APPLIED MECHANICS APPLIED MECHANICS AN ELEMENTARY GENERA!,- INBUVTWF 7 G THE THEORY OF STRUCTURES AND MACHINES WITH DIAGRAMS, ILLUSTRATIONS, AND EXAMPLES BY JAMES H. COTTERILL, RR.S. */ HONORARY VICE-PRESIDENT OF THE INSTITUTION OF NAVAL ARCHITECTS ; ASSOCIATE MEMBER OF THE INSTITT7TION OF CIVIL ENGINEERS ; SOMETIME PROFESSOR OF APPLIED MECHANICS IN THE ROYAL NAVAL COLLEGE, GREENWICH SIXTH EDITION MACMILLAN AND CO, LIMITED NEW YORK : THE MACMILLAN COMPANY 1906 All rights reserved First Edition, 1884. Second ,, 1890. Third , 1892. Fourth ,, 1895. Fifth 1900. Sixth 1906. GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO LTD. PEEFACE. ON the author's appointment to lecture on Mechanics in the Royal Naval College, a course of elementary lessons was commenced, based on RANKINE'S well-known treatise, with such assistance as could be obtained from other sources. After some years this course assumed a tolerably permanent form, and it was thought desirable to print it, partly from the inconvenience to students of being exclusively dependent on oral instruction, and partly from an idea that it might be useful to others besides those who were immediately addressed. The place which these lectures occupy in the programme of the College will be found explained in an Appendix. The preparation of the work for the press has extended over a considerable period, and has been subject to many interruptions. There is therefore not always the unity desirable in a scientific treatise ; nor is it by any means complete, even when due account is taken of the stringent limitations explained in the Introduction. It is, however, hoped that these deficiencies may be partly com- pensated for by the fact that the book is the product of a great deal of experience in teaching the subject, and a great deal of con- sideration as to the matter which ought to find a place in a general elementary treatise. Nearly the whole has been delivered in the form of lectures, and some part has actually been printed from notes taken throughout one session by a member of the junior class (Mr. H. J. Oram, R.N.) at that time, which were afterwards transcribed for the press by the author's assistant. Everything, however, of any importance has been re-written, with alterations and additions, to make it better fit for publication. Throughout, the 677667 vi PREFACE. object has been to give reasons, not rules, and details of application -are consequently subordinated to the principles on which the theory is based. Especially has the author endeavoured to distinguish as clearly as possible between those parts of the subject which are universally and necessarily true, and those parts which rest on hypotheses more or less questionable. The book is intended to give that general knowledge of the mechanics of structures and machines which should accompany the detailed study either of naval architecture or of any special branch of engineering to which a student proposes to devote himself. Much, therefore, is excluded which might naturally be expected to form part of the work, simply because, however important, it is required only by a special class of students. The introduction of descriptive details is not necessary to the plan of this work, except in certain parts of the theory of mechanism, nor, indeed, in a general treatise would it be possible to include them systematically within any reasonable compass. In the chapters on mechanism, however, they are required, and elsewhere it has been thought advisable to introduce them occasionally. Care has been taken to select working examples almost exclusively, the plates representing which have mostly been drawn by Mr. T. A. Hearson, to whom the author is indebted for many suggestions and portions of the descriptive matter, together with some assistance in revising proof sheets and transcribing lecture notes for the press. The proofs have been read by Professor W. C. Unwin, M.I.C.E., to whose great technical knowledge some corrections are due. In a general elementary work there is not room for much that is new : in the references at the end of each chapter and in the Appendix the various sources of information have been stated fully. GREENWICH, May, 1884. PREFACE TO THE FOURTH EDITION. THE origin and object of the present work are so fully explained in the original preface that it is unnecessary for the author to do more than express his gratification that it has been found in some degree, however imperfectly, to fulfil its purpose. In this, as in the third edition, a considerable amount of additional matter has been intro- duced, partly on subjects which have acquired additional importance since the book was originally written, and partly where further explanation appeared urgently required. The whole, so far as circumstances permitted, has been revised and brought up to date. The method of treatment originally employed has been as far as possible adhered to. To some readers familiar with modern treatises on theoretical mechanics it may appear in many respects unduly conservative, but the author is convinced that it is that which is best suited to the work on hand. It is too often forgotten/ that the mechanics of the engineer has a history of its own, and has developed in its own way. His fundamental idea the idea of work was long ignored in academic lecture rooms, and has only recently been appro- priated for the most part without acknowledgment by writers of elementary text books. Apart, therefore, from the special nature of the subject-matter, the distinction between "applied" and "theoretical" mechanics, the mecamque industrielle and the mtcanique rationelle of the French, though happily fast disappearing, is even at the present day much more real than many persons are disposed to admit. It is hardly necessary to say that the units of measurement employed in the physical laboratory would be entirely out of place here ; the system is not used by engineers either at home or abroad, nor is there viii PREFACE TO THE FOURTH AND FIFTH EDITIONS. any reason to think that it could be practically introduced without great modifications. Metric gravitation measure stands on a different footing ; it is recognized in all countries, and its universal adoption for the purposes of ordinary life is clearly only a question of time. In the present edition it has therefore been explained in due course, though for some time to come the system in common use must continue to be that which is principally employed. The note in the Appendix on the resistance and propulsion of ships introduced in 1892 has been retained. The subject is one of great importance, and though only a summary in skeleton of leading facts relating to this intricate question, it is hoped that the note may be of service to some readers in directing their attention to the principles on which any sound treatment of the subject must be based, and its connection with other branches of mechanics. As in preceding editions the author has pleasure in acknowledging the services of his assistant for the time being. In the present case Mr. R. B. Dixon, R.N., has read the greater part of the proofs, and to his care many corrections are due. GREENWICH, July, 1895. PREFACE TO THE FIFTH EDITION. THE changes in this edition are few. For reasons already sufficiently indicated additional matter has been introduced on the effects of the inertia and obliquity of connecting rods (pp. 229, 285), on the torsion of tubes (pp. 358, 422), and on the centrifugal whirling of shafts (p. 396). Some other changes of slight importance have been made in the text and further notes have been added, especially on the representation of a curve of crank effort by a Fourier Series (p. 615) and on certain points in the theory of turbines (p. 629). The author hopes that the book may still continue to be found useful, KENSINGTON, October, 1900. CONTENTS. GENERAL INTRODUCTION. PART I. STATICS OF STRUCTURES. CHAPTER I. FRAMEWORK LOADED AT THE JOINTS. ART1CLE PAGE 1. Preliminary Explanations and Definitions, ... 1 SECTION I. TRIANGULAR FRAMES. 2. Diagram of Forces for a Simple Triangular Frame, . .3 3. Triangular Trusses, ..... 5 4. Cranes and Derricks, ...... g 5. Sheer Legs and Tripods, ....... 8 6. Effect of the Tension of the Chain in Cranes, ... 9 Examples, ...... 10 SECTION II. INCOMPLETE FRAMES. 7. Preliminary Remarks, ....... 11 8. Simple Trapezoidal or Queen Truss, ...... 12 9. General Case of a Funicular Polygon under a Vertical Load, . . ]3 10. Suspension Chains, Arches, Bowstring Girders, . . .15 11. Suspension Chains (continued), Bowstring Suspension Girder, . . 19 Examples, ....... 21 SECTION III. COMPOUND FRAMES. 12. Compound Triangular Frames for Bridge Trusses, .... 22 13. Roof Trusses in Timber, ....... 25 14. Queen Truss for Large Iron Roofs, ...... 27 15. Concluding Remarks, Diagrams of Forces in General, ... 29 Examples and References, ....... 31 CHAPTER II. STRAINING ACTIONS ON A LOADED STRUCTURE. 16. Preliminary Explanations, ....... 33 SECTION I. BEAMS. 17. Straining Actions on a Beam, .... .34 13. Example of a Balanced Lever, General Rules, .... 35 19. Beam supported at the Ends and Loaded at an Intermediate Point, . 36 C.M. b CONTENTS. 20. Beam supported at the Ends and Loaded Uniformly, 21. Beam Loaded at the Ends and supported at Intermediate Points, . 22. Application of the Method of Superposition, Examples, ..... 41 SECTION II. FRAMEWORK GIRDERS. 23. Preliminary Explanations, 24. Warren Girders under various Loads, . . . . .44 25. N Trusses, .... 48 Examples, ..... 49 SECTION III. GIRDERS WITH REDUNDANT BARS. 26. Preliminary Explanations, . . . . 49 27. Lattice Girders, Flanged Beams, ... 51 Examples and References, . . 53 CHAPTER III. STRAINING ACTIONS DUE TO ANY VERTICAL LOAD. 28. Preliminary Remarks, ....... 55 29. Relation between the Shearing Force and Bending Moment, . 55 30. Application to the Case of a Loaded Beam, . . . .60 31. Application to a Vessel Floating in Water, . . . ... 61 32. Maximum Straining Actions, . , 65 Examples, . . . . . . 65 33. Travelling Loads, . . ... 66 34. Counter-Bracing of Girders, . . . - -; 69 Examples, . . . . . . . 70 35. Method of Sections applied to Incomplete Frames, Culmann's Theorem, . 71 36. Method of Sections in General, Ritter's Method, . t 72' Examples and References, ....... 74 CHAPTER IV. FRAMEWORK IN GENERAL. 37. Straining Actions on the Bars of a Frame. General Method of Reduction, 75 38. Hinged Girders, Virtual Joints, ...... 77 39. Hinged Arches, '. 79 40. Structures of Uniform Strength, . . . . . .80 41. Stress due to the Weight of a Structure, ..... 82 42. Straining Actions on a Loaded Structure in General, ... 84 43. Framework with Redundant Parts, ...... 85 44. Concluding Remarks, ........ ^6 Examples, ......... 87 PART II. KINEMATICS OF MACHINES. 45. Introductory Remarks, ....... 91 CHAPTER V. LOWER PAIRING. SECTION I. ELEMENTARY PRINCIPLES. 46. Definition of Lower Pairs, ....... 93 47. Definition of a Kinematic Chain, ... . . 95 48. Mechanism of a Direct- Acting Engine, Position of Piston, ... 97 49. ,, ,, ,, Velocity of Piston, . . 99 Examples, ..... . 102 CONTENTS. XI SECTION II. EXAMPLES OP CHAINS OF LOWER PAIRS. ARTICLE PACK 50. Mechanisms derived from the Slider Crank Chain, . . 104 51. Double Slider Crank Chains, .... . 109 5lA. Wedge Chain, .... .... Ill 52. Crank Chains in General, ....... 112 53. Screw Chains, . . . . . . . . .113 54. Parallel Motions derived from Crank Chains, .... 113 SECTION III. KINEMATICS OF LINK WORK MECHANISMS. 55. Combination of a Sliding and a Turning Pair, .... 118 56. Diagram of Velocities, . . . 121 56A. Closure of Kinematic Chains, ....... 123 Examples and References, ..... 124 CHAPTER VI. CONNECTION OF TWO LOWER PAIRS BY HIGHER PAIRING. SECTION I. TENSION AND PRESSURE ELEMENTS. 57. Preliminary Remarks, Tension Elements, ..... 126 58. Simple Pulley Chain, Blocks and Tackle, . . 127 59. Wheel and Axle, . . . . . . .129 60. Pulley Chains with Friction Closure, Belts, . . . 131 61. Shifting of Belts, Fusee Chain, ...... 133 62. Simple Hydraulic Chain, .... . . 133 Examples, ........ . 134 SECTION II. WHEELS IN GENERAL. 63. Higher Pairing of Rigid Elements, ... . . 135 64. Rolling Contact, ........ 136 65. Augmentation of a Chain, Trains of Wheels,. . . . 139 66. Wheel Chains involving Screw Pairs, ... . 140 Examples, ..... ... 141 SECTION III. TEETH OF WHEELS. 67. Preliminary Explanations, ....... 142 68. Involute Teeth, ...... .144 69. Path of Contact, the Pitch Circle, ... .146 70. Path of Contact, any Circle, . . . . .148 71. Addendum and Clearance of Teeth, ...... 150 72. Endless Screw and Worm Wheel, . ... 151 Examples, .... . . 151 SECTION IV. CAMS AND RATCHETS. 73. Reduction of a Crank Chain by omission of the Coupling Link, . . 151 74. Cams with Continuous Action, . . . . . . 154 75. Mechanisms with Intermittent Action, . . . 155 76. Ratchets, ..... ... 156 77. Other Forms of Ratchet Mechanisms, . . 158 78. Screw Cams, .... 158 Examples, ... . . 159 xii CONTENTS. CHAPTER VII. MECHANISM IN GENERAL. ARTICLE PAGE 79. Plane Motion in General, Centrodes, . 80. Axoids, Elementary Examples of Centrodes, . : . 162 81. Profiles for given Centrodes, .... 82. Centrodes for a Higher Pair connecting Lower Pairs, 83. Construction of Centres of Curvature of Profiles, Willis's Method, . 167 84. Sphere Motion, ..... '-'.-'' '"'. 169 85. Screw Motion, ... .169 86. Classification of Simple Kinematic Chains, . . '-";'' . 170 87. Compound Kinematic Chains, . . . "'* 171 PART III. DYNAMICS OF MACHINES. CHAPTER VIII. PRINCIPLE OF WORK. SECTION I. BALANCED FORCES (STATICS). 88. Preliminary Explanations, Definition of Work, . .179 89. Oblique Resistance, ...... .180 90. Variable Resistance, . . . . . .182 91. Resistance to Rotation, Stability of a Vessel, . . . 183 92. Internal and External Work, . . . . .185 93. Energy, Principle of Work . .186 94. Machines, . . . . . . , . ' . 188 95. Verification of the Principle of Work in Special Examples, . 190 96. Periodic Motion of Machines, . . 192 97. Power, Sources of Energy, . , . . . 193 98. Reversibility, Conservation and Storage of Energy, .... 194 Examples, . . . . . . . 195 SECTION II. UNBALANCED FORCES (KINETICS). 99. Kinetic Energy of Translation, Sliding Pair, . .196 100. Partially Unbalanced Forces, Principle of Work, . . . 198 101. Kinetic Energy of Rotation, Turning Pair, . . . 199 102. Kinetic Energy of the Moving Parts of a Machine, . . . 202 103. Conservation of Energy, ..... .203 Examples and References, . . . . . . . 207 CHAPTER IX. DYNAMICS OF THE STEAM ENGINE. 104. Construction of Polar Curves of Crank Effort, .... 210 105. Construction of Linear Curves of Crank Effort, . . . 212 106. Ratio of Maximum and Minimum Crank Effort to Mean, . . . 213 107. Fluctuation of Energy, . . ... . .214 108. Fluctuation of Speed, Fly Wheels, . . . . . .219 109. Correction of Indicator Diagram for Inertia of Reciprocating Parts, . 223 110. Construction of Curves of Crank Effort for any given Indicator Diagram, 230. 111. Pumping Engines, ........ 231 lllA. Periodic Motion of Machines in General, ..... 234 Examples, ....... . 234 CHAPTER X. FRICTIONAL RESISTANCES. 112. Preliminary Remarks, .... 237 CONTENTS. xiii SECTION I. EFFICIENCY OF LOWER PAIRING. ARTICLE PAOE 113. Ordinary Laws of Sliding Friction, ...... 237 114. Friction of Bearings, ..... . . 238 115. Friction of Pivots, ..... . 240 116. Friction and Efficiency of Screws, . . . ... . 242 117. Efficiency of Mechanism by Exact Method, . . . 244 118. Efficiency of Mechanism by Approximate Method, .... 245 119. Experiments on Sliding Friction (Morin), ..... 246 120. Limits of the Ordinary Laws, . . .... 247 121. Experiments on Axle Friction, . ... 249 SECTION II. EFFICIENCY OF HIGHER PAIRING. 122. Rolling Friction, ........ 250 123. Friction of Ropes and Belts, . .... 252 124. Driving Belts, .... .... 25$ 125. Slip of Belts, . . .254 126. Stiffness of Ropes, ... .... 255 127. Friction of Toothed Wheels and Cams, ... . . . 256 SECTION III. FRICTION AL RESISTANCES IN GENERAL. 128. Efficiency of Mechanism in General, ...... 258 Examples and References, .... 260 CHAPTER XL MACHINES IN GENERAL. 129. Preliminary Remarks, . . . . . . . . 263 SECTION' I. ELEMENTARY PRINCIPLES OF DYNAMICS. 130. Quantity of Matter, Mass, ... . . 263 131. Equation of Momentum, Centrifugal Force, ..... 265 132. Principle of Momentum, ....... 26f> 133. Internal and External Kinetic Energy, ... . 267 134. Examples of Incomplete Constraint Case I., Recoil of a Gun, . . . .268 Case II., Collision of Vessels,. ...... 269 Case III., Free Rotation, ..... . 270 SECTION II. REGULATORS AND METERS. 135. Preliminary Remarks, Revolving Pendulum, . . . . 271 136. Speed of a Governor to overcome given Frictional Resistances, Loaded Governors, ......... 272 137. Variation of Height of a Governor by a Change of Position of the Arms, . 274 138. Parabolic Governors, ........ 275 139. Stability of Governors, . .... 276 140. Brakes, ... .277 141. Dynamometers, ........ 278 SECTION III. STRAINING ACTIONS ON THE PARTS OF A MACHINE. 142. Transmission of Stress in Machines, ...... 280 143. Reversal of Stress, . .... 281 144. Stability of Machines, Balancing, ...... 282 144A. Stability of Machines (continued), . . . 285 145. Straining Actions on the Parts of a Machine due to their Inertia, . . 288 146. Virtual Machines, ... . . .289 Examples and References, ....... 290 xiv CONTENTS. PART IV. STIFFNESS AND STRENGTH OF MATERIALS. ARTICLE PAGE 147. Introductory Remarks. . 295 CHAPTER XII. SIMPLE TENSION COMPRESSION, AND BENDING OF PERFECTLY ELASTIC MATERIALS. SECTION I. TENSION AND COMPRESSION. 148. Simple Tension, .... .298 149. Work done in Stretching a Rod, .... . 300 150. Thin Pipes and Spheres under Internal Fluid Pressure, . 301 151. Remarks on Tension, ... 303 152. Simple Compression, . 304 Examples, ... . 304 SECTION II. SIMPLE BENDING. 153. Proof that the Stress at each Point varies as its Distance from the Neutral Axis, ..... . . 305 154. Determination of the Position of the Neutral Axis, . . . 308 155. Determination of the Moment of Resistance, . . 308 156. Remarks on the Theory of Bending, . . .310 157. Calculation of Moments of Inertia, . . . 311 158. Beams of I Section with Equal Flanges, . . 312 159. Ratio of Depth to Span in I Beams, . . .313 160. Proportion of I Beams for Equal Strength, . . 313 161. Beams of Uniform Strength, . . . .316 162. Unsymmetrical Bending, . . . . . 317 Examples, . . . . . 318 CHAPTER XIII. DEFLECTION AND SLOPE OF BEAMS. 163. Deflection due to the Maximum Bending Moment, .... 321 164. General Equation of Deflection Curve, . . .322 165. Elementary Cases of Deflection and Slope, . ... 323 166. Beam propped in the Middle, ... . .324 167. Beam fixed at the Ends, .... .325 168. Stiffness of a Beam, ... . . 327 169. General Graphical Method, . . . 328 170. Examples of Graphical Method, . . , , 329 171. Elastic Energy of a Bent Beam, . ... 331 172. Concluding Remarks, ........ 332 Examples, ... . . . 333 CHAPTER XIV. TENSION OR COMPRESSION COMPOUNDED WITH BENDING. CRUSHING BY BENDING. 173. General Formula for the Stress due to a Thrust or a Pull in Combination with a Bending Moment, ....... 334 174. Strut or Tie under the action of a Force parallel to its Axis in cases where Lateral Flexure may be neglected, ..... 334 175. Remarks on the Application of the General Formula, . . . 336 176. Straining Actions due to Forces normal to the Section, . . . 337 177. Maximum Crushing Load of a Pillar, ..... 337 178. Manner in which a Pillar crushes. Formula for Lateral Deviation, . 340 CONTENTS. xv ARTICLE PAGE 179. Actual Crushing Load, . . .342 180. Gordon's Formula, ........ 343 180A. Effects of Partial Fixture of the Ends of a Pillar, . .345 181. Collapse of Flues, ........ 347 Examples, ...... . 349 CHAPTER XV. SHEARING AND TORSION OF ELASTIC MATERIAL. SECTION I. ELEMENTARY PRINCIPLES. 