside diameter, inches Average thickness of wall, inches Actual length of tube, feet Collapsing pressure, pounds per square inch 1 8.643 0.185 20.026 536 2 8.653 0.184 15.010 548 3 8.656 0.178 10 . 002 548 4 8.658 0.180 5.006 592 5 8.656 0.176 2.512 977 6 8.642 0.215 13.140 847 7 8.663 0.219 11.801 835 8 8.669 0.214 10.007 845 9 8.661 0.212 4.997 907 10 8.657 0.212 2.507 1,314 11 8.666 0.267 19.995 1,438 12 8.652 0.272 14.996 1,540 13 8.668 0.267 9.993 1,533 14 8.656 0.268 4.993 1,636 15 8.662 0.268 2.494 1,784 16 8.657 0.273 19.387 1,347 17 8.659 0.275 14.995 1,421 18 8.671 0.271 10.003 1,541 19 8.672 0.280 4.997 1,731 20 8.653 0.269 2.505 1,961 21 8.656 0.294 19.999 1,686 22 8.654 0.308 14.987 1,791 23 8.649 0.305 9.989 1,810 24 8.654 0.306 4.993 2,073 25 8.646 0.311 2.509 2,397 26 6.017 0.128 20.000 519 27 6.017 0.131 20.000 529 28 6.022 0.167 20.000 969 29 6.026 0.166 20.000 924 30 6.032 0.163 20.000 917 31 6.033 0.170 20.000 ,007 32 6.023 0.189 20 . 000 ,318 33 6.021 0.212 20.000 ,457 34 6.015 0.206 20 . 000 ,555 35 6.022 0.186 20.000 ,188 36 6.032 0.263 20.000 2,139 STEEL TUBES 65 TABLE XIX (Continued) COLLAPSING PEESSURE OF TUBES Test number Average outside diameter, inches Average thickness of wall, inches Actual length of tube, feet Collapsing pressure, pounds per square inch 37 6.034 0.264 20.000 2,381 38 6.654 0.164 20.000 678 39 6.684 0.200 20.000 1,184 40 6.666 0.253 20.000 2,081 41 7.044 0.160 20 . 000 563 42 7.050 0.242 20.000 1,680 43 6.661 0.154 20.000 563 44 6.655 0.269 20.100 2,214 45 6.681 0.249 20.100 1,745 46 6.049 0.266 20.110 2,528 47 8.643 0.185 20.000 536 48 8.642 0.215 14.133 847 49 8.666 0.267 19.995 1,438 50 8.657 0.273 19.550 1,347 51 8.656 0.293 20.000 1,686 52 8.663 0.305 20.100 1,756 53 8.673 0.354 20.080 2,028 54 6.987 0.279 20.170 2,147 55 7.011 0.160 20.170 621 56 5.993 0.271 20.180 2,487 57 10.041 0.165 20.180 225 58 10.026 0.194 20.110 383 59 10.001 0.316 20.180 1,319 60 3.993 0.119 20.170 964 61 4.014 0.175 20.190 2,280 62 4.026 0.212 20.190 3,170 63 4.014 0.327 20.100 5,560 64 3.000 0.109 20.000 1,733 65 2.994 0.113 20.000 1,962 66 2.992 0.143 20.000 2,963 67 2.995 0.188 20.100 4,095 68 10.779 0.512 19.470 2,585 69 12.790 0.511 19.960 2,196 70 13.036 0.244 20.000 463 66 MACHINE DESIGN This stress is uniform from end to end as is the case with the loaded straight column in Fig. 18. Furthermore, the character- istic shape assumed by the collapsed tube, as shown in dotted lines in Fig. 17, has its tangents at a' and c' parallel to their original position at a and c, corresponding to the conditions for buckling of a column with fixed ends shown by dotted lines in Fig. 18. V \ \ \ \ \ \ \ \ \ \ I / I I I I Let / = length of equivalent column r = radius of gyration of section of column Then will l = ^(d-t) (where d = outer diameter of tube) and IV _J_ \12 3.464 By Professor Stewart's formula (B) P = 86,670^-1386 From (a) and (B) by equating: =86,670^-1386 (b) (c) STEEL TUBES 67 and =43,335-693^ (d) From (b) : 1--I+ 1 t Tit Substituting value of t from (c) : d . = ~?L +1=0.1838 ~ + l (e) t 3.4647ZT r Substituting this value of - in (d) and reducing: L S = 42,642- 127.4 - (24) corresponding to the straight line formula for columns (see Table la). Professor Stewart suggests as a substitute for formula (A) p. 62, the following: P = 50,210,000 (G) 26. Tube Joints. The failure of boiler tubes, especially of those having water or steam pressure inside, is frequently due to slipping of the tube in the plate or fitting to which it joins. Such tubes are expanded in the plate by the use of a roller or Dudgeon expander and are sometimes flared or beaded on the outside for additional security. Under pressure, the tubes often slip in the holes so as to cause failure of the joint or at least leakage of the contained fluid. Some experiments made by Professors O. P. Hood and G. L. Christiansen were reported by them in 1908 and give the most reliable information on this subject. 1 The tests were made on 3-in., twelve-gage, cold drawn Shelby tubes rolled into holes in plates of various thicknesses and reamed in various shapes. Some of the tubes were flared outside the plate and some not. Initial slip occurred at total pressures of from 5000 to 10,000 Ib. or from one-sixth to one-third the elastic limit of the material 1 Trans. A. S. M. E., Vol. XXX. 68 MACHINE DESIGN of the tube. The ultimate holding power was usually about double the slipping load. The coefficient of friction varied from 26 to 35 per cent, assum- ing the elastic limit to vary between 30,000 and 40,000 Ib. The total friction per square inch of bearing area was about 750 Ib. Various degrees of rolling and various forms of tapered hole did not seem to affect the initial slipping load materially. Serrating the bearing surface of the hole had a very marked effect, raising the initial slipping load in some instances as high as 40,000 to 45,000 Ib., or more than the elastic limit of the tube. The slipping point of the tube bears a certain analogy to the yield-point in metals and the diagrams of pressure and slip much resemble the stress-strain diagrams of soft steel. It is apparent from these experiments that overrolling has no advantages and that flaring the tubes will not prevent leakage. The fact that ordinarily slipping will occur at a pressure well inside the elastic limit of the material shows that timely warning will be given by leakage before there is any danger of failure. 27. Tubes under Concentrated Loads. In 1893, the author made some experiments on steel hoops to determine the strength and stiffness under a concentrated load applied in the direction of a diameter. 1 Large steel tubes with relatively thin walls are sometimes exposed to external compression at the point of support causing distortion and occasionally permanent injury. The hoops tested were made of mild steel boiler plate, having a tensile strength of 60,000 Ib. and a modulus of elasticity of 30,000,000, cut into strips 2.5 in. wide, bent to a circular form and welded. Each hoop was compressed laterally in a testing machine until failure occurred, vertical and horizontal diameters being measured at regular intervals. Regarding the hoop as composed of two semi-circular columns fixed at the ends and each having a constant deflection of one- half the mean diameter, it is evident that a treatment is allowable similar to that used in Rankine's formula for columns (for- mula (12)). The increase in deflection for loads inside the elastic limit is small compared with the length of the hoop radius. l Jour. Assoc. Eng. Soc., Dec., 1893. STEEL HOOPS 69 Let P = load in pounds at elastic limit D = inner horizontal diameter in inches 6 = breadth of hoop in inches t = thickness of ring S stress on inner fibers at extremity of horizontal diameter. Then as in Rankine's formula: Where M is the bending moment at extremity of horizontal diameter. Assume M = kPD. Then p where q = empirical constant. The average value of q as determined by experiment was q = 0.946. Substituting this value in (b) and solving for P } we have: 2btS "1 + 0.946.D' t Table XX gives the principal data and results of experiment. In determining the value of q from the experiments, S was assumed to be the same as the elastic limit in compression of a straight specimen of the same metal. The limited number of hoops tested and the method of their construction forbids the application of formula (25) to general cases of this character. It is offered here merely as a guide in design. 28. Pipe Fittings. Steam pipe up to and including pipe 2 in. in diameter is usually equipped with screwed fittings, including ells, tees, couplings, valves, etc. Pipe of a larger size, if used for high pressures, should be put together with flanged fittings and bolts. One great advantage of 70 MACHINE DESIGN o a* W o 3| ll o OS i i IO 00 ' to -G 0) QS CO Q ^ oo ^ S * 10 (N C^ i I CO CO TjH rG .S "JJH o O o O o o 3 O OH bfi^ 7! (^ to 'Z 4 v"b* . is proportional to *~- and 3 in the two cases. t t But Solving in these equations for p' and p" /Y\* p ~ Substituting these values in (a) and (b) : (32) (33) As l>b usually, equation (33) is the one to be used. If the plate is square l = b and '=^/| (34) If the plate is merely supported at the edges then formulas (32) and (33) become: For rectangular plate : For square plate: (36) A round plate may be treated as square, with side diameter, without sensible error. The preceding formulas can only be regarded as approximate. Grashof has investigated this subject and developed rational formulas but his work is too long and complicated for introduc- tion here. His formulas for round plates are as follows: FLAT PLATES 85 Round plates: Supported at edges: / ' I^P fQ.'7\ ~2\6S Fixed at edges: where t and p are the same as before, d is the diameter in inches and S is the safe tensile strength of the material. Comparing these formulas with (34) and (36) for square plates, they are seen to be nearly identical if allowance is made for the difference in the value of S. Experiments made at the Case School of Applied Science in 1896-97 on rectangular cast-iron plates with load concentrated at the center gave results as follows: Twelve rectangular plates planed on one side and each having an unsupported area of 10 by 15 in. were broken by the application of a circular steel plunger 1 in. in diameter at the geometrical center of each plate. The plates varied in thickness from J in. to 1 J in. Numbers 1 to 6 were merely supported at the edges, while the remaining six were clamped rigidly at regular intervals around the edge. To determine the value of S, the modulus of rupture of the mateiial, pieces were cut from the edge of the plates and tested by cross-breaking. The average value of S from seven experi- ments was found to be 33,000 Ib. per square inch. In Table XXV are given the values obtained for the breaking load W under the different conditions. If we assume an empirical formula: and substitute given values of S, I and 6 we have nearly: w = mw. (b) Substituting values of W and t from the Table XXI we have the values of k as given in the last column. If we average the values for the two classes of plates and substitute in (a) we get the following empirical formulas: 86 MACHINE DESIGN For breaking load on plates supported at the edges and loaded at the center: ^ (39) = and for similar plates with edges fixed: tf/ 2 TF-442 *P+b* S in both formulas is the modulus of rupture. TABLE XXV CAST-IRON PLATES 10X15 IN. (40) No. Thickness t Breaking load W Constant k 1 .562 7,500 237 2 .641 11,840 288 3 .745 14,800 267 4 .828 21,900 320 5 1.040 31,200 289 6 1.120 31,800 254 7 .481 9,800 424 8 .646 17,650 422 9 .769 26,400 446 10 .881 33,400 430 11 1.020 47,200 454 12 1.123 59,600 477 Those plates which were merely supported at the edges broke in three or four straight lines radiating from the center. Those fixed at the edges broke in four or five radial lines meeting an irregular oval inscribed in the rectangle. Number 12, however, failed by shearing, the circular plunger making a circular hole in the plate with several radial cracks. Some tests were made in the spring of 1906 at the Case School laboratories by Messrs. Hill and Nadig on the strength of flat cast-iron plates under uniform hydraulic pressure. Table XXVI gives the results of the investigation. The low value of S is explained by the fact that the material was a soft rather coarse gray iron, having an average tensile strength of about 12,000 Ib. FLAT PLATES 87 TABLE XXVI CAST-IRON PLATES, UNIFORM LOAD, FIXED EDGES Breaking load in pounds per Size of plate, Thickness Modulus square inch inches Inches S By formula Actual 12X12 0.75 20,440 (34) 320 375 12X12 1.00 27,900 (34) 777 675 12X18 0.94 26,600 (33) 390 450 12X18 1.25 24,000 (33) 622 650 Further experiments are needed to establish any general conclusions. 32. Steel Plates. Mr. T. A. Bryson of Rensselaer Polytechnic Institute has recently made some tests on steel plates under hydrostatic pressure and published a monograph on the subject. The material tested was medium steel boiler plate from J to \ in. thick and the sizes used were 18 by 18 in. and 24 by 24 in. Two plates separated by a cast-iron distance piece were clamped at the edges by cast-iron frames bolted together. Hydrostatic pressures from to 225 Ib. per square inch were applied and deflections were measured at five points. Both working deflections and permanent sets were noted. The characteristics of the material were determined from test pieces cut off the edge of each plate. Mr. Bryson develops formulas similar to Morley's, 1 which differ from those just given in the values of the constants. All the formulas for square plates can, however, be reduced to the general form : t = bj k j- (See formula 34) \ o or S = k* (41) where S is the maximum stress in the plate. The value of k, as determined by the average of eight tests 1 Morley's Strength of Materials. 88 MACHINE DESIGN with different values of b and t, was 0.141 at the elastic limit of the material, the maximum value being 0.156 and the mini- mum 0.131. This value of k may then be used for steel plates with fixed edges without serious error. Mr. Bryson after discussing the experiments of Bach on square and rectangular plates recom- mends the following general formula for rectangular steel plates fixed at the edges and uniformly loaded: 0.5 . Vp ~ *~ where r = b/l Where l = b this reduces to formula (41). The value of S for plates merely supported may be assumed to be 50 per cent greater than in formula (42) . The value of k in formula (32) is determined by substituting r = and reducing: . r 2 2(1 + r 4 ) or for a square plate: -J <*> These values of k are much larger than those just given. In Mr. Bryson' s tests it was found that suspension stresses gradually supplanted those due to bending and that this change reduced the value of k. This would not be true of cast-iron plates and the formulas given on page 84 would be preferable. The values of k for the four experiments detailed in Table XXVI would be respectively: fc=.213-.287-.363-.400 which shows that formula (42) is not applicable to cast-iron plates. The most comprehensive experiments on flat plates are those by Professor Bach, and Grashof's formulas are largely controlled FLAT PLATES 89 by them. 1 Table XXVII gives the derived formulas for some of the more usual cases. The notation is the same as that of the previous formulas. The strength of the plates depends also on the manner of fastening at the edges, the number and size of bolts, the nature of gasket used, if any, etc., etc. TABLE XXVII STRESSES IN FLAT PLATES Shape Edges Load Value of fiber stress S = Value of coefficient fc- Remarks Circle Fixed . . Uniform .... pr* Cast iron, 0.8 r = radius. k ~w Steel, 0.5 Circle . pr2 Cast iron 1 2 T = radius. k -p Steel 7 Ellipse.... Fixed Uniform. . . . pb* i K lt*(l + n*) Cast iron, 1.34 Steel 84 Estimated. Ellipse.... Support. . . Uniform .... pb* i 4 2 (Z 4- n 2 ) Cast iron, 2.26 Steel, 1.41 Estimated. Rect Fixed At center. . . Wlb t 2 (l*+b*) Cast iron, 2.63 Rect Wlb Cast iron 3 k t*(i*+b*) Rect Fixed Uniform .... k -* l - b m i*+ 6 2 ) Cast iron, 0.38 Steel 24 Estimated. Rect Support. . . Uniform .... P Z 2 & 2 P(l 2 +b 2 ) Cast iron, 0.57 Steel, 0.36 Estimated. Fixed At center. . . .W Cast iron, 1.32 k T* Square.. . . Support. . . At center. . . k^ p Cast iron, 1.50 Square.. . . Fixed Uniform .... k ptf Cast iron, 0.19 Steel, 0.12 Estimated. Square. . . . Support. . . Uniform .... 4 Cast iron, 0.28 Steel, 0.18 Estimated. NOTE. n minor axis major axis 1 See Am. Mach., Nov. 25, 1909. 90 MACHINE DESIGN It will be interesting to compare values of S in Table XXVII with those obtained by experiment so as to determine whether S corresponds to the tensile strength of the metal or to the modulus of rupture in cross breaking. PROBLEMS 1. Calculate the thickness of a steam-chest cover 12X16 in. to sustain a pressure of 90 Ib. per square inch with a factor of safety = 10. 2. Calculate the thickness of a circular manhole cover of cast iron 18 in. in diameter to sustain a pressure of 200 Ib. per square inch with a factor of safety = 8, regarding the edges as merely supported. 3. Determine the probable breaking load for a plate 18X24 in. loaded at the center, (a) when edges are fixed. (6) When edges are supported. 4. In experiments on steam cylinders, a head 12 in. in diameter and 1.18 in. thick failed under a pressure of 900 Ib. per square inch. Determine the value of S by formula (34). REFERENCES Mechanics of Materials, Merriman, Chapter XIV. Strength of Materials, Slocum and Hancock, Chapters VII and VIII. Details of High-pressure Piping. Cass. June, 1906. Design and Construction of Piping. Eng. Mag., April, 1908. Piping for High Pressures. Power, Sept. 22, 1908. Flanges for High Pressures. Power, July, 1905; Dec., 1905. High-pressure Tests of Large Pipes. Eng. News, Apr. 15, 1909. CHAPTER IV FASTENINGS 33. Bolts and Nuts. Tables of dimensions for U. S. standard bolt heads and nuts are to be found in most engineering hand- books and will not be repeated here. These proportions have not been generally adopted on account of the odd sizes of bar required. The standard screw-thread has been quite generally accepted as superior to the old V-thread. Roughly the diameter at root of thread is 0.83 of the outer diameter in this system, and the pitch in inches is given by the formula (45) where d = outer diameter. TABLE XXVIII SAFE WORKING STRENGTH OF IRON OR STEEL BOLTS Safe load in tension, Safe load in shear, Diam. of bolt, inch Threads per inch, No. Diam. at root of thread, inches Area at root of thread, sq. in. pounds pounds 5,000 Ib. 7,500 Ib. 4,000 Ib. 6,000 Ib. per sq. in. per sq. in. per sq. in. per sq. in. i 20 .185 .0269 135 202 196 294 A 18 .240 .0452 226 340 306 . 460 i 16 .294 .0679 340 510 440 660 i 7 s 14 .344 .0930 465 695 600 900 * 13 .400 .1257 628 940 785 1,175 91 92 MACHINE DESIGN TABLE XXVIII (Continued) SAFE WORKING STRENGTH OF IRON OR STEEL BOLTS Safe load in tension, Safe load in shear, Diam. of bolt, i_ Threads per inch, XT Diam at root of thread, Area at root of thread, pounds pounds | men .No. inches sq. in. 5,000 Ib. 7,500 Ib. 4,000 Ib. 6,000 Ib. per sq. in. per sq. in. per sq. in. per sq. in. A 12 .454 .162 810 1,210 990 1,485 1 11 .507 .202 1,010 1,510 1,230 1,845 i 10 .620 .302 1,510 2,260 1,770 2,650 I 9 .731 .420 2,100 3,150 2,400 3,600 i 8 .837 .550 2,750 4,120 3,140 4,700 H 7 .940 .694 3,470 5,200 3,990 6,000 U 7 1.065 .891 4,450 6,680 4,910 7,360 if 6 1.160 1.057 5,280 7,920 5,920 7,880 H 6 1.284 1.295 6,475 9,710 7,070 10,600 If 54 1.389 1.515 7,575 11,350 8,250 12,375 if 5 1.490 1.744 8,720 13,100 9,630 14,400 U 5 1.615 2.049 10,250 15,400 11,000 16,500 2 4i 1.712 2.302 11,510 17,250 12,550 18,800 The shearing load is calculated fiom the area of the body of the bolt. Bolts may be divided into three classes which are given in the order of their merit. 1. Through bolts, having a head on one end and a nut on the other. 2. Stud bolts, having a nut on one end and the other screwed into the casting. 3. Tap bolts or screws having a head at one end and the other screwed into the casting. The principal objection to the last two forms and especially to (3) is the liability of sticking or breaking off in the casting. Any irregularity in the bearing sui faces of head or nut where they come in contact with the casting, causes a bending action and consequent danger of rupture. This is best avoided by having a slight annular projection on the casting concentric with the bolt hole and finishing the flat surface by planing or counter-boring. Counter-boring without the projection is a rather slovenly way of over coming the difficulty. BOLTS AND NUTS 93 When bolts or studs are subjected to severe stress and vibration, it is well to turn down the body of the bolt to the diameter at root of thread, as the whole bolt will then stretch slightly under the load. A check nut is a thin nut screwed firmly against the main nut to prevent its working loose, and is commonly placed outside. As the whole load is liable to come on the outer nut, it would be more correct to put the main nut outside. (Prove this by figure.) After both nuts aie firmly screwed down, the outer one should be held stationary and the inner one reversed against it with what force is deemed safe, that the greater reaction may be between the nuts. Numerous devices have been invented for the purpose of hold- ing nuts from working loose under vibration but none of them are entirely satisfactory. Probably the best method for large nuts is to drive a pin or cotter entirely through nut and bolt. A flat plate, cut out to embrace the nut and fastened to the casting by a machine screw, is often Used. Machine Screws. A screw is distin- guished from a bolt by having a slot- ted, round head instead of a square or hexagon head. The head may have any one of four F IG> 26. shapes, the round, fillister, oval fillister and flat as shown in Fig. 26. A committee of the American Society of Mechanical Engineers has recently recommended certain standards for machine screws. The form of thread recommended is the U. S. Standard or Sellers type with provi- sion for clearance at top and bottom to insure bearing on the body of the thread. The sizes and pitches recommended are shown in Table XXIX. In designing eye-bolts it is customary to make the combined sectional area of the sides of the eye one and one-half-times that of the bolt to allow for obliquity and an uneven distribution of stress. 94 MACHINE DESIGN TABLE XXIX MACHINE SCREWS Standard diam. .070 .085 .100 .110 .125 .140 .165 .190 .215 .240 .250 .270 .320 .375 Threads per in. 72 64 56 48 44 40 36 32 28 24 24 22 20 16 Reference is made to the report itself for further details of heads, taps, etc. 34. Crane Hooks. Heretofore, the large wrought-iron or steel hooks used for crane service have usually been designed by con- sidering the fibers on the inside of a hook to be subjected to a tension which was the resultant of the direct load and of the bending due to the eccentricity of the loading. Experiments made by Professor Rautenstrauch in 1909 1 show that such methods do not give correct results. Ten hooks of various capacities were tested by direct loading and their elastic limits determined. The following table gives the leading data and results. The dimensions are those of the principal cross-section: TABLE XXX ELASTIC LIMIT OF CRANE HOOKS Nominal Cross-section dimensions Elastic capacity, Material limit, tons Ib. A / I y 30 C. steel . . . 23.35 111.6 7.25 3.36 56,000 20 C. steel . . . 14.48 5.90 2.75 30,000 15 C. steel . . . 13.92 5.13 2.23 48,000 15 W. iron... 8.40 11.9 5.00 1.87 16,000 10 C. steel . . . 8.72 4.30 2.05 18,000 10 W. iron... 6.08 6.5 4.00 1.50 16,000 5 C. steel. . . 5.69 3.25 1.42 18,000 5 W. iron . . . 4.80 3.8 3.47 1.35 14,000 3 C. steel... 3.50 2.89 1.16 8,500 2 C. steel. . . 2.03 2.03 0.88 4,700 1 Am. Mach., Oct. 7, 1909. CRANE HOOKS 95 A area in square inches 1 = moment of inertia about gravity axis 1= distance from load line to gravity axis y = distance from inner fiber to gravity axis. It will be noticed that the nominal capacity of the hook is in several cases greater than the elastic limit as shown by experi- ment. This is particularly true ot the larger sizes. The standard cross-section of crane hooks is that of a trapezoid with curved bases as shown in Fig. 27. The wider base corre- sponds to the inner side of the hook where the tension is greatest. The dimensions given are approxi- mately those of a 20-ton steel hook. Professor Rautenstrauch finds that the values of the load at elastic limit, as de- termined by the ordinary formula above alluded to, are entirely erroneous, being in many cases more than twice that found by the actual tests. He recommends in- stead the so-called Andrews-Pearson for- mula which takes into account the curva- ture of the neutral axis and the lateral distortion of the metal. The discussion is too long for reproduc- tion here and reference is made to his paper and to the original presentation of this formula. 1 A similar condition exists in large chain links. The bending moment in this case is, however, usually eliminated by the insertion of a cross piece or strut. 