182. Distinction between Tangential Stress and Normal Stress. Equality of Tangential Stress on Planes at Right Angles, .... 350 183. Tangential Stress equivalent to a Pair of Equal and Opposite Normal Stresses. Simple Distorting Stress, ..... 351 SECTION II. TORSION. 184. Torsion of a Tube, Round Shafts, ...... 353 185. Elliptic and other Sections, . ..... 357 186. Crank Shafts, ... ..... 362 187. Spiral Springs, . . . . .363 SECTION III. SHEARING IN GIRDERS. 188. Web of a Beam of I Section, ....... 365 189. Distribution of Shearing Stress on the Section of a Beam, . . . 367 190. Deflection due to Shearing, ..... .370 191. Effects of Insufficient Resistance to Shearing, . . . .371 192. Economy of Material in Girders, . . 372 193. Joints and Fastenings, ... . . 373 Examples, . . . . . . . . . 375 CHAPTER XVI. IMPACT AND VIBRATION. 194. Preliminary Remarks. General Equation of Impact, . . . 377 195. Augmentation of Stress by Impact in Perfectly Elastic Material, . . 378 196. Sudden Application of a Load, ...... 380 197. Action of a Gust of Wind on a Vessel, . ... 381 198. Impact at High Velocities. Effect of Inertia, ... 382 199. Impact when the Limits of Elasticity are not exceeded. Resilience, . 384 ^ 200. Free Vibrations of an Elastic Structxire, . ... 386 201. Forced Vibrations, ..... '390 202. Examples of Fluctuating Loads, . . .392 202A. Augmentation of Stress by Fluctuation, . ... 395 203. Compound Vibrations, . . .395 203A. Centrifugal Whirling of Shafts, . ... 396 Examples, . . . .399 CHAPTER XVII. STRESS, STRAIN, AND ELASTICITY. SECTION I. STRESS. 204. Ellipse of Stress, ... .401 205. Principal Stresses. Axes of Stress, . . .402 206. Varying Stress. Lines of Stress. Bending and Twisting of a Shaft, . 403 207. Straining Actions on the Web of an I Beam, . .404 208. Remarks on Stress in General, ...... 405 Examples, ....... .406 SECTION II. STRAIN. 209. Simple Longitudinal Strain. Two Strains at Right Angles, comparison between Stress and Strain, . ... 407 xvi CONTENTS. SECTION III. CONNECTION BETWEEN STRESS AND STRAIN. ARTICLE P AGE 210. Equations connecting Stress and Strain in Isotropic Matter, . 409 211. Elasticity of Form and Volume, . . , 411 212. Modulus of Elasticity under Various Circumstances. Elasticity of Flexion, 412 213. Remarks on Shearing and Bending, . .414 214. Thick Hollow Cylinder under Internal Pressure, .... 416 214A. Strengthening of Cylinder by Rings. Effect of great Pressures, . . 420- 215. Elastic Energy of a Solid, . .421 216. Rigidity of Shafts, . . . .422 Examples, , , . . . .423 CHAPTER XVIII. MATERIALS STRAINED BEYOND THE ELASTIC LIMIT. STRENGTH OF MATERIALS. 217. Plastic Materials, . .... 425 218. Flow of Solids, . . . . . . . .427 219. Preliminary Remarks on Materials. Stretching of Wrought Iron and Steel, 427 220. Breaking down Point, . . . . . . .430 221. Real and Apparent Strength of Ductile Metals, . . . 432 222. Increase of Hardness by Stress beyond the Elastic Limit, . . . 433 223. Compression of Ductile Materials, .... . 434 224. Bending beyond the Elastic Limit, ... .436 225. Stretching of Cast Iron, ... 438 226. Crushing of Rigid Materials, ... .439 227. Breaking of Cast Iron Beams, . .440 228. Shearing and Torsion, ........ 441 229. Connection between Co efficients of Strength, . . . 443 230. Wohler's Experiments on Fluctuating Stress, . . . 446 231. Influence of Repetition on Elastic Limit, . ... 449 232. Impact, ...... . 450 233. Factors of Safety and Co-efficients of Working Strength, . . 451 234. Values of Co-efficients, ... .453 235. Fibrous Materials. Ropes, . ... .454 236. Tables of Strength, . . ... 455 237. Principle of Similitude, . 459 238. Expansion and Contraction, . . 459 Examples, ... . . . 460 Description of Plate VIII., . . ... 463 PAET V. TEANSMISSION AND CONVERSION OF ENERGY BY FLUIDS. 239. Introductory Remarks, ....... 467 CHAPTER XIX. ELEMENTARY PRINCIPLES OF HYDRAULICS. SECTION I. INTRODUCTORY. 240. Velocity due to a given Head, . . . .469 241. Hydraulic Resistances in General, ...... 472 242. Discharge from Small Orifices, . . . . . 472 243. Steady Flow through Pipes. Conservation of Energy, . . . 475 SECTION II. MOTION OF AN UNDISTURBED STREAM. 244. Distribution of Energy in an Undisturbed Stream. Vortex Motion, . 477 245. Viscosity, ....... . . 479 246. Discharge from Large Orifices in a Vertical Plane, . . . .480 247. Similar Motions, . 482 CONTENTS. xvij. SECTION III. HYDRAULIC RESISTANCES. ARTICLE pAOE 248. Surface Friction in Genera], ... . . 4g3_ 249. Surface Friction in Pipes, . . .... 486 250. Discharge of Pipes, . . 437 251. Open Channels, . . . . . 488 252. Virtual Slope of a Pipe, . . 439 253. Loss of Energy by Eddies and Broken Water, . . . 489 254. Bends in a Pipe. Surface Friction, ...... 494 255. Summation of Losses of Head, ...... 496 SECTION IV. PRINCIPLE OF MOMENTUM. 256. Direct Impulse and Reaction, . .... 497 257. Oblique Action. Curved Surfaces, . ... 498 258. Impulse and Reaction of Water in a Closed Passage, . . . 500 SECTION V. RESISTANCE OF DEEPLY-IMMERSED BODIES. 259. Eddy Resistance, ....... 501 260. Oblique Moving Plate, . . . 595 261. Pressure of a Current against an Obstacle, .... 507 Examples, First Series, . .... 508 Second Series, . . .... 510 CHAPTER XX. HYDRAULIC MACHINES. 262. Preliminary Remarks, ........ 511 SECTION I. WEIGHT AND PRESSURE MACHINES. 263. Weight Machines, ..... . . 511 264. Hydraulic Pressure Machines in Steady Motion, .... 513 265. Hydraulic Pressure Machines in Unsteady Motion, . . . 515 266. Examples of Hydraulic Pressure Machines, ..... 517 267. Hydraulic Brakes, . ....... 518 268. Transmission of Energy by Hydraulic Pressure, . . . .521 269. Pumps,. . ....... 522 Description of Plates IX. and X., . . . .525 Examples, ........ 526 SECTION II. IMPULSE AND REACTION MACHINES. 270. Impulse and Reaction Machines in General, . . . . . 527 271. Angular Impulse and Momentum, .... . 530 272. Reaction Wheels, ... .... 531 273. Turbine Motors, . ..... 533 274. Turbine Pumps, . . . . . . . 537 275. Approximate Investigation of the Efficiency of a Centrifugal Pump, . 540 276. Limitation of Diameter of Wheel, . . . .542 277. Impulse Wheels, , . . 544 278. Stress due to Rotation, ....... 545- SECTION III. SOME GENERAL PRINCIPLES. 279. Equation of Steady Flow in a Rotating Casing, . . . 545 279A. Similar Hydraulic Machines, . . . . 547 Examples and References, .... . 548 CHAPTER XXL ELASTIC FLUIDS. 280. Preliminary Remarks, ........ 550 SECTION I. MACHINES IN GENERAL. 281. Expansive Energy, ... . 550 282. Transmitted Energy, ....... 552 xviii CONTENTS. ARTICLE PAGE 283. Available Energy, . . 553 284. Cycle of Operations in a Pneumatic Motor. Mechanical Efficiency, . 554 285. Pneumatic Pumps, ........ 557 286. Indicator Diagrams, ........ 558 287. Brake Efficiency, ..... .560 SECTION II. THERMODYNAMIC MACHINES. 288. Cycle of Thermal Operations in a Heat Engine, . . . .561 289. Mechanical Equivalent of Heat, . . . . . .562 290. Mechanical Value of Heat, ....... 562 291. Available Heat of Steam, ... . .565 292. Thermal Efficiency, . . . . . . .566 293. Compound Engines, . . . . . 568 294. Useful Work of Steam, . . . . . . . 569 295. Efficiency and Performance of Steam Engines, .... 570 296. Reversed Heat Engines, . . . . . 571 SECTION III. TRANSMISSION OP ENERGY. FLOW OF GASES. 297. Internal Energy. Internal Work, ...... 573 298. Transmission of Energy by Compressed Air, . . . . 575 299. Steady Flow through a Pipe. Conservation of Energy, . . . 577 300. Velocity of Efflux of a Gas from an Orifice, . 578 301. Discharge from an Orifice, . . . . . . . 580 302. Flow of Gases through Pipes, . . . . * . .582 303. Flow of Gases under Small Differences of Pressure, . . 585 304. Varying Temperature. Chimney Draught, . . . 1 . 586 Examples, First Series, . . . '' . 588 Second Series, ....... 591 Third Series, ....... 593 APPENDIX. A. NOTES AND ADDENDA. I. Statics of Structures, ... . .597 II. Kinematics of Machines, . 600 III. Dynamics of Machines, . . . . . - . 603 IV. Stiffness and Strength of Materials, ... 609 V. Hydraulics and Hydraulic Machines, . . . . 620 VI. Elastic Fluids, ..... ... 631 VII. Resistance and Propulsion of Ships, ....... 633 B. ORGANIZATION OF THE CLASSES IN ENGINEERING AND NAVAL ARCHI- TECTURE IN THE ROYAL NAVAL COLLEGE, . . . ,645 INDEX TO PLATES. PLATE I. Mechanisms derived from a Slider Crank Chain, . . . .109 II. Mechanisms derived from a Double Slider Crank Chain, . . . Ill III. Mechanisms derived from Wheel Chains, ..... 141 IV. Cam and Ratchet Mechanisms, . . ... 159 V. Curves of Crank Effort, . ... . . . . 230 VI. Governors, ....... . 272 VII. Brakes and Dynamometers, ....... 279 VIII. Sections of Girders. Pin Joints, ... 463 IX., X. Hydraulic Pressure Machines, ...... 525 XI. Crosby's Indicator, ..... 559 INTRODUCTION. WHAT are the conditions of a science? and when may any subject be said to enter the scientific stage ? When the facts of it begin to resolve themselves into groups ; when phenomena are no longer isolated experiences, but appear in connection and order ; when, after certain antecedents, certain consequences are uniformly seen to follow; when facts enough have been collected to furnish a basis for conjectural explanation; and when conjectures have so far ceased to be utterly vague that it is possible in some degree to foresee the future by the help of them. FKOUDE. A competent view of the world can never be bad as a gift ; we must acquire it by hard work. MACH. INTRODUCTION. THE province of the Engineer and Architect is to control the forces of nature and apply them to useful purposes, an object which is effected by means of pieces of material suitably connected and arranged. The protection of life and property from destructive forces is accomplished by pieces rigidly connected with one another which transmit their action to bodies to which they are not injurious; while the utilization of such forces in moving weights, changing the form of bodies, and other similar operations, is effected by a set of moving pieces which transmit their action to the required place and modify it in some given way. In the first case the pieces are called, collectively, a STRUCTURE, in the second, a MACHINE. The object of the present work is to give an outline of the principles on which structures and machines are designed. The actual form of such a construction is almost always the final result of a process of evolution by which it has been gradually per- fected by adaptation from some previously existing construction. To meet new wants the engineer selects some arrangement, suggested by experience of some nearly similar case, which appears likely to answer the purpose by its simplicity, facility of construction, and adaptation to the forces which it is proposed to control and utilize. If the new arrangement is merely a copy of the old, this may be suffi cient and the construction may be at once proceeded with, but if there be any important difference it is necessary, before incurring the expense and risk of actual construction, to ascertain that the design is in conformity with those general laws governing the action of natural forces upon matter which reason and experience alike show to be necessarily true in all cases. To a certain extent this has already been considered by the designer, whose knowledge and experience enable him to avoid at once arrangements which are obviously inad- missible, but complete conformity can only be secured by comparison with results deduced by reasoning and verified by experiment. xxii INTRODUCTION. In any branch of knowledge the explanation of a set of facts by a general principle, from which new results can be obtained, is properly described as a Theory of the phenomena to which they relate. When its principles are well established it enables us to predict the results of experiment ; when they are not, it is even more necessary, to direct the course which experiment should take for more perfect knowledge. The systematic study of structures and machines with a view to discover the theoretical principles on which their construction is based, and the deduction from those principles of results which may be useful to the designer, forms a branch of science which, following RANKINE, we may describe as Applied Mechanics. In some cases the subject may have been so exhaustively studied, and may be in its nature so limited, that all the arrangements which can be employed for a given purpose may be foreseen and the best determined by a priori considerations. The process of invention itself then becomes a problem in science. This, however, is the rare exception ; in general, the use of theory is limited to the answering of certain questions relating to an arrangement which has already been proposed. Among the most important of these are (1.) What should be the dimensions of the parts of the construction that they may be strong enough to resist the action of the forces to which it is exposed? (2.) Will the construction be sufficiently stable and rigid 1 (3.) Are the natural forces, which it is proposed to utilize, sufficient for the proposed purpose and are they under proper control 1 ? It is only in the very simplest cases that these and similar questions can be answered completely, without reference to the direct results of experience in order to interpret theoretical reasoning and render it applicable. Even, therefore, after the general plan of a construction is decided on, the work of the practical designer includes much which cannot be reduced to a mere process of deduction from given data. Nevertheless the part of theory in controlling and directing inventive power is of great and constantly-increasing importance, by furnishing principles of universal application, in conformity with which every mechanical construction must be designed, and by which the researches of the experimentalist must be guided. The mechanics of structures and machines is based on the properties of materials, and on those general laws connecting matter and motion, the study of which is the object of Abstract Mechanics, but the special nature of the subject-matter occasions a certain difference in the methods employed. In the elementary branches of purely abstract INTRODUCTION. xxiii mechanics the number of bodies considered seldom exceeds two ; if more are introduced the questions to be considered become impractic- ably complex considered as abstract mathematical problems. In applied mechanics a number of pieces are connected with comparatively little freedom so as to form an organic whole, and the results of experience or of mathematical investigations too complex for ordinary use are ad- mitted freely for the purpose of simplification. Hence the calculations employed are of a coarser type, and, in particular, graphical methods are everywhere employed when possible, not only to exhibit, but also to obtain, numerical results. On the other hand, no investigation is considered as complete until it has been checked by reference to experience, and unless its errors are approximately known. The elementary principles of abstract statics, dynamics, and hydrostatics must be supposed already known, and some practical knowledge of machines and structures is presupposed. The classification of mechanical constructions depends in great measure on the number of pieces connected and on the mode of connection. We have first the broad distinction between structures, in which the pieces have no movements except such as may be due to changes in their form and dimensions consequent on the forces to which they are exposed, and machines in which the object is attained by means of such movements. This distinction is so fundamental that there is no word in common use which includes both. Structures may be ranged in order of simplicity according to the degree of constraint with which their parts are connected as follows : (1.) Structures with pin joints without redundant parts. (2.) Structures with pin joints which include redundant parts. (3.) Blockwork and earthwork structures. (4.) Structures with riveted or other forms of fastened joints. A pin joint, such as is shown in a simple form in Figs. 1 and 2, Plate VIII., page 463, is one in which the pieces connected are united by a single pin fitting into holes in the pieces, and, in consequence, neglect- ing friction, the mutual action between the pieces connected necessarily passes through the axis of the pin. A redundant part is one which may be removed without destroying the structure if the remaining parts be sufficiently strong. The first class of structures therefore possess a peculiar characteristic which renders their theory much more simple than that of any other, namely, that the forces acting on each piece depend only on the external forces acting on the whole and not on xxiv INTRODUCTION. the material or the dimensions of the pieces. In the theory of struc- tures, then, this class is first considered, and the answer to the first of the general questions propounded above consists in the solution of two general problems. (1.) Being given the load on the structure, it is required to find the forces acting on each part. (2.) Being given the forces acting on a piece of material, it is required to find its dimensions that it may be sufficiently strong and stiff. The first forms a part of the subject which may be properly described &s the " Statics of Structures " ; while the second, which depends on the properties of the materials of construction, is known as the ''Strength and Stiffness of Materials." The results obtained are in continual requisition in the theory of the more complex structures, but require to be supplemented by further investigations and by results derived from direct experience, peculiar to each class. The present treatise, being simply introductory, refers to the more complex struc- tures only incidentally. A Machine is a structure the parts of which are in motion. The motion introduces new forces, often of great magnitude and importance, which must be taken into account in its design ; but we have, in addi- tion, to consider the third general question mentioned above, namely, the adaptation of the natural forces available, to the work which the machine has to do. The simplest machines consist chiefly of a number of rigid pieces, and their theory is divided into two parts one con- cerned with the motion of the machine, the other with the work it does. In many of the most important machines fluids are used, and their theory forms a distinct branch of the subject not less important than the rest, some account of which is indispensable. Thus the whole subject is divided into five parts. Since the parts of structures as well as machines possess, though to a very limited extent, freedom to move, and since such movements often have to be supposed for the purposes of an investigation, the most natural arrangement perhaps would be to commence with the first part of the theory of machines, and then pass on to the statics of structures. In the present treatise it has, however, been found convenient to invert this order, and we now, therefore, commence with structures. PART I.-STAT1CS OF STRUCTURES. CHAPTER I. FRAMEWORK LOADED AT THE JOINTS. 