2 PROBLEMS 1. Discuss the effect of the initial tension caused by the screwing up of the nut on the bolt, in the case of steam fittings, etc.; i.e., should this tension be added to the tension due to the steam pressure, in determining the proper size of bolt? 1 Technical Series 1, Draper Company's Research Memoirs, 1904. See also Slocum and Hancock's Strength of Materials. 2 See Univ. of Illinois, Bulletin No. 18. " The Strength of Chain Links," by G. A. Goodenough and L. E. Moore. 3.25-4 FIG. 27. 20-ton steel hook. 96 MACHINE DESIGN i II 2. Determine the number of J-in. steel bolts necessary to hold on the head of a steam cylinder 18 in. diameter, with the internal pressure 90 Ib. per square inch, and factor of safety = 12. 3. Show what is the proper angle between the handle and the jaws of a fork wrench. (1) If used for square nuts. (2) If used for hexagon nuts; illustrate by figure. 4. Determine the length of nut theoretically necessary to prevent stripping of the thread, in terms of the outer diameter of the bolt. (1) With U. S. standard thread. (2) With square thread of same depth. 5. Design a hook with a swivel and eye at the top to hold a load of 10 tons with a factor of safety 5, the center line of hook being 8 in. from line of load, and the material soft steel. 35. Riveted Joints. Riveted joints may be divided into two general classes: lap joints where the two plates lap over each other, and butt joints where the edges of the plates butt ___ together and are joined by j A Q Oi^ over-lapping straps or welts. If the strap is on one side only, the joint is known as a butt joint with one strap: if straps are used inside and out the joint is called a butt joint with two straps. Butt joints are generally used when the material is more than \ in. thick. Any joint may have one, two or more rows of rivets and hence be known as a single riveted joint, a double riveted joint, etc. Any riveted joint is weaker than the original plate, simply because holes cannot be punched or drilled in the plate for the introduction of rivets without removing some of the metal. Fig. 28 shows a double riveted lap joint and Fig. 29 a single riveted butt joint with two straps. ^^m ^ %l^SS <;yj^ -^ %f^ S FIG. 28. c c FIG. 29. RIVETED JOINTS 97 Riveted joints may fail in any one of four ways: 1. By tearing of the plate along a line of rivet holes, as at A B, Fig. 28. 2. By shearing of the rivets. 3. By crushing and wrinkling of the plate in front of each rivet as at C, Fig. 28, thus causing leakage. 4. By splitting of the plate opposite each rivet as at D, Fig. 28. The last manner of failure may be prevented by having a suffi- cient distance from the rivet to the edge of the plate. Practice has shown that this distance should be at least equal to the diameter of a rivet. Experience has shown that lap joints in plates of even moderate thickness are dangerous on account of the liability of hidden cracks. Several disastrous boiler explosions have resulted from the presence of cracks inside the joint which could not be detected by inspection. The fact that one or both plates are out of the line of pull brings a bending moment on both plates and rivets. Some boiler inspectors have gone so far as to condemn lap joints altogether. Let = thickness of plate d = diameter of rivet hole p = pitch of rivets n = number of rows of rivets T = tensile strength of plate C = crushing strength of plate or rivet S= shearing strength of rivet. Average values of the constants are as follows: Material T C s Wrought iron Soft steel 50,000 56000 80,000 90000 40,000 45 000 The values of the constants given above are only average values and are liable to be modified by the exact grade of material used and the manner in which it is used. The tensile strength of soft steel is reduced by punching from 98 MACHINE DESIGN 3 to 12 per cent according to the kind of punch used and the width of pitch. The shearing strength of the rivets is diminished by their tendency to tip over or bend if they do not fill the holes, while the bearing or compression is doubtless relieved to some extent by the friction of the joint. The values given allow roughly for these modifications. 36. Lap Joints. This division also includes butt joints which have but one strap. Let us consider the shell as divided into strips at right angles to the seam and each of a width = p. Then the forces acting on each strip are the same and we need to consider but one strip. The resistance to tearing across of the strip between rivet holes is (p-d)tT (a) and this is independent of the number of rows of rivets. The resistance to compression in front of rivets is ndtC (b) and the resistance to shearing of the rivets is ^nd'S. (c) If we call the tensile strength T = unity then the relative values of C and S are 1.6 and 0.8 respectively. Substituting these relative values of T, C and S in our equations, by equating (b) and (c) and reducing we have d = 2.55Z (46) Equating (a) and (c) and reducing we have (47) t Or by equating (a) and (b) p = d + l.Gnd (48) These proportions will give a joint of equal strength throughout, for the values of constants assumed. 37. Butt Joints with Two Straps. In this case the resistance to shearing is increased by the fact that the rivets must be sheared RIVETED JOINTS 99 at both ends before the joint will fail. Experiment has shown this increase of shearing strength to be about 85 per cent and we can therefore take the relative value of S as 1.5 for butt joints. This gives the following values for d and p d = lMt (49) t?r/ 2 p = d + 1.18 (50) 6 p = d + l.6nd. . (51) In the preceding formulas the diameter of hole and rivet have been assumed to be the same. The diameter of the cold rivet before insertion will be y 1 ^ in. less than the diameter given by the formulas. Experiments made in England by Prof. Kennedy give the following as the proportions of maximum strength : Lap joints d = 2.33t p = d + Butt joints d = l.St 38. Efficiency of Joints. The efficiency of joints designed like the preceding is simply the ratio of the section of plate left between the rivets to the section of solid plate, or the ratio of the clear distance between two adjacent rivet holes to the pitch. From formula (48) we thus have: Efficiency = . (52) This gives the efficiency of single, double and triple riveted seams as .615, .762 and .828 respectively. Notice that the advantage of a double or triple riveted seam over the single is in the fact that the pitch bears a greater ratio to the diameter of a rivet, and therefore the proportion of metal removed is less. 100 MACHINE DESIGN 39. Butt Joints with Unequal Straps. One joint in common use requires special treatment. It is a double riveted butt joint in which the inner strap is made wider than the outer and an extra row of rivets added, whose pitch is double that of the original seam; this is sometimes called diamond riveting. See Fig. 30. This outer row of rivets is then exposed to single shear and the original rows to dou- ble shear. Consider a strip of plate of a width = 2p. Then the resis- tance to tearing along the outer row of rivets is (2p-d)tT As there are five rivets to compress in this strip the bearing resistance is __ 5dtC As there is one rivet in single shear and four in double shear the resistance to shearing is + (4X1.85) | 7t 4 d 2 S = QM 2 S Solving these equations as in previous cases, we have for this particular joint d = 1.52t (53) 1 1 > } 1 1 1 ) o o o' ii o o o 1- o o o it )" o o o 1 ^> 1 o o 1 J J iHI Vffi . 2p-d 8 Efficiency ==-^= - (54) (55) 40. Practical Rules. The formulas given above show the proportions of the usual forms of joints for uniform strength. In practice certain modifications are made for economic reasons. To avoid great variation in the sizes of rivets the latter are graded by sixteenths of an inch, making those for the thicker plates con- RIVETED JOINTS 101 siderably smaller than the formula would allow, and the pitch is then calculated to give equal tearing and shearing strength. Table XXXI shows what may be considered average practice in this country for lap joints with steel plates and rivets. TABLE XXXI RIVETED LAP JOINTS Thick- Diam. Diam. Pitch Efficiency of plate ness of of of plate rivet hole Single Double Single Double t * A If If .59 .68 A f H If 2* .58 .68 I 1 If 2J .57 .67 T 7 ff it 1 2 2f .56 .68 i 1 if 2 2f .53 .67 The efficiencies are calculated from the strength of plate between rivet holes and the efficiencies of the rivets may be even lower. Comparing these values with the ones given in Art. 38 we find them low. This is due to the fact that the pitches assumed are too small. The only excuse for this is the possibility of getting a tighter joint. TABLE XXXII RIVETED BUTT JOINTS Pitch Thickness of Diam. of Diam. of plate rivet hole Single Double Triple t f U 2f 4 5i f 13 16 ' 1 2| 3| 5t 1 1 11 2| 3| 6* 1 If 1 2f 31 5 1 1 IA 2f 3f 5 102 MACHINE DESIGN Table XXXII has been calculated for butt joints with two straps. As in the preceding table the values of the pitch are too small for the best efficiency. The tables are only intended to illustrate common practice and not to serve as standards. There is such a diversity of practice 'among manufacturers that it is advisable for the designer to proportion each joint according to his own judgment, using the rules of Arts. 36-39 and having regard to the practical considerations which have been mentioned. A committee of the Master Steam Boiler Makers' Association has made a number of tests on riveted joints and reported its conclusions. The specimens were prepared according to generally accepted practice, but on subjecting them to tension many of them failed by tearing through from hole to edge of plate. The committee recommends making this distance greater, so that from the center of hole to edge of plate shall be perhaps 2d instead of 1.5d. The committee further found the shearing strength of rivets to be in pounds per square inch of section. Single shear Double shear Iron rivets Steel rivets 40,000 49 000 78,000 84 000 Compare these values with those given in Art. 35. Also note that the factor for double shear is 1.95 for iron rivets and only 1.71 for steel rivets as against the 1.85 given in Art. 37. The committee found that machine-driven rivets were stronger in double shear than hand-driven ones. PROBLEMS 1. Calculate diameter and pitch of rivets for J-in. and -in. plate and compare results with those in Table XXXI. Criticise latter. 2. Show the effect in Prob. 1 of using iron rivets in steel plates. 3. Criticise proportions of joints for J-in. and 1-in. plate in Table XXXII by testing the efficiency of rivets and plates. 4. A cylinder boiler 5X16 ft. is to have long seams double riveted laps and ring seams single riveted laps. If the internal pressure is 90 Ib. gage pressure and the material soft steel, determine thickness of plate and pro- portion of joints. The net factor of safety at joints to be 5. RIVETED JOINTS 103 5. A marine boiler is 13 ft. 6 in. in diameter and 14 ft. long* The long seams are to be diamond riveted butt joints and the ring seams ordinary double riveted butt joints. The internal pressure is to be 175 Ib. gage and the material is to be steel of 60,000 Ib. tensile strength. Determine thick- ness of shell and proportions of joints. Net factor of safety to be 5, as in Prob. 4. r\ ^///////7////////////////////y//// r\ FIG. 31. 6. Design a diamond riveted joint such as shown in Fig. 31 for a steel plate f in. thick. Outer cover plate is f in. and inner cover plate is T V in. thick; the pitch of outer rows of rivets to be twice that of inner rows. Determine efficiency of joints. 7. The single lap joint with cover plate, as shown in Fig. 32, is to have pitch of outer rivets double that of inner row. Determine diameter and pitch of rivets for |-in. plate and the efficiency of joint. 41. Riveted Joints for Narrow Plates. The joints which have been so far described are continuous and but one strip of a width equal to the pitch or the least common multiple of several pitches, has been investigated. When narrow plates such as are used in structural work are to be joined by rivets, the joint is designed as a whole. Diamond riveting similar to that shown in Fig. 30 is generally used and the joint may be a lap, or a butt with double straps. The diameter of rivets may be taken about 1.5 times the thickness of plate [see equation (53)], and enough rivets used so that the total shearing strength may equal the tensile strength of the plate at the point of the diamond, where there is one rivet hole. It may be neces- sary to put in more rivets of a less diameter in order to make the figure symmetrical. 104 MACHINE DESIGN The efficiency of the joint may be tested at the different rows of rivets, allowing for tension of plate and shear of rivets in each case. PROBLEMS 1. Design a diamond riveted lap joint for a plate 12 in. wide and f in. thick, and calculate least efficiency for shear and tension. 2. A diamond riveted butt joint with two straps has rivets arranged as in Fig. 33, the plate being 12 in. wide and f in. thick, and the rivets being 1 in. in diameter. If the plate and rivets are of steel, find the probable ultimate strength of the following parts: (a) The whole plate. (6) All the rivets on one side of the joint. (c) The joint at the point of the diamond. (d) The joint at the row of rivets next the point. o o o 000 o o o o o o FIG. 33. 42. Joint Pins. A joint pin is a bolt exposed to double shear. If the pin is loose in its bear- ings it should be designed with allowance for bending, by adding from 30 to 50 per cent to the area of cross-section needed to resist shearing alone. Bending of the pin also tends to spread apart the bearings and this should be prevented by having a head and nut or cotter on the pin. If the pin is used to connect a knuckle joint as in boiler stays, the eyes forming the joint should have a net area 50 per cent in excess of the body of the stay, to allow for bending and uneven tension (see Eye-bolts, Art. 33). Fig. 34 shows a pin and angle joint for attaching the end of a boiler stay to the head of the boiler. 43. Cotters. A cotter is a key which passes diametrically through a hub and its rod or shaft, to fasten them together, and is so called to distinguish it from shafting keys which lie parallel to axis of shaft. Its taper should not be more than 4 degrees or about 1 in 15, unless it is secured by a screw or check nut. The rod is sometimes enlarged where it goes in the hub, so that the effective area of cross-section where the cotter goes through may be the same as in the body of the rod. (See Fig. 35.) COTTERS 105 Let: d = diameter of body of rod d^ = diameter of enlarged portion t thickness of cotter, usually =- b = breadth of cotter 1 = length of rod beyond cotter. Suppose that the applied force is a pull on the rod causing tension on the rod and shearing stress on the cotter. The effective area of cross-section of rod at cotter is nan FZG. 34. FIG. 35. nd* d*_ Q Snuds osoio 0} ptK>T pasoio - H 30 uai\[ 30 CO rH rt (M (M r-l i-l St Ot J-t 7-H T-H r-c ^< ^1 i-l t- T-I TH T-H rH OTtMecccinoti-^'eomeceocceo-'i'cococo! dnoj) HELICAL SPRINGS 111 The apparent variation of G in the experiments 'is probably due to differences in the quality of steel and to the fact that the formula for G in the case of helical springs is an approximate one. The same may be said of the values of S, but if these values are used in designing similar springs one error will off-set the other. In some few cases, as in No. 18, it was necessary to use an abnormally high value of S to meet the conditions. This neces- sitated a special grade of steel, and great care in manufacture. Such a spring is not safe when subjected to sudden and heavy loads, or to rapid vibrations, as it would soon break under such treatment; if merely subjected to normal stress, it would last for years. Springs of a small diameter may safely be subjected to a higher stress than those of a larger diameter, the size of bar being the same. The safe variation of S with R cannot yet be stated. There is an important limit which should be here mentioned. Springs having two small a diameter as compared with size of bar are subjected to so much internal stress in coiling as to weaken the steel. A spring, to give good service, should never have R less than 3. The size of bar has much to do with the safe value of S; the probable explanation is this: A large bar has to be heated to a higher temperature in working it, and in high carbon steel this may cause deterioration; when tempered, the bath does not affect it so uniformly, as may be seen by examining the fracture of a large bar. The above facts must always be taken into consideration in designing a spring, whatever the grade of steel used. A safe value of S can be determined only by one having an accurate knowledge of the physical characteristics of the steel, the pro- portions of the spring, and the conditions of use. For a good grade of steel the values of S on p. 112 have been found safe under ordinary conditions of service, the value of G being taken as 14,500,000. For bars over 1| in. in diameter a stress of more than 100,000 should not be used. Where a spring is subjected to sudden shocks a smaller value of S is necessary. As has been noted, the springs referred to in this paper were all compression springs. Experience has shown that in close 112 MACHINE DESIGN coil or extension springs the value of G is the same, but that the safe value of S is only about two-thirds that for a compression spring of the same dimensions. VALUES OF S ... d in. or less i in \jQ 3 in 112,000 110,000 85,000 80,000 f in. to \\ in 105,000 75,000 47. Spring in Torsion. If a helical spring is used to resist torsion instead of tension or compression, the wire itself is subjected to a bending moment. We will use the same notation as in the last article, only that P will be taken as a force acting tangentially to the circumference of the spring at a distance -~- from the axis, and S will now be the safe transverse strength of the wire, having the following values: $ = 60,000 for spring brass wire = 90,000 to 125,000 for cast steel tempered # = 9,000,000 for spring brass wire = 30,000,000 for cast steel tempered. Let 6 = angle through which the spring is turned by P. The bending moment on the wire will be the same throughout PD and = - This is best illustrated by a model. t To entirely straighten the wire by unwinding the spring would require the same force as to bend straight wire to the curvature of the helix. To simplify the equations we will disregard the obliquity of the helix, then will l = nDn and the radius of curvature D ~ 2" Let M = bending moment caused by entirely straightening the wire; then by mechanics . El 2EI HELICAL SPRINGS 113 and the corresponding angle through which spriifg is turned is 2/m. But it is assumed that a force P with a radius -= turns the t spring through an angle 6. PD 2EI -0 X 2 D _ EIO _EId Solving for 0: ' PDl V = ~-( 3n' X PI* W + 2n 7 V Enbt 3 The remaining deflection or that due to the application of the load P is: But 3ri + 2n"I Enbt 3 6 Pl3 where P'+P^^load at each end of spring and (P'+P")Z = bending moment at band. Substituting in (j) and reducing: 2 SZ 2 , PQ . - (69) If all the leaves are full length : ELLIPTIC SPRINGS 119 If all the leaves are graduated: r = and A = -=- Hit PROBLEMS 1. A spring balance is to weigh 50 Ib. with an extension of 2 in., the diameter of spring being f in. and the material, tempered steel. Determine the diameter and length of wire, and number of coils. 2. Determine the safe twisting moment and angle of torsion for the spring in example 1, if used for a torsional spring. 3. Test values of G and S from data given in Table XXXIII. 4. By using above table design a spring 8 in. long to carry a load of 2 tons without closing the coils more than half way. 5. Design a compound flat spring for a locomotive to sustain a load of 16,000 Ib. at the center, the span being 40 in., the number of leaves 12 and the material steel. 6. Determine the maximum deflection of the above spring, under the working load. 7. A semi-elliptic spring has 9 leaves in all and 6 graduated leaves, and the load on each end is P=4000 Ib. Develop formulas for the fiber stress in each set of leaves if there is no initial stress. Determine proper breadth and thickness of leaves if length of span is 42 in. 8. In Prob. 7 develop a formula for the necessary gap to equalize the fiber stresses. 9. In Prob. 8 determine the pull on the band due to the initial stress. 10. A semi-elliptic spring has 4 leaves 36 in. long, and 12 graduated leaves. The leaves are all 4 in. wide and f in. thick, and the band at the center is 4 in. wide. If there is no initial stress find the share of the load and the fiber stress on each set of leaves when there is a load of 6 tons on the center. Also determine deflection. 11. In Prob. 10, determine the amount of gap needed to equalize the stresses in the two sets of leaves, and the pull on the band at the center. Determine the deflection under the load. 12. Measure various indicator springs and determine value of G from rating of springs. 13. Measure various brass extension springs, calculate safe static load and safe stretch. 14. Make an experiment on torsion spring to determine distortion under a given load and calculate value of E. REFERENCES Vibration of Springs. Am. Mach., May 11, 1905. Tables of Loads and Deflections. Am. Mach., Dec. 20, 1906. The Constructor. Reuleaux. CHAPTER VI SLIDING BEARINGS 50. Slides in General. The surfaces of all slides should have sufficient area to limit the intensity of piessure and prevent forcing out ol the lubricant. No general rule can be given for the limit of pressure. Tool marks parallel to the sliding motion should not be allowed, as they tend to start grooving. The sliding piece should be as long as practicable to avoid local wear on stationary piece and for the same reason should have sufficient stiffness to prevent springing. A slide which is in continuous motion should lap over the guides at the ends of stroke, to prevent the wearing of shoulders on the latter and the finished surfaces of all slides should have exactly the same width as the surfaces on which they move for a similar reason. Where there are two parallel guides to motion as in a lathe or planer it is better to have but one of these depended upon as an accurate guide and to use the other merely as a support. It must be remembered that any sliding bearing is but a copy of the ways of the machine on which.it was planed or ground and in turn may reproduce these same errors in other machines. The interposition of h and -sci aping is the only cure for these hereditary complaints. In designing a slide one must consider whether it is accuracy of motion that is sought, as in the ways of a planer or lathe, or accuracy of position as in the head of a milling machine. Slides may be divided according to their shapes into angular, flat and circular slides. 51. Angular Slides. An angular slide is one in which the guiding surface is not normal to the direction of pressure. There is a tendency to displacement sideways, which necessitates a second guiding surface inclined to the first. This oblique pressure constitutes the principal disadvantage of angular slides. 120 ANGULAR SLIDES 121 Their principal advantage is the fact that they are 'either self- adjusting for wear, as in the ways of lathes and planers, or require at most but one adjustment. Fig. 39 shows one of the V's of an ordinary planing machine. The platen is held in place by gravity. The angle between the two surfaces is usually 90 degrees but may be more in heavy machines. The gro.oves g, g are intended to hold the oil in place; oiling is sometimes effected by small rolls recessed into the lower piece and held against the platen by springs. The principal advantage of this form of way is its ability to hold oil and the great disadvantage, its faculty for catching chips and dirt. Fig. 40 shows an inverted V such as is common on the ways of engine lathes. The angle is about the same as in the preceding form but the top of the V should be rounded as a precaution against nicks and bruises. FIG. 39. FIG. 40. The inverted V is preferred for lathes since it will not catch dirt and chips. It needs frequent lubrication as the oil runs off rapidly. Some lathe carriages are provided with extensions filled with oily felt or waste to protect the ways from dirt and keep them wiped and oiled. Side pressure tends to throw the carriage from the ways; this action may be prevented by a heavy weight hung on the carriage or by gibbing the carriage at the back (see Fig. 46). The objection to this latter form of con- struction is the fact that it is practically impossible to make and keep the two V's and the gibbed slide all parallel. Fig. 41 shows a compound V sometimes used on heavy ma- chines. The obtuse angle (about 150 degrees) takes the heavy 122 MACHINE DESIGN vertical pressure, while the sides, inclined only 8 or 10 degrees, take any side pressure which may develop. 