1. Preliminary Explanations and Definitions. A frame is a structure composed of bars, united at their extremities by joints, which offer no resistance to rotation. In the first instance we may suppose the centre lines of the bars all in one plane, and in that case the joints may consist simply of smooth pins passing through holes at the ends of the bars, which are to be imagined forked, if necessary, so as to allow the centre lines to meet in a point. A large and important class of structures, known to engineers as "trusses," approach so closely to frames that calculations respecting them may be conducted by treating them as if they were frames. The difference between a truss and a frame will appear as we proceed. The frame may be acted on by forces applied at points in one or more of its bars, or at the joints which unite the bars together. An important simplification, however, is effected by supposing, in the first instance, that the joints only are loaded, an assumption which will be made throughout this chapter, except in a few simple examples. It will be shown hereafter (p. 75) that all other cases may be derived from this by means of a preliminary reduction. Assuming, then, that the frame is acted on by forces at the joints, due either to weights or other external causes, or to the reaction of supports on which the frame rests, the problem to be solved is to find the forces called into play on each of the bars of which it is constructed. These forces are caused by the pressure of the pins on the sides of the holes through which they pass, and it at once follows, since no other C.M. A STATICS OF STEUCTUEES. [PART i. forces kct on \.i i'o.^i'** ^ti.. ** centres of the holes. There are two Pigrs.ia,ib. possible cases shown in Figs, la, Ib; r^/r\\_> in tne first tlie bar is acted on ky a i ~\^s pair of equal and opposite forces tend- ing to lengthen it, and in the second to shorten it. The pairs of forces are called a Pull and a Thrust respectively, while the bars subjected to their action are called Ties and Struts respectively. Between a pull and a thrust there is no statical difference but that of sign ; the constructive difference, however, between a tie and a strut is great. The first may theoretically be a rope or chain, and the second may be made up of pieces simply butting against one another without fastening, while a rigid bar will serve either purpose,, though its powers of resistance are generally entirely different in the two cases. It often happens that it is unknown whether a bar be a strut or a tie, and the pair of forces are then called a STRESS on the bar. This word " stress " was introduced by Rankine to denote the mutual action between any two bodies, or parts of a body, and here means, in the first instance, the mutual action between the parts of the frame united by the bar we are considering. If, however, we imagine the bar cut into two parts, A and B, by any transverse section, as shown in Figs, la, Ib, those parts are held together in the case of a pull, or thrust away from each other in the case of a thrust, by internal molecular forces called into play at each point of the transverse section, and acting one way on A and the other way on B. As A and B must both be in equilibrium, it is obvious that these internal forces must be exactly equal to the original forces, and thus it appears that the stress on the bar may also be regarded as the internal molecular action between any two parts into which it may be imagined to be divided. Stress regarded in this way, will be fully considered in a subsequent division of this work; it will be here sufficient to say that its intensity is measured by dividing the total amount by the sectional area of the bar, and is limited to a certain amount, depending upon the nature of the material of which the bar is constructed. It is further manifest from what has been said, that the stress on a bar may likewise be regarded as a mutual action between the bar and either of the pins at its ends which are pulled towards the middle of the bar in the case of a pull, or thrust away from it in the case of a thrust ; each pin is therefore acted on, in addition to any load which CH. i. ART. 2.] FRAMEWORK LOADED AT THE JOINTS. 3 may be suspended from it, by forces, the directions of which are the lines joining the centres of the pins, from which it follows at once that every joint may be regarded as a point kept in equilibrium by the load at that joint and by forces of ichich the bars of the frame are the lines of application. This principle enables us to find the stress on each bar of a frame loaded at the joints whenever such stress can be determined by statical considerations alone, without reference to the material or mode of construction, that is to say, in all cases which properly belong to the present division of our work. Forces are measured in pounds, or, when large, in tons of 2240 Ibs. They are often distributed over an area or along a line, and are then reckoned per square foot or per " running " foot, the last expression being commonly abbreviated to "foot-run." The bars need not be connected by simple pin joints as has been supposed for clearness, provided that their centre lines if prolonged meet in a point through which passes the line of action of the load on the joint. This point may be called the centre of the joint, and we may replace the actual joint by a simple pin, or, if the bars are not in one plane, by a ball and socket which has the same centre. We shall return to this hereafter, but now pass on to consider various kinds of frames, commencing with the simplest. SECTION I. TRIANGULAR FRAMES. 2. Diagram of Forces for a Simple Triangular Frame. The simplest kind of frame is a triangle. In Fig. 2a, AGE is such a triangle; it is supported at AB so that AB is horizontal, and loaded at C with a weight W. Then evidently the effect of the weight is to compress AC, BC, and to stretch AB, which is conveniently indicated by drawing AC, BC in double lines, and AB in a single line. Also the weight produces certain vertical pressures on the supports A, B, which will be balanced by corresponding reactions P and Q. To find the magnitude of the thrust on AC, BC, the pull on AB, and the reactions, the diagram of forces Fig. 2b is drawn ; ab is a vertical line representing W on any convenient scale, while aO, bO are lines drawn through a, b respectively, parallel to AC, BC to meet in 0, and finally On is drawn parallel to AB, or, what is the same thing, perpen- dicular to ab. Now, applying the fundamental principle laid down above, we observe that C is a point kept in equilibrium by three forces, the load at C, namely W, the thrust of AC which we will call S, and the thrust of BC which we will call 1L In the second figure the triangle STATICS OF STRUCTURES. [PART i. Oab has its sides parallel to these forces, and hence it follows that Oa, Ob represents S, R on the same scale that ab represents W . Again, A is a point kept in equilibrium by three forces, the thrust of AC, the pull of the tie AB, which we will call H, and the upward reaction P of the support A. But referring to the figure 2b, On, an are respectively parallel to the two last forces, so that, by the triangle of forces, they represent H, P on the same scale that Oa represents S. The same reasoning applies to the point B, and therefore bn represents the other supporting force Q, as is also obvious from the consideration that P+Q=PP. We thus see that all the forces acting upon and within the triangular frame ACB are represented by corresponding lines in Fig. 2b, which is thence called the "diagram of forces" for the triangular frame. Such a diagram can be drawn for any frame, however complicated, and its construction to scale is the best method of actually determining the stresses on the several parts of the frame. The force H requires special notice : it is called the " thrust" of the frame. In the present case the thrust is taken by the tension of the third side of the triangle, but this may be omitted, and the supports A and B must then be solid and stable abutments capable of resisting a horizontal force H. In many structures such a horizontal thrust exists; and its amount and the mode of providing against it are among the first things to be considered in designing the structure. Besides the graphical representation just given, which enables us to obtain the thrust of a triangular frame by constructing a simple diagram, it may also be calculated by a formula which is often convenient. Let AC be denoted by b and BC by a, as is usual in works on trigonometry, and let AN, BN their projections on AB be called b', a', and let the height of the triangle be h and its span I, then by similar triangles, P an CN h H~On~AN~b" ______ H~0n~ BN~a" en. i. ART. 3.] FRAMEWORK LOADED AT THE JOINTS. Therefore, by addition, or H= W -Tr- ill In practical questions it often happens that a', b', h are known by the nature of the question, whence H is readily determined. The case when the load bisects the span may be specially noticed : then a' = b' = ^l and Wl Th' H When the height of the frame is small compared with the span, this calculation is to be preferred to the diagram, which cannot then be constructed with sufficient accuracy. The simple frame here considered may be inverted, in which case the diagram of forces and the numerical results are unaltered, the only change being that the two struts have become ties and the tie a strut. 3. Triangular Trusses. Triangular frames are common in practice, and the rest of this section will be devoted to some of the commonest forms in which they appear. Fig. 3a shows a simple triangular truss consisting of a beam, AB, supported by a strut at the centre, the lower extremity of which is carried by tie roads, AC, BC, attached to the ends of the beam. If now a weight, W, be placed at ~~w~| a a the centre, immediately over Fig.Sa. the strut, it does not bend the beam (sensibly) as it would do if there were no strut, but is transmitted by the strut to the joint (7, so that the truss is equivalent to the simple tri- angular frame of the last article. This, however, supposes that the strut has exactly the proper i n i * length to prevent any bending of the beam ; if it be too short or too long the load on the frame will be less or greater than W, a point which will be further considered presently. It should be noticed that D is not necessarily at the centre. Fig. 3b shows the same construction inverted. CD is a tie by which D is suspended from C ; we will suppose this rod to pass through AB and a nut applied below, by means of which D may be raised or lowered. 6 STATICS OF STRUCTURES. [PART i. Let AB now be uniformly loaded with a given weight, then the bending of AB is resisted by CD, which supports it and carries a part of the load, which may be made greater or less by turning the nut. If, how- ever, we imagine AB, instead of being continuous through D, to be jointed at D, then the tie CD necessarily carries half the weight of AD and half the weight of BD, that is to say, half the whole load, whatever be its exact length. This simple example illustrates very well the most important difference between a truss and a mathematical frame ; namely, that in the truss one or more of the bars is very often con- tinuous through a joint. Such cases can only be dealt with on the principles of the present division of our work, by making the supposition that the bar in question, instead of being continuous, is jointed like the rest. The error of such a supposition will be considered hereafter ; it is sufficient now to say that in order that it may be exact in the particular case we are considering, the nut must be somewhat slackened out so that D may be below the straight line AB, and that being dependent on accuracy of construction, temperature, and other varying circumstances, such errors cannot be precisely stated, but must be allowed for in designing the structure by the use of a factor of safety. The supposition is one which is usual in practical calculations, and will be made throughout this division of our work. The foregoing is one of the simplest cases where, as is very common in practice, the bars of the frame are loaded and not the joints alone. When such bars are horizontal and uniformly loaded, the effect is evidently the same as if half the load on each division of the loaded bar were carried at each of the joints through which it passes. This is also true if the loaded bars be not horizontal, but the question then requires a much more full discussion, which is reserved for a later chapter (see Ch. IV.). When one of the joints of the loaded bar is a point of support, like A in Fig. 3, the supporting force is due partly to the half weight of one or more divisions of the loaded bar, and partly to the downward pull or thrust of other bars meeting there : the first of these causes -does not aifect the stress on the different parts of the truss, and the calculations are therefore made without any regard to it. The explanations given in this article should be carefully considered, as they apply to many of the examples subsequently given. The triangular truss in both the forms given in this article is frequently employed in roofs and bridges of small span, as well as for other purposes. 4. Cranes. The arrangements adopted for raising and moving en. i. ART. 4.] FRAMEWORK LOADED AT THE JOINTS. Fig.4a. weights furnish many interesting examples of triangular frames. Fig. 4a shows one of the forms of the common crane, a machine the essential members of which are the jib, BC, supported by a stay, CE, attached to the crane-post, BE, which is vertical. In cranes proper this third mem- ber rotates, carrying EC and CE with it, but in the sailors' derrick a fixed mast, plays the part of a crane-post and the stay, CE, is a lashing of rope frequently capable of being lengthened and shortened by suitable tackle, so as to raise and lower the jib, a motion very common in cranes and hence called a derrick motion. The weight is generally also capable of being raised and lowered directly by blocks and tackle, but for the present will be supposed directly suspended from C. The diagram of forces now assumes the form shown in Fig. 4b, in which the lettering is the same as in Fig. 2b, page 4, the only difference in the diagrams being that in the present case AC, which is now a tie, is divided into two parts, AE and EC, inclined at an angle. The stress on AE is therefore not the same as on EC, but is got by drawing a third line, Ca, parallel to AE. The perpendicular On gives us in this instance not only the stress on AB and the horizontal thrust of CB at B, but also the horizontal pull of CE at E we may call this H as before. There is an upsetting moment on the structure as a whole which is equal to the product of the weight W by its horizontal distance from B (often called the radius of the crane) arid also to the force H, multiplied by the length of the crane-post, BE. One principal difference between different types of cranes lies in the way in which this upsetting moment is provided against. (a.) In portable cranes, such as shown in Fig. 4a, there is a horizontal platform, AB, supported by a stay. AE, and carrying a counterbalance' weight, P, sometimes capable of being moved in and out so as to provide for different loads. The right magnitude of counterbalance weight and the pull on the stay, AE, are shown by the diagram, P corresponding to the supporting force at A in the previous case. (ft.) In the pit crane, the post is prolonged below into a well and the Fig.4l3. 8 STATICS OF STRUCTURES. [PART i. Fig.5. lower end revolves in a footstep, the upper bearing being immediately below B. In this instance the post has to be made strong enough to resist a bending action at B, equal to the upsetting moment, and the bearings have to resist a horizontal force equal to H multiplied by the ratio of the length of the crane-post, BE, to that of its. prolongation below the ground. (y.) The upper end of the crane-post may revolve in a headpiece,, which is supported by a pair of stays anchored to fixed points in the ground. The upright mast of a derrick frequently requiring support in the same way, this arrangement is known as a derrick crane. It is (C shown in Fig. 5, ED, ED' being the stays. To find the stress on the stays it is necessary to pro- long the vertical plane through EC, to intersect the line DD\ joining the feet of the stays in the point A, and imagine the two stays, ED, ED replaced by a single stay, EA : then a diagram of forces, drawn as in the previous case, determine S', the pull on this stay. But it is clear that S' must be the resultant pull on the two original stays, and may be considered as a force applied at E in the direction of AE to the simple triangular frame DED'. A second diagram of forces therefore will determine the pull on each stay, just as in the next following case. 5. Sheer Legs and Tripods. Instead of employing an upright post to give the necessary lateral stability to the triangle, one of its members may be separated into two. Thus in moving very heavy weights sheer legs are used, the name being said to be de- rived from their resemblance to a gigantic pair of scissors (shears) partly opened and standing on their points. In Fig. 6, CD, CD are spars, or D tubular struts, often of great length, resting on the ground at DD' and united at C, so as to be capable of turning together about DD as an axis. The load is carried at C and the legs are supported by a stay, CA, which is sometimes replaced by a rope and tackle, capable of being lengthened or shortened Fig.6. CH. i. ART. 6.] FKAMEWOKK LOADED AT THE JOINTS. Fig.7a,7b. so as to raise or lower the sheers. Drawing AB to the middle point of DD', the pair of legs are to be imagined replaced by a single one, CB, then the diagram of forces may be constructed just as in Fig. 4b, and we shall obtain the tension of the rope 8 and the resultant thrust on the pair of legs R. Now draw the triangle ODD', as in Fig. 7a, and imagine it loaded at C with a weight, , , R, then drawing the diagram of forces, Fig. 7b, we get R' the thrust on each leg. The horizontal force, H', in this second diagram represents the tend- ency of the feet of the legs to spread outwards laterally, while the force H of the original diagram represents their tendency to move inwards per- pendicular to DD'. In some cases the D guy rope and tackle, CA, are replaced by a third leg called the back leg, and the sheers are then raised and lowered by moving A by a large screw ; the force H is then also the force to be overcome in turning the screw. Instead of having only two legs, as in sheers, we may have three forming a tripod. This arrangement is frequently used to obtain a fixed point of attachment for the tackle required to raise a weight, and is sometimes called a "gin," or as military engineers prefer to spell the word, a "gyn." The thrust on each leg and the tendency of the legs to move outwards can be obtained by a process so similar to that in the preceding examples that we need not further consider it. 6. Effect of the Tension of the Chain in Cranes. In most cases the load is not simply suspended from C as has been hitherto supposed, but is carried by a chain passing over pulleys and led to a chain barrel, generally placed somewhere on the crane-post. The tension of the chain in this case is Wjn t where n is a number depending on the nature of the tackle, and this tension is to be considered as an addi- tional force applied at C to be compounded with the load W, the effect of which has been previously considered. Fig. 8 shows the form the diagram of forces assumes in this case. Drawing ba as before to repre- sent W) and aa' parallel to the direction in which the chain is led off from the pulley at C and equal to the tension Wjn t the third side of the triangle, ba', must be the resultant force at C due to both forces, whence drawing a'O parallel to the stay and bO parallel to the jib, and reasoning as before as to the equilibrium of the forces at C, we see that these lines must be the tension of the stay and the thrust on the jib. 10 STATICS OF STRUCTURES. [PART i. The effect of the tension of the chain is generally to diminish the pull on the stay arid increase the thrust on the jib, sometimes very consider- ably, as for example in certain older types of crane still used for light loads under the name of " whip " cranes. In these cranes the chain passes over a single fixed pulley at the end of the jib, and is attached directly to the weight, so that the tension of the chain is equal to the weight. The other end of the chain is led off along a Fig.8. / I w horizontal stay to a wheel and axle at the top of the crane-post, a chain from the wheel of which passes to a windlass below. This arrangement, the double windlass of which facilitates changes in the lifting power corresponding to the load to be raised, is a develop- ment of the primitive machine in which the wheel was a tread wheel worked by men or animal power. In this case the pull on the stay is diminished by the whole weight lifted, and is thus reduced very much. Where a crane has to be constructed of timber only, this is a consider- able advantage, from the difficulty of making a strong tension joint in this material. EXAMPLES. 