52. Gibbed Slides. All slides which are not self-adjusting for wear must be provided with gibs and adjusting screws. Fig. 42 shows the most common form as used in tool slides for lathes and planing machines. g FIG. 41. FIG. 42. The angle employed is usually 60 degrees; notice that the corners c c are clipped for strength and to avoid a corner bearing; notice also the shape of gib. It is better to have the points of screws coned to fit gib and not to have flat points fitting recesses in gib. The latter form tends to spread the joint apart by forcing the gib down. If the gib is too thin it will spring under the screws ^^ and cause uneven wear. \~ f^5\ The cast-iron gib, Fig. 43, is free from this latter defect but makes the slide rather clumsy. The screws, however, are more accessible in this form. Gibs are sometimes made slightly tapering and adjusted by a screw and nut giving endwise motion. FIG. 43. 53. Flat Slides. This type of slide requires adjustment in two directions and is usually provided with gibs and adjusting screws. Flat ways on machine tools are the rule in English practice and are gradually coming into use in this country. Although more expensive at first and not so simple they are more durable and usually more accurate than the angular ways. Fig. 44 illustrates a flat way for a planing machine. The other FLAT SLIDES 123 way would be similar to this but without adjustment. The normal pressure and the friction are less than with angular ways and no amount of side pressure will lift the platen from its position. FIG. 44. FIG. 45. Fig. 45 shows a portion of the ram of a shaping machine and illustrates the use of an L gib for adjustment in two directions. Fig. 46 shows a gibbed slide for holding down the back of a lathe carriage with two adjustments. LnJ FIG. 46. FIG. 47. The gib g is tapered and adjusted by a screw and nuts. The saddle of a planing machine or the table of a shaper usually has a rectangular gibbed slide above and a taper slide below, this form of the upper slide being necessary to hold the weight of the 124 MACHINE DESIGN overhanging metal (see Fig. 47). Some lathes and planers are built with one V or angular way for guiding the carriage or platen and one flat way acting merely as a support. B 54. Circular Guides. Examples of this form may be found in the column of the drill press and the overhanging arm of the milling machine. The cross heads of steam engines are sometimes fitted with circular guides; they are more frequently flat or angular. One advantage of the circular form is the fact that the cross F IG 48< head can adjust itself to bring the wrist pin parallel to the crank pin. The guides can be bored at the same setting as the cylinder in small engines and thus secure good alignment. Fig. 48 illustrates various shapes of cross head slides in common use. 55. Stuffing Boxes. In steam engines and pumps the glands for holding the steam and water packing are the sliding bearings FIG. 49.' which cause the greatest friction and the most trouble. Fig. 49 shows the general arrangement. B is the stuffing box attached to the cylinder head; R is the piston rod; G the gland adjusted by STUFFING BOX 125 nuts on the studs shown; P the packing contained in, a recess in the box and consisting of rings, either of some elastic fibrous material like hemp and woven rubber cloth or of some soft metal like Babbitt metal. The pressure between the packing and the rod, necessary to prevent leakage of steam or water, is the cause of considerable friction and lost work. Experiments made from time to time in the laboiatories of the Case School have shown the extent and manner of variation of this friction. The results for steam packings may be summarized as follows: 1. That the softer rubber and graphite packings, which are self- adjusting and self -lubricating, as in Nos. 2, 3, 7 7 8, and 11, con- sume less power than the harder varieties. No. 17, the old braided flax style, gives very good results. (See Table (XXXIV.) 2. That oiling the rod will reduce the friction with any packing, 3. That there is almost no limit to the loss caused by the injudicious use of the monkey-wrench. 4. That the power loss varies almost directly with the steam pressure in the harder varieties, while it is approximately con- stant with the softer kinds. The diameter of rod used 2 in. would be appropriate for engines from 50 to 100 horse-power. The piston speed was about 140 ft. per minute in the expeiiments, and the horse-power varied from .036 to .400 at 50 Ib. steam pressure, with a safe average for the softer class of packings of .07 horse-power. At a piston speed of 600 ft. per minute, the same friction would give a loss of from .154 to 1.71 with a working average of .30 horse-power, at a mean steam pressure of 50 Ib. In Table XXXIV Nos. 6, 14, 15, and 16 are square, hard rubber packings without lubricants. Similar experiments on hydraulic packings under a water pressure varying from 10 to 80 Ib. per square inch gave results as shown in Table XXXVI. The figures given are for a 2-in. rod running at an average piston speed of 140 ft. per minute. 126 MACHINE DESIGN TABLE XXXIV Average Horse- Total horse- power Kind of packing No trials time of run in power con- sumed con- sumed at 50 Remarks on leakage, etc. minutes by each Ib. pres- box sure 1 5 22 .091 .085 Moderate leakage. 2 8 40 .049 .048 Easily adjusted; slight leakage. 3 5 25 .037 .036 Considerable leakage. 4 5 25 .159 .176 Leaked badly. 5 5 25 .095 .081 Oiling necessary; leaked badly. 6 5 25 .368 .400 Moderate leakage. 7 5 25 .067 .067 Easily adjusted and no leakage. 8 5 25 .082 .082 Very satisfactory; slight leakage. 9 3 15 .200 .182 Moderate leakage. 10 3 .275 Excessive leakage. 11 5 25 .157 .172 Moderate leakage. 12 5 25 .266 .330 Moderate leakage. 13 5 25 .162 .230 No leakage; oiling necessary. 14 5 25 .176 .276 Moderate leakage; oiling necessary 15 5 25 .233 .255 Difficult to adjust; no leakage. 16 5 25 .292 .210 Oiling necessary; no leakage. 17 5 25 .128 .084 No leakage. TABLE XXXV Kind of packing Horse-power consumed by each box, when pressure was applied to gland nuts by a 7-in. wrench Horse-power before and after oiling rod 51b. 81b. 10 Ib. 12 Ib. 14 Ib. 16 Ib. Dry Oiled 1 3 4 5 6 7 8 9 11 12 13 15 16 17 .120 .136 .021 .123 .055 .154 .248 .220 .348 .126 .363 .666 .430 .228 .500 .303 .260 .535 .330 .520 .390 .340 .533 .323 .067 .533 .666 .454 .454 .194 .053 .236 .636 .176 .122 .405 .161 .317 526 .454 .242 .394 .359 .582 .454 .327 .198 .860 .277 .380 ! STUFFING BOXES TABLE XXXVI 127 No. of packing Av. H. P. at 20 Ib. Av. H. P. at 70 Ib. Max. H. P. Min. H. P. Av. H. P. for entire test 1 .077 .351 .452 .024 .259 2 .422 .500 .512 .167 .410 3 .130 .178 .276 .035 .120 4 .184 .195 .230 .142 .188 5 .146 .162 .285 .069 .158 6 .240 .200 .255 .071 .186 7 .127 .192 .213 .095 .154 8 .153 .174 .238 .112 .165 9 .287 .469 .535 .159 .389 10 .151 .160 .226 .035 .103 11 .141 .156 .380 .064 .177 12 .053 .095 .143 .035 .090 Packings Nos. 5, 6, 10 and 12 are braided flax with graphite lubrication and are best adapted for low pressures. Packings Nos. 3, 4 and 7 are similar to the above but have paraffine lubri- cation. Packings Nos. 2 and 9 are square duck without lubri- cant and are only suitable for very high pressures, the friction loss being approximately constant. PROBLEMS Make a careful study and sketch of the i following machines and analyze as to (a) Adjustment, (d) Lubrication. 1. An engine lathe. 2. A planing machine. 3. A shaping machine. 4. A milling machine. 5. An upright drill. 6. A Corliss engine. 7. A locomotive engine. 8. A gas-engine. 9. An air-compressor. liding bearings on each of the Purpose. (6) Character, (c) CHAPTER VII JOURNALS, PIVOTS AND BEARINGS 56. Journals. A journal is that part of a rotating shaft which rests in the bearings and is of necessity a surface of revolution, usually cylindrical or conical. The material of the journal is generally steel, sometimes soft and sometimes hardened and ground. The material of the bearing should be softer than the journal and of such a quality as to hold oil readily. The cast metals such as cast iron, bronze and Babbitt metal are suitable on account of their poious, granular character. Wood, having the grain normal to the bearing surface, is used where water is the lubiicant, as in water wheel steps and stern bearings of propellers. Bearing materials may naturally be divided into soft and hard metals. The standard soft metal is so-called " genuine Babbitt," of the following composition: Tin, 85 to 89 per cent. Copper, 2 to 5 per cent. Antimony, 7 to 10 pei cent. The substitution of lead for tin and the omission of the copper makes a cheaper and softer metal suitable for low pressures and speeds. The addition of more antimony hardens the metal. The hard metals include the various brasses and bronzes tanging from soft yellow brass to phosphor and aluminum bronzes. Professor R. C. Carpenter recommends as suitable for a bearing an aluminum bronze whose composition is: Aluminum, 50 per cent. Zinc, 25 per cent. Tin, 25 per cent. This metal is light, fairly hard, and will not melt readily. 1 1 Trans. A. S. M. E., Vol. XXVII, p. 425. 128 JOURNAL BEARINGS 129 57. Adjustment. Bearings wear more or less lapidly with use and need to be adjusted to compensate for the wear. The adjustment must be of such a character and in such a direction as to take up the wear and at the same time maintain as far as possible the correct shape of the bearing. The adjustment should then be in the line of the greatest pressure. Fig. 50 illustrates some of the more common ways of adjusting a bearing, the arrows showing the direction of adjustment and presumably the direction of pressure; (a) is the most usual where the principal wear is vertical, (d) is a form frequently used on the main journals of engines when the wear is in two directions, FIG. 50. FIG. 51. horizontal on account of the steam pressure and vertical on account of the weight of shaft and fly-wheel. All of these are more or less imperfect since the bearing, after wear and adjust- ment, is no longer cylindrical but is made up of two or more approximately cylindrical surfaces. A bearing slightly conical and adjusted endwise as it wears, is probably the closest approximation to correct practice. Fig. 51 shows the main bearing of the Porter-Allen engine, one of the best examples of a four part adjustment. The cap is adjusted in the normal way with bolts and nuts; the bottom can be raised and lowered by liners placed underneath; the cheeks can be moved in or out by means of the wedges shown. Thus it is possible not only to adjust the bearing for wear but to align the shaft perfectly. A three part bearing for the main journal of an engine is 130 MACHINE DESIGN shown in Fig. 52. In this bearing there is one horizontal adjust- ment instead of two as in Fig. 51. The main bearing of the spindle in a lathe, as shown in Fig. 53, offers a good example of symmetrical adjustment. The head- FIG. 52. FIG. 53. stock A has a conical hole to receive the bearing B, which latter can be moved lengthwise by the nuts FG. The bearing may be split into two, three or four segments or it may be cut as shown in e, Fig. 50, and sprung into adjustment. A careful distinction FIG. 54. must be made between this class of bearing and that before mentioned, where the journal itself is conical and adjusted end- wise. A good example of the latter form is seen in the spindles of many milling machines. Fig. 54 shows the spindle of an engine lathe complete with its two bearings. The end thrust is taken by a fiber washer backed JOURNAL BEARINGS 131 by an adjusting collar and check nut. Both bearings belong to the class shown in Fig. 53. A conical journal with end adjustment is illustrated in Fig. 55, which shows the spindle of a milling machine. The front journal is conical and is adjusted for weai by drawing it back into its bearing with the nut. The rear journal on the other hand is cylindrical and its bearing is adjusted as are those just described. The end thrust is taken by two loose rings at the front end of the spindle. FIG. 55. 58. Lubrication. The bearings of machines which run inter- mittently, like most machine tools, are oiled by means of simple oil holes, but machinery which is in continuous motion as is the case with line shafting and engines requires some automatic system of lubrication. There is not space in this book for a detailed description of all the various types of oiling devices and only a general classification will be attempted. Lubrication is effected in the following ways: 1. By grease cups. By oil cups. By oily pads of felt or waste. By oil wells with rings or chains for lifting the oil. By centrifugal force through a hole in the journal itself. Grease cups have little to recommend them except as auxiliary safety devices. Oil cups are various in their shapes and methods of operation and constitute the cheap class of lubricating devices. They may be divided according to their operation into wick oilers, needle feed, and sight feed. The two first mentioned are nearly obsolete and the sight-feed oil cup, which drops the oil at regular 132 MACHINE DESIGN intervals through a glass tube in plain sight, is in common use. The best sight-feed oiler is that which can be readily adjusted as to time intervals, which can be turned on or off without dis- turbing the adjustment and which shows clearly by its appear- ance whether it is turned on. On engines and electric machinery which are in continuous use day and night, it is very important that the oiler itself should be stationary, so that it may be rilled without stopping the machinery. A modern sight-feed oiler for an engine is illustrated in Fig. 56. T is the glass tube where the oil drop is seen. The feed is regulated by the nut N t while the lever L shuts off the oil. Where the lever is as shown the oil is | dropping, when horizontal the oil is shut off. The nut can be adjusted once for all, and the position of the lever shows immediately whether or not the cup is in use. In modern engines particular attention has been paid to the problem of continuous oiling. The oil cups are all stationary and various ingenious devices are used for catching the drops of oil from the cups and distributing them to the bearing surfaces. For continuous oiling of stationary bearings, as in line shafting and electric machinery, an oil well below the bearing is preferred, with some automatic means of pumping the oil over the bearing, when it runs back by gravity into the well. Porous wicks and pads acting by capillary attraction are un- certain in their action and liable to become clogged. For bearings of medium size, one or more light steel rings running loose on the shaft and dipping into the oil, as shown in Fig. 57, are the best. For large bearings flexible chains are employed which take up less room than the ring. Cases have been reported, however, where suction oilers on line shafting have proved, more efficient than ring oilers. One instance is quoted where a suction oiler has been in continuous FIG. 56. FIG. 57. LUBRICATION 133 use for nearly thirty years and has worked perfectly during that entire period. 1 Much depends on the care of such devices, the prevalence or absence of dust, and the quality of oil used. Centrifugal oilers are most used on parts which cannot readily r\ FIG. 58. be oiled when in motion, such as loose pulleys and the crank pins of engines. Fig. 58 shows two such devices as applied to an engine. In A the oil is supplied by the waste from the main journal; in B an external sight-feed oil cup is used which supplies oil to the central revolving cup C. FIG. 59. Loose pulleys or pulleys running on stationary studs are best oiled from a hole running along the axis of the shaft and thence out radially to the surface of the bearing. See Fig. 59. A loose bushing of some soft metal perforated with holes is a good safety device for loose pulleys. 1 Trans. A. S. M. E., Vol. XXVII, p. 488. 134 MACHINE DESIGN Note: For adjustable pedestal and hanging bearings see the chapter on shafting. 59. Friction of Journals : Let TP = the total load of a journal in pounds =the length of journal in inches d = ihe diameter of journal in inches N = number of revolutions per minute v = velocity of rubbing in feet per minute F=friction at surface of journal in pounds = W tan W nearly, where W is the angle of repose for the two materials. If a journal is properly fitted in its bearing and does not bind, the value of F will not exceed W tan W and may be slightly less. The value of tan W varies according to the materials used and the kind of lubrication, from .05 to .01 or even less. See experiments described in Art. 62. The work absorbed in friction may be thus expressed : ndN TidNWtanW ., ,, . /rrrv . - C^ CO rH O5 CO CO : 9 ,__, 8 rH CO rH 1 ^ ^ r-T O" CO^ rHCOrH s (M : . r3 f N 55 rH CO CO O O 2$^ oo g (N Jc rH rH C^ rH c^ o^ o^ rH CO CO N C^l . ^^ s 1 O TJH r . 3 >O *" *~^ J>^ !> rH T^H CO ^^ *O C^ rH CO O5 CO rH "* rH _ CD^ rH rH^ 10 55 01 CO CO rH oo rH jl 'o 10 CO rH b. 8 CO 00 rH rH 00 O5 rJH (M 1> O Tfl IO rH l> rH rH (M CO : 5 s o 5 CO iO (N ^ | 00 H rH iT r-T CO (M ** 8 8 rH rH rH 2 O5 *O O O rH CO CO C^ |> r-T TH 05 IO CO (M 00 CO rH rH rH ^^ CO 8 B rH CO oo o o co r-T CO 00 CQ IO IO rH (M CO rH 1 OS rH co CO 02 03 H ^ to i> (N t^ to to "0 (M tO (N (M CO to TH O 00 (N to r^ CO 8 CO OS to to 2% t>. 0^ rH 00 CO rH CO O 00 rH (M I>- OS OS TH CO to oo oo - rH TH TH OS tO TH rH 00 rH TH CO 00 to 10 i (M I> CO (N X l> OS (N 00 OS (M CO TH rH 00 OS CO t^ OS (^ (N r ^ TH CO OS CO O t^ CO CO CO co co IO rH TH tO rH TH TH TH rH TH CO CO CO "o 00 T 1 t- ' t^ OS OS TH to t^ to t^ o g 00 S3 : TH rH rH s rH CO cq TH TH (N CO to ' (N CO TH 05 OS 00 g I s " O (M >o (M OS (N CO CO O rH CO (M >> (N TH TH CO rH CO o> TH 00 10 TH rH CO (M rH rH IO rH to +9 co o to (N CO 3 OS 00 CO TH 00 CO CO rH OS OS (N 'B r- : CO CO (N rH IO IO tO O TH O t>. t> TH r-~ CO TH CO ^f Q 55 (M to i_o rH TH jt C 00 rH OS (M rH TH TH (N rH rH (M CQ : A i a -P 'S S a 5 : 73 hf) ^ 2 ^H f^ p/* f_( rt ^j o3 v_x TTJ OQ O J3 o "^ o f- ^ H-^ bC E 4-2 i ! 1 | .2 'E 1 bO 15 B P 2 o T*. CQ u PH 2 d O ta^^ ^ Q i o .2 73 3.8 g & -a PH n, S S o ! PH *fi 73 ? bD ', H "' o < $ 4 1 't 1 5 S i ! += fc g s o *= 1 H 5 O 2 * PH PH ^ ^ 142 MACHINE DESIGN 63. Strength and Stiffness of Journals. A journal is usually in the condition of a bracket with a uniform load, and the bending * Wl moment M-- Therefore by formula (6) = 3/102M ./5.1TTZ " \ tf~ \ !o~~ or d= 1.721 V^- (75) The maximum deflection of such a bracket is Wl 3 A = SEI Wl 3 64 8#A 64W 2.547W1 3 If as is usual A is allowed to be T ^ in., then for stiffness E w or approximately d = 4 \/- (77) E The designer must be guided by circumstances in determining whether the journal shall be calculated for wear, for strength or for stiffness. A safe way is to design the journal by the formulas for heating and wear and then to test for strength and deflection. Remember that no factor of safety is needed in formula for stiffness. Note that W in formulas for strength and stiffness is not the average but the maximum load. 64. Caps and Bolts. The cap of a journal bearing exposed to upward pressure is in the condition of a beam supported by the holding down bolts and loaded at the center, and may be designed either for strength or for stiffness. JOURNALS 143 Let: P = max. upward pressure on cap L = distance between bolts b= breadth of cap at center h = depth of cap at center A greatest allowable deflection. Sbh 2 PL Strength: Af= __ = __ /Q~PT (78) \ ALftO TT7 T 3 Stiffness: A = (79) If A is allowed to be T | ir in. and E for cast iron is taken 18,000,000 then: fc=.01115L . (80) The holding down bolts should be so designed that the bolts on one side of the cap may be capable of carrying safely two- thirds of the total pressure. PROBLEMS 1. A flat car weighs 20 tons, is designed to carry a load of 40 tons more and is supported by two four-wheeled trucks, the axle journals being of wrought iron and the wheels 33 in. in diameter. Design the journals, considering heating, wear, strength and stiffness, assuming a maximum speed of 30 miles an hour, factor of safety = 10 and C = 300,000. 2. The following dimensions are those generally used for the journals of freight cars having nominal capacities as indicated: V Capacity Dimensions of journal 100,000 Ib ................................... 4. 5 by 9 in. 60,000 Ib ................................... 4 .25 by 8 in. 40,000 Ib ................................... 3.75 by 7 in. 144 MACHINE DESIGN Assuming the weight of the car to be 40 per cent of its carrying capacity in each instance, determine the pressure per square inch of projected area and the value of the constant C { Formula (71)}. 3. Measure the crank pin of any modern engine which is accessible, calculate the various constants and compare them with those given in this chapter. 4. Design a crank pin for an engine under the following conditions: Diameter of piston =28 in. Maximum steam pressure =90 Ib. per square inch. Mean steam pressure =40 Ib. per square inch. Revolutions per minute = 75. Determine dimensions necessary to prevent wear and heating and then test for strength and stiffness. 5. Design a crank pin for a high-speed engine having the following dimensions and conditions: Diameter of piston . = 14 in. Maximum steam pressure = 100 Ib. per square inch. Mean steam pressure =50 Ib. per square inch. Revolutions per minute =250. 6. Make a careful study and sketch of journals and journal bearings on each of the following machines and analyze as to (a) Materials, (b) Adjust- ment, (c) Lubrication. a. An engine lathe. b. A milling machine. c. A steam engine. d. An electric generator or motor. 7. Sketch at least two forms of oil cup used in the laboratories and explain their working. 8. The shaft journal of a vertical engine is 4 in. in diameter by 6 in. long. The cap is of cast iron, held down by 4 bolts of wrought iron, each, 5 in. from center of shaft, and the greatest vertical pressure is 12,000 Ib. Calculate depth of cap at center for both strength and stiffness, and also the diameter of bolts. 9. Investigate the strength of the cap and bolts of some pillow block whose dimensions are known, under a pressure of 500 Ib. per square inch of projected area. 10. The total weight on the drivers of a locomotive is 64,000 Ib. The drivers are four in number, 5 ft. 2 in. in diameter, and have journals 1\ in. in diameter. Determine horse-power consumed in friction when the locomotive is running 50 miles an hour, assuming tan ^ = .05. 65. Step -Bearings. Any bearing which is designed to resist end thrust of the shaft rather than lateral pressure is denomi- STEP BEARINGS 145 nated a step or thrust bearing. These are naturally most used on vertical sh'afts, but may be frequently seen on horizontal ones as for example on the spindles of engine lathes, boring machines and milling machines. Step-bearings may be classified according to the shape of the rubbing surface, as flat pivots and collars, conical pivots, and conoidal pivots of which the Schiele pivot is the best known. When a step-bearing on a vertical shaft is exposed to great pres- sure or speed it is sometimes lubricated by an oil tube coming up from below to the center of the bearing and connecting with a stand pipe or force-pump. The oil entering at the center is distributed by centrifugal force. 66. Friction of Pivots or Step -bearings. Flat Pivots. Let W = weight on pivot d^ = outer diameter of pivot p = intensity of vertical pressure T = moment of friction /= coefficient of friction = tan S; = resultant tensile stress Ss^ resultant shearing stress. Then is it shown in treatises on the mechanics of materials that the maximum values of the resultant stresses are as follows: 1 p 2 (a) 1 Merriman's Mechanics of Materials, p. 151. Slocum and Hancock's Strength of Materials, p. 116. 170 MACHINE DESIGN ^ (b) 2 Let M bending moment on shaft T = twisting moment on shaft. Then by formulas (5) and (8) p. 3, = 10 1 2M _5.17 7 Substituting these values in (a) and (b) and reducing, we have: (o) (d) But the bending moment which would produce a stress = St is: and the twisting moment which would produce a stress = S 8 is: l 5.1 Combining these equations with (c) and (d) respectively and reducing: (95) (96) The method of designing a shaft subjected to both bending and twisting moments may thus be stated : Determine the diameter of shaft necessary to withstand safely a bending moment M lt Equation (95) ; also, calculate the diameter to safely resist a twisting moment 7\ (Equation (96)). The larger diameter would then be used, i.e., Equations (a) and (b) in this article may be used in combining shearing and tensile or shearing and compressive stresses, in whatever manner produced. COUPLINGS 171 Other examples of combined stresses are furnished by columns and by machine frames having eccentric loads (see Art. 17). In the case of columns, where the load is assumed to be central, the empirical formulas (12) and (12-a) given on pp. 4 and 5 are recommended. Where a material like cast iron is concerned, as in the case of machine frames, no theoretical analysis is of much value and reliance can be placed only on experimental determinations of stresses and breaking loads. 84. Couplings. The flange or plate coupling is most commonly used for fastening together adjacent lengths of shafting. Fig. 73 shows the proportions of such a coupling. The flanges are turned accurately on all sides, are keyed to the shafts and the two are centered by the projection of the shaft from one part into the other as shown at A. The bolts are turned to fit the holes loosely so as not to interfere with the alignment. The projecting rim as at B pre- vents danger from belts catching on the heads and nuts of the bolts. The faces of this coupling should be trued up in a lathe after being keyed to the shaft. Jones and Laughlins in their shafting catalogue give the following proportions for flange couplings. FlG Diam. of shaft Diam. of hub Length of hub Diam. of coupling 2 4* 3i 8 2* 5f 4f 10 3 6| 5* 12 3| 8 6* 14 4 9 7 16 5 m 8! 20 172 MACHINE 'DESIGN There are five bolts in each coupling. The sleeve coupling is neater in appearance than the flange coupling but is more complicated and expensive. In Fig. 74 is illustrated a neat and effective coupling of this type. It consists of the sleeve S bored with two tapers and two threaded ends as shown. The two conical, split bushings BB B FIG. 74. are prevented from turning by the feather key K and are forced into the conical recesses by the two threaded collars CC and thereby clamped firmly to the shaft. The key K also nicks slightly the center of the main sleeve S, thus locking the whole combination. Couplings similar to this have been in use in the Union Steel Screw Works, Cleveland, Ohio, for many years and have given good satisfaction. The Sellers coupling is of the type illustrated in Fig. 74, but is tightened by three bolts running parallel to the shaft and taking the place of the collars CC. In another form of sleeve coup- ling the sleeve is split and clamped to the shaft by bolts passing through the two halves as illus- trated in Fig. 75. The "muff" coupling, as its name implies is a plain sleeve slipped over the shafts at the point of junction, accurately fitted and held by a key running from end to end. It may be regarded as a permanent coupling since it is not readily removed. ( "" r "\ i j i rAi t rfh 1 r iijn [ L W J ] L L J FIG. 75. CLUTCHES 173 85. Clutches. By the term clutch, is meant a coupling which may be readily disengaged so as to stop the follower shaft or pulley. Clutch couplings are of two kinds, positive or jaw clutches and friction clutches. The jaw clutch consists of two hubs having sector shaped projections on the adjacent faces which may interlock. One of the couplings can be slid on its shaft to and from the other by means of a loose collar and yoke, so as to engage or disengage with its mate. This clutch has the serious disadvantage of not being readily engaged when either shaft is in motion. Friction clutches are not so positive in action, but can be engaged without difficulty and without stopping the driver. Three different classes of friction clutches may be distinguished according as the engaging members are flat rings, cones or cylinders. The Weston clutch, Fig. 76, belongs to the first-named class. A series of rings inside a sleeve on the follower B in- terlocks with a similar series outside a smaller sleeve on the driver A somewhat as in a thrust bearing (Art. 70). Each ring can slide on its sleeve but must rotate with it. When the parts A and B are forced together the rings close up and engage by pairs, producing a considerable turning moment with a moderate end pressure. Let: P = pressure along axis n = number of pairs of surfaces in contact f= coefficient of friction r = mean radius of ring T = turning moment Then will: T = Pfnr. (97) If the rings are alternately wood and iron, as is usually the case, /will have values ranging from 0.25 to 0.50. The cone clutch consists of two conical frustra, one external -KB FIG. 76. 174 MACHINE DESIGN and one internal, engaging one another and driving by friction. Using the same notation as before, and letting a = angle between element of cone and axis, the normal pressure between the two * T.I * i * Ml 1 * J surfaces will be: sin a. and the friction will be: sin a Therefore : T= Pfr_ sin a (98) a should slightly exceed 5 degrees to prevent sticking and /will be at least 0.10 for dry iron on iron. Substituting /= 0.10 and sin a =0.125 we have T = 0.8 Pr as a convenient rule in designing. Fig. 77 illustrates the type of clutch more generally used on shafting for transmitting moderate quantities of power. As shown in the figure one member is attached to a loose pulley on the shaft, but this same type can be used for connecting two independent shafts. The ring or hoop H, finished inside and out, is gripped at intervals by pairs of jaws JJ having wooden faces. These jaws are actuated as shown by toggles and levers connected with the slip ring R. The toggles are so adjusted as to pass by the center and lock in the gripping position. These clutches are convenient and durable but occupy con- siderable room in proportion to their transmitting power. The Weston clutch is preferable for heavy loads. Cork inserts in metal surfaces have been used to some extent, as the coefficient of friction is much greater for cork than for wood. The cork may be boiled to soften it and forced into holes in one of the members. When pressure is applied, the projecting cork takes the load and carries it with good efficiency. As the FIG. 77. CLUTCHES 175 normal pressure is increased, the cork yields, finally becoming flush with the metal surface and dividing its load with the latter. Cork in its natural state is liable to wear quite rapidly under hard service. It may be hardened by being heated under heavy pressure and in this condition is much more durable. Professor I. N. Hollis gives the coefficients of friction for dif- ferent materials used in clutches, as follows: Cast iron on cast iron 0.16 Bronze on cast iron 0.14 Cork on cast iron . 33 Professor C. M. Allen in experiments on clutches for looms found that cork inserts gave a torque nearly double that of a leather face on iron. The roller clutch is much used on automatic machinery as it combines the advantages of positive driving and friction engage- ment. A cylinder on the follower is embraced by a rotating ring carried by the driver. The ring has a number of recesses on its inner surface which hold hardened steel rollers. These recesses being deeper at one end allow the rollers to turn freely as long as they remain in the deep portions. The bottom of the recess is inclined to the tangent of the circle at an angle of from 9 to 14 degrees. When by suitable mechanism the rollers are shifted to the shallow portions of the recesses they are immediately gripped between the ring and the cylinder and set the latter in motion. A clutch of this type is almost instantaneous in its action and is very powerful, being limited only by the strength of the materials of which it is composed. Several small rolls of different materials and diameters were tested by the writer in 1905 with the following results: Material Diameter Length Set load Ultimate load Cast iron 0.375 1.5 5,500 12,400 Cast iron 75 1.5 6,800 19,500 Cast iron Cast iron 1.125 0.4375 1.5 1.5 7,800 8,800 29,700 20,000 Soft steel 0.4375 1.5 11,100 Hard steel .... 0.4375 1.5 35,000 176 MACHINE DESIGN 86. Automobile Clutches. The development of the automobile industry has created a demand for clutches of small size and considerable power; these clutches must also be capable of picking up a load gently and of holding it firmly; they must be durable and reliable under peculiarly severe conditions and for consider- able periods. Mr. Henry Souther contributes to the literature of this subject an interesting paper from which some of the following data are quoted. Reference is made to the paper itself for more complete information. 1 Automobile clutches may be roughly classified as (a) conical; (b) disc or multiple disc; (c) band either expanding or contracting. The clutch is located between the engine and gear box, usually near the fly-wheel and sometimes forming a part of it. Conical clutches are in some respects the most satisfactory for automobile use. They require but slight motion for engagement and slight pressure to hold them in place. No lubricant is necessary and therefore there is no trouble from gumming and sticking. The materials used for the rubbing surfaces are generally aluminum covered with leather for one, and gray cast iron for the other. Castor or neatsfoot oil may be used to keep the leather soft. To render the engagement more gradual, springs are sometimes placed under the leather at six or eight points on the circumference; these permit some slipping until the whole surface of the leather is brought into contact. The angle of the cone is about 8 degrees in ordinary practice, but some manufacturers are using 10 or 12 degrees. (This is the angle on one side.) The principal difficulty with conical clutches is that of poor alignment. Unless the axes of the two cones coincide, engage- ment is uncertain and irregular. This coincidence can only be secured by the use of two universal joints insuring perfect flexibility. Mr. Souther gives the following table as representing three typical clutches in successful use: 1 Trans. A. S. M. E., May, 1908. CLUTCHES 177 TABLE XL VII POWER OF CLUTCHES 1 2 3 Area of surface (square inches) 113 1 78 7 73 6 Angle (one side) (degrees) Maximum radius (inches) . . 8 8 8 8 8 7 Spring pressure (pounds) 375 320 250 Horse-power 48 42 40 Fig. 78 illustrates the conical clutch in its simplest form. The disc dutch consists of a disc on the driven member clamped between two discs on the driver, which latter is generally the fly-wheel. Springs are used to insure separation when dis- engaged and other springs furnish the pressure for engagement. A multiple disc clutch similar to the Weston is also used. In this case the discs are alter- nately of bronze and steel. All disc clutches must be lubricated and upon the type and quantity of lubri- cation depends the character of the service. Copious lubrication means gradual engagement and slight driv- ing power; scanty lubrica- tion gives more power and quick seizure. The principal disadvan- tage of the disc clutch is the heavy spring pressure necessary to insure driving power. The multiple disc clutches cause some trouble in lubrication and are complicated and difficult of access. Band clutches depend for their driving power on the friction between the case and an adjustable" band or ring which can be expanded or contracted by suitable mechanism. FIG. 78. 178 MACHINE DESIGN The more usual construction has a band which is expanded against the inside surface of the enclosing case by means of internally operated levers and springs. Centrifugal force at high speeds has a disturbing effect on the levers and sometimes causes the clutch to release auto- matically. This difficulty has been overcome in some clutches by an improved arrangement of levers and springs. 87. Coupling Bolts. The bolts used in the ordinary flange couplings are exposed to shearing, and the combined moment of the shearing forces should equal the twisting moment on the shaft. Let n = number of bolts d 1 = diameter of bolt D = diameter of bolt circle. We will assume that the bolt has the same shearing strength as the shaft. The combined shearing strength of the bolts is and their moment of resistance to shearing is This last should equal the torsion moment of the shaft or o.l Solving for d l and assuming D=3d as an average value, we have ^==, (79) In practice rather larger values are used than would be given by the formula. 88. Shafting Keys. The moment of the shearing stress on a key must also equal the twisting moment of the shaft. Let 6 = breadth of a key 1 = length of key h = total depth of key S' = shearing strength of key. SHAFTING KEYS 179 The moment of shearing stress on key is and this must equal -^r Usually b = ~.- O.I 4 For shafts of machine steel S = S', and for iron shafts S = %S' nearly, as keys should always be of steel. Substituting these values and reducing: For iron shafting l = 1.2d nearly. For steel shafting l = 1.6d nearly as the least lengths of key to prevent its failing by shear. If the keyway is to be designed for uniform strength, the shear- ing area of the shaft on the line AB, Fig. 79, should equla the shearing area of the key, if shaft and key are of the same material and AB = CD = b. These proportions will make the depth of keyway in shaft about =ffr and would be appropriate for a square key. To avoid such a depth of keyway which might weaken the shaft, it is better to use keys longer than required by preceding for- mulas. In American practice the total depth of key rarely exceeds J6 and one-half of this depth is in shaft. To prevent crushing of the key the moment of the compressive strength of half the depth of key must equal T. dlh^ Sd* or 2 X ^ X c = lU (a) where S c is the compressive strength of the key. For iron shafts S C = 2S and for steel shafts S c = = S Substituting values of S c and assuming h = %b = -fad we have Iron shafts l = 2.5d nearly. Steel shafts l = 3^d nearly, as the least length for flat keys to prevent lateral crushing. 180 MACHINE DESIGN The above refers to parallel keys. Taper keys have parallel sides, but taper slightly between top and bottom. When driven home they have a tendency to tip the wheel or coupling on the shaft. This may be partially obviated by using two keys 90 degrees apart so as to give three points of contact between hub and shaft. The taper of the keys is usually about | in. to 1 ft. The Woodruff key is sometimes used on shafting. As may be seen in Fig. 80 this key is semi-circular in shape and fits a recess sunk in the shaft by a milling cutter. 89. Strength of Keyed Shafts. Some very interesting experiments on the strength of shafts with keyways are reported by Professor H. F. Mo^re * The material of the shafts was "soft steel some being turned and some cold-rolled. The diameters varied from 1 to 2\ in. Keyways of ordinary pro- portions, both for straight keys and for Woodruff keys, were cut in the specimens and the latter were then subjected to twisting and to combined twisting and bending. So far as the ultimate strength was concerned, the keyways seemed to have little effect, the shaft with a single keyway having about the same strength as a shaft without the keyway. After the elastic limit was passed, the keyways gradually closed up and were entirely closed at rupture. The elastic limit, however, was noticeably affected by the presence of a keyway. The ratio of the strength at elastic limit with keyway to the strength at elastic limit without keyway is called the efficiency and is denoted by -e-. The corresponding ratio of angles of twist inside the elastic limit is denominated k. According to Professor Moore, the following equations repre- sent fairly well the values of e and k : e = l-Q.2w-l.lh (99) h (100) FIG. 80. University of Illinois Bulletin No. 42, 1909. SHAFTING KEYS 181 where w = and h = width of keyway diameter of shaft depth of keyway diameter of shaft Two values of w were used in the experiments: w = Q.25 and 0.50 and two values of h: h = Q.125 and 0.1875. Table XLVIII gives the values of e as obtained by the experi- ments: TABLE XLVIII EFFICIENCY OF SHAFTS WIFH KEYWAYS ^ . elastic strength of shaft with keyway Efficiency = -r r , , . r-r -^ elastic strength of shaft without keyway Dimensions of keyway 17 = 0.50 A = 0.125 W = 0.25 h = 0.1875 PF = 0.25 A = 0.125 Woodruff System 1 Under simple torsion: Cold-rolled shaft, diameter, 11 in. Cold-rolled shaft, diameter, 1 9/16 in. 0.762 0.803 0.758 0.760 0.846 0.817 0.820 0.900 0.889 0.840 0.860 0.815 Cold-rolled shaft, diameter, 1 15/16 in. 0.748 0.764 0.710 0.750 0.860 0.824 0.826 0.835 Cold-rolled shaft, diameter, 21 in. 0.848 0.705 0.775 0.689 0.839 0.825 0.943 0.861 Under combined torsion and bending: 1. Twisting moment = bend- ing moment. Cold-rolled shaft, diameter, 11 in. 0.630 0.680 0.636 0.698 0.791 0.803 0.716 0.750 Cold-rolled shaft, diameter, 1 15/16 in 0.584 0.671 0.697 775 0.854 0.858 840 2. Twisting moment = 5/3 bending moment. Cold-rolled shaft, diameter, 11 in. 0.895 0.870 0.670 0.735 0.940 0.888 0.930" 0.880 Cold-rolled shaft, diameter, 1 15/16 in. 0.740 0.815 0.832 0.840 0.856 0.810 General average 0.752 0.735 0.850 0.845 l ln 1 1/4-in. shafts keyways were cut for No. 15 Woodruff keys. In 1 9/16-in. shafts keyways were cut for No. 25 Woodruff keys. In 1 15/16-in. shafts keyways were cut for No. S Woodruff keys. In 2 1/4-in. shafts keyways were cut for No. U Woodruff keys. 182 MACHINE DESIGN The average value of the fiber stress of the cold-rolled shafting at the elastic limit was 38940 Ib. and the average modulus of elasticity 11,985,000. It would appear that, considering the factor of safety usually allowed in shafting, the effect of ordinary keyways can safely be neglected. 90. Hangers and Boxes. Since shafting is usually hung to the ceiling and walls of buildings it is necessary to provide means FIG. 83. FIG. 84. for adjusting and aligning the bearings as the movement of the building disturbs them. Furthermore as line shafting is contin- uous and is not perfectly true and straight, the bearings should be to a certain extent self-adjusting. Reliable experiments HANGERS 183 have shown that usually one-half of the power developed by an engine is lost in the friction of shafting and belts. It is important that this loss be prevented as far as possible. The boxes are in two parts and may be of bored cast-iron or lined with Babbitt metal. They are usually about four diam- eters of the shaft in length and are oiled by means of a well and rings or wicks. (See Art. 58.) The best method of support- ing the box in the hanger is by the ball-and-socket joint; all other contrivances such as set screws are but poor sub- FIG. 85. stitutes. Fig. 81 shows the usual arrangement of the ball and socket. A and B are the two parts of the box. The center, is cast in the shape of a partial sphere with C as a center as shown by the dotted lines. The two sockets S S can be adjusted vertically in the hanger by means of screws and lock nuts. The horizontal adjustment of the hang- er is usually effected by moving it bodily on the support, the bolt holes being slotted for this purpose. Counter shafts are short and light and are not subject to much bending. Consequently there is not the same need of adjustment as in FlG - 86 - line shafting. In Fig. 82 is illustrated a simple bearing for counters. The solid cast-iron box B with a spherical center is fitted directly in a socket in the hanger H and held in position by the cap C and a set screw. There is not space here to show all the various forms of hangers and floor stands and reference is made to the catalogues of manufacturers. Hangers should be symmetrical, i.e., the center of the box should be in a vertical line with center of base. They should have relatively broad bases and should have the 184 MACHINE DESIGN metal disposed to secure the greatest rigidity possible. Cored sections are to be preferred, Fig. 83 illustrates the proportions of a Sellers line-shaft hanger. This type is also made with the lower half removable so as to facilitate taking down the shaft. Fig. 84 shows the outlines of a hanger for heavy shafting as manufactured by the Jones & Laughlins Company while Fig. 85 illustrates the design of the box with oil wells and rings. The open side hanger is sometimes adopted on account of the ease with which the shaft can be removed, but it is much less rigid than the closed hanger and is suitable only for light shafting. The countershaft hanger shown in Fig. 86 is simple, strong and symmetrical and is a great improvement over those using pointed set screws for pivots. Hangers similar to this are used by the Brown & Sharpe Mfg. Co. with some of their machines. PROBLEMS 1. Calculate the safe diameters of head shaft and three line shafts for a factory, the material to be rolled iron and the speeds and horse-powers as follows : Head shaft 100 H. P. 200 rev. per min. Machine shop 30 H. P. 120 rev. per min. Pattern shop 50 H. P. 250 rev. per min. Forge shop 20 H. P. 200 rev. per min. 2. Determine the horse-power of at least two lines of shafting whose speeds and diameters are known. 3. Design and sketch to scale a flange coupling for a 3-in. line shaft including bolts and keys. 4. Design a sleeve coupling for the foregoing, different in principle from the ones shown in the text. 5. A 4-in. steel head shaft makes 100 rev. per min. Find the horse-power which it will safely transmit, and design a Weston ring clutch capable of carrying the load. There are to be six wooden rings and five iron rings of 12-in. mean diame- ter. Find the moment carried by each pair of surfaces in contact and the end pressure required. 6. Find mean diameter of a single cone clutch for same shaft with same end pressure. 7. Find radial pressure required for a clutch like that shown in Fig. 77, the ring being 24 in. in mean diameter and there being four pairs of grips. Other conditions as in preceding problems. HANGERS 185 8. Select the line-shaft hanger which you prefer among ^hose in the laboratories and make sketch and description of the same. 9. Do. for a countershaft hanger. 10. Explain in what way a floor-stand differs from a hanger. REFERENCES Machine Design. Low and Be vis, Chapter VIII. Efficiency of Shafting. Tr. A. S. M. E., Vol. VI, p. 461; Vol. VII, p. 138; Vol. XVIII, p. 228; Vol. XVIII, p. 861. Shafting Clutches. Tr. A. S. M. E., Vol. XIII, p. 236. Ball Bearing Hangers. Tr. A. S. M. E., Vol. XXXII, p. 533. Test of Clutch Coupling. Tr. A. S. M. E., Vol. XXXII, p. 549. CHAPTER X GEARS, PULLEYS AND CRANKS 91. Gear Teeth. The teeth of gears may be either cast or cut, but the latter method prevails, since cut gears are more accurate and run more smoothly and quietly. The proportions of the teeth are essentially the same for the two classes, save that more back lash must be allowed for the cast teeth. The circular pitch is obtained by dividing the circumference of the pitch circle by the number of teeth. The diametral pitch is obtained by dividing the number of teeth by the diameter of the pitch circle and equals the number of teeth per inch of diameter. The reciprocal of the diametral pitch is sometimes called the module. The addendum is the radial projection of the tooth beyond the pitch circle, the dedendum the corresponding- distance inside the pitch circle. The clearance is the difference between the dedendum and addendum; the back lash the differ- ence between the widths of space and tooth on the pitch circle. Let circular pitch =p module =-=m 71 diametral pitch = -= p m addendum =a dedendum or flank =f clearance =fa = c height =a+f=h width =w. (See Fig. 88.) The usual rule for standard cut teeth is to make w = ^, allowing 2i g m no calculable back-lash, to make a = m and f=-^- or h = 2^m o and clearance = =-' o There is, however, a marked tendency at the present J^ime toward the use of shorter teeth. The reasons urged for their 186 GEAR TEETH 187 adoption are: first, greater strength and less obliquity of action; second, less expense in cutting. 