1. The slopes of a simple triangular roof truss are each 30. Find the thrust of the roof and the stress on each rafter when loaded with 250 Ibs. at the apex. Thrust of roof =216 '5 Ibs. Stress on rafters =250 ,, 2. A beam 15 feet long is trussed with iron tension rods, forming a simple triangular truss 2 feet deep. Find the stress on each part of the frame when loaded with 2 tons in the middle. Thrust on strut 2 tons. Pull on tension rods =3 '88 ,, Thrust on beam =3 '75 ,, 3. The platform of a foot bridge is 20 feet span, and 6 feet broad, and carries a load of 100 Ibs. per sq. ft. of platform. It is supported by a pair of triangular trusses each 3 feet deep, one on each side of the bridge. Find the stress on each part of one of the trusses. The whole load of 12,000 Ibs. rests equally on the two trusses, there is therefore 6000 Ibs. distributed uniformly along the horizontal beam of each truss. Thrust on strut =3,000 Ibs. Tension of tie rods =5,220 ,, Thrust on horizontal beam = 5,000 ,, 4. The slopes of a simple triangular roof truss are 30 and 45 and span 10 ft. The rafters are spaced 2^ feet apart along the length of the wall, and the weight of the roofing material is 20 Ibs. per sq. ft. Find by graphical Construction the thrust of the roof. Each rafter carries a strip of roof 2 feet wide, the load on rafter =50 Ibs. per foot length of rafter. Find the lengths by construction or otherwise. The virtual load at apex=| weight on the two rafters=311 Ibs. Thrust of roof = 198 Ibs. en. i. ART 7.] FRAMEWORK LOADED AT THE JOINTS. 11 5. The jib AC of a ten-ton crane is inclined at 45 to the vertical, and the tension rod BC at an angle of 60. Find the thrust of the jib, and the pull of the tie rod when fully loaded, the tension of the chain being neglected. If a back stay BD be added inclined at 45, and attached to the end of a horizontal strut AD, find the counterbalance weight required at D to balance the load on the crane, and find also the tension of the back stay. Thrust on jib AC =33 '5 tons. Tension of tie rod =27'o ,, Counterbalance weight =23 '5 ,, Tension of back stay =33 '5 ,, 6. A pair of sheer legs are 40 feet high when standing upright, the lower extremities rest on the ground 20 feet apart, the legs stand 12 feet out of the perpendicular, and are supported by a guy rope attactied to a point 60 feet distant from the middle point of the feet. Find the thrust on each leg, and the tension of the guy rope under a load of 30 tons. Thrust on each leg =19 '5 tons. Tension of guy rope = 12 '8 , , 7. In example 5 the tension of the chain is half the load, and the chain barrel is so placed that the chain bisects the crane-post AB. Find the stress on the jib and tie rod. Thrust of jib =36 tons. Pull of tie rod =25 8. In a derrick crane the projections of the stays on the ground form a right-angled triangle, each of the equal sides of which is equal to the crane-post. The jib is inclined at 45 and the stay at 60 to the vertical. Find the stress on all the parts (1) when the plane of the jib bisects the angle between the stays ; (2) when it is moved through 90 from its first position. Load 3 tons. Answer. Case 1. Pull on each stay = 7'1 tons. Case 2. Pull on one = thrust on other =7 '1 9. A load of 7 tons is suspended from a tripod, the legs of which are of equal length and inclined at 60 to the horizontal. Find the thrust on each leg. If the load be removed and a horizontal force of 5 tons be applied at the summit of the tripod in such a way as to produce the greatest possible thrust on one leg, find that thrust and deter- mine the stress on the other two legs. Answer. Case 1. Thrust on each leg =2 '7 tons. Case 2. Thrust on one leg =6'7 ,, Pull on each of the others =3^ SECTION II. INCOMPLETE FRAMES. 7. Preliminary Remarks. A frame may have just enough bars and no more to enable it to preserve its shape under all circumstances, or the number of bars may be insufficient or there may be redundant bars. The distinction between these three classes of frames is very important : in the first the structure will support any load consistent with strength, and the stress on each bar bears a certain definite relation to the load, so that it can be calculated without any reference to the material or mode of construction ; in the second, the frame assumes different forms according to the distribution of the load, but the stress on each bar can still be calculated by reference to statical considerations alone ; in the third, where the frame has redundant bars, the stress on some or all of the bars depends on the relative yielding of the several bars of the 12 STATICS OF STRUCTURES. [PARTI. frame. It is to the second class, which may be called " incomplete "" frames, that the present section will be devoted. In incomplete frames the structure changes its form for every distri- bution of the load, and, strictly speaking, therefore, such constructions cannot be employed in practice, because the distribution of the load is always variable to a greater or less extent. But when the greater part of the load is distributed in some definite way the principal part of the structure may consist of an incomplete frame, designed for the parti- cular distribution in question, and subsequent moderate variations of distribution may be provided for either by stiffening the joints or by subsidiary bracing. Such cases are common in practice, and investiga- tions relating to incomplete frames are therefore of much importance. 8. Simple Trapezoidal or Queen Truss. We will first consider a frame which is composed of four bars. The most common case is that in b c D Fig.9a. which of the bars are horizontal and the other two equal to one another, thus forming a trapezoid. The structure is called a trapezoidal frame. It is suitable for carrying weights applied at the joints CD, either directly or by transmission through vertical suspending rods from the beam AB. From the symmetry of the figure it is evidently necessary for stability that the loads at C and D should be equal. This fact will also appear from the investigation. Consider first the joint C, and draw the triangle of forces, Oan, for that point ; an being taken to represent W, aO will represent the thrust on AC and On that along CD. The triangle Obn will represent the forces at the joint D, Ob representing the thrust of BD ; bn will represent the load at D, and from the symmetry of the figure must equal an, and hence weight at D must for equilib- rium equal that at C. Now let us proceed to joint A, where there are also three forces acting, one along A C is now known and represented by aO, thus On will represent the tension of AB, and an will be the necessary supporting force at A equal to W, as might be expected. The tension of A B is equal to the thrust on CD. We observe that the diagram of forces is the same as that of a triangular frame, carrying 2ir at the vertex and of span equal to the difference between AB and CD. Trapezoidal frames are employed in practice for various purposes. ni. i.ART.9.] FRAMEWORK LOADED AT THE JOINTS. 13 (a.) A beam, AB (Fig. lOa), loaded throughout its length may be strengthened by suspending pieces, CN, DM, transmitting a part of the weight to the arch of bars, AC, CD, BD, an arrangement common in small bridges. (/?.) As a truss for roofs, in which case there will be a direct load at C and D due to the weight of the roofing material, while vertical mem- bers serve partly as suspending rods by which part of the weight of tie beam and ceiling (if any) is transmitted to CD, and partly to enable the structure to resist distortion under an unequal load. When made of wood, this is the old form of roof called by carpenters a " Queen Truss," CN, DM, being the " queen posts " (see Section III. of this chapter). This name is constantly used for all forms of trapezoidal truss erect or inverted which include the vertical "queens." (7.) Not less common is the inverted form, Fig. lOb, applied to the beams carrying a traversing crane, the cross girders which rest on the main girders of a railway bridge and carry the roadway, and many other purposes. The bars AC, CD, BD are now iron tie rods. In this case also if the two halves of the beam are unequally loaded there will be a tendency to distortion, to resist which completely, diagonal braces, CM, DN, must be provided, as shown in the figure by dotted lines. Such diagonal bars occur continually in framework, and their function will be fully considered in the next chapter. But in the present case they are quite as often omitted, the heavy half of the beam then bends downwards and the light half bends upwards (see Ex. 4, p. 87), but the resistance of the beam to bending is found to give sufficient stiffness. 9. General case of a Funicular Polygon under a Vertical Load. Example of Mansard Hoof. We next take a general case. In Fig. 1 la, 1 2 3 ... 6 is a rope or chain attached to fixed points at its ends and loaded with weights, W^ W^,.., suspended from the points 1, 2, etc. The figure shows 5 weights, but there may be any number. The rope hangs in a polygon, the form of which depends on the proportions between the weights. It is often called a "funicular polygon" and possesses very important properties. We shall find it convenient to distinguish the sides of this polygon by letters a, b, c, etc. We are about to determine the proportions between the weights when the rope hangs in a given form, and, conversely, the form of the rope when the weights are given. STATICS OF STRUCTURES. [PART i. In Fig. lib draw ab vertical to represent W^ the load suspended at the angle of the polygon where the sides a and b meet, then draw aO, 10 parallel to a, b respectively to meet in 0, thus forming a triangle Oab, which we distinguish by the number 1, which represents the forces- w, W W 4 i\ Fig.iib. acting on the point 1, so that the tensions of the sides a, b are thus determined. Now draw Oc parallel to the side c to meet the vertical in c; we thus obtain a triangle distinguished by the number 2, which represents the forces acting at that point, and as Ob is already known to be the tension of b it follows that be must be the weight W v and Oc the tension of the side c. Proceeding in this way we get as many triangles as there are weights, and the sides of these triangles must represent the weights and the tensions of the parts of the rope to which they are respectively parallel. Thus, if the form of the rope is known and one of the weights, all the rest can be determined. Conversely, to find the form of the funicular polygon when the weights are given in magnitude and line of action, we have only to set downwards on a vertical line the weights in succession and join the points a b..., which will now be known, to any given point 0, then the funicular polygon must have its angles on the lines of action of the weights and its sides parallel to the radiating lines Oa, Ob, Oc, etc., so that the sides can be drawn in succession, starting from any point we please. In the diagram of forces, Fig. lib, if ON be drawn horizontal to meet the vertical a, b, c... in N, this line must represent the horizontal tension of the rope. The rope may be replaced by a chain of bars which may be inverted, thus forming an arch resting on fixed points of support, the diagram of forces will be unaltered, and ON will represent the thrust of the arch. As an elementary example of an arch of bars we will consider a truss n. i. ART. 10.] FRAMEWORK LOADED AT THE JOINTS. 15 used for supporting a roof of double slope called a Mansard roof. We will take the usual case in which the truss is symmetrical about the centre. Suppose it is loaded at the joints. There is one proportion of load which the truss is able to carry without any bracing bars being added. From symmetry the weights at 2 and 2' (see Fig. 12a) must be equal. To find the portion between the weights at 1, and at 2 2', together with the stresses on the bars of the frame, in Fig. 1 2b set down ad to represent fFat 1, and draw aO and a' parallel to a and a', the thrusts along these bars will be determined. Then, considering the equilibrium of either 2 or 2', say 2, one of the three forces acting at the joint, namely aO, along the bar a being known, the other two forces may be W 2 determined by drawing ab and Ob parallel to them, ba parallel to IV.^ and Ob to the bar b. If ON be drawn horizontally it will give the amount of the horizontal thrust of the roof or the tension of a tie bar 3 3', if there is such a bar. If the proportion of W. 2 to W l is greater than ab to aa' the structure will give way by collapsing, 2 and 2' coming together ; and if the proportion is less, the structure will give way by 2 and 2' moving outwards and 1 falling down between. In practice it is impossible to secure the necessary proportion of loads, on account of variation of wind pressure and other forces, and therefore stiffening of some kind is always needed. If bracing bars be placed as shown by the dotted lines 2 3', 2' 3, 2 2', the structure will stand whatever be the pro- portion between the loads. The truss may be partially braced by the horizontal bar 2 2' only. Then the proportion between the loads W l and 7F 2 may be anything we please, but the loads at 2 and 2' must be equal, at least theoretically, but in practice the stiffness of the joints will generally be sufficient for stability, especially if vertical pieces be added connecting these points to the tie beam as in a queen truss. 10. Suspension Chains. Arches. Bowstring Girders. We now go on to consider another important example, in which the number of bars 16 STATIC'S OF STKlVTrilKs. [PART i. composing the frame is very much increased, as found in the common suspension bridge. Let AB (Fig. 13a) be the platform of a bridge of some considerable span, which has little strength to resist bending. Suppose it divided into a number of equal parts, an odd number for convenience, say nine, and each point suspended by a vertical rod from a chain of bars secured at the end to fixed points, D and E, in a horizontal line. In the figure only half the structure is shown. Suppose the platform loaded with a uniformly distributed weight : we require to know the stress on W W W each bar and the form on which the chain will hang. Equal weights on each division of the platform will produce equal tensions in the vertical suspending rods, and if we neglect the differences of weight of the rods and bars themselves, the load at each joint of the chain of bars will be the same. (Cornp. Art. 11.) Let 7f=load at each joint. Now the centre link KK\ since there is an odd number and the chain is symmetrical, will be horizontal. Let us consider the equilibrium of the half chain between C and D. The four weights, /IF, hanging at A', L, J/, A r , are sustained in equilibrium by the tension of the bars KK' and NI). The resultant of the four /Fs will act at the middle of the third division from the left end, and since this resultant load together with the tensions of the middle and extreme links maintain the half chain in equilibrium, the three forces must meet in a point, the point Z shown in the figure. Thus the direction of the extreme link DN may be drawn. The direction and position of the other links may be found also. Considering the portion of the chain XC carrying three weights, the resultant of which is in the line through , the link JO/ must be in such a direction as to pass through the point where this resultant cuts KK' produced. Having drawn AW, ML may be drawn in a similar way, and then LK. Returning to the consideration of the half chain, the three forces which keep it in equilibrium may be represented by the three sides of a triangle. Set down an (Fig. 13b) to represent H. I.AUT. 10.] FKAMKWOKK LOADED AT THK JOINTS, 17 4//~. and d: ul n<> parallel to /'/and /( ' : /(> will be the tension of /).V and /<(> of A'A". If an be divided into I equal parts, and the points b, c, (/ joined to ('. these lines will represent the tensions of links .V.I/, .I/A. and 7.A". It may be easily shown that they will be parallel to those links. \Vo see that the tension increases as we pass from link to link, from the centre to the ends. In many cases in practice, the number of vertical suspending rods and links in the chain is very great. We may then, in what follows, without sensible error, regard the chain as forming a continuous curve. D i E In such a case, (7, the Jowest point of the chain (Fig. 14a), is over the middle of the platform. The tangent at (\ which is horizontal, will nuet the tangent to the chain at /), in a point Z, a which will be over the middle of the half platform, for that will be a point in the line of action of the resultant load on the half chain. We can now draw a triangle of forces anO, for the half chain as before ; On will represent the tension of the chain at the lowest point, or the horizontal component of the tension of the chain at any point. We can ea^ily obtain a convenient expression for this horizontal tension thus: Let / = span of the bridge, and tr = load per foot-run. Then Jir/ = weight on the half chain repre- sented by an. Let H= horizontal tension, then H = 0n \ wl ~ an ' But if we drop a perpendicular from 7) to cut the horizontal tango m in a point /' (not shown in the figure), DV will be the dip of the chain or ' smce H => therefore x 2 is proportional to y. CH. i. ART. 11.] FRAMEWORK LOADED AT THE JOINTS. 21 Now the curve whose co-ordinates have this relation one to another is called a parabola. If the load, instead of being uniformly distributed on a horizontal platform, were simply due to the weight of the chain itself, then the curve in which the chain would hang would deviate somewhat from the parabola ; for in that case, since the slope increases as we approach the piers, the load also, per horizontal foot, would increase as we approach the piers, causing the chain near the piers to sink and become more rounded, and at the centre to rise and become more flattened. The curve in which the chain hangs by its own weight is called the catenary. In the catenary, as in the parabola, the tension increases as we approach the piers. This may be taken account of by proportioning the section of the chain to the tension at the various points ; this would tend still more to make the weight of chain, per horizontal foot, increase as we approach the piers, and cause the chain to deviate still further from the parabolic form. Such a curve is called a catenary of uniform strength. In an actual suspension bridge, where there is a uniformly loaded platform, as well as a heavy chain, the true curve in which it hangs will lie somewhere between the parabola and the catenary ; but since in most cases the deviation from uniformity of the weight of chain is small compared with the load it carries, the deviation from the parabola is not great. The error involved in assuming the curve to be parabolic is generally greatest in bridges of large span; in such cases a prelimi- nary calculation of approximate weights may be necessary so as to be able to apply the general process of article 9. EXAMPLES. 