1 Several systems have been proposed in which the height of tooth h varies from 0.425p to According to the latter system a Q.25p,f=Q.3p, and c = .05p. In modern practice the diametral pitch is a whole number or a common fraction and is used in describing the gear. For instance, a 3-pitch gear is one having 3 teeth per inch of diameter. The following table gives the pitches in common use and the proportions of long and short teeth. 7) If the gears are cut, w ^> if cast gears are used, w 0.46p to QASp. TABLE XLIX PROPORTIONS OP GEAR TEETH Pitch Standard teeth Short teeth Diametral Circular Addend, a Height h Clear- Addend, a Height h Clear- ance c ance c i 6.283 2. 4.25 0.25 1.571 3.456 0.314 1 4.189 1.33 2.82 0.167 1.047 2.303 0.209 1 3.142 1. 2.125 0.125 0.785 1.728 0.157 H 2.513 0.8 1.7 0.1 0.628 1.383 0.125 li 2.094 0.667 1.415 0.083 0.524 1.152 0.105 H 1.795 0.571 1.212 0.071 0.449 0.988 0.09 2 1.571 0.5 1.062 0.062 0.392 0.863 0.078 21 1.396 0.445 0.945 0.056 0.349 0.768 0.070 2i 1.257 0.4 0.85 0.05 0.314 0.691 0.063 2f 1.142 0.364 0.775 0.045 0.286 0.629 0.057 3 1.047 0.333 0.708 0.042 0.262 0.576 0.052 3J 0.898 0.286 0.608 0.036 0.224 0.494 0.045 4 0.785 0.25 0.531 0.031 0.196 0.432 0.039 5 0.628 0.2 0.425 0.025 0.157 0.345 0.031 6 0.524 0.167 0.354 0.021 0.131 0.288 0.026 7 0.449 0.143 0.304 0.018 0.112 0.246 0.022 8 0.393 0.125 0.266 0.016 0.098 0.216 0.020 9 0.349 0.111 0.236 0.014 0.087 0.191 0.017 10 0.314 0.1 0.212 0.012 0.079 0.174 0.016 11 0.286 0.091 0.193 0.011 0.071 0.156 0.014 12 0.262 0.0834 0.177 0.010 0.065 0.143 0.013 13 0.242 0.077 0.164 0.010 0.060 0.132 0.012 14 0.224 0.0715 0.152 0.009 0.056 0.123 0.011 15 0.209 0.0667 0.142 0.008 0.052 0.114 0.010 16 0.196 0.0625 0.133 0.008 0.049 0.108 0.010 1 See Am. Mach. Jan. 7, 1897, p. 6. 188 MACHINE DESIGN 92. Strength of Teeth. Let P = total driving pressure on wheel at pitch circle. This may be distributed over two or more teeth, but the chances are against an even distribution. Again, in designing a set of gears the contact is likely to be confined to one pair of teeth in the smaller pinions. Each tooth should therefore be made strong enough to sustain the whole pressure. Rough Teeth. The teeth of pattern molded gears are apt to be more or less irregular in shape, and are especially liable to be thicker at one end on account of the draft of the pattern. In this case the entire pressure may come on the outer corner of a tooth and tend to cause a diagonal fracture. Let C in Fig. 87 be the point of application of the pressure P, and A B the line of probable fracture. B Drop the _LCD on AB Let AB = x A and CD=y angle CAD = a \~7 1 X- i \ WM^ : ' 'WS> FIG. 87. The bending moment at section AB is M = Pfy, aiid the moment of resistance is M' = ^Sxw 2 where $ = safe transverse strength of material. and QPy W 2 X (a) If P and w are constant, then S is a maximum when - is a maximum. GEAR TEETH 189 But y = h sina and x = cos a - sina cosa which is a maximum x when a -=45 and ^ = 1 x 3P Substituting this value in (a) we have S = ^ 3 P But in this case w = A7p and therefore S= 2 and p = 3.684 \^ (101) 1 fs" diametral pitch, - = .853 \^. (102) //& x^ Unless machine molded teeth are very carefully made, it may be necessary to apply this rule to them as well. Cut Gears. With careful workmanship machine molded and machine cut teeth should touch along the whole breadth. In such cases we may assume a line of contact at crest of tooth and a maximum bending moment. M = Ph. The moment of resistance at base of tooth is when 6 is the breadth of tooth. In most teeth the thickness at base is greater than w, but in radial teeth it is less. Assuming standard proportions for cut w = .5p and substituting above: .6765 Pp= P = .06166Sp. (103) For short teeth having h = .55p formula (103) reduces to: P = .075SbSp. (104) The above formulas are general whatever the ratio of breadth 190 MACHINE DESIGN to pitch. The general practice in this country is to make b=3p. Substituting this value of b in (103) and (104) and reducing: Long teeth : p = 2.326\^ (105) Short teeth: p= 2.098 \L- (106) The corresponding formulas for the diametral pitch are: 1 l~Cf Long teeth: - = 1.35 \~ (107) Short teeth: ~ = 1.49 \^ (108) 93. Lewis' Formulas. The foregoing formulas can only be regarded as approximate, since the strength of gear teeth depends upon the number of teeth in the wheel; the teeth of a rack are broader at the base and consequently stronger than those of a pinion. This is more particularly true of epicycloidal teeth. Mr. Wilfred Lewis has deduced formulas which take into account this variation. For cut spur gears of standard dimensions the Lewis formula is as follows: (109) where n = number of teeth. This formula reduces to the same as (103), for n = 14 nearly. Formula (103) would then properly apply only to small pin- ions, but as it would err on the safe side for larger wheels, it can be used where great accuracy is not needed. The same criticism applies to the other formulas in Art. 92. The value of S used should depend on the material and on the speed. The following safe values are recommended for cast iron and cast steel. GEAR TEETH 191 Linear velocity ft. per min. 100 200 300 600 900 1200 1800 2400 Cast iron 8000 6 000 4,800 4,000 3,000 2,400 2,000 1 700 Cast steel 24,000 15,000 12,000 10,000 7,500 6,000 5,000 4,250 For gears used in hoisting machinery where there is slow speed and liability of shocks a writer in the Am. Mach. recommends smaller values of S than those given above 1 and proposes the following for four different metals: Linear velocity ft. per min. 100 200 300 600 900 1200 1800 2400 Gray iron 4,800 4,200 3,840 3,200 2,400 1,920 1,600 1,360 Gun metal 7,200 6,300 5,760 4,800 3,600 2,880 2,400 2,040 Cast steel 9,600 8,400 7,680 6,400 4,800 3,840 3,200 2,720 Mild steel 12,000 10,500 9,600 8,000 6,000 4,800 4,000 3,400 The experiments described in the next article show that the ultimate values of S are much less than the transverse strength of the material and point to the need of large factors of safety. 94. Experimental Data. In the Am. Mach. for Jan. 14, 1897, are given the actual breaking loads of gear teeth which failed in service. The teeth had an average pitch of about 5 in., a breadth of about 18 in. and the rather unusual velocity of over 2000 ft. per minute. The average breaking load was about 15,000 Ib. there being an average of about 50 teeth on the pinions. Substi- tuting these values in (109) and solving we get = 1575 Ib. This very low value is to be attributed to the condition of pressure on one corner noted in Art. 92. Substituting in formula for such a case. op This all goes to show that it is well to allow large factors of safety for rough gears, especially when the speed is high. 1 Am. Mach., Feb. 16, 1905. 192 MACHINE DESIGN Experiments have been made by the author on the static strength of rough cast-iron gear teeth by breaking them in a test- ing machine. The teeth were cast singly from patterns, were two pitch and about 6 in. broad. The patterns were constructed accurately from templates representing 15 degrees involute teeth and cycloidal teeth drawn with a describing circle one-half the pitch circle of 15 teeth; the proportions used were those given for standard cut gears. There were in all 41 cycloidal teeth of shapes corresponding to wheels of 15-24-36-48-72-120 teeth and a rack. There were 28 involute teeth corresponding to numbers above given omitting the pinion of 15 teeth. The pressure was applied by a steel plunger tangent to the surface of tooth and so pivoted as to bear evenly across the whole breadth. The teeth were inclined at various angles so as to vary the obliquity from to 25 degrees for the cycloidal and from 15 degrees to 25 degrees for the involute. The point of applica- tion changed accordingly from the pitch line to the crest of the tooth. From these experiments the following conclusions are drawn : 1. The plane of fracture is approximately parallel to line of pressure and not necessarily at right angles to radial line through center of tooth. 2. Corner breaks are likely to occur even when the pressure is apparently uniform along the tooth. There were fourteen such breaks in all. 3. With teeth of dimensions given, the breaking pressure per tooth varies from 25,000 Ib. to 50,000 Ib. for cycloids as the num- 'ber of teeth increases from 15 to infinity; the breaking pressure for involutes of the same pitch varies from 34,000 Ib. to 80,000 Ib. as the number increases from 24 to infinity. 4. With teeth as above the average breaking pressure varies from 50,000 Ib. to 26,000 Ib. in the cycloids as the angle changes from degrees to 25 degrees and the tangent point moves from pitch line to crest; with involute teeth the range is betw r een 64,000 and 39,000 Ib. 5. Reasoning from the figures just given, rack teeth are about twice as strong as pinion teeth and involute teeth have an advan- tage in strength over cycloidal of from 40 to 50 per cent. The GEAR TEETH 193 advantage of short teeth in point of strength can also be seen. The modulus of rupture of the material used was about 36,000 Ib. Values of S calculated from Lewis' formula for the various tooth numbers are quite uniform and average about 40,000 Ib. for cycloidal teeth. Involute teeth are to-day generally preferred by manufacturers. 95. Modern Practice. Two tendencies are quite noticeable to-day in the practice of American manufacturers, one toward the use of shorter teeth and the other toward a larger angle of obliquity. The effect of these two changes upon the action of gear teeth is the subject of a comprehensive paper read by Mr. R. E. Flanders in 1908. 1 Those readers who desire a detailed mathematical discussion of these points are referred to Mr. Flander's paper. In brief, the effects of shorter teeth are: (a) To reduce the evils of interference with involute teeth; (b) to diminish the arc of action; (c) to increase the strength; (d) to increase the durabil- ity; (e) to reduce the price of the gear. The effects of an increase in the angle of obliquity are (/) to diminish interference; (g) to diminish the arc of action; (h) to strengthen the teeth; (/) to increase side pressure on bearings; (/c) to increase the lost work; (I) to distribute the wear more evenly. It will be noticed from the above that the effects of the two changes are mainly the same. To reduce interference, to strengthen the teeth, and to secure durability and uniform wear are all desirable and important. The question of side pressure is not important nor that of lost work. The efficiency of accurately cut spur gearing, according to experiments by Lewis and others, is between 95 and 98 per cent. Some examples of modern practice will show the present tendencies. (See next page.) Mr. Flanders states that seven out of eleven automobile manufacturers questioned are using the stub form of tooth and like it. 1 Trans. A. S. M. E., 1908. 194 MACHINE DESIGN "\TQTY->O r\f fi-r-m Involute teeth iMame 01 nrm Addendum Pressure angle Remarks Wm. Sellers & Co 942 m 20 degrees C. W Hunt Co 785 m 14^ degrees Well man-Sea ver-M organ Co. Fellows Gear Shaper Co. ... 0.785m / 0.70 m \0.80m 20 degrees 20 degrees Steel mill. NOTE. m = module = - n A committee has recently been appointed by the American Society of Mechanical Engineers to investigate the subject of interchangeable involute gearing and if practicable to recommend a standard form. Mr. Wilfred Lewis, the chairman of the com- mittee, is on record as approving a pressure angle of 22J degrees and an addendum of 0.875 m. 1 Mr. Fellows, another member of the committee, recommends an angle of 20 degrees and a = 0.75 m. Either of these plans will do away with interference and allow of the use of 12-tooth pinions. Mr. Gabriel of the Brown & Sharpe Manufacturing Company is in favor of retaining the present angle of 14^ degrees and a = m. One objection to the present standard is that it is necessary to empirically modify the addendum near the crest of the tooth to prevent interference. It is claimed on the other hand that this " easing off" of the point of the tooth is a help in bringing the teeth together without shock. Teeth cut with a milling cutter or planed by a form can be made of empirical shape without difficulty, but teeth generated or "hobbed" must correspond in all ways to some theoretical curve which matches the rack used as a basis for the system. At the present writing, the problem of choosing an acceptable standard for involute gears seems far from solution. 1 Trans. A. S. M. E., 1910. GEAR TEETH 195 96. Teeth of Bevel Gears. There have been many formulas and diagrams proposed for determining the strength of bevel gear teeth, some of them being very complicated and incon- venient. It will usually answer every purpose from a practical standpoint, if we treat the section at the middle of the breadth of such a tooth as a spur wheel tooth and design it by the foregoing formulas. The breadth of the teeth of a bevel gear should be about one-third of the distance from the base of the cone to the apex. One point needs to be noted; the teeth of bevel gears are stronger than those of spur gears of the same pitch and number of teeth since they are developed from a pitch circle having an ele- ment of the normal cone as a radius. To illustrate, we will sup- pose that we are designing the teeth of a miter gear and that the number of teeth is 32. In such a gear the element of normal cone is V2 times the radius. The actual shape of the teeth will then correspond to those of a spur gear having 32V2 = 45 teeth nearly. NOTE. In designing the teeth of gears where the number is unknown, the approximate dimensions may first be obtained by formula (105) or (106) and then these values corrected by using Lewis' formula. PROBLEMS 1. The drum of a hoist is 8 in. in diameter and makes 5 revolutions per minute. The diameter of gear on the drum is 36 in. and of its pinion 6 in. The gear on the countershaft is 24 in. in diameter and its pinion is 6 in. in diameter. The gears are all cut. Calculate the pitch and number of teeth of each gear, assuming a load of two tons on drum chain and $ = 6000. Also determine the horse-power of the machine. 2. Calculate the pitch and number of teeth of a cut cast-steel gear 10 in. in diameter, running at 350 revolutions per minute and transmitting 20 horse-power. 3. A cast-iron gear wheel is 30 ft. 6f in. in pitch diameter and has 192 teeth, which are machine-cut and 30 in. broad. Determine the circular and diameter pitches of the teeth and the horse- power which the gear will transmit safely when making 12 revolutions per minute. 4. A two-pitch cycloidal tooth, 6 in. broad, 72 teeth to the wheel, failed under a load of 38,000 Ib. Find value of S by Lewis' formula. 5. A vertical water-whe'el shaft is connected to horizontal head shaft by cast-iron gears and transmits 150 horse-power. The water-wheel makes 200 revolutions per minute and the head shaft 100. 196 MACHINE DESIGN Determine the dimensions of the gears and teeth if the latter are approxi- mately two pitch. 6. Work Problem 1, uisng short teeth instead of standard. 97. Rim and Arms. The rim of a gear, especially if the teeth are cast, should have nearly the same thickness as the base of tooth, to avoid cooling strains. It is difficult to calculate exactly the stresses on the arms of the gear, since we know so little of the initial stress present, due to cooling and contraction. A hub of unusual weight is liable to contract in cooling after the arms have become rigid and cause severe tension or even fracture at the junction of arm and hub. A heavy rim on the contrary may compress the arms so as actually to spring them out of shape. Of course both of thesa errors should be avoided, and the pattern be so designed that cooling shall be simultaneous in all parts of the casting. The arms of spur gears are usually made straight without curves or taper, and of a flat, elliptical cross-section, which offers little resistance to the air. To support the wide rims of bevel gears and to facilitate drawing the pattern from the sand, the arms are sometimes of a rectangular or T section, having the greatest depth in the direction of the axis of the gear. For pulleys which are to run at a high speed it is important that there should be no ribs or projections on arms or rim which will offer resistance to the air. Experiments by the writer have shown this resistance to be serious at speeds frequently used in practice. A series of experiments conducted by the author are reported in the Am. Mach. for Sept. 22, 1898, to which paper reference is here made. Twenty-four pulleys having 3j in. face and diameters of 16, 20 and 24 in. were broken in a testing machine by the pull of a steel belt, the ratio of the belt tensions being adjusted by levers so as to be two to one. Twelve of the pulleys were of the ordinary cast-iron type having each six arms tapering and of an elliptic section. The other twelve were Medart pulleys with steel rims riveted to arms and having some six and some eight arms. Test pieces cast from the same iron as the pulleys showed an average modulus of rupture of 35,800 for the cast iron and 50,800 for the Medart. In every case the arm or the two arms nearest the side of the belt PULLEY ARMS 197 having the greatest tension, broke first, showing that the torque was not evenly distributed by the rim. Measurements of the deflection of the arms showed it to be from two to six times as great on this side as on the other. The buckling and springing of the rim was very noticeable especially in the Medart pulleys. The arms of all the pulleys broke at the hub showing the greatest bending moment there, as the strength of the arms at the hub was about double that at the rim. On the other hand, some of the cast-iron arms broke simultaneously at hub and rim, showing a negative bending moment at the rim about one-half that at the hub. The following general conclusions are justified by these experiments: (a) The bending moments on pulley arms are not evenly distributed by the rim, but are greatest next the tight side of belt. (6) There are bending moments at both ends of arm, that at the hub being much the greater, the ratio depending on the relative stiffness of rim and arms. The following rules may be adopted for designing the arms of cast-iron pulleys and gears: 1. Multiply the net turning pressure, whether caused by belt or tooth, by a suitable factor of safety and by the length of the arm in inches. Divide this product by one-half the number of arms and use the quotient for a bending moment. Design the hub end of arm to resist this moment. 2. Make section modulus at the rim ends of arms one-half as strong as at the hub ends. 98. Sprocket Wheels and Chains. Steel chains connecting toothed wheels afford a convenient means of getting a positive speed ratio when the axes are some distance apart. There are three classes in common use, the block chain, the roller chain and the so-called " silent" chain. Mr. A. Eugene Michel publishes quite a complete discussion of the design of the first two classes in Mchy., for February, 1905, and reference is here made to that journal. Block chain is that commonly used on bicycles and small motor cars, so named from the blocks with round ends which are 198 MACHINE DESIGN used to fill in between the links. The sprocket teeth are spaced to a pitch greater than that of the chain links and the blocks rest on flat beds between the teeth, Fig. 89. Roller chains have rollers on every pin and have inside and outside links. The sprocket teeth have the same pitch as the chain Links, the rollers fitting circular recesses between the sprockets, Fig. 90. The most serious failing of the chain is its tendency to stretch with use so that the pitch becomes greater than that of the sprocket teeth. To obviate this difficulty in a measure considerable clearance should be given to the sprocket teeth as indicated in Fig. 90 As the pitch of the chain increases it will then ride higher upon FIG. 89. FIG. 90. the sprockets until the end of the tooth is reached. The teeth are rounded on their side faces, that they may easily enter the gaps in the chain and have side clearance. Mr. Michel gives the following values for the tensile strength of chains as determined by actual tests. ROLLER CHAIN Pitch inches. 4 f i 1 H 14 If 2 ' Tensile strength Ib . 1,200 1,200 4,000 6,000 9,000 12,000 19,000 25,000 BLOCK CHAIN 1 inch pitch 1200 to 2500 Ib. 1 inch pitch 5000 Ib. CHAIN DRIVES 199 Mr. Michel further recommends a factor of safety of from 5 to 40 according to the severity of the conditions as to speed and shocks. The tendency is to use short links and double or triple width chains to increase the rivet bearing surface, as it is this latter factor which really determines the life of a chain. Roller chains may be used up to speeds of 1000 to 1200 ft. per minute. The sprocket should be so designed that 'one tooth will carry the load safely with the pressure near the crest since these con- ditions obtain as the chain stretches. Use values of S as in Art. 93. 99. Silent Chains. The weak points in the ordinary chain, whether it be made with blocks or rollers, are the rivet bearings. It is the continual wear of these, due to insufficient area and lack of proper lubrication, that shortens the life of a chain. The so-called " silent chain" with rocker bearings, is com- paratively free from this defect. Fig. 91 illustrates the shapes of links, rivets and sprockets for this kind of chain as manufac- tured by the Morse Chain Com- P an y- FIG. 91. The chain proper is entirely outside of the sprocket teeth so that the latter may be contin- uous across the face of the wheel, save for a single guiding groove in the center. Projections on the under side of the links engage with the teeth of the sprocket, E being the point of contact for the driver and I a similar point for the follower when the rotation is as indicated. Each rivet consists practically of two pins called by the makers the rocker pin and the seat pin. Each pin is fastened in its particular gang of links and the relative motion is merely a rocking of one pin on the other without appreciable friction. The pins are of hardened tool steel with softened ends. The combination of this freedom from rubbing contact with the adap- 200 MACHINE DESIGN tation of the engaging tooth profiles, gives a chain which can be safely run at high speeds without objectionable vibration or appreciable wear. The chains can be made of almost any width from J- in. up to 18 in., the width depending upon the pitch of the chain and the power to be transmitted. The following are the working loads -(and limiting speeds) of chains 2 in. in width and of different pitches, taken from a table published by the makers: Pitch in inches 2 f 1 .9 1.2 1.5 Working load in pounds 130 190 236 380 520 760 Limiting speed revolutions per minute. 2,000 1,600 1,200 1,100 800 600 The number of teeth in the small sprocket may vary from 15 to 30 according to the conditions. Assuming 17 teeth and the number of revolutions given in the above table the speed of chain would be 1420 ft. per minute for the ^-in. pitch and 1275 ft. per minute for the 1.5 in. Chains of this character have been run successfully at 2000 ft. per minute. PROBLEMS 1. Design eight arms of elliptic section for a gear 54. in. pitch diameter, to transmit a pressure on tooth of 800 Ib. Material, cast iron having a work- ing transverse strength of 6000 Ib. per square inch. 2. Two sprocket wheels of 75 and 17 teeth respectively are to transmit 25 horse-power at a chain speed of about 800 ft. per minute, with a factor of safety of 12- Determine the proper pitch of roller chain, the pitch diameters of the sprockets, and the numbers of revolutions. 3. Suppose that in Problem 2, a " silent " chain is to be used and the chain speed increased to 1200 ft. per minute. Determine the proper pitch of chain to be used if the width of chain is 3 in. Determine diameters and revolutions of sprockets as before. 100. Cranks and Levers. A crank or rocker arm which is used to transmit a continuous or reciprocating rotary motion is in CRANKS AND LEVERS 201 the condition of a cantilever or bracket with a load At the outer end. If the web of the crank is of uniform thickness theory requires that its profile should be parabolic for uniform strength, the vertex of the parabola being at the load point. A convenient approximation to this shape can be attained by using the tangents to the parabola at points midway between the d FIG. 92. hub and the load point. See Fig. 92. The crank web is designed of the right thickness and breadth to resist the moment at A B, and the center line is produced to Q, making PQ = ^PO. Straight lines drawn from Q to A and B will be tangent to the parabola at the latter points and will serve as contour lines for the web. Assume the following dimensions in inches: 1 = length of crank = OP t = thickness of web h breadth of web = A B d = diameter of eye = cd d = diameter of pin 6 = breadth of eye D = diameter of hub = CD Dj= diameter of shaft B = breadth of hub. If the pressure on the crank pin is denoted by P then will the oment at A B section will be: PI moment at A B be - and the equations of moments for the cross- Pl StW 2 6 [See Formula (3)] and from this the dimensions at AB may be calculated. 202 MACHINE DESIGN The moment at the hub will be PI and will tend to break the iron on the dotted lines CD. The equation of moments for the hub is therefore: From this equation the dimensions of the hub may be calcu- lated when DI is known. The eye of a crank is most likely to break when the pressure on the pin is along the line OP, and the fracture will be along the dotted lines cd. The bending moment will be P multiplied by the distance from center of pin to center of eye measured along axis of pin. If we call this distance x, then will the equation of moments be: Sb 2 ,, Px = -^-(d-d l ) It is considered good practice among engine builders to make the values of x, b and B as small as practicable, in order to reduce the twisting moment on the web of the crank and the bending moment on the shaft. In designing the hub, allowance must be made for the metal removed at the key-way. PROBLEM Design a cast-steel crank for a steam engine having a cylinder 12 by 30 in. and an initial steam pressure of 120 Ib. per square inch of piston. The shaft is 6 in. and the crank pin 3 in. in diameter. The distance x may be assumed as 4 in. Calculate, 1. Dimensions of web at AB. 2. Dimensions of hub allowing for a key 1 Xf in. 3. Dimensions of eye for pin and make a scale drawing in ink showing profile of crank complete. S may be assumed as 6000 Ib. per square inch. REFERENCES Modern American Machine Tools, Benjamin. Chapter VIII. Proportions of Arms and Rims. Am. Mack., Sept. 30, 1909. Efficiency of Gears. Am. Mach., Jan. 12, 1905; Aug. 19, 1909. Strength of Gear Teeth. Mchy., Jan., 1908; Am. Mach., Feb. 16, 1905; May 9, 1907; Jan. 16, 1908. Proposed Standard Systems of Gear Teeth. Am. Mach., Feb. 25, 1909; July 1, 1909. Roller Chains. Mchy., Feb., 1905. Tests of Short Bearings in Chains. Am. Mach., Dec. 28, 1905. CHAIN DRIVES 203 Progress in Chain Transmission. Am. Mach., Nov. 7, 1907. English Chain Drives. Cass., May, 1908. Lewis' Experiments on Gears. Tr. A. S. M. E., Vol. VII, p. 273. Strength of Gear Teeth. Tr. A. S. M. E., Vol. XVIII, p. 766. Chain Gearing. Tr. A. S. M. E., Vol. XXIII, p. 373. Symposium on Gearing. Tr. A. S. M. E., Vol. XXXII, p. 807. CHAPTER XI FLY-WHEELS 101. In General. The hub and arms of a fly-wheel are designed in much the same way as those of pulleys and gears, the straight arm with elliptic section being the favorite. The rims of such wheels are of two classes, the wide, thin rim used for belt trans- mission and the narrow solid rim of the generator or blowing engine wheel. Fly-wheels up to 8 or 10 ft. in diameter are usually cast in one piece; those from 10 to 16 ft. in diameter may be cast in halves, while wheels larger than the last mentioned should be cast in sections, one arm to FIG. 93. each section. This is a matter, not of use, but of convenience in casting and in transportation. The joints between hub and arms and between arms and rim need not be specially considered here, since wheels rarely fail at these points. The rim and the joints -in the rim cannot be too carefully designed. The smaller wheel cast in one piece is more or less subject to stresses caused by shrinkage. The sectional wheel is generally free from such stresses but is weakened by the numerous joints. Rim joints are of two gen- eral classes according as bolts or links are used for fastenings. Wide, thin rims are usually fastened together by internal flanges and bolts as shown in Fig. 93, while the stocky rims of the fly-wheels proper are joined directly by links or TMiead "prisoners" as in Fig. 94. 204 FIG. 94. FLY WHEELS 205 As will be shown later, the former is a weak antf unreliable joint, especially when located midway between the arms. The principal stresses in fly-wheel rims are caused by centrifu- gal force. 102. Safe Speed for Wheels. The centrifugal force developed in a rapidly revolving pulley or gear produces a certain tension on the rim, and also a bending of the rim between the arms. We will first investigate the case of a pulley having a rim of uni- form cross-section. It is safe to assume that the rim should be capable of bearing its own centrifugal tension without assistance from the arms. Let D = mean diameter of pulley rim = thickness of rim b = breadth of rim w = weight of material per cubic inch = .26 Ib. for cast iron = .28 Ib. for wrought iron or steel n = number of arms N = number revolutions per minute t) = velocity of rim in feet per second. First let us consider the centrifugal tension alone. The cen- trifugal pressure per square inch of concave surface is Wv 2 where W is the weight of rim per square inch of concave surface = wt, and r = radius in feet = 7^7- The centrifugal tension produced in the rim by this force is by formula (15) _L' < 115 ) tn 2 10 In a pulley with a thin rim and small number of arms, the stress due to this bending is seen to be considerable. It must, however, be remembered that the stretching of the arms due to their own centrifugal force and that of the rim will diminish this bending. Mr. Stanwood recommends a deduction of one-half from the value of S in (d) on this account. Prof. Gaetano Lanza has published quite an elaborate mathe- matical discussion of this subject. (See Vol. XVI, Trans. A. S. M. E.) He shows that in ordinary cases the stretch of the arms will relieve more than one-half of the stress due to bending, perhaps three-quarters. 103. Experiments on Fly-wheels. In order to determine experimentally the centrifugal tension and bending in rapidly revolving rims, a large number of small fly-wheels have been tested to destruction at the Case School laboratories. In all ten wheels, 15 in. in diameter and twenty-three wheels 2 ft. in diameter have been so tested. An account of some of these experiments may be found in Trans. A. S. M. E., Vol. XX. The wheels were all of cast iron and modeled after actual fly- wheels. Some had solid rims, some jointed rims and some steel spokes. To give to the wheels the speed necessary for destruction, use was made of a Dow steam turbine capable of being run at any speed up to 10,000 revolutions per minute. The turbine shaft 208 MACHINE DESIGN was connected to the shaft carrying the fly-wheels by a brass sleeve coupling loosely pinned to the shafts at each end in such a way as to form a universal joint, and so proportioned as to break or slip without injuring the turbine in case of sudden stoppage of the fly-wheel shaft. One experiment with a shield made of 2-in. plank proved that safety did not lie in that direction, and in succeeding experiments with the 15-in. wheels a bomb-proof constructed of 6Xl2-in. white oak was used. The first experiment with a 24-in. wheel showed even this to be a flimsy contrivance. In subsequent experiments a shield made of 12Xl2-in. oak was used. This shield was split repeatedly and had to be re-enforced by bolts. A cast-steel ring about 4 in. thick, lined with wooden blocks and covered with 3-in. oak planking, was finally adopted. The wheels were usually demolished by the explosion. No crashing or rending noise was heard, only one quick, sharp report, like a musket shot. The following tables give a summary of a number of the experiments. TABLE L FIFTEEN-INCH WHEELS Bursting speed Centrifugal No. tension 0 = T!-T, ^ \ /= coefficient of friction between *^"V / belt and pulley \f B 6 = arc of contact in circular ^~ FIG. 98. measure. The tension T at any part of the arc of contact is intermediate between T l and T 2 . Let AB Fig. 98 be an indefinitely short element of the arc of contact, so that the tensions at A and B differ only by the amount dT. dT will then equal the friction on A B which we may call dR. Draw the intersecting tangents OT and OT' to represent the tensions and find their radial resultant OP. Then will OP represent the normal pressure on the arc A B which we will call P. Single belts ^=4200 (137) TT T) Double belts w = 2500-^- (138) The most convenient rules for belting are those which give the horse-power of a belt in terms of the surface passing a fixed point per minute. In formula (136) we will substitute the following: IV W = width of belt in 7i DN V = velocity in ft. per mm. = 1 HP - 265& Substituting values of C and T t as before and solving for WV = square feet per minute we have approximately: Single belts WV = WHP. (139) Double belts WV = 55#P. (140) 118. Speed of Belting. As in the case of pulley rims, so in that of belts a certain amount of tension is caused by the centrifugal force of the belt as it passes around the pulley. 12wv 2 From equation (110) S = - i/ where v = velocity in feet per second w = weight of material per cubic inch S tensile stress per square inch. 228 MACHINE DESIGN To make this formula more convenient for use we will make the following changes in the contsants : Let 7 = velocity of belt in ft. per minute = 60v w = weight of ordinary belting = .032 lb. per cubic inch $! = tensile stress per inch width, caused by centrifugal force = about T 3 g- S for single belts. Then v = L Substituting these values in (110) and solving for S 1 V 2 Sl= 1610000 The speed usually given as a safe limit for ordinary belts is 3000 ft. per minute, but belts are sometimes run at a speed exceeding 6000 ft. per minute. Substituting different values of V in the formula we have: 7 = 3000 S,= 5.591b. 7=4000 S,= 9.941b. 7 = 5000 ^ = 15.53 lb. 7 = 6000 5 1 =22.361b. The values of S^ for double belts will be nearly twice those given above. At a speed of 5000 ft. per minute the maximum tension per inch of width on a single belt designed by formula (137), if we call C = .5, will be: (30x2)+15.=751b. giving a factor of safety of eight or ten at the splices. In a similar manner we find the maximum tension per inch of width of a double belt to be: (50x2)+30 = 1301b. and the margin of safety about the same as in single belting. A double belt is stiffer than a single one and should not be ROPE DRIVES 229 used on pulleys less than 1 ft. in diameter. Triple feelts can be used successfully on pulleys over 20 in. in diameter. 119. Manila Rope Transmission. Ropes are sometimes used instead of flat belts for transmitting power short distances. They possess the following advantages: they are cheaper than belts in first cost; they are flexible in every direction and can be carried around corners readily. They have, however, the disadvantage of being less efficient in transmission than leather belts and less durable; they are also somewhat difficult to splice or repair. There are two systems of rope driving in common use: the English and the American. In the former there are as many separate ropes -as there are grooves in one pulley, each rope being an endless loop always running in one groove. In the American system one continuous rope is used passing back and forth from one groove to another and finally returning to the starting-point. The advantage of the English \ 1 system consists in the fact that FIG. 99. one of the ropes may fail with- out causing a breakdown of the entire drive, there usually being two or three ropes in excess of the number actually neces- sary. On the other hand the American system has the ad- vantage of a uniform regulation of the tension on all the plies of rope. The guide pulley, which guides the last slack turn of rope back to the starting-point, is usually also a tension pulley and can be weighted to secure any desired tension. The English method is most used for heavy drives from engines to head shafts ; the American for lighter work in distributing power to the different rooms of a factory. The grooves in the pulleys for manila or cotton ropes usually have their sides inclined at an angle of about 45 degrees, thus wedging the rope in the groove. The Walker groove has curved sides as shown in Fig. 99, the curvature increasing toward the bottom. As the rope wears and 230 MACHINE DESIGN stretches it becomes smaller and sinks deeper in the groove; the sides of the groove being more oblique near the bottom, the older rope is not pinched so hard as the newer and this tends to throw more of the work on the latter. 120. Strength of Manila Ropes. The formulas for transmis- sion by ropes are similar to those for belts, the values for S and W being changed. The ultimate tensile strength of manila and hemp rope is about 10,000 Ib. per square inch. To insure durability and efficiency it has been found best in practice to use a large factor of safety. Prof. Forrest R. Jones in his book on Machine Design recommends a maximum tension of 200 d 2 pounds where d is the diameter of rope in inches. This corresponds to a tensile stress of 255 Ib. per square inch or a factor of safety of about 40. The value of /for manila on metal is about 0.12, but as the normal pressure between the two surfaces is increased by the wedge action of the rope in the groove we shall have the apparent value of/: f 1 =f-s-sm -= where a = angle of groove, For a = 45 to 30 f l varies from 0.3 to 0.5 and these values should be used in for- mula (134). \l e * j in this formula, for an arc of contact of 150 degrees, becomes either .54 or .73 according as/ 1 is taken 0.3 or 0.5. If 7\ is assumed as 250 Ib. per square inch, the force R trans- mitted by the rope varies from 135 Ib. to 185 Ib. per square inch area of rope section. The following table gives the horse-power of manila ropes based on a maximum tension of 255 Ib. per square inch. ROPE DRIVES 231 TABLE LVIII Table of the horse-power of transmission rope, reprinted from the trans- actions of the American Society of Mechanical Engineers, Vol. XII, page 230, Article on "Rope Driving" by C. W. Hunt. The working strain is 800 Ib. for a 2-in. diameter rope and is the same at all speeds, due allowance having been made for loss by centrifugal force. Diameter Speed of the rope in feet per minute Smallest diameter pulleys, 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 6,000 7,000 inches 1 3.3 4.3 5.2 5.8 6.7 7.2 7.7 7.7 7.1 4.9 30 i 4.5 5.9 7.0 8.2 9.1 9.8 10.8 10.8 9.3 6.9 36 1 5.8 7.7 9.2 10.7 11.9 12.8 13.6 13.7 12.5 8.8 42 li 9.2 12.1 14.3 16.8 18.6 20.0 21.2 21.4 19.5 13.8 54 H 13.1 17.4 20.7 23.1 26.8 28.8 30.6 30.8 28.2 19.8 60 if 18.0 23.7 28.2 32.8 36.4 39.2 41.5 41.8 37.4 27.6 72 2 23.1 30.8 36.8 42.8 47.6 51.2 54.4 54.8 50.0 35.2 84 121. Cotton Rope Transmission. Cotton rope is more expen- sive than manila in its first cost, but has a greater efficiency and a longer life than its rival. Instances are given where cotton ropes have been in continuous service for periods of fifteen, twenty-five and even thirty years. The rope of three strands without a core is most flexible and durable as there is good contact between the working strands and no waste room. Mr. Edward Kenyon gives the following values for the power which can be safely transmitted by good three-strand cotton ropes running on pulleys not less than thirty times their respec- tive diameters (English system). 1 (See next page.) The horse-power at any other speed will be in proportion to the speed. It will also be noticed that the horse-power is proportional to the square of the diameter of the rope. Mr. Kenyon gives figures for the speed as high as 7000 ft. per minute, and reports actual installations where ropes are running success- fully at this speed. He makes no allowance for centrifugal force and denies that this has any appreciable effect on the driving power or the durability. Mr. Kenyon's figures have reference only to ropes used in single plies as in the English system. 1 Am. Mach., July 8, 1909. 232 MACHINE DESIGN TABLE LIX HORSE-POWER OF COTTON ROPES VELOCITY 1000 FT. PER MINUTE Diameters in inches Horse-power Rope Smallest pulley 1 30 3.3 1* 34 4.1 H 38 5.1 If 41 6.1 H 45 7.4 if 49 8.6 if 53 10. 11 57 11.5 2 60 13. He calls especial attention to the use of casing in high-speed pulleys to reduce the air resistance. 122. Wire Rope Transmission. Wire ropes have been used to transmit power where the distances were too great for belting or hemp rope transmission. The increased use of electrical transmission is gradually crowding out this latter form of rope driving. For comparatively short distances of from 100 to 500 yd. wire rope still offers a cheap and simple means of carrying power. The pulleys or wheels are entirely different from those used with manila ropes. Fig. 100 shows a section of the rim of such a pulley. The rope does not touch the sides of the groove but rests on a shallow depression in a wooden, leather or rubber filling at the bottom. The high side flanges prevent the rope from leaving the pulley when swaying on account of the high speed. The pulleys must be large, usually about 100 times the diam- FIG. 100. ROPE DRIVES 233 eter of rope used, and run at comparatively high speeds. The ropes should not be less than 200 ft. long unless some form of tightening pulley is used. Table LX is taken from Roebling. Long ropes should be supported by idle pulleys every 400 ft. TABLE LX TRANSMISSION OF POWER BY WIRE ROPE Showing necessary size and speed of wheels and rope to obtain any desired amount of power. Diameter of wheel in feet Number of revolu- tions Diameter of rope Horse- power Diameter of wheel in feet Number of revolu- tions Diame- ter of rope Horse- power 4 80 5-8 3.3 10 80 11-16 58.4 100 5-8 4.1 100 11-16 73. 120 5-8 5. 120 11-16 87.6 140 5-8 5.8 140 11-16 102.2 5 80 7-16 6.9 11 80 11-16 75.5 100 7-16 8.6 100 11-16 94.4 120 7-16 10.3 120 , 11-16 113.3 140 7-16 12.1 140 11-16 132.1 6 80 1-2 10.7 12 80 3-4 99.3 100 1-2 13.4 100 3-4 124.1 120 1-2 16.1 120 3-4. 148.9 140 1-2 18.7 140 3-4 173.7 7 80 9-16 16.9 13 80 3-4 122.6 100 9-16 21.1 100 3-4 153.2 120 9-16 25.3 120 3-4 183.9 8 80 5-8 22. 14 80 7-8 148. 100 5-8 27.5 100 7-8 185. 120 5-8 33.0 120 7-8 222. 9 80 5-8 41.5 15 80 7-8 217. 100 5-8 51.9 100 7-8 259. 120 5-8 62.2 120 7-8 300. PROBLEMS 1. Design a main driving belt to transmit 200 horse-power from a belt wheel 18 ft. in diameter and making 80 revolutions per minute. The belt to be double leather without rivets. 2. Investigate driving belt on an engine and calculate the horse-power it is capable of transmitting economically. 234 MACHINE DESIGN 3. Calculate the total maximum tension per inch of width due to load and to centrifugal force of the driving belt on the motor used for driving machine shop, assuming the maximum load to be 10 horse-power. 4. Design a manila rope drive, English system, to transmit 400 horse- power, the wheel on the engine being 20 ft. in diameter and making 60 revo- lutions per minute. Use Hunt's table and then check by calculating the centrifugal tension and the total maximum tension on each rope. Assume v 2 *^ = en wnere v= f ee "k P er second. oU 5. Design a wire rope transmission to carry 150 horse-power a distance of one-quarter mile using two ropes. Determine working and maximum ten- sion on rope, length of rope, diameter and speed of pulleys and number of supporting pulleys. REFERENCES Manufacture of Belting. Power, Feb., 1903. Steel Belts. Am. Mach., Jan. 24, 1908; Eng. News, Oct. 14, 1909. Belts vs. Ropes. Power, Dec. 1, 1908. Lewis' Experiments on Belts. Tr. A. S. M. E., Vol. VII, p. 549. Transmission of Power by Belting. Tr. A. S. M. E., Vol. XX, p. 466; Vol. XXXI, p. 29. Various Systems of Rope Transmission. Am. Mack., July 8, 1909. Rope Driving. (Hunt.) Tr. A. S. M. E., Vol. XII, p. 230. Working Load for Ropes. Tr. A. S. M. E., Vol. XXIII, p. 125. CHAPTER XIII DESIGN OF TOGGLE-JOINT PRESS 123. Introductory Statement. In discussing the subject of Machine Design much time may be saved by assuming some simple machine and illustrating methods in design by a fairly complete analysis of all the important theoretical calculations. Such a layout at once gives the scope of the work and protects the beginner from so much " working in the dark.' 7 An assignment may then be made, differing in a lesser or greater degree from the illustrated design and a complete analysis required of all parts of the machine. After the student's experience with the first design he will need the second one developed less elaborately and possibly the third one not at all. Design No. 1 is meant especially to cover static forces, i.e., simple applications of members in tension, compression, flexure and shear. A good illustration of this is the toggle-joint press. Machines of this class are sometimes used in forming thin sheets of copper and brass into articles for ornamental purposes, conse- quently it is a useful tool. Plates C-l, C-2, and C-3 show a design of a small machine and are inserted to give an idea as to the arrangement of the drawings. The design was worked up on three 12 in. Xl8 in. sheets; two of details and one assembly view. It is urged that the designer regard these sheets merely as illustrative of a good drawing room job and that, from the stand- point of ideas, he will cultivate originality and make a design as nearly independent as possible. Alternative Designs will be found at the end of this chapter. These may be substituted for the regular designs if preferred. In designing a complete machine each part should be worked up as an independent unit but with all available information as to its relation to the other parts composing the machine. Be- fore attempting to develop any individual part, the designer should have a good idea of what the machine looks like. Free-hand 235 236 MACHINE DESIGN sketching should be insisted upon. These sketches when satis- factory should become a part of the report and be handed in with the finished drawings. Calculate by rational formula every part that will admit of such treatment. Where the conditions of stresses are not well known apply empirical rules and the best approximations possible. In any case the judgment of the designer must be used to modify and check even the best rational or empirical rules. Theoretical deductions should not be mini- mized but good j udgment should be emphasized. All calculations should be saved until the entire design is finished and these should be kept in the exact order of development. Sometimes a part that is at first considered wrong may later be found to be correct and recalculation is avoided. Occasionally it is necessary to review part of the calculations to prove some part of the design. Where the theoretical work is neatly made and logically arranged this may be done without much loss of time. In the analysis of the forces and the calculations of the various parts a high degree of refinement should be aimed at for the sake of showing how prin- ciples are applied, even though the illustrative piece may not demand a very thorough analysis. The object sought is not so much that a machine be designed by the student as that he be fortified with the ability to analyze a problem and that he be able to apply to it the correct principles of design. 124. Drawings. The following dimensions are suggested for the cutting sizes of the sheets. The designer is at liberty to make his own selection from these sizes. It is suggested, however, that the sheets be taken as small as will admit of a clear and distinct set of drawings. 24 in.X36 in. Size A 18 in.X24 in. Size B 12 in.XlS in. Size C 9 in.Xl2 in. Size D Scale. Any scale may be taken which will show clearly all the details and give a good arrangement on the sheet. Details may have different scales on the same sheet if so desired. When this is done each detail should have the scale given. 