1 . A trapezoidal truss is 16 feet span and 4 feet deep, the length of the upper bar is 6 feet. Find the stress on each part when loaded with 2 tons at each joint. Stress on sloping bars =3*2 tons. ,, horizontal bars =2 '5 tons. 2. The platform of a bridge, 8 feet broad and 27 feet span, is loaded with 150 pounds per square foot. It is supported on each side by an inverted queen truss placed below, the queen posts, each 3 feet deep, dividing the span into three equal portions. Find the stress on each part. Load on each truss = half the whole load on platform =16, 200. g 16,200=5,400 is the load at each of the two joints of one of the queen trusses. Tension of sloping bars = 17,074 Ibs. Tension and thrust of horizontal bars = 16, 200. 3. The height of a Mansard roof without bracing is 10 feet and span 14 feet. The height of the triangular upper portion is 4 feet and span 8 feet. The load being 1 ton at the ridge, find the necessary load at each intermediate joint and the thrust of the roof. By the construction described in the text, load at each intermediate joint = ton, and the thrust of the roof= ton. 22 STATICS OF STRUCTURES. [PART i. 4. If the roof in the last question be partly braced by a bar joining the intermediate joints, find the stress on the bar when the load at each intermediate joint is 1 ton. Thrust on bar = i ton. 5. The load on the platform of a suspension bridge, 600 feet span, is \ ton per foot-run, inclusive of chains and suspending rods. The dip is T Vth the span. Find the greatest and least tensions of one of the chains. Least tension = horizontal tension =243f tons. Greatest tension = 255 tons. 6. The load on a simple parabolic arch, 200 feet span and 20 feet rise, is 360 tons, determine the thrust and greatest stress on the arch. Thrust =450 tons ; greatest stress = 484 tons. 7. The rise of a bowstring bridge is 15 feet and span 120 feet, find the thrust when the load on each girder is 2,000 Ibs. per foot-run. Thrust 240,000 Ibs. =107f tons. 8. In example 5 the ends of the chain are attached to saddles resting on rollers on the tops of piers 50 feet high, and prolonged to reach the ground at points 50 feet distant from the bottom of the piers, where they are anchored. Find the load on the piers and the pull on the anchors. Load on the pier = 637^ tons ; Pull on each anchor =344 '6 tons. 9. A light suspension bridge is to be constructed to carry a path 8 feet broad over a channel 63 feet wide by means of 6 equidistant suspending rods, the dip to be 7 feet. Find the lengths of the successive links of the chain. Supposing a load of 1 cwt. per square foot of platform, find the sectional areas of the links of the chain, allowing a stress of 4 tons per square inch. f of the whole load is carried by the chains and the remaining portion by the piers directly. Tension of each suspending rod =36 cwt. Links. Tensions. Areas. Lengths. Centre, 2777 3-47 9' 2nd, 280' 3-5 9-08 3rd, 287' 3-6 9-3 4th, 298- 372 9-66 10. Construct a parabolic arch, the thrust of which is half the total load. Span = four times the rise. 11. If a weight of a uniformly loaded platform be suspended from a chain by vertical rods, show that the corners of the funicular polygon lie on a parabola. SECTION III. COMPOUND FRAMES. 12. Compound Triangular Frames for Bridge Trusses. By a compound frame is meant a frame formed from two or more simple frames by uniting two or more bars. Many frames of common occurrence in practice may conveniently be considered as combinations of the simpler CH. i. ART. 12.] FRAMEWORK LOADED AT THE JOINTS. 23 examples already described. They are generally dealt with by use of what we may call the principle of superposition, which may be thus stated : The stress on any bar due to any total load is the algebraical sum of the stresses due to the several parts of the load. We will now consider some examples of compound frames which are used in bridge trusses. In these structures the object is to carry a distributed load by means of a comparatively slender beam. A prop in the centre may still leave the halves too weak to carry the weight on them, and the beam may be strengthened by supporting it in more than one point. (1) Suppose the beam supported by a number of equidistant struts, the lower ends of which are carried by tension rods attached to the ends of the beam, we then have a structure called a Bollman truss. There may be any number of struts 2, 3, 4, or more; the structure has been used for bridges of comparatively large span. If the actual load is distributed in some manner over the beam, we must first reduce the case to that of a structure loaded at the joints only. The loads on the struts are due to the weights resting on the adjacent divisions of the beam, and may be determined by supposing the beam broken or jointed at the points where the struts are applied. Let us suppose the beam has three divisions, and that the load on the two struts are W l and W^. These loads will be transmitted Fig.i?. down the struts to the apices A c *w, w 2 D o (Fig. 17) E and F, and will be ' ^^^ independently supported, each ^X^-^*^^ by its own pair of tension rods. We may thus separately determine the stress on each part of either of the elementary triangular frames, AEB or AFB. AB will be in com- pression on account both of the load at E and also at F. On account of W v using the formula previously obtained, the 'horizontal thrust J1' E W-jj-, and on account of W z at F, H F = ^zjr- Tension of AE, T AX =H E secEAB, T FB = H F secFBA ; EB, T EB = H, sec EBC, T Af = H F sec FAD. The actual tensions of the sloping rods are simply as written, but since AB is a part of both triangular frames, the total thrust along it is found by summing the thrusts due to each : so This is an example of the principle of superposition stated above. (2) Suppose the beam which carries the distributed load to be 24 STATICS OF STRUCTUKES. [PART. i. supported by a central strut forming a simple triangular truss, and further let the halves of the beam, not being strong enough to carry the load on them, each be subdivided and trussed by a simple tri- angular truss, the tension rods from the bottom of the subdividing struts proceeding only to the ends of each half beam. If the quarter spans are still too great, they may each of them be trussed in a similar way, and so on. Such a structure is called a Finck truss. Suppose, for example, we have three struts. (Fig. 18.) We must first determine the load at the joints that is, in this case, the load on the struts due to the distributed load on the beam. Suppose that on account of the weights on the adjacent subdivisions those loads are W^, W^ W^. If the load is uniformly distributed over the beam the PF's are each of them equal to J total weight on beam. Fig 13 We may now separately consider the triangular frame, AFC, carrying the load, W v On account of it there will be a thrust on AC. IT _ w^ G - w JL W ^h~ ^m The tensions of AF and FC are each = H F sec FAE. We get similar results from the triangle CH B. Just in the same way we may consider the principal triangle frame, ADB, but in this case the thrust down the strut, CD, which is the load at _D, is not simply W^ but greater by the amount of the downward pull of the two tension rods, CF and CH. The vertical components of these tensions are \W^ and \W^ so that the total thrust down the strut = W% + \(W^ + W^. This is the load which must be taken to act at I) in determining the stresses on the members of ADB. Hence it appears that and the tensions of AD and DB are each equal to H t) sec DAB. It will be seen that the thrust on the central strut and tensions of the longer rods are the same as if the secondary trusses had not been introduced. For example, if the W'& each = J whole load on beam, then the virtual load at D = \ weight on beam. The mere strengthening of each half the beam by trussing it can no more relieve the central strut of the load it has to carry, than the fact of strengthening a structure of any kind can relieve the two points of support from the CH. i. ART. 13.] FRAMEWORK LOADED AT THE JOINTS. 25 duty each must have of bearing its own proper share of the weight. In stating the thrust on the beam we must divide it into two portions, JC and CB. The portion AG is subjected to the thrust of the triangles AFC and ADB ; .'. H AC = H F -\- H D , and CB being a portion of the triangles CHB and ADB, H CB = H n + H n . When W.^ is not equal to W v the thrusts on the two portions will be different. This' is quite possible although the beam AB may be a continuous one. Both these simple forms of truss have been used for bridges of considerable span. As an example of the first may be mentioned the bridge at Harper's Ferty, U.S., destroyed during the war. It was 124 feet span in 7 divisions. The great length of the tension rods and their inequality appears objectionable. The second in 8 or 16 divisions has been much used in America; but in England other forms mentioned in a later chapter are much more common. 13. Roof Trusses in Timber. In roofs of small span, 10 or 12 feet only, the roofing material, slates or tiles, rests on a number of laths set lengthways to the roof, and these laths rest on sloping rafters spaced 1 or 2 feet apart, with their feet resting on the walls of the building ; the stability of the walls being depended on for taking the thrust. When we come to larger and more important roofs we find additional members added for strength and security. The closely spaced rafters just mentioned are called common rafters. These being too long and slender to carry the weight of the roofing material and transmit it to the walls, are supported, not only at the ends by the walls and ridge piece, but also at the middle by a longitudinal beam of wood called a purlin, and the purlin is supported at intervals of its length by principal rafters. The principal rafters again are supported by struts at their central points, immediately below the purlins. To carry the lower ends of the struts, a vertical tension piece is introduced, by which they are suspended from the apex of the principals, while the thrust is taken by a, tie beam connecting the feet of the rafters. In such a roof, a ceiling or floor may frequently be required to be supported by the tie beam, and to prevent it from sagging under the weight an additional tension will come on the vertical suspending rod. This rod is then a very important member of the structure, and is called the king post, and the whole structure, consisting of the principal rafters, king post, etc., is called a King post truss. This truss is often constructed entirely of wood. The sloping struts then for constructive reasons (Ch. XV.) butt on an enlarged part at the bottom of the king post above the point where the horizontal tie beam is attached, but for calculation STATICS OF STRUCTURES. [PART i. purposes may be regarded as meeting at that point as shown in Fig. 19. Pig. 19. By means of the purlins and the ridge piece the weight of the roofing material will produce loads at the joints EOF '= W^W^^ suppose. Now treat the structure as made up of three simple triangular frames, A ED, DFB, and ACB. First consider AED with the load W^ at vertex E. The horizontal thrust of this frame AD = 'W- t rT - where 1 4:fl h is the height of point E above AD. Also the thrust along AE and ED due to the load E = H E secEAD. In an exactly similar manner we may consider the triangle DFB ; the results for this will be to those for AED in the proportion of W% to W v Next as to the primary triangle AGE. There is at G a direct load of W^ due to the weight between E and (7, and F and C. But beside this, the king post pulls the point C downwards, so that the total load at C = W z + tension of king post. In addition to a portion of the weight of the ceiling (if any) the post has to support D against the downward thrust of the two struts ED and FD. The vertical components of these thrusts are \W^ and \W^ therefore, neglecting the weight of ceiling, the virtual load at C=W< L + \(W^ + W^. Let us call this total load W, then H c the horizontal thrust of ACB= ^777 and the thrusts along AC and CB due to load C=H c secA. Now in the complete structure, since AD is a member both of the triangular frame AED and ACB, the total tension of AD = H E + H c . For the same reason tension of DB= H F + H C , and thrust of AE = (H E + H c ) sec A, The other members of the structure are portions of one elementary frame only, and the stress is due only to the load at the apex of that frame. The king post truss serves for roofs of spans under 30 feet, but for spans greater than this trusses of more complicated construction are required. If the span is from 30 to 50 feet, then instead of supporting the common rafters by a purlin at the centre of its length only, as in CH. i. ART. 14.] FRAMEWORK LOADED AT THE JOINTS. 27 the king post truss, two supporting purlins may be used, dividing the length of the rafter into three equal portions. These purlins may be carried by a queen truss, the sloping members of which are -supported in the middle by struts, as shown in the figure (Fig. 20). Fig.20. The vertical queen posts, DN and FK, serve to sustain the downward thrust of the struts, EN and GK, and also to support the weight of a ceiling, if there is one. Supposing the weight of the ceiling omitted, let W be the weight of roofing material on one side for a length of roof equal to the spacing of the trusses, then \W will, through the common rafters and purlins, act at E, and \ at D ; and similarly for the other side. At the ridge C there will also be \W acting ; but this will be distributed equally amongst the common rafters which are carried by the truss, and will produce compression in those rafters without directly affecting the truss. The part of the thrust of the roof arising from this will, however, generally, like the rest, ultimately come on the principal tie beams. To find the stresses on the different members of the truss. Consider first the small triangles AEN and BGK, each carrying JJFat the vertex. We then consider the trapezoidal truss ADFB. The loads at D and F will be J^+ tension of queen post. Since the tension of the queen post D N = the vertical component of the thrust alojig EN it will equal . \W=\W^ and the total load at each joint of the trapezoidal truss will be \W+\W=\W^ the same as would have acted if there had been no purlin at E and no strut EN. After having determined the respective stresses due to the triangles and trapezoid separately, we must add the results for any bar which is a part of both. Were it not for the friction at the joints and the power of resistance of the continuous rafters AC, CB to bending, this structure would be stable only under a symmetrical load. In practice, however, it is able to sustain an unsymmetrical load, such as roofs are frequently subjected to. 14. Queen Truss for large Iron Roofs. As the span of the roof is still further increased we find other kinds of trusses employed to support them. A common form in iron roofs is constructed, as shown in 28 STATICS OF STRUCTURES. [PART i. Fig. 21. It is in reality a further development of the wooden queen truss, and is known by the same name. AC and CB are divided into L a number of equal parts, and sloping struts and vertical suspending rods are applied as shown. Suppose the load the same at each joint on one side of the roof, the load on the right, however, riot being necessarily equal to that on the left. Let the upward supporting force at A = P. P will be half the total weight if the loading is symmetrical, but in any other case it may be found by taking moments of the loads about B. We might solve the problem of finding the stress on each member of the structure by treating separately each elementary triangle into which the structure may be divided, and summing the stresses for any bar which may form a part of two or more triangular frames. But we will describe another method. First, to find the tension of the vertical suspending rods consider ^12' as an independent triangle, carrying a load W at its vertex. The slope of 12' being the same as that of A\, the tension rod 22' must supply a supporting force to the joint 2' ' = \W. Considering next the triangle ^23' and its equilibrium about the point A. The forces along 23 and 3'4' have no moment about A, so that the moment of the two weights //' at 1 and 2 about A must be balanced by the upward pull of the tension rod 33'. .*. tension of 33' = W. In a similar way we can see that the tension of 44' = f JF. However many divisions of the roof there may be, the tensions of the vertical suspending rods will increase in arithmetical progression, with the same difference between each. The road 11', except so far as may be due to the weight of the rod A '2', will have no tension on it. Calling this the 1 st tension rod, the tension of the n th = ^ W. We must notice that the rod 55' is common to both sides of the roof, and we must add the two tensions to get the total. Now consider any joint, say 4' in the tie bar AB, and resolve vertically and horizontally. If R = thrust of 34', (9 its inclination to the horizontal, and T the pull on that division of AB which is indicated by the numerical suffix placed below it, CH.I.ART. 15.] FRAME WOHK LOADED AT THE JOINTS. 29 But from figure cot = J cot A \ Whichever joint we select we should find the same result namely, that the difference between the tensions of two consecutive portions of the tie rod is a constant quantity = \W cot A. So that these tensions are in arithmetical progression diminishing towards the centre. If we call A'2' the 1 st division of the rod, then for the joint between the n - 1 th and n ih we have 2tcos6 = r M _! - T HJ and cot = cot A ; If A\ is the 1 st division of the rafter, then the thrust on the n th Now, the tension of the tie rod in the 1 st division = P cot A, nd =>- The thrust on the /i th division of rafter = (P -- ^^ The thrust on any strut may best be found by squaring and adding the two equations of equilibrium of the lower joint of it. We get W Thrust of n ih strut = --Jri 2 15. Concluding Remarks. General Method of Constructing Diagrams of Forces. Cases of framework often occur which are much more com- plicated than those which we have hitherto considered, but if there are no redundant bars the stress on each part depends on statical principles only, without reference to the relative yielding of the several parts of the structure. Such cases may always be treated by use of the general principle stated in Art. 1, and we shall conclude this chapter by ex- plaining briefly a graphical method of applying that principle invented by the late Professor Clerk Maxwell. The forces will be supposed all in one plane, and each of them will be supposed known, that is to say, if there be any unknown reactions at points of support they will be supposed previously found by a graphical or other process, from the consideration that the whole must form a set of forces in equilibrium. In Fig. 22a a frame is shown acted on by known forces PQB..., an ideal example is chosen which is better suited for the purpose of 30 STATICS OF STRUCTURES. [PART i. explaining the method than any case of common occurrence in practice. First seek out a joint where only two bars meet : there will usually be two such joints if there be no redundant bars in the frame, and in the present instance we will choose the joint where P acts. Distinguish all the triangles, making up the frame by letters A, B, C, etc., and place numbers or letters outside the frame, one for each bar. In Fig. 22b draw 18 parallel to the force P and representing it in magnitude, Sa parallel to 8, la parallel to 1, to intersect in the point a ; then, as in previous examples, Sa, la represent the stress on the two bars to which they are parallel. Pass now to the joint where Q acts : this joint is chosen because only three bars meet there, on one of which we have just determined the stress; draw 12 parallel to Q and representing it, then ab parallel to the bar lying between the triangles A and B, and 26 parallel to the bar 2 ; we thus get a polygon 1 2ba, the sides of which are parallel to the four forces acting at the joint where Q acts, while two of them represent two forces already known, the other two, there- 8 P \ W fore, will represent the remaining two forces. Proceed now to the joint where W acts and complete in the same way the polygon Sabcl, then to the joint where R acts, and so on. We at length arrive at the triangle 4/5, the third side of which, if we have performed the con- struction accurately, and if the forces be really in equilibrium, must be parallel to the last force T. On examination of the diagram of forces (Fig. 22b) it will be seen that to every joint of the frame corresponds a polygon representing the forces at that joint, while each line, such as ab or 7c, gives the stress on the bars separating those letters or numbers in the frame-diagram. The polygon 12... 8 is the polygon of external forces, each side representing the force to which it is parallel. The method here described is easy to understand in the general case we have considered, and with a little practice the transformations the diagram of forces undergoes will offer no difficulty. Some joints are usually unloaded, and the corresponding lines in the polygon of external CH. i. ART. 15.] FRAMEWORK LOADED AT THE JOINTS. 31 forces vanish ; the forces may be parallel, in which case the polygon becomes a straight line, while not unfrequently the sides of two of the polygons representing the forces at the joints coincide. The figure, however, always possesses the same properties. In Mr. Bow's excellent work referred to at the end of this chapter over 200 examples will be found of the application of this method^ including almost all known forms of bridge and roof trusses. EXAMPLES. 1. A Bollman truss of three divisions is 21 feet span, and is loaded uniformly with 1 tori per foot. The depth of 'the truss is 3 feet. Find the stress on each part. Load on each strut = 7 tons. Tension of short rods =10 '4 ,, longer rods= 9 '6 ,, Total thrust on beam = 18 ,, being 9J due to each triangle. 2. A Finck truss of 4 divisions, 20 feet span and 3 feet deep, is loaded with 1 ton per foot, find the stress on each part. Thrust on 26 and 48 =5 tons. 37 =10 Tensions of 16, 63, 38, and 85= 4 '86 17 and 75 =17 '4 Thrust on 13 and 35 = 4 + 16=2Q tons. 3. In the last question suppose one half the truss loaded with an additional 1 ton per foot. Find the stress on each part. Suppose the additional load on the right-hand side. Tensions. On 16 and 63= 4 '86 tons. 38 85= 9'72 17 75 = 261 Thrusts. On 26= 5 tons. ; 37 = 15 , 48=10 ,, 13= 4j + 25 35= 8J + 25 4. A roof 28 feet span, height 7 feet, rests on king-post trusses spaced 10 ft. apart. The weight of roof is 20 Ibs. per square foot. Find the stress on each part. Also obtain results when an additional load of 40 Ibs. per square foot rests on one side. Load at each joint. 1st case = 1566 '6 Ibs. Stress in Ibs. Stress. Bars. Equal Load. Additional Load. Bars Equal Load. Additional Load. 1 5254 8756 1' 5254 12261 2 3503 7006 2' 3503 7006 3 4700 7833 3' 4700 10966 4 1752 1752 4' 1752 5255 5 1566-6 3113 5. A roof 48 feet span, 12 feet high, rests on queen trusses 8 feet high, spaced 10 feet apart. Find the stresses for a load of 20 Ibs. per square foot. 7155 6367 5. An A roof, braced as in the figure, is 40 feet span, and 10 feet high ; the horizontal STATICS OF STEUCTUEES. [PART i. tie bar is 8 feet below the vertex. Find the stresses on each part when loaded with 2 tons at each joint by constructing a diagram of forces or otherwise. Bars. Stress. 1 10-4 2 9-4 3 9-4 4 47 5 1-8 6 5" 7. In the last question suppose an accumulation of snow on one side equivalent to an additional load of 2 tons at the middle of the rafter, and 1 ton at the ridge. Find the stress on each part. Bars. Stress. Bars. Stress. 1 13-9 1' 17-3 2 12-5 2' 157 3 12-8 3' 15-4 4 5'n 4' 8'6 5 1-8 5' 3'6 6 7-5 8. Suppose there are 11 suspending rods in iron roof shown in the figure, the height of which is ^th the span. Find the stress on each part 1st, when loaded with ton at each joint on both sides, and, 2nd, when loaded with an additional | ton at each joint on one side, not including the ridge. I5| I4| I3 llff 13s I5| 185 20| 23s Additional load is on right-hand side, and the figures on the diagram refer to case 2. 9. The roadway of a bridge, 80 feet span, is carried by a pair of compound trapezoidal trusses, each consisting of three simple trapezoids of the same height, the six "queens" of which are equidistant, forming seven divisions of length four thirds the height of the truss. Find the stress on all the bars due to ^ ton per foot-run on the bridge. 10. Find the stress on each part of a "straight-link suspension" bridge formed by inverting the truss of the last question, assuming the pull at the centre of the platform aero. REFERENCES. For further information on the subjects treated of in the present chapter the reader may refer amongst other works to GLYNN Construction of Cranes. Weale's series. HURST Carpentry. Spon, 1871. Bow Economics of Construction. Spon, 1873. CHAPTER II. STRAINING ACTIONS ON A LOADED STRUCTURE. 16. Preliminary Explanations. In the preceding chapter we have considered only those structures in which the parts are subject to compression and tension alone, except by way of anticipation in a few special cases. But the parts of a structure are generally subject to much more complex forces, and besides, although the forces acting on each bar have been determined, we should, if we stopped here, have a most imperfect idea of the way in which the load affects the structure as a whole. If we imagine a structure to be made up of any two parts, A and B, united by joints, or distinguished by an ideal surface cutting through the structure in any direction, the whole of the forces acting on the structure may be separated into two sets, one of which acts on A, the other on B. Since the structure is in equilibrium as a whole the two sets of forces must balance one another, and must therefore produce equal and opposite effects on A and B, effects which are counteracted by the union existing between the parts. The two sets of forces taken together constitute a STRAINING ACTION of which each set is an element, and the object of this and the next two chapters is to consider the straining actions to which loaded structures and parts of structures are subject. Straining actions differ in kind, according to the nature of the effects which they tend to produce. Four simple cases may be distinguished : (1) The parts A and B may tend to move towards each other or away from each other perpendicular to a given plane. This effect is called Compression or Extension, and the corresponding straining action is a thrust or a pull. (2) A and B may tend to slide past each other parallel to a given plane. This effect is called Shearing. (3) A and B may tend to rotate relatively to each other about an axis lying in a given plane. This is called Bending. C.M. C 34 STATICS OF STRUCTURES. [PART. i. (4) A and B may tend to rotate relatively to each other about an axis perpendicular to a given plane. This is called Twisting. In the first two cases the straining action reduces to two equal and opposite forces, and in the second two to two equal and opposite couples. In general, straining actions are compound, consisting of two or more simple straining actions combined. The given plane with reference to which the straining actions are reckoned may always be considered as an ideal section separating A and B even when the actual dividing surface is different. We shall commence by con- sidering the straining actions on a beam of small transverse section, SECTION I. BEAMS. 17. Straining Actions on a Beam. The action of a simple thrust or pull on a bar has already been sufficiently considered in Chapter I. They are usually considered as separate cases, and the simple straining actions on a bar are therefore reckoned as five in number. The other three are (1) shearing, (2) bending, and (3) twisting, of which the last rarely occurs, except in machines, and will, there- fore, be considered in a later division of this work, under that head. Shearing and bending are due to the action of forces, the directions of which, are at right angles to* the bar : in structures, the forces usually lie in one plane passing through the axis of the bar. A bar loaded in this way is called a beam. Simple shearing is due to a pair of equal and opposite forces, F (Fig. 23), applied to points very near together, tending to cause the two parts A and B to slide past one another, as shown in the figure (Figs. 23a, 23b). Either element is called the shearing force, and is a measure of the magnitude of the shearing action, but in considering the sign we must consider both together. In this work, if the right hand portion, A, tends to move upwards, and B downwards, as in Fig. 23b, the shearing action will usually be reckoned negative, while in the converse case (Fig. 23a) it will be reckoned positive. | P Flg.24. *P Fig.23a. I Simple bending is due to a pair of equal and opposite couples applied to the bar, one acting on A, the other on B, as in Fig. 24, tending to- en. u. ART. 18. STRAINING ACTIONS. 35 make A arid B rotate in opposite directions. The magnitude of the bending is measured by the moment of either couple, which is called the bending moment. In this work bending moments will usually be reckoned positive when the left-hand half, B, rotates with the hands of a watch, and the right-hand half in the opposite direction ; that is to say, when the beam tends to become convex downwards, as in the ordinary case of a loaded beam supported at the ends. In loaded beams shearing and bending generally exist together, and vary from point to point of the beam. We shall now consider various special cases. 18. Example of a Balanced Level'. General Rules for calculating S.F. and B.M. First take the case of a beam, AB, supported at C (Fig, 25), and loaded with weights, PQ, at its ends. If the weights are such % that P.ACQ.BC the beam will be in equilibrium, but the two parts, AC, BC, . P+Q Fig.25. tend to turn about C in opposite direc- B |o A tions ; there is therefore a bending action ^ K~ at C, of which the equal and opposite Q p moments P .AC, Q. BC are the elements. Either of these is the bending moment usually denoted by M, so that we write M C = P.AC=Q.BC. Not only is there a bending action at C, but if we take any point, K, and consider the forces acting on AK, BK separately, we see that AK tends to turn about K under the action of the force P, while BK tends to turn about K under the action of the forces P + Q at C and Q at B. The first tendency is immediately seen to be simply the moment P.AK, while the second is Q. BK - (P + Q)CK. The last quantity reduces to Q.BC-P.CK, or, remembering that Q.BC = P.AC, to P . AK. The two moments, then, as before, are equal and opposite, and constitute a bending action at K, measured by the bending moment M K = P.AK. This example will sufficiently explain the general rule for calculating the bending moment at any point, K, of a beam. Divide the forces into two sets, one acting to the right and the other to the left of K, and estimate the moment of either set about K, then the result mil be the bending moment at K. The example shows that the calculation of one of the two moments will generally be more simple than that of the other, and cases constantly occur, as where a beam is fixed at one end in a wall, where nothing is known about one set of forces except that they balance the other set. In each case the simplest calculation is of course to be preferred. Moments are measured numerically by unit weight acting at unit 36 STATICS OF STRUCTURES. [PART i. leverage, as, for example, 1 ton acting at a leverage of 1 foot, for which the expression " foot-ton " is commonly employed. This phrase, however, is used also for a wholly different quantity, namely, the unit of mechanical work, and for this reason it would be preferable to call the unit of moment a " ton-foot" for the sake of distinction. The peculiar action called shearing will be better understood when we come to consider the action of forces on a framework girder in the next section ; it will here be sufficient to say that if the sum of the forces acting on AK, EK are not separately equal to zero, they must tend to cause AK, BK to move past each other in the vertical direction, thus constituting a shearing action measured by the mag- nitude of the shearing force, which may be thus calculated for any point K. Divide the forces into two sets, one acting to the right of K, and the other to the left of K, the algebraical sum of either set is the shearing force at K. As before, either set may be chosen, whichever gives the result most simply. In the example just given the shearing force at any point of AC is P ; and at any point of BC, Q. 19. Beam Supported at the Ends and Loaded at an Intermediate Point. We will next consider the case of a beam supported at the ends and Fig.26. loaded at some intermediate point. Before we can apply the rules previously enunciated, to find the shearing force and bending moment at any point, we must first determine the supporting forces at the two ends. We find the force P acting at A (Fig. 26), by taking moments about B, thus, CH. ii. ^RT. 19.] STRAINING ACTIONS. 37 Wn and similarly Q=-^r a + b First as to the shearing force. Taking any point K in AC, and considering the forces acting on AK, of which there is only one, F -P- Wb **- P -^Tb' At any point K' between C and B we have It will be noticed that at K the tendency is for the left-hand portion to slide upwards relatively to the right, whereas at K' the tendency is for the right-hand portion to slide upwards relatively to the left. It is advantageous to distinguish between these two tendencies, as previously stated, by calling the one positive and the other negative. We may draw a diagram to represent the shearing force at any point thus. Let A'E' be drawn parallel to and below AB to represent the length of the beam, and let CC'L be the line of action of the weight. If we set up an ordinate A'F=P, and downwards an ordinate B'M= Q, and draw FE and ML parallel to A'B to meet the vertical EC'L ; the shearing force at any point will be represented by the ordinates of the shaded figure A'FELMB, measured from the base line A'B. Not only should the magnitude of the shearing force be represented, but also the direction of the sliding tendency. This is why the ordinate was set downwards on the right-hand side of C'. In this example the supporting forces may be found by construction, and thus the whole operation of determining and representing the shear- ing force performed graphically. For, set down B'K=W> join A'K, and where the vertical through C' cuts A'K, draw LM horizontal, then BM = Q and MK= P. Then set up A'F= MK, and draw FE horizontal. Next as to the bending moment at any point. Take any point K in AC distant x from A, then and similarly at K' in CB distant x' from B, so for either side of (7, the bending moment is greater the greater the distance of the point from the end of the beam. Thus the greatest bending moment is at C. If in the value of M K we put x = a, or M x' = b 38 STATICS OF STRUCTURES. [PARTI. ing moment at C' = , on some convenient scale, on such a scale for we get the same result, viz., that M c = - 7 = greatest bending moment. The graphical representation of the bending moment at any point is very useful and instructive. We may construct the diagram thus : A'E' representing the length of the beam, set up from C", C'N' the bend , instance as 1 inch = 20 ft.-lbs. Then joining AN' and B'N' 9 the ordinate of the figure A'N'B', measured from the base line A'E ', will express on the scale chosen the bending moment at any point of the beam. If a = b = % span, so that the load is applied at the centre of the beam, then M c = \W x span = greatest bending moment. 20. Beam Supported at the Ends and Loaded Uniformly. The next example for consideration is that of a beam supported at the ends and loaded uniformly throughout its length with w Ibs. per foot (Fig. 27). Fig.27. Let the span = 2a. Take any point, K, distant x from the centre The load on AK is wAK, and therefore the shearing force at K. reckoning the forces on the left-hand side, must be F K = wa - wAK= wa - w(a -x) = wx. That is, the shearing force is proportional to the distance of the point from the centre of the beam. At the end A where x = a, F A = wa, and at B where x -a, F B = -wa. If from A'B, below AB in the diagram, we set up and down ordinates at A and B' = wa on some scale, and join LM, the ordinates of the -CH. ii. ART. 21.] STRAINING ACTIONS. 39 sloping line will represent the shearing force at any point. The shearing force at the centre of the beam is zero. In finding the bending moment at K, reckoning still from the left- hand side, we must clearly take account not only of the supporting force at A, but also of the effect of the load which rests on the portion of the beam AK. The moment of this load about K is the same as if it were all collected at its centre of gravity, namely, at the centre of AK. Thus M = wa . AK- wAK. That is to say, the bending moment at any point is proportional to the product of the segments into which the beam is divided by the point. Putting AK=a-x and BK=a + x, M K = $w(a?-x*\ which is greater the less x is. At the centre z = 0, and we have the maximum bending moment If we put Iwa = W, the total load on the beam M = \W* span. This is only one half the bending moment due to the same load when concentrated at the centre of the beam. If ordinates be set up from A'B' = ^w(a?-x 2 ), at all points, the extremities of the ordinates will lie on a curve which may easily be seen to be a parabola with its axis vertical and vertex above the middle point of the beam. For SZ = SK - KZ = %wa* -\w (a 2 - x*) = %wx*. So that SZ is proportional to SN' 2 , showing that the curve is a parabola. 21. Beam Loaded at the Ends and Supported at Intermediate Points. Next, suppose a beam (Fig. 28) supported at A, B, and loaded with weights P, Q, at the ends C, D, which overhang the supports. If AC 9 A B, BD are denoted by a, /, b respectively, the supporting force S at A (by taking moments about B) is given by Similarly B, the supporting force at B, is given by l=Q(b + l}-Pa. Take now a point K distant x from A then where M A , M B are the bending moments at A, B. 40 STATICS OF STRUCTURES. Also for the bending moment at K, or, as we may write it, [PART i.. These formulae show that the shearing force is constant while the bending moment varies uniformly. In the diagram this is indicated by setting up ordinates Aa, b, to represent the bending moments at A, B, and joining a, b ; the ordinate Kk of this line corresponding to an intermediate point K, will represent the bending moment there. Fig.28. The moments are in this example reckoned positive for upward bending if the ordinates are considered as drawn upwards from the base line CD, and it is therefore better to suppose them drawn downwards from the broken line C, a, b, D, An important special case is when M A = M B ; then the bending moment is constant and the shearing force zero. We have then no- shearing but only bending. Simple bending is unusual in practice, but an instance occurs in the axle of a carriage. The ordinates of the straight lines Ca, Db, represent the bending moment at any point of the overhanging parts of the beam. 22. Application of the Method of Superposition. When a beam is acted on by several loads, the principle of superposition already stated in Chapter I. is often very useful in drawing diagrams and writing down formulae for the straining action at any point. Thus, for example, in the preceding case, if there be many weights on the overhanging end of a beam, the bending moment and shearing force at each point must be the sum of that due to each taken separately ; and hence it follows that, whatever be the forces acting on a beam, if there be a part AB under the action of no load, and the bending moments at the ends of that part be M A , M s , the straining actions at any intermediate point K will always be given by the formulae just written down. And, CH. ii. ART. 22.] STRAINING ACTIONS. 