'IT PLATE C 1 50"- Note: See details on drawings C 2 and C 3. TOGGLE JOINT PRESS ASSEMBLY Scale Fin-due UoivrsitV L by by INSTRUCTIONS CONCERNING DRAWINGS 237 Border Line. A margin of J in. should be left between the border line and the edge of the finished sheet on the top, bottom and right end, and J in. on the left end to allow for punching and fastening. Name Plate. Make the name plate or title at the lower right- hand corner to cover a space about 2J in. X3^ in. If any other standard corner is preferred other dimensions may be sub- stituted. No border line need be drawn around the name plate. It would be well for each designer to make a standard corner plate to be used below the various tracings when working up this part. All drawings will be carefully worked up in pencil and turned in to the instructor. The instructor will give them to another designer who will be responsible for the checking. Checking will be done in the form of notes on a separate paper and attached to the drawings. These notes and drawings will then be returned to the designer for approval and corrections. When the designer traces his drawings, or such part of them as may be selected by the instructor, he will obtain the signature of the checker to them and submit the same with the checker's notes to the instructor for approval. Each designer should have experience not only in planning and executing well his own designs, but he should take up designs of other men and offer suggestions and criticisms upon their work. One way to obtain this experience has been suggested above. In checking up the work of another man the following points should be observed: (1) General appearance of the design relative to workman- ship and execution, arrangement of drawings, notes, dimen- sions, etc. (2) General design relative to proportion, strength and arrangement of parts. This is to be merely the checker's im- pression and need not require the checking of the ' original calculations. No drawing should be retained longer than one exercise and at the completion of the checking should be returned to the designer. It is estimated that any set of drawings may be checked in this way within two hours' time. No notes or marks will be made on the drawings but special paper will be provided 238 MACHINE DESIGN for this purpose. In looking over the drawings finally, the in- structor will give credit to the work of the checker as well as to that of the designer. In all this work some standard text on mechanical drawing should be adopted as reference concerning arrangement of views, sectioning, cross-hatching, lettering, and the like. Every dimension should be clearly shown so that no measure- ments need be taken by scale from the drawing. All dimensions should be given in round vertical figures, heavy enough to, print well. No diagonal-barred fractions, thin or doubtful figures should be accepted. All dimensions should be given in inches. All dimension lines should be made as light as will insure good printing and should have a central space for figures. 36' All dimensions should read in FIG. 101. ,, -,. ,. ,. ,, the direction 01 the arrows. Avoid crowding the dimensions to the center of any detail. A much better way is by the use of projected lines as shown by Fig. 101. All detailed pieces should be accompanied by a shop note or call as "C. I. One wanted"; " M. S. Two wanted"; "Finish all over;" "Turned for a shrinking fit;" etc., etc. The following abbreviations will be considered satisfactory in these calls: C. S Cast steel. f Finish (see sheets of details). C. I Cast iron. B. b. t.. Babbitt metal. W. I Wrought iron. D Diameter. M. S Machine steel. R Radius. 125. Calculations. Each designer is expected to draw up a report in parallel with the design. This report should contain such free-hand sketches as relate to the calculations, also a full report of the calculated sizes and accepted sizes of the different parts of the design, and be submitted in a manila cover with the finished tracings and drawings. n^/ '54.^.,. . -llil PLATE C 2 4- hole.* - i"x la" OQp 3CTOW *** uiCO Drill i" For Pi ? 'V 8" MS n frl Die Head C.I. C.5. Per .'sc. 2 react. -8' Drill y For Pi 7 w -i i -2 ' r- i V .r o" Q. - & -6 1 '? ?Utr" 4 "-t fe ^rrrr T~ 'i M.S. t S Teq.'d LJ 1 C.I. M-* ^ ^ 4 J 1 " i P ! ' _T T i Not*-.- TDEELE JOINT PRESS DETAILS Scale. i^ Q ?d e size RiTdue U^versity LaFayette. \i Drctwrp . TOGGLE-JOINT DETAILS 239 Design No. 1 TOGGLE-JOINT PRESS 126. Analysis of the Forces Involved. Referring to Plate C-l, it will be seen that the acting forces can be represented in the following force diagram, with the direction of the forces repre- sented by arrows. FIG. 102. Each designer will be given a value for W, I, V and 0. In all the designs < may be taken at 10, assuming that the maximum load will be carried at this position and that the lever arm will then be horizontal. In the assignments for a number of designs the range of values may be as follows: TF = 200, 300, 400 1000 Ib. 1 = 4, 4.5, 5, 5.5 10 ft. V =for large sizes, 6, 8, 10 12 in. for small sizes 6, 6.5, 7 8 in. Selecting for our analysis the following values: TF = 100 Ib.; 1 = 5 ft. 3 in.; I' = 7 in.; and < = 10 degrees, we have from the force diagram Wl 100X63 W = V 7 100X56 = 9001b. = 8001b. W 900 = 2591.4 Ib. 2 sin. cf) .3473 W 3 = W^ cos = 2591.4 X. 98481= 2552 Ib. 240 MACHINE DESIGN 127. Lever. The formula for calculating beams in flexure, Art. 4, is M = SZ, where M= bending moment in pounds- inches, S = working fiber stress in pounds and Z = resistance of the section or modulus. In any section of the lever transversely across the axis let b and h be the breadth and the height of the section respectively. The designer must here decide if the beam is to have parallel sides, in which case b will be constant for all sections, or taper sides, in which case a certain ratio of b to h would be used. The best way to decide which to use is to find the size of the sections at two critical points as g and c, Fig. 103 (c is the fulcrum and g is any point near the handle), for each case and select between them. Assuming iS 8000 for 22- m FIG. 103. wrought iron or mild steel, 6 = 1, and disregarding the hole at c, which has little effect since the fiber stress of any section ap- proaches zero at the center, our formula M = SZ gives (section at g) 100 X 6 = 8000x1 X/> 2 ^6; h= .67 in. (section at c) 100X56 = 8000x1 Xh 2 -=-6; h = 2.05 in. This beam would have a better shape and would also be lighter if the thickness be reduced below 1 in., say to .75 in. With this value the formula gives (At g) 100X 6 = 8000X.75X& 2 ^6; h= .77 in. (At c) 100 X 56 - 8000 X. 75 X h 2 -6; h -2.37 in. These values give a well-shaped beam, having a section .75 in. X.77 in. at g and .75 in. X 2.37 in. at c. On the other hand, suppose a ratio of 6 to h = \, to be desired, the problem becomes (At g) 100X6 = 8000 ft 3 -=-24; h = 1.22 in. and 6 = 1.22 -=-4 = .3 in. L_ . i -^H ..--,- *. PLATE C 3 2." ! Alsorec^'eJ - O 7 O ^. bolt, SQ '?*J all over, tfireoded I" Wit^two TDGGLE3 JOINT PRESS DETAILS Scale, size. locd-i}utft . Pfcs:&< r ; TOGGLE-JOINT DETAILS ' 241 (At c) 100X56 = 8000 h 3 -r- 24; h = 2.56 in. and 6 = 2.56 -r-4 = . 64 in. section at g= .3 in. X 1.22 in. section at c = .64 in. X 2. 56 in. The above gives the method of determining the size of the section at any point of the beam. Sections should be taken at regular intervals of length and a diagram plotted from the results. One section only need be taken between a and c, say at o midway between. This diagram when completed will show the beam to take the form of a curve similar to Fig. 104. It may be found convenient, however, to approximate this curve with a straight line as x y. This would be satisfactory for strength and would be more easily constructed. It will be noticed that the bending moment becomes zero at the points a and p where the loads are applied. This would theoret- ically give no size to the handles and make it impossible of FIG. 104. construction. Some satisfactory design of handle or hub must be made at these points with sufficient size to carry the pins or bolts, each hub to have the sides and edges of the beam filleted into it in a neat manner. See Plate C-3. A handle can be placed at p for all loads of 300 Ib. or less and a drilled hub for larger loads so that a small air or steam cylinder can be attached. A similar hub will be added at a, for connection to the post at the rear. 128. The following shapes may be found useful in designing the lever. Shapes at p. The size and shape of the handle or hub at this end will be largely a question of neatness, since the load carried is very small. The pin, if one is used, may be calculated for 242 MACHINE DESIGN \ FIG. 106. TOGGLE-JOINT DETAILS 243 double shear to get the minimum size allowable, but this size will probably be so small that it will be necessary to increase the size of both pin and hub to add symmetry to the design. Such points as this call for special investigation. Any piece of a machine may be made extra strong, if necessary to harmonize with the other parts of the machine, but the reverse is not the case. Construction of the Joint at a. Referring to Fig. 106, shapes A and B would be preferred. In most cases the standard would be made of cast iron and could easily be cored out to fit over the lever arm end rather than to fit the arm end over the stand- ard as at C. The only calcu- lations necessary for this end of the lever, besides figuring the pin, are those that deter- mine the diameter of the hub and the length of the hub. It is reasonable to assume that the diameter of this hub should be made equal to the diameter of the cast hub of the standard. To illustrate: at a FIG. 107. a tensional force of W" is acting upward and this force is resisted by four cast iron areas on the section, RS, equal in total area to R'S', of the standard (Fig. 107). These four areas are produced by passing a plane through the standard along the line RS. Each area is equal to bh and should be figured for cast iron in direct tension by the formula W = SA. In making this calculation the ratio of b to h may be assumed. Having figured the pin for double shear by the formula W" = 2SA. find the diameter of the pin and add to it 2h, which will give the diameter of the cast hub and consequently the diameter of the lever end. If S for shear in wrought iron be taken at 5000 Ib. per square inch, the diameter of the pin will be .33 in. or, say f in. If S for tension in cast iron be taken at 1500 Ib. per square inch, the area bh will be .133 square inch, from which, if b be taken at J in., h becomes .53 in. This would make the diameter of the hubs at a, If in. It will be next in order to find the length of the hub at the lever end, also the corresponding values of the standard top. These 244 MACHINE DESIGN are determined largely from the crushing strength of the pin. First examine b of the standard to see if the assumed J in. is sufficient. The part of the pin in the casting and thepart in the lever are both subjected to a crushing force. The resistance of the pin to crushing is in proportion to the projected area of that part of the pin involved. In Fig. 108 let the pin be cut by a horizontal plane through its diameter 1, 6, 7, 4, corresponding to the plane along RS of the standard. 1, 2, 3, 4 and 5, 6, 7, 8 are the projections of the parts FIG. 108. included within the arms of the standard and 2, 5, 8, 3 is the pro- jection of that part included within the lever end. The diameter of the pin has previously been figured to resist shearing along the two planes 2, 3 and 5, 8. Now it is necessary to find the length 1, 2 and 5, 6 such that these parts will be safe from crush- ing. For the part in the casting, 26d = 2Xi Xf = A sq. in. = areas 1, 2, 3, 4 + 5, 6, 7, 8. If now the factor of safety for the wroughtiron pin be taken so that 5000 is a safe value for shear, S s , the pin will sustain W = /S 8 A = 1 \X5000 = 938 Ib. safely. This we find is greater than the load W" actually pulling on the standard so that part of the pin within the castiron standard is safe. If it had been found that 2bd was so small that the load it was capable of sustaining safely before crushing was less than the load applied, then either b or d or both would be increased. If d were increased without changing 6 then the hub diameter would be increased this amount above the calculated size of If in., but if b were increased, the areas bh would be stronger than the calculated value and h could be reduced accordingly, if it were considered necessary. TOGGLE-JOINT DETAILS 245 By the same line of reasoning the length of the pin within the lever would determine the mimimum length of the lever hub to resist crushing. This would be 2b = % in. From inspection it is readily seen that the thickness at a must be necessarily increased to that of the lever section. This at c is .64 in. In every fastening of this kind, investigation may be made for shearing of the pin, the strength of the sections around the pin, and the crushing of the pin, within both lever and standard. o 129. In calculating the size of the section at c the hole was not considered. ( The error introduced by this is very slight and in most cases may be neg- lected. The fiber stress in the cross-section of the arm varies from zero near the center to a maximum at the edge as shown in diagram B, Fig. 109, where by proportion we can readily obtain the relative resistance offered by the metal at the center as com- pared to that at the edge of the section. The loss at the center is more than taken up by the addition of a fraction of an inch at the edge or a very small boss around the hole. If the hole in any case should be large, a modulus could be selected for this hollow section, and the exact sizes obtained. The pin would be calculated in double shear as at a. FIG. 109. 246 MACHINE DESIGN The size of the boss, if any be added, is largely optional and is put on for finish. 130. Screw Fastening for Standard. In deciding upon the kind of fastening between the standard and the bed, it would be well to first examine it regarding the turning moments about a, Fig. 110, where W"b + W 3 h 3 - W'Jb' = W x l' + W y l" . Assume b = b'=3 in., h 3 = 2 in., Z' = 5 in. and Z" = l in. then with TP" = 800, TF 3 = 2552, and TF' 4 = 450 lb. We have 5 TF z + TFy = 6154 inch- pounds. If W x = Wy then 6 W x = 6154 or W x = 1026 lb. This is equiva- lent to a ^-in. bolt. Suppose W y , because of its location, to be of little value in resisting turning about a, then 5 TF X = 6154 and 17* = 1231 lb. = approximately iV m - bolt. If more than one bolt is used along the line W x or W y then the total bolt area at the root of the threads may be the equivalent of that given above. Next examine the joint for a sum-^ mation of all vertical forces. W" W' 4 = force holding stand- ard to bed = W x + W y . IiW x = W y then, 2 W x = 800 450 = 350 lb. and FIG. 110. W x = 175 lb. Since this force is less than that obtained by moments it need not be considered. Next examine the joint for a summation of the horizontal forces. In this the force W 3 tends to shear the bolts off in a plane with the top of the bed. It also acts upon the flanges to shear the casting inside the bolt holes. Considering the bolts first W 3 = SA . If we take S = 5000, then 2552 = 5000A; ,4 = .51 sq. in. of bolt area. If the bolt shears at the root of the thread, as would be the case with a cap screw, we need at least four f-in. cap screws. TOGGLE-JOINT DETAILS 247 In the second case, if the flange is, say 6 in. long and t in. thick, we have for the two sides 2552 = 2X6 tS. Let = 1500 for cast iron and find t = approximately .15 in. This would, of course, be made thicker, say J to J in., for the appearance and good proportion of the casting. In the above discussion of the standard fastening, the part most liable to fail would be the shearing of the bolts. This might not be true in every case; for example, if h 3 were very great when compared to I' , the failure of the joint would probably be by moments about a. The above calculations would be modified, also by the arrangement of the bolts or cap screws. It is well in every case to examine a joint from all standpoints and design for the greatest requirement. 131. Standard. The design of the standard would depend largely upon the magnitude of the force to be resisted. In the smaller machines it would undoubtedly be made of cast iron and FIG. 111. as such the upper end would be as shown in the preceding paragraph. In the larger machines the standard would be made of wrought iron or steel plates, in which case the sizes of the standard and lever end would be calculated from different values of S than those used for cast iron. The cross-section of the body of a cast iron standard may be shaped as in Fig. 111. Assuming the areas to be equal, the strongest section to resist any bending action that may come upon it, is D. The lower end of the standard would be planned to receive the rod W 2 , and would have a flange for fastening to the top of the bed. Fig. 112 shows some of the shapes that may be used. The pin at the base is figured for double shear by the formula 248 MACHINE DESIGN NOTE. When the constant 2 is used in the formula for double shear the result is the single cross-section of the piece. When this constant is omitted as in W = SA the result is the combined cross-sectional area. C FIG. 112. 132. Toggle. There are three ways in which the toggle may fail at the central joint: by shearing the pin, by bending the pin and by crushing the pin. In Fig. 113, B shows a very simple arrangement of this point. To obtain the size of the pin in this case to resist shearing Wi Wi 2 FIG. 113. W, = W 2 = 2SA . IiS = 5000, then 2591.4 = 2X50004.; A = .26 sq. in. and d = .5S say f in. It is readily seen that the pin would be found to be the same W size if the load -~ were figured for single shear as if W were figured for double shear. To obtain the size of the pin to resist bending assume some TOGGLE-JOINT DETAILS 249 length of pin between the outer forces --, as 2 in. or 3 in., and solve by the formula W'l + 8 = SZ. See Art. 4. There might be a question raised here concerning the proper formula to use for the bending moment, i.e., fixed ends or free ends. With the two ends of the pin held somewhat rigidly between the two sets of resisting forces, it is in about the same condition as a beam FIG. 114. fixed at the ends and loaded at the middle. If $ = 8000 and 1 = 2, then 900X2-8 = 8000X7rd 3 -^-32; d = .65 say \\ in. The toggle action on the pin at the center requires that the smaller force W should come at the center of the length of the pin as shown in A and B, Fig. 113. If the heavier force W 1 or W 2 acts at the center of the pin it would cause an unnecessary bend- ing as shown in C, and would require too large a pin to resist this stress. Fig. 114 shows other methods of designing the toggle. Concerning the crushing of the pin see Art. 128. 250 MACHINE DESIGN 133. Fig. 115 gives some shapes of toggle members. A, B, C, and D are usual shapes of the horizontal members. A and B have split ends and are necessarily hard to forge and machine. C is the simplest form. This form is sometimes modified by adding bosses to one or both sides as shown in D. The vertical member may be constructed solid as at E or adjustable as at F. FIG. 115. 134. Die Heads. The sliding head receives the thrust W 1 from the toggle and moves along the top of the bed toward or from the stationary head. It must be a good fit to the bed top having a free sliding contact but no side motion. The stationary head must be planned for longitudinal adjustment and for fasten- TOGGLE-JOINT DETAILS 251 ing rigidly to the bed top when desired. Suggestions for attach- ing these heads are shown in Fig. 116. Rectangular and V-- shaped ways are used, some having adjusting gibs and some plain. A is the simplest form and may be grooved from the solid or held down by plates. In such a design the overlap below the top of the bed should be made sufficiently strong to resist the turning action from W s . B and C show the application of gibs between the sliding head and the bed to take up side slack. In some classes of machines gib arrangements are essential. If, FIG. 116. however, heavy side thrusts are involved the form C is question- able unless made very heavy and strong. With the bed planed to an angle as at C and D, the latter would be considered the stronger. Sliding Head. Since the sliding head cannot be rigidly fastened to the bed, it must be fitted to a set of guides. The most common fastening is shown in Fig. 117. Having the forces W 3 and W 4 (resultant forces from WJ acting on the pin and allowing all the reaction from the die to fall at the upper point of the head, say 4 in. above the bed, we have a cantilever beam projecting upward from the bed top and acted upon by three forces tending to break the beam at some section as ao. Any leverages may be 252 MACHINE DESIGN selected other than 4 and 2 but these are given for the sake of argument, the actual values used depending largely upon the kind of die used between the two heads. It is evident that the two forces opposing each other (W 3 , action and reaction) will have the same value. These moments will cause stresses in any section under investigation. Suppose the line ao to be the weakest section in the beam. The tendency to break here is resisted by two metal sections, each 6 inches in width and h inches in height, or, by one section 2 b inches by h inches. The fiber stress caused by the moments from the two W 3 forces will cause a maximum tension at o which becomes less as it approaches a. This tensional fiber stress at o will be partially neutralized by a downward force W 4 distributed more or less uniformly over the area 2 bh; the final stress at o being the algebraic sum of the two. Let S p = pressure per square inch acting perpendicularly kl j w * *t I r - (7 + X __^ 2 I < } ^4 FIG. 117. to the bed top over section 2 bh, S m = tensional fiber stress at o due to the moments and S t = resulting fiber stress at o. Taking W 3 in two moments about ao and W 4 in direct pressure we have W 4 = S P A and M = S m Z, from- which we obtain S p = 450 , , _ 2552X2X6 15312 pressure and o m = due to direct f-v 7 7 \JL *-4.Vy \J\J \A.JH.\s\j\J l_/ J. \_/OO IAJ. V/ CVAAVA ^-'7/2- OLL2 7^ 1*2 due to the summation of the moments. Now if S m S p = S t ', also if h = 5 in. and $ = 1500 we have fo^approx. f in. If the fiber strength of tension and shear in cast iron be taken the same, then b' = b approximately. In like manner the reaction TF 3 from the die may be taken at the bottom instead of the top of the sliding head, and the turning moment figured in this way to see if there is greater danger to the section than when taken at the top. TOGGLE-JOINT DETAILS 253 Other investigations may be made for this fastening. If the projection b' were fastened on with screws the calculations would be worked up in a similar way to the fastening at the base of the standard. Stationary Head. As in the sliding head, it is assumed tha,t the stationary head is properly desig- ned above the bed top and that the fastening only is in question. Fastenings for small machines will not be difficult but those for large machines will call for extreme care. The simplest fastening is shown in Fig. 118 and acts as a frictional resistance only. If W t = tension on the bolt in pounds, TF 3 = 2552 lb., y = 4 in., and x = 6 in., we have by moments, disregarding the benefit obtained from the overlap of the block around the frame, 2552X4 = 6TF*, or, TF f = 1702 lb. This will hold the block to the bed. It is now necessary to determine if the block will slip with this force binding the frame between these FIG. 118. FIG. 119. two friction surfaces. Let the coefficient of friction be- tween the block and the frame also between the washer and the frame be = say .3, then the resistance due to friction is, by formula, 2fiW t = F and when applied to our problem is 1021.2 lb. That is, with the conditions as stated, if W 3 were 254 MACHINE DESIGN only 40 per cent as large as it now is the block would just slip. Since the bolt as figured from moments proves to be too small to keep the block from slipping, let us reverse the process and find how large a bolt will be necessary to hold the block against the force TF 3 . By substituting as above we have 2 X .3 X W t = 2552, from which 1^ = 4253.5 Ib. This force is being exerted at the root of the thread tending to elongate the bolt. With $ = 8000, this will give slightly greater area than .5 sq. in. and will require a bolt of approximately 1 in. diameter. It is evident from this that more than one bolt should be used, or that some other arrangement be substituted for the friction surfaces. In Fig. 119, A is very similar to Fig. 118, excepting that the lower surface is notched to protect it ^ from slipping. The upper block may slip slightly, but this will cause a greater grip and a conse- quent increase of frictional resis- tance. A possible improvement on this, if the construction of the machine would permit it, would FIG. 120. be to have the bolt at an angle as shown in the dotted lines. Let this angle be, say 30 degrees with the horizontal, then from Fig. 120, A, .3T sin a = resistance due to friction, and T cos a = horizontal component of the bolt tension. Combining we have, .866r + .3x.5T = 2552, or, 7 7 = 25121b. This will require a lf-in. bolt. Fig. 120, B, will cause a tension on the bolt (disregarding friction) of T'=2552 tan (P + 2). Let /? = 90 degrees then I" = 2552 Ib., requiring a if-in. bolt. It is very evident that if friction were included in this it would reduce the bolt size somewhat. Let the student investigate this with friction included. C, Fig. 119 is probably not as strong in the shape of the tooth as A and B, but with a large tooth area the unit shear becomes small enough so that the teeth are not endangered. The vertical faces on the teeth reduce the vertical thrust on the bolt to a TOGGLE-JOINT DETAILS 255 minimum and permit the use of a bolt just sufficiently strong to protect the block from turning as in Fig. 118. D is arranged to have pins to fasten into the frame either through the block, or behind it. These pins keep the block from sliding and are calculated for shear, while the bolt is cal- culated to resist turning as in Fig. 118. Another way in which these fastenings may fail is by shearing the bolt. Assume W 3 Fig. 118 entirely acting to shear, we have 2552 = ' 51 sq< i bolt area< taken S = say 5000 full area of the bolt it would be if -in. diameter. This shows a requirement about equal to that for tension. In any form of fastening it is well to investigate both tension and shear and take the larger requirement. It should be understood that, if the block clamps over the edges of the frame on planed ways, this will assist the bolt in holding the block down and a smaller bolt may be used. Let the student investigate this as in the case of the sliding head. 135. Frame or Bed. The calculations for the frame will be found somewhat more complicated. Assume a simple type, say FIG. 121. of the same general shape and cross-section as Fig. 121. Assume also the force W 3 acting at some point along the block face, say at the middle of the block, a distance of 2 in. above the top of the frame. Any other height may be taken but in all probability if the dies should not be parallel and they should strike hard at the top, this inequality would be accounted for by a slight 256 MACHINE DESIGN springing of the bed. It may be assumed that the force W 3 will act somewhere near the center of the die before it reaches such a magnitude as to endanger the frame. This force tends to break the bed along some line as rs, and produces combined ten- sional and compressional stresses in the fibers of the section. Considering the part to the right of the section as free we have, Fig. 122, the fibers on the upper or weak side subjected to two tensional stresses, the sum of which should not exceed the safe fiber stress of the metal, i.e., S 1 +S 2 = S t ' t and the fibers on the lower side, subjected to a tensional and a compressional stress, the algebraic sum of which should not exceed the safe com- pressional fiber stress of the metal, i.e., S 1 + ( S 2 ) =S C where $!= uniform tensional stress $2 = stress due to bending St and S c = combined stresses. To obtain Sj^ and S 2 on the tension side use the formulas W 3 = S^A and M = S?Z and obtain W p = S 1 where A = area of section in square inches and - -- - = S where Z = modulus of section. It will be seen 2 that the moment arm is the perpendicular distance between the force and the center of the section. The value ~ would be t changed for any other than a uniform section. See Art. 144. Having selected the section of the bed as Fig. 121, we find the modulus to be Z .ty-yv x2 SeeArt . 4 . b h It will be necessary here to select some values for 6, b', h and h' and make a trial solution. Take b=2 in.; b' = 1^ in.; h = Q in. and h'=4: in. With these values we find A = 12 sq. in. and 2552 S l = -^r- =212. 7 Ib. per square inch LZ , t also Z = 18.8 and 2552 (3 + 2) . , ^2 = -- TOO - = 679 Ib. per square inch. io.o , = ^+2 = 679+212.7 = 891.7 Ib. per square inch. TOGGLE-JOINT DETAILS 257 Since the usual value of S t for cast iron is 1500 to 2000, this shape and size of section would be stronger than necessary. Now, if the figures of the section be changed to read 6 = 2 in.; &' = l^in.; h = 5 in.; and h' =4 in. the value becomes 2552 2552 i = o~~ = 319 Ib. per square inch, and S 2 = -- = 1126 Ib. per square inch. = Si = 319 + 1126 = 1445 Ib. per square inch. FIG. 122. This seems to agree very well with the safe value of cast iron in tension, and may be used. Since this is a symmetrical section and since cast iron is much weaker in tension than in compression, the latter will not need to be investigated and the above figures can be accepted for the sizes of the bed. With a section that is unsymmetrical it would be necessary to investigate both sides of the section. See Art. 144. Having found the shape of the simple section it is possible to modify it to a certain degree without affecting the calculations seriously. To illustrate, the portion abed, Fig. 123, may be lopped off and added to the inner side at a'b'c'd' without affecting the modulus. Metal may be moved paralled to the axis of the section so long as the section is not distorted to such an extent that it will break by twisting. Any change of metal, however, toward or from the axis of the section, changes the modulus and hence the resisting power of the section. Fillets may be added at the interior corners giving a shape similar to most frame tops. For the bottom, a slight deflection or slope of the web, as shown by the dotted lines, gives a result very similar to a plain cast iron 258 MACHINE DESIGN engine or lathe bed. Other minor changes such as slight curves instead of straight sides might be made without any loss of rigidity. In any case where the shape of the simple section is found and the designer wishes to increase the thickness of any part he may do so and the result is merely to increase the factor of safety. Suppose some other than a uniform section is desired, the same process would be employed in finding the stresses as given above. The modulus, Z, however, would be obtained as shown in Art. 144. If under very heavy loads it is advisable to specify one or more steel I beams or channels from Cambria, this may be done by making a trial selection of a section and substituting the value of Z and A in the formulas as before. If this value $, +S 2 = St = 8000 to 16,000, the exact value depending upon the rigidity of the beam, the condition is fulfilled as in the case of the cast frame. 136. The final determination in this design is to obtain the length of the frame to prevent overturning when the load is FIG. 123. FIG. 124. applied. Let W 5 Fig. 124 = the weight of the frame, then from the force diagram we have the following moments about the end at b but W'J, 3 + WJ,t = W"l 3 and Wx = W 5 l 5 . The length may then be obtained by adjusting the values of x and 1 5 such that the equation will be satisfied. TOGGLE-JOINT DETAILS 259 To obtain the length, however, in a more direct way the fol- lowing can be used: If x = l-l 3 andZ 5 =Z 2 -^2 then W(l-l s )=WJ, 2 +2. Knowing the cross-section of the bed in square inches, the weight of 1 in. in length would be .26 A; the total weight of the bed being .26Z 2 A approximately. 1 Then LJ> = W(l -1 3 ) ^.13A. Let I 3 = l 2 a where a is the offset as shown, then Z, 2 = TF (I l 2 + a) -T-.13A, from which we obtain the formula W W W^ 2 ,= - 3.85-^ \ 7.7 (l + a) -"- + 14.82 weight of the frame W 5 would be greater than here shown be- cause of the metal in the ends of the frame and the attached mechanisms, all of which would be effective. The error, whatever it may be, is toward that of safety. 260 MACHINE DESIGN First Alternate, Design No. 1 w FIG. 125, A THE TOGGLE JOINT PRESS 137. Assignment. W= . . . .; Z=. . . .; l'= ....; 6 (min.) = . . . . In this design the lever is placed within the bed rather than above it. It will be noticed that the end of the bed is slotted to allow for a movement of the lever arm between the points x and x' '. The weakening of the bed due to this slot need not be con- sidered a serious matter. With a long and shallow bed, however, the movement of the arm will be small and will give a very slight movement to the sliding block. For our purpose this machine may be designed merely to exert a pressure between the two sliding blocks, in which case a very slight movement is all that is necessary and the form shown will be satisfactory. FIG. 125, B. In case the movement of the sliding block is desired greater than that allowed here, the lever may be arranged as shown in Fig. 125, B. ALTERNATE DESIGNS 261 Second Alternate, Design No. 1 FIG. 126. VERTICAL HAND-POWER PRESS 138. Assignment. - W = . . . . ; 1= . . . . ; I' = ....; 6 (mm.) = . . . This design follows the principles laid down in No. 1, with two exceptions. First, the length I' here becomes so small that a separate crank cannot be used and a bent shaft or an eccentric is substituted. In the eccentric the length I' is the distance between the center of the shaft and the center of the eccentric. Second, the thrust of the sliding block is received through a screw directly against the base of the frame. A hollow rectangular section is suggested as the best shape of the frame. Investigate also for the screw and nut to resist the thrust. 262 MACHINE DESIGN Third Alternate, Design No. 1 FIG. 127. THE VERTICAL FOOT-POWER PRESS 139. Assignment. W = (100 or less) Ib. I = (60 to 72) in. V = (3 to 6) in. 6 (min.) = degrees. This machine can be used for all kinds of light press work where but a small movement of the ram is needed. Where this movement is desired as great as possible, increase I' and decrease Z, also reduce the length of the toggle members. The ram may be made rectangular in section and the forming dies need not be developed. The frame is hollow and the lever I is fastened on the plane of the toggle. ALTERNATE DESIGNS 263 Fourth Alternate, Design No. 1 FIG. 128. SMALL HAND-POWER PUNCH Fig. 128 shows a small bench tool, used for punching sheet iron and other thin metals. Because of its simplicity only two parts of the assignment will be given. All other necessary assumptions may be made by the designer and a complete set of calculations and drawings made. The diagram to the right shows the mechanism. 140. Assignment. W = (at end of lever I, 50 to 100) Ib. T = (length of throat) in. 264 MACHINE DESIGN Fifth Alternate, Design No. 1 FIG. 129. HAND-POWER PUNCH AND SHEAR The hand-power punch and shear is strictly a bench tool for operating on light work. The force at the end of the lever arm I should not be greater than 100 lb.; V is the eccentricity of the cam, a is the distance from the pivot point of the shear arm to the point where the cam force is applied, and b is the distance from the pivot point to the point of greatest shearing resistance. 141. Assignment. (See Design No. 2 for methods.) Kind of material to be cut ........................ Length of cut or diameter of punch ................ in. Thickness of plate to be cut (up to |) .............. in. Depth of throat ............................ in. CHAPTER XIV DESIGN OF BELT-DRIVEN PUNCH OR SHEAR 142. General Statement. A belt driven punch or shear is the machine selected to represent the second general design. In- cluded within this one machine are problems covering the design of frame, levers, gears, fly-wheel, pulleys, bearings, shafts, sliding head, punch, die, clutch, stripper and cam. The fact that this machine finds such general use in manufacturing plants and that it embodies such a variety of designs makes it an ideal subject for analysis. Fig. 130 shows a motor driven shear of FIG. 130. late .design. It is not expected that the required design will be for a motor drive, but that the distance between the bearings" be shortened and pulleys used instead. In giving out the design the following requirements will be made: first, the work to be accomplished, i.e., diameter and depth of hole to be punched or the cross-section of the piece to be sheared; second, the maximum distance from the edge of the plate to the center of the cutter, i.e., the depth of the throat of the machine; third, the average cutting velocity of the punch or knife in inches per second, ^or the r.p.m. of the cam shaft. 265 266 MACHINE DESIGN In the analysis of the methods employed in working up such a design, the frame sections will be carried out somewhat in detail because of the advanced character of the work; the rest of the machine will be dealt with more briefly. In making the assign- ments, the members of the class should be given values that differ materially from those worked out here. The five sample plates at the end of the design show a complete set of drawings of such a machine. 143. Requirements of the Design. A machine to punch a 1-in. hole through f-in. mild steel plate, the center of the hole to be not greater than 7 in. from the edge of the plate. The velocity of the punch during cutting may be taken in this case as approxi- mately 1 in. per second. 144. Frame. The material used in the frame of such a machine is either close grained cast iron or steel casting. The general shape is about as shown in Fig. 130 and the sections of the frame, Fig. 131, are either hollow cast iron as shown in B and C or web-shaped steel as shown in A. Of the three sections, B and C are the most common. Fig. 137 represents the outline of the n yf - I < w * ~jj-* a l \^/ / ^ / |p2fc y- "C -y WT"~ =-1-" -* '^y^SM A b i t g FIG. 131. assembly drawing as finally worked out about x x, the center line of the frame. To plan the general shape of the frame about the punch, begin by laying off the throat depth, G, say 8 in., along the line x x. Find H of the same figure by assuming some shape of frame section and calculating the sizes for the various parts of the section as described in this article. Find also other safe sections at various angles to the horizontal and trace the outer curve 'ri E rrnJ SLIDINC . HEA'JD. vn-n T! m V, PUNCH < ->^SMiLAT . ] j 1 1 | I ! ^ 1 ij . 1 \ 1 ! " " ! i : TTJ LIT D Uuj jib ' i 1 o D i a FIG. 144. action of the punch on the material is shown in A, Fig. 144, the hole tapering from the size of the punch on one side to the size of the die on the other. This taper is slight and is considered of no consequence in rough work, but in finished work it is a difficulty that can easily be remedied by reaming the hole afterward. For reference see " Dies, Their Construction and Use. " Woodworth. 284 MACHINE DESIGN There are various methods of fastening the punch to the sliding head; B shows the bottom of the sliding head fitted with the square ended socket and punch. A screw ended socket is sometimes used as at E. C shows the bottom of the head flanged and drilled for the attachment of either punches or shears. In single machines it is desirable that both punching and shearing be done. Where such is the case this is a good form. Side adjustment of the punch may easily be made if the head be slotted as at D and fitted with a tee block as E. Dies are made from high carbon steel and are held in a holder; the holder in turn is bolted to the horizontal face of the frame. A certain amount of adjustment is necessary in locating the die, conse- quently the holder is made in two parts. Other Types of Shearing and Punching Machines The smallest sizes of punching and shearing machines are oper- ated by hand power or foot power, medium sized machines are operated almost exclusively by belt and the largest machines are operated by belt, steam, water or electricity as shown in Figs. 145, 146, 147, and 148 respectively. These designs show present practice and are added to enable the designer to become more familiar with the form of the parts and the make-up of the machines in general. It will be noticed that in the larger machines the frame is of such a size as to project below the floor, the weight being carried on legs or lugs cast on the side of the frame. It will also be noticed that arrangements are made at the top of the frame for the attachment of a crane to assist in handling the material. Most single machines have the lower end of the ram so con- structed that either punches or shear blades may be attached. This requires some little time in changing and adjusting the tools. Double machines avoid the necessity of such changes. Machines such as are here represented require more work than should be expected of one assignment. They may, however, be assigned to two men. This is especially true of the double machines, in which case the frames may be worked up independ- ently, and the driving mechanism, jointly. Electric motor sizes and capacities may be obtained from any standard catalog of electric machinery. LARGE PUNCHING AND SHEARING MACHINES 285 FIG. 145. FIG. 146. 286 MACHINE DESIGN FIG. 147 FIG. 148. TYPICAL DRAWINGS 287 I 288 MACHINE DESIGN TYPICAL DRAWINGS 289 11 pi n c;- 1 f?" 63>F U 75 >*x ^ < ^ ^ Ibl] I y Q ^ t & I&- ^ z If - en C s ^ o O H cu 290 MACHINE DESIGN \ . PLATE G 7 SINGLE POWER PUNCH DETAILS Pvrdue University Lafayette Ind I . . . . ALTERNATE DESIGNS First Alternate, Design No. 2 291 B FIG. 149 THE BEVEL SHEAR (Niles-Bement-Pond Catalog) 156. Assignment. Kind of material to be sheared Width of plate to be sheared (6 to 12) in. Thickness of plate to be sheared (J to 1) in. Depth of throat (6 to 18) in. Strokes of the ram per minute (15 to 20) The frame sections may be calculated, if desired, to a regular outline as shown in the dotted lines, after which modifications in this outline may be made by approximation. A better way, however, would be to sketch the approximate longitudinal frame section as above and figure for each of the several irregular sections, as A, B and C. 292 MACHINE DESIGN Second Alternate, Design No. 2 K,\'Y\' 2 go .2 y l c3 O 3 5 3 s & & I* o -2 .Soil, S^ ^2 lisi C 03 _Jj w | fl O a CD -e .5 I ?-l O * I I 2 .. > 03 o -a bJO o 81 .2 a CD ^ -3 g o> IS bJO 10 PJ > i! L a* --> rj s 5 s^ ml Go S 9 *w o HM KINEMATIC PROBLEMS Mechanism of the Straight Line Governor 337 O -fi -M ^ a a L3 O V DQ s s bO bJD O fl " rH rt} PH *w qs s i ^ ^8 pb S3 c3 "r" 1 o -' g 3 g eS O * S ; j rt 0) a, < SP PH O . r^ i -^ s r.- ;-.:.' . " : v - -' '.'. : tfii . . . Mechanism of the (See "Auchinclc 207. Assignment. In this analysis assign the lead and the cu' this cut-off and draw in position of cut-off. Finally draw valv ;haert Valve Gear Valve Motions) Kf. C*nfrL>r* Kalvsc 3O CUT off &. V^\LSCMALRT NAIVE. ANALYSIS (the latter varies from 20 to 80 per cent). Set the link to give ipse and fill in table of events. A ol ' .*?> ^q 08 c5 (^ ^motl feeh'7 .1 k> eIJ>a-J 1 \\ \\ II go a M S g II *r ~ -3 o a a " ^ o ^ I GTO3 *2S2 silt Sf.9 v ~"' O T3 >-: te J> fl /" f N 5 { * \ j I i S H^ J \ 1 I i s \ I 9 y \ ^ sj \ * \ \ ' i \ <3 ' \^3tta J *J O ^ I o Jg Ssdj s 1 1 & 6 0^ 00 T3 X X INDEX Abbreviations, 1 Air hoist, 299 Alloy steels, 14 Alloys, 15 Arms of gears, 196 Automobile clutches, 176 Ball bearings, design, 157 endurance, 157 journal, 153 materials, 156 step, 155 Beams, cast iron, 28 formulas, 7 uniform strength, 8 Bearings, adjustment, 129 ball, 153 experiments, 138 friction, 134, 145 heating, 136 Hyatt, 160 journal, 128 lubrication, 131 Mossberg, 164 multiple, 149 pressure, 134 roller, 159 sliding, 120 step, 144 . Belting, friction, 221^ slip, 223 speed, 227 strength, 225 width, 226 Bevel shear, plain, 287 j rotary, 296 Bobbin winder, 313 Boiler, shells, 50 tubes, 55 Bolts and nuts, 91 Brass, 16 Bronze, 16 Bulldozer, 293 Butt joints, 99 Cams, accelerating, 324 conical, 321 crosshead, 320 lever, 319 planer, 311 sewing machine, 312 steamboat, 314 writing, 332 Caps and bolts, 142 Castings, iron, 10 steel, 14 Chain drives, 197 Clip former, 324 Clutches, 173 automobile, 176 press, 280 Columns, 4 Cotters, 104 Cotton ropes, 231 Coupling bolts, 178 Couplings, 171. Crane hooks, 94 Cranks and levers, 200 Crucible steel, 13 Curved frames, design, 47 tests, 42 Cylinders, steam, 78 tests, 80 thick, 51 Die heads, punch, 283 sliding, 251 stationary, 253 Discs, conical, 216 logarithmic, 217 339 340 INDEX Discs, plain, 215 tests, 218 Drawings, size and scale, 236 Elliptic springs, 114 Factors of safety, 17 Fittings, pipe, flanged, 71 screwed, 69 Flanged fittings, 71 Flanging machine, 297 Flat plates, 83 springs, 113 Fly wheel, experiments, 207 press, 274 rims, 204 safe speeds, 205 Formulas, 3 Frames, curved, 42 design, 21 machine, 26 press, 255 riveter, 40 shape, 38 shear, 45, 266 stresses, 39 Friction, belting, 221 journals, 139 , 145 Gears, bevel, 195 cut, 189 design, 310 rim and arms, 196 speed, 212 teeth, 186 Governor, centrifugal, 336 shaft, 337 Hangers, 182 Heating of journals, 136 Helical springs, 107 Hoist, air, 299 Hooks, crane, 94 Hoops, steel, 68 Hyatt bearings, 160 Iron, cast, 10 malleable, 11 wrought, 12 Joint pins, 104 Joints, rim, 210 Joints, riveted, butt, 99 diamond, 103 lap, 98 tube, 67 Journals, 128 strength of, 142 Keys, shafting, 178 Kinematics of machines, 309 Lap joints, 98 Lever design, 240 Link motion, Stephenson, 338 Walschaert, 339 Lubrication, 131 Machine frames, 26 screws, 93 supports, 25 Malleable iron, 11 Manganese, bronze, 16 steel, 14 Manila ropes, 229 Materials, 9 Metals, strength of, 18 Mossberg bearings, 165 Mushet steel, 15 Nickel steel, 14 Notation, 2 Open hearth steel, 13 Phosphor bronze, 16 Pipe, fittings, 69 tables, 56 Pivots, conical, 146 flat, 145 . - Schiele's, 147 Plates, flat, 83 steel, 87 INDEX 341 Power press, 294 Press, foot power, 262 hand power, 261 power, 294 shear, 295 toggle joint, 235 Pulleys for press, 273 Punch, hand power, 263 horizontal, 288 Quick return, 315 Riveted joints, 96 Riveter, Allen, 300 alligator, 304 frames, 40 Hanna, 302 hydraulic, 307 lever, 306 mudring, 305 Riveting machine, hydraulic, 308 Roller bearings, 159 design, 163 step, 162 Rope transmission, cotton, 231 Manila, 229 strength, 230 wire, 232 Rotary shear, 295 Schiele's pivot, 147 Screw, design, 246 machine, 93 Sections, cast iron, 27 Shaft for press, 276 Shafting, bending, 169 clutches, 173 couplings, 171 diameter, 168 hangers, 182 . v keys, 178 Shapes of frames, 38 Shear press, clutches, 280 die head, 283 fly wheel, 274 forces, 272 frame, 45, 266 Shear press, gears, 277 pulleys, 273 shaft, 276 sliding head, 279 types, 284 Shear, rotary, 295 Shells, thin, 50 thick, 51 Silent chains, 199 Slides, angular, 120 circular, 124 flat, 122 gibbed, 122 Slip of belts, 223 Springs, elliptic, 114 experiments, 109 flat, 113 helical, 107 torsion, 112 Sprocket wheels, 197 Standard for press, 247 Steam cylinders, 78 Steel, alloys, 14 Bessemer, 13 castings, 14 crucible, 13 mushet, 15 open hearth, 13 plates, 87 Step bearings, 144 Stephenson link motion, 338 Strength of materials, 17 Stuffing boxes, 124 Supports, machine, 25 Tests of gears, experiments, 191 formulas, 190 practice, 193 proportions, 187 strength, 188 Thrust bearings, 150 Toggle joint press, alternate design, 260 analysis, 239 design No. 1, 235 die head, 250 frame, 255 342 INDEX Toggle lever, 240 Units, 1 screw, 246 , , n . Vanadium steel, 15 standard, 247 toggle, 248 Walschaert link motion, 339 Torsion springs, 112 Wire ropes, 232 Tubes, boiler, 55 Wooden pulleys, 212 joints, 67 Writing cam, 332 tests on, 62 Wrought iron, 12 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $t.OO ON THE SEVENTH DAY OVERDUE. 6 193} 261931 IMA MAY 7 1941 M MAY 35 1948 LD 21-2m-l,'33 (52m) UNIVERSITY OF CALIFORNIA LIBRARY