41 further, if there be a load of any kind on AB, and m be the bending moment, on the supposition that the beam simply rests on supports at A, B, then the actual bending moment must always be given by M=M A . j- + M B . I + m, a general formula of great importance. The result is shown graphically in the diagram, where the curve represents the bending moment m, and the straight line ab the effect of the bending moments at the ends, supposed, as is frequently the case, to be in the opposite direction to m ; then the intercept between the curve and the straight line represents the actual bending moment. If several weights act on a beam, triangles may readily be constructed showing the bending moment due to each weight ; then adding the ordinates of all the triangles at the points of application of the weights, and joining the extremities by straight lines, a polygon is obtained which is the polygon of bending moments for the whole load. This process may also be applied to shearing forces. It is simple, but some- what tedious when there are many weights, and other methods of construction will be explained hereafter. In superposing two loads the artifice just employed in Fig. 28 is very useful. A propped beam (Ex. 11, p. 42) is an important example. EXAMPLES. 1. A beam, AB, 10 feet long, is fixed horizontally at A, and loaded with 10 tons dis- tributed uniformly, and also with 1 ton at B. Find the bending moment in inch-tons at A, and also at the middle of the beam. M=7'20 inch-tons at A. =210 ,, at the centre. 2. In the last question find the shearing forc.e at the two points mentioned. F=ll tons at A. = 6 ,, at the centre. 3. A beam, AB, 10 feet long, is supported at A and B, and loaded with 5 tons at a point distant 2 feet from A. Find the shearing force in tons, and the bending moment in inch-tons at the centre of the beam. Find also the greatest bending moment. F &t the centre = 1 ton. M at the centre = 60 inch-tons. Maximum bending moment =96 ,, 4. In the last question suppose an additional load of 5 tons to be uniformly distributed.. Find the shearing force and bending moment at the centre of the beam. .Fat centre = 1 ton as before. M at centre = 114 foot-tons =135 inch-tons. 5. A beam, AB, 20 feet long, is supported at C and D, two points distant 5 feet from A arid 6 feet from B respectively. A load of 5 tons is placed at each extremity. Find the bending moment at the middle of CD in inch-tons. Moment =330 inch-tons. 6. In the example just given draw the diagrams of shearing force and bending moment at each point of the beam. 42 STATICS OF STRUCTURES. [PARTI. 7. A foundry crane has a horizontal jib, AC, 21 feet long, attached to the top of a crane-post 14 feet high, which turns on pivots at A and B. The crane carries 15 tons, which may be considered as suspended at the extremity of the jib. The jib is supported by a strut attached to a point in it 7 feet from A, and resting on the crane-post at B. Find the stress on crane-post and strut, and the shearing force and bending moment at Any point of the jib. Tension of crane-post =30 tons. Thrust on strut =50 ,, 8. A rectangular block of wood, 20 feet long, floats in water ; it is required to draw the curves of shearing force and bending moment when loaded (1) with 1 cwt. in the middle, (2) with J cwt. at each end, and (3) ^ cwt. placed at two points equidistant from the middle and each end. 9. A beam, AB, 20 feet long, is supported at the ends, and loaded at two points distant 6 feet and 11 feet respectively from one end with weights of 8 tons and 12 tons ; employ the method of superposition to construct the polygons of shearing force and bending moment. Find the maximum bending moment in inch-tons. Maximum moment =972 inch-tons. 10. A beam is supported at the ends and loaded uniformly throughout a part of its length ; show that the diagram of moments for the part of the beam outside the load is ihe same as if the load had been concentrated at the centre of the loaded part, and for the remainder is a parabolic arc. Construct this arc. Also, draw a diagram of shearing force. 11. Abeam is supported at the ends and uniformly loaded, The beam is also sup- ported in the middle by a prop which carries a given fraction of the total load ; employ the method of superposition to draw diagrams of shearing force and bending moment. Find the fraction when the beam is strongest. Ans. Fraction = '586. 12. A beam is supported at the ends and uniformly loaded : if the span be divided into any number of equal parts, and half the weight on each division be concentrated at the dividing points, show that the corners of the polygon of moments lie on the parabola due to the uniform load. SECTION II. FRAMEWORK GIRDERS WITH BOOMS PARALLEL, AND WEB A SINGLE TRIANGULATION. 23. Preliminary Explanations. Hitherto we have only considered beams of small transverse section, but the part of a beam may be played by a framework or other structure under the action of transverse forces. Such a structure, when employed as a beam, is called a Girder, and consists essentially of an upper and a lower member called the Booms of the girder, connected together by a set of diagonally placed bars, called collectively the Web. The web consists sometimes of several triangulations of bars crossing each other, and may even be continuous. In the present section the booms will be supposed straight and parallel, and the web a single triangulation. The action of a load on such a girder furnishes the simplest and best illustration of the nature of the straining actions we have just been considering. Suppose, in the first place, we have a rectangular beam of considerable transverse dimensions, which has one end fixed horizontally, and the other end loaded with a weight W. Now let a part of the length, CD CH. II. ART. 23." STRAINING ACTIONS. 43 (see Fig. 29), be cut away, and replaced by three bars, CD, EF, DE, jointed at their ends to the two parts of the beam CD, EF, forming a rectangle of which DE is a diagonal. With this construction the load If will be sustained, as well as by the original beam, but the three bars will be subject to stresses which we shall now determine. To do this, suppose each of the three bars (in succession) removed, and examine the effect on the structure an artifice which often enables us to see very clearly the nature of the stress on a given part of the structure. In the first place, suppose CD removed ; then the portion EB will turn about the joint E, as shown in the lower part of the diagram, so that the function of the bar CD must be to prevent this turning, which is exactly what we have previously described as bending. The tendency to turn round E that is, the bending moment at E is in this case simply = W+ CB. But if there is a system of loads, the bending moment at E may be found by methods previously described. Now let H= stress on CD. It may readily be seen to be a tensile stress, be- cause, on the removal of the bar, the ends C and D separate from one another. Also, let h = CE or DF, the depth of the beam. The power of CD to prevent EB from turning about E is measured by the moment about E of the force H which acts along it. Therefore , ^ T jtin = J\! E . And dividing the bending moment at E by the depth of the beam, Ave obtain the magnitude of the tension of CD. Next, let the bar EF be removed. The structure will yield by turning round the joint D, the point F approaching E. Thus the bar EF is in compression, and by its thrust, =H' say, towards F, it prevents FB from turning round D. The tendency to turn round D, due to the action of the external forces = M In will be equal to the resisting moment H'h. Therefore, if we divide the bending moment for the joint D opposite to the bar, by the depth of the beam h, we obtain the magnitude of the cornpressive force H'. Lastly, let us suppose the diagonal bar ED to be removed, the effect is quite different from the two former cases ; for instead of the over- 44 STATICS OF STRUCTURES. [PART i. hanging portion of the beam turning about some point, it now gives way by sliding downwards (as shown in the centre of the diagram), remaining horizontal all the time. CD and EF turn about C and E, remaining parallel to one another. The rectangle CDFE becomes distorted by the shortening of the diagonal ED and the lengthening of OF. In the structure then the function of the diagonal bar ED is to prevent the sliding, by resisting the tendency to shorten. Thus the bar ED must be in compression, and by its thrust upon the point D it maintains FB from sliding downwards. Let S = thrust along ED and 6 = angle it makes with the vertical. The force S may be resolved into two components, a horizontal one, S sin 0, and a vertical one, ScosO. It is the vertical component alone which resists the sliding action, and maintains D in its proper position. Now the tendency to slide is no other thing than the shearing force on the structure, which we have previously been investigating. In this example the shearing force is simply W for all sections between A and E. But in other cases of loading the shearing force may be estimated by previously given methods. Since the downward tend- ency of the shearing force is balanced by the upward thrust of the vertical component of S, we have in all cases Instead of the points E and D being joined there might have been a bar CF, which, by the resistance to lengthening which it would offer, would have sustained the portion FB from sliding downwards. Such a bar would be in tension just as the bar ED is in compression, and in finding the stress on it we should use exactly the same equation. Now, instead of having three bars only, the whole structure may be built up of horizontal and diagonal bars. The same principles will apply. On removing any one of the horizontal bars, we see that the structure yields by turning round a joint opposite : so we say the function of the horizontal bars is to resist bending. This is expressed by the equation Hh = M. On the other hand, the function of the diagonal bars being to resist the shearing tendency, we have always S cos = F. 24. Warren Girders under various Loads. Fig. 30 shows a Warren Girder, so called from the name of the inventor, Captain Warren, a type of girder much used for bridges since its first introduction about the year 1850. It consists of a pair of straight parallel booms connected together by a triangulation of bars inclined to each other, generally at 60, so that the triangles formed are equilateral. The booms in the actual structure are generally continuous through the junctions with the diagonal bars, but, if well constructed, there is no sensible error in CH. 11. ART. 24.] STRAINING ACTIONS. 45 regarding the structure as a true frame, in which the several divisions are all united by perfectly smooth joints. Any three bars forming a parallelogram and its diagonal may be considered as playing the same part as regards the rest of the structure as in the case just considered. Fig.30. When a Warren girder is used, it is generally supported at the ends, and the loads are applied at one or more joints in the lower boom. We will examine some examples. (1) Suppose there is a single load applied at a joint in the centre of the span. First as to the diagonal bars. It was shown above that the duty of these bars was to prevent the structure yielding under the action of the shearing force ; the vertical component of the stress on either of the diagonal bars being equal to the shearing force for the interval of the length of the girder within which the diagonal bar lies. This is expressed by the equation Now, in the example which we are considering with the load in the centre, the shearing force will be the same at all sections to the right and left, namely, =-^W. Therefore the stress on all the diagonal bars is of the same magnitude, W = W_ 2 cos 30 ^3' If we consider the effect of removing either of the bars we shall find that commencing from one end they prevent alternately the shortening and lengthening of the diagonals which they join, so that, commencing with one end, the bars are alternately in compression and tension. The compression bars are shown in double lines. Next, as to the several portions of the length of the top and bottom booms. As was shown above, the stress on any division of the horizontal bars has the effect of preventing a bending round the joint opposite : so that the moment of the stress about the joint is equal to the bending moment at the joint, due to the external forces. This is expressed by the equation Hh = M. Let a = length of a division. 46 STATICS OF STRUCTURES. [PART i. Then, since the supporting force at the joint is \W, the bending moment at the joints numbered 1, 2, 3, etc., are W a Wa ^i = Y2 = -f W Wa M -2 = ^ a = -p W3 3Wa M *= 2 9^ = 4 ' and so on, the bending moments increasing in arithmetical progression. Since the depth of the girder h is the same at all parts of the length ._ if we divide the M'a each of them by h, we obtain the magnitude of the stress on the bars opposite the respective joints. Thus Wa , Wa . ZWa H, 2 = ^,H^^H^ - u -,and so on. We see, then, that the stress on the several divisions increases in arithmetical progression as we proceed from the ends towards the centre. By observing the effect of removing either of the bars, we see that all the divisions of the upper boom are in compression. This is expressed by drawing them with double lines in the figure.. All the divisions of the lower boom are in tension. (2) Next suppose the load is applied at some other joint not in the centre the joint 4 for example. We must first calculate the supporting forces. Suppose they are P at and Q at 12. For the portion of the girder to the left of 4 the shearing force will be the same at all sections and be equal to P. So the stress on all the diagonals between and 4 will be equal to Psec30 c . To the right of joint 4 the shearing force = Q, and the stress on all the diagonal bars from 4 to 12 will be Q sec 30. Proceeding from either end towards the joint where the load is applied, we observe that the diagonal bars are alternately in com- pression and tension so that the bar 56 is now in compression, whilst the bar 54 is in tension. On these bars the nature of the stresses is just opposite to that to which they were exposed when the load was at the centre joint. Thus by varying the position of the load, we not only vary the magnitude of the stress, but we may in some cases change the character of the stress, requiring a diagonal bar to act sometimes as a strut and sometimes as a tie. For the divisions of the horizontal booms on the left of W the stresses are Pa 2Pa 3Pa CH. ii. ART. 24.] STRAINING ACTIONS. 47 in arithmetical progression up to the bar opposite the joint to which the load is applied ; and to the right of W, Qa 2Qa 3Qa M W ~2P in arithmetical progression also up to the bar opposite the load. The upper bars are all in compression and the lower in tension as before. When there are a number of loads placed arbitrarily at the different joints, the simplest way of determining the stresses is often to find the stress on the bars due to each load taken separately, and then apply the principle of superposition. In applying the principle due regard must be paid to the nature of the stress. A compressive stress must be considered as being of opposite sign to a tens-ile stress, and, in compounding, the algebraical sum of the stresses for each load will be the total stress on the bars. (3) There is one particular case, that in which the girder is uniformly loaded, which it is advisable to examine separately. In general, the load on the platform of the bridge is by means of transverse beams or girders transferred to 'the joints of the lower boom. The transverse beams may be the same in number as the joints in the lower boom. In that case the girder will be loaded with equal weights at all the bottom joints. If the transverse beams are more numerous their ends will rest on the bottom booms, and tend to produce a local bending action in each division, in addition to the tensile stress which, as the bottom member of the girder, it will have to bear. In some cases, to lessen or get rid of this bending action, vertical suspending rods are introduced, by which means the middle points of the lower divisions are supported, and the loads transmitted to the upper joints of the girder. In such a case we may take all the joints both in the upper and lower booms to be uniformly loaded. We will, however, suppose equal weights applied to the joints of the lower boom only. First as to the shearing forces. Between the end and the first weight the shearing force = the supporting force = half the total load = P say. In the next division the shearing force is less by the amount of the load at the first lower joint = P- W, In the third division of the lower boom from the end the shearing force = P - 2 W, and so on. The stresses on the diagonals can now be found by multiplying the shearing force in the division within which any one diagonal lies by the secant of the angle which the diagonal makes with the vertical. The stresses will diminish in arithmetical progression as we pass inwards from the ends towards the centre. It will be observed that on the first and second diagonals from the end the stress is of the same magnitude. On the third and fourth 48 STATICS OF STRUCTURES. [PART i. it is alike also, and so on. The stresses are alternately compression and tension, commencing with compression on the first bar. To find the stresses on the booms we must determine the bending moments at all the joints. Division of the M's by A, the depth of the girder, will give the several horizontal stresses. They will be found to increase as we pass from the ends towards the centre. 25. N Trusses. The web of the girder, instead of consisting of bars sloping both ways, forming a series of equilateral triangles, may be constructed of bars placed alternately vertical and sloping at an angle, so forming a series of right-angled triangles, looking like a succession of capital letters N. (See Fig. 31.) For this reason it is sometimes called an N girder. The ordinary practice is to divide the girder into a number of squares by means of the vertical bars, so that the diagonals slope at an angle of 45. It is advantageous to place the Flg.31. diagonals so as to be in tension For a load in the centre, or a uniformly distributed load, they should slope upwards from the centre towards the ends. The vertical bars will then be in compression. A short bar is better able to resist compression than a long one, whereas a tension bar is of the same strength whether short or long ; so it is manifestly economical of material, and a saving of weight, to place the long bars, that is the sloping bars, so as to be in tension. The same methods will apply to find the stresses on the bars, since, as before, the web resists the shearing action, and the booms the bending. The simple queen truss, considered in Chapter I., Section II, is another example of a web consisting of alternate vertical and diagonal bars, but the diagonal is not usually inclined at 45 to the vertical. H. II. ART. 26.] STRAINING ACTIONS. EXAMPLES. 1. A trapezoidal truss is 24 feet span and 3 feet deep. The central part is 8 feet long and is braced by a diagonal stay so placed as to be in tension. Find the stress on each part when loaded with 4 tons at one joint and 5 tons at the other. Stress on diagonal stay= '95 ton. 2. A bridge is constructed of a pair of Warren girders, with the platform resting on the lower booms, each of which is in 6 divisions. The bridge is loaded with 20 tons in the middle. Find the stress on each part. 3. In example 2 obtain the result when the load is supported at either of the other joints. 4. From the results of examples 2 and 3 deduce the stress on each part of the girder when the bridge is loaded with 60 tons, divided equally between the three pairs of joints from one end to the centre. Results for questions 2, 3, 4, the bars being numbered 'as in Fig. 30. Stress 01 i Boom. Stress on Diagonals. Bars. Load at 6. at 4. at 2. at 6, 4, and 2. Bars. Load at 6. at 4. at 2. at 6, 4, and 2. 02 2-88 3'85 4-8 11-53 01 -5-76 -77 -9-6 -23-06 13 -576 -77 -9-6 -23-06 12 5-76 7-7 9-6 23-06 24 8-64 11-55 8-64 28-83 23 -576 -7-7 1-92 -11-54 35 -11-52 -15-36 -7-68 -34-56 34 5-76 77 -1-92 11-54 46 14-4 13-44 6-72 34-56 45 -576 3-85 1-92 57 -17-28 -11-52 -5-76 -34-56 56 5-76 -3-85 -1-92 68 14-4 9-6 4-8 28-8 67 576 3-85 1-92 11-54 79 -11-52 -7-68 -3-84 -23-04 78 -576 -3-85 -1-92 -11-54 8,10 8-64 5-76 2-88 17-28 89 576 3-85 1-92 11-54 9,11 -5-76 -3-84 -1-92 -11-52 9, 10 -5-76 -3-85 -1-92 -11-54 10,12 2-88 1-92 96 5-66 10, 11 5-76 3-85 1'92 11-54 11, 12 -5-76 -3-85 -1-92 -11-54 5. A bridge of 80 feet span is constructed of a pair of N girders in 10 divisions, the plat- form resting on cross girders supported by the lower booms, and the diagonals so arranged as to be all in tension. A load of 80 tons is tiniformly distributed over the platform. Find the stress on each bar. Draw polygons of shearing force and bending moment. SECTION III. GIRDERS WITH REDUNDANT BARS. 26. Preliminary Explanations. Again, returning to the (p. 43) beam out of which a portion has been cut and replaced by bars, let us sup- pose that instead of one diagonal bar only, there are two. We require to find the stresses on the bars. First, on the diagonal bars. In this case also the stress on these bars will be due to the shearing force. Together they prevent the structure yielding under the shearing action, but the amount each one bears is indeterminate until we know how the diagonals are constructed and attached to the rest of the structure. Suppose, for example, the diagonals are simple struts placed across the corners of the rectangle, but not secured at the ends. The struts will be incapable of taking tension ; and the diagonal ED, which slopes in the direction, to be subject to compression will have to bear the whole shearing force. The other diagonal is ineffective. Secondly, suppose C.M. D 50 STATICS OF STRUCTURES. [PART i. the diagonals to be simple ties, such as a chain or slender rod, and so incapable of withstanding compression. Then the bar CF will carry the whole shearing force. We may have any number of intermediate cases between these extremes according to the material of the diagonals and the method of attachment. In all cases one diagonal tends to lengthen, and the other to shorten, and according to their powers of resistance to these tendencies they offer resistance to the shearing. If 5 1 and $ 2 be stresses on the two bars, then in all cases If the diagonals are exactly similar rigid pieces similarly secured at the ends, equal changes of length will produce the same stress whether in compression or tension, so that each will bear an equal share of the shearing force. We shall then have S^S^^FsecO. The foregoing is one of the simplest examples of a frame with redundant bars, and shows clearly why, in such cases, the stress on each bar cannot be determined by statical considerations alone, but depends upon the materials and mode of construction. In structures such as those considered in Chapter I., Section II., in which the principal part is an incomplete frame, stiffened by bracing or other means to provide against variations of the load, the bracing is usually redundant, and the stress on it cannot be calculated with certainty. Allowance has to be made for this in designing the structure by the use of a larger factor of safety. Redundant material is often no addition at all to the strength of the structure, and may even be a source of weakness, as will appear hereafter. When framework girders were first introduced, it was objected by eminent engineers that failure of a single part would destroy the structure. Experience appears to have shown that risks of this kind are not serious, and the tendency of modern engineering design appears to be rather towards the employment of structures with as few parts as possible. Next, as to the horizontal bars. These still sustain the bending moment, but not precisely in the same way as when there is only one diagonal. To find the magnitude of the forces we employ a method similar to that used before, but instead of removing a bar we suppose the girder cut through one or more bars at any place convenient to our purpose ; then the principle which we make use of is, that the action of each of the two halves on the other must be in equilibrium with the external forces which are applied to either half. In Fig. 32 let us take a vertical section through the point of intersection of the CH. II. ART. 27.] STEAINING ACTIONS. 51 diagonals, four bars are cut by the section, and through the medium of these four bars the structure to the left will act on the portion of the structure to the right of the section, and sustain it against the action of the external loads which rest on it. First, there is the force H^ pulling at K, and the force H 2 thrusting at L, and at there are the two forces S l and S 2 on the two diagonals. Now, if we Fig.32. consider the tendency for the external forces to bend the right hand portion round 0, we see that the diagonal bars offer no resistance to this bending action, and must so far be left out of account. The whole resistance to m E" !L bending is due to the bars CD and EF along which the forces H l and H 2 act, so that if M be the bending moment at due to the external forces, c H '!K This will be true whatever be the proportion between S 1 and S 2 , and H^ and H 2 . Instead, therefore, of taking the bending moment about a joint, as we did previously, we have in this case to take the moment about the point where the two diagonals cross. But besides the balancing of the bending moment, there are other conditions to which the forces are subject, in order that the right-hand portion may be in equilibrium. One is, that all the forces which act on this portion must balance horizontally. There are no external forces which have any horizontal action, so that it is only the four internal forces which act along the bars cut, of which we have to take any account, and these must, on the whole, have no resultant hori- zontal action. The two thrusts must equal the two pulls ; that is, This also is true whatever be the distribution of the shearing force between the two diagonals. If, now, we suppose S 2 = ti v then R^ = H^ = H, say. And the above formula becomes Hh = M , the same as we had before ; but it must be applied* a little differently, the moment now being taken about the point of intersection of the diagonals. If S l is not equal $ 2 , then H will be the mean of H l and H.,. 27. Lattice Girders, Flanged Beams. Constructions with a double set of diagonals are common in practice. If, for example, in the N girder 52 STATICS OF STRUCTURES. [PART i. (Fig 31) we place in each division two diagonals instead of one only, the construction is called a lattice or trellis girder. When employed for heavy loads, the diagonals are generally inclined at an angle of 45 to the vertical. In light structures, or when used for giving stiffness, they are often inclined at a much greater angle. To determine the stresses, it will be necessary to make an assump- tion for the distribution of the shearing force between the two diagonals for each division of the girder, and it will generally be sufficiently correct to suppose each to carry half, and to write =^sec 0, and Hh = M for the points where the diagonals intersect. In lattice girders we more frequently find the double set of sloping bars introduced, but the vertical bars omitted. In this case it will not be true that the two diagonals in any one division are exposed to the same stress. We can determine the stresses otherwise. The structure may be divided into two elementary girders, each with its own system of diagonal bracing, and each with its own set of loads. Suppose, for simplicity, the number of divisions in the complete girder even, and each half girder loaded with equal weights applied to all the lower joints. Then if we make the simple, and in most cases safe, assumption that the thrusts on the two end vertical bars are equal, the forces on Fig.SS. all the bars of the structure will be determinate. In the example shown in Fig. 33 the thrusts on the vertical end bars will be 2/ J . After we have calculated the stresses on each bar in each elementary girder, then, for any bar which is a portion of both, we must compound to obtain the total stress. We may further increase the number of diagonal bars and obtain a girder, the web of which is a network of bars. In this case it will not be exactly, but will be very nearly, true that the horizontal bars take the bending, and the sloping bars the shearing action, the shearing force being regarded as equally distributed between all the cfiagonals cut by any one vertical section. We may go on adding diagonal bracing bars until the space between the booms is practically filled up, and even then assume that the bending is taken by the horizontal bars and the shearing by the web. The numerous bracing bars may then be replaced by a vertical plate, CH. ii. ART. 27.] STRAINING ACTIONS. 53 which will form a continuous web to the girder. Such a construction is a very common one in practice, the horizontal members are called the top and bottom flanges of what is still a girder, and often called so, but more often a flanged or I beam. In the smallest class of these beams, they are rolled or cast in one piece ; but for large spans they are built up of plates and angle irons riveted together. For figures showing the transverse sections of such beams see Part IV. In taking the depth of such a girder, to make use of in the equation Hh = M, we ought to measure the vertical distance between the centres of gravity of the parts which we consider to be the flanges of the beam or girder. In the simple rolled or cast beam this will be the distance from centre to centre of depth of flanges. In the built-up beam account must be taken of the effect of the angle irons. It must be remembered that this method of determining the strength of an I beam is only approximate. Its strength will be determined in a more exact way hereafter, when it will be found that the web itself assists in resisting the bending moment, but, area for area, to the extent only of about one-third that borne by the flange. On the other hand, the effective depth is less than the distance from centre to centre of the flanges. In rough preliminary calculations we may often neglect this, and employ the same formula as for lattice girders. Girders are often of variable depths, so that the booms are not parallel ; when this is the case the booms assist in resisting the shearing action of the load, as will be seen hereafter. EXAMPLES. 1. A beam of I section is 24 feet span and 16 inches deep ; the weight of the beam is 1,380 Ibs. It is loaded in the centre with 5 tons. Assuming the resistance to bending to be wholly due to the flanges, find the maximum total stress on each flange and the sectional area of each the resistance to compression being taken to be 3 tons and to tension 4 tons per square inch. Maximum total stress =53,505 Ibs. =23*88 tons. Sectional area of upper flange = 8 square in. ,, ,, bottom ,, =6 ,, 2. A trellis girder, 24 feet span and 3 feet deep, in three divisions, separated by vertical bars, with two diagonals in each division, is supported at the ends and loaded (1) with 20 tons symmetrically distributed over the middle division of the top flange, (2) with 20 tons placed over one of the vertical bars. Find the stress on each part of the girder, assuming each diagonal to carry half the corresponding shearing force. Stress on diagonals Case 1. 14 '2 14 '2 Case 2. 18f 9 9 Remark. These results show the unsuitability of this construction for carrying a heavy load on account of the great inclination of the diagonals to the vertical. 3. A water tank, 20 feet square and 6 feet deep, is wholly supported on four beams, each carrying an equal share of the load. The beams are ordinary flanged ones, 2 feet 54 STATICS OF STRUCTURES. [PART i.] deep. Find approximately the maximum stress on each flange, assuming that the weight of the tank is one-fourth the weight of water it contains. Distributed load on one beam =^^-=46, 875 Ibs. Ibs. =261 tons. 4. The Conway tubular bridge is 412 feet span. Each tube is 25 feet deep outside and 21 inside. The weight of tube is 1,150 tons, and the rolling load is estimated at f ton per foot-run. Find approximately the sectional areas of the upper and lower parts of the tube, the stress per square inch being limited to 4 tons. ^max =3,267 tons. Area =817 square in. 5. In the girder shown in Fig. 33, p. 52, suppose the weights P and Q are each 1 ton. Find the stress on each member. If the girder be stiffened by the addition of vertical members at each joint of the beams ; find the stress on each member, making the usual assumption. 6. A rectangular tank with vertical sides and flat bottom is filled with water to a depth of 15 feet. The sides of the tank are constructed of iron plates riveted together and stiffened by vertical J_ irons spaced 4 feet apart. Assuming these stiffening pieces to take the whole bending action due to the water pressure : find the maximum bending moment on one of the stiffening pieces. REFERENCES. For details of construction of girders the reader is referred to Girder Making . . . in Wrought Iron. E. HUTCHINSON. Spon, 1879. CHAPTER III. STRAINING ACTIONS DUE TO ANY VERTICAL LOAD. 28. Pelimiruwy Remarks. The preliminary discussion in the pre- ceding chapter of the straining actions to which loaded beams and framework girders are subject will have given some idea of the import- ance of the effect of shearing and bending on structures, and we shall now go on to consider the question somewhat more generally. Let us suppose any body or structure possessing, as it usually will, a longitudinal vertical plane of symmetry, to be acted on by a set of parallel forces in equilibrium symmetrically disposed with respect to this plane, as, for example, gravity combined with suitable vertical supporting forces. Then these forces will be equivalent to a set of parallel forces in the plane of symmetry in question. Let the structure now be divided into two parts, A arid B, by an ideal plane section, parallel to the forces and perpendicular to their plane. Then the forces acting on A may be reduced to a single force F lying very near the section considered and a couple M, while the forces acting on B may be reduced to an equal and opposite force F lying very near the section and an equal and opposite couple M. The pair of forces are the elements of the shearing action on the section, and the pair of couples are the elements of the bending action on the section. As the nature of the structure is immaterial, we may consider these straining actions for a given vertical section quite independently of any particular struc- ture, and describe them as the Shearing Force and Bending Moment (hie to the given Vertical Load. We shall first consider the connection which exists between the two kinds of straining action and the method of determining them for any possible load. CONNECTION BETWEEN SHEARING AND BENDING. 29. Relation between the Shearing Force and the Bending Moment. Figure 34 shows the lines of action of weights W^ W.^ etc., placed at the successive intervals .,, a.,, etc. 56 STATICS OF STRUCTURES. [PART u w. vv In the first division the shearing force is in the second F 2 = W^ + W^ = F l + W^ , in the third F^W^ + W^+W^F^+W^, and so on for all the divisions, so that in the n ih division We express this in words by saying that the difference between the shearing forces on two consecutive intervals is equal to the load applied at the point between the two intervals; or it may be written By setting down ordinates to a horizontal base line we obtain the stepped figure as the graphical representation of the shearing force at any point of the beam. It is drawn by first setting downwards at 1 an ordinate for the shearing force on the first interval, and then passing along the beam to the other end, on meeting the lines of action of the successive weights the length of the ordinates is increased by the amount of the weights. In so doing we make use of the proposition which has just been proved. This is called the Polygon of Shearing Force, or more generally, when the loads are continuous, the Curve of Shearing Foixe. Next as to the bending moment. At the first point where W l is. applied M l = 0, at the second point M 2 = W^ F^ ; CH. in. ART. 29.J STRAINING ACTIONS. 57 at the third point M 3 = W^ + a. 2 ) + W^ = W^ + ( W^ + W^a z at the fourth point 7If 4 - ^( a i + % + %) + ^ 2 ( a 2 + a s) + and generally, M M - !/_! = /' T , l _ 1 (X w _ 1 . We may express this in words by saying that the difference between the bending moments at the two ends of an interval is equal to the shearing force y multiplied by the length of the interval. Or the result may be written We will now take a numerical example and see how we may make use of this property to determine a series of bending moments. Let AB be a beam fixed at one end, and loaded with weights of 2, 3, 5, 11, 13, 7 tons, placed at intervals of 3, 2, 3, 5, 4, 6 feet,. w. F. a. Fa. M. 2 2 3 6 3 6 5 2 10 5 16 10 3 30 11 46 21 5 105 13 151 34 _| 136 7 287 41 6 246 533 commencing from the free end. We adopt a tabular method of carrying out the work of calculation. First set down a column of weights applied, as shown by the figures in the column headed W. In the next column write the shearing forces. Since the shearing forces are uniform over the intervals between the weights, it will be best to write the F's opposite the spaces between the weights. Any F is found by adding to the F' above it the adjacent W. In the third column we set down the lengths of the intervals. Then multiplying the F's and corresponding a's together, set the results in column 4. Lastly, we can write down the column of bending moments by the repeated addition of the Fa's the bending moment at any point being found by adding to the bending, moment at the point above the value of Fa between the points. 58 STATICS OF STRUCTURES. [PART i. If instead of all the forces acting one way some of them act upwards, a minus sign should be set opposite, and all the operations performed algebraically. The method is equally applicable however the beam is supported. For example, let a beam 23 feet long be supported at the ends and loaded with 3, 2, 7, 8, 9 tons, placed at intervals of 2, 2, 3, 4, 5, 7 feet, reckoning from one end. First calculate one supporting force, say at the left-hand end by w. F. a. Fa. M. 16-17 16-17 2 32-34 -3 32-34 13-17 2 26-34 -2 58-68 11-17 3 33-51 7 92-19 417 4 16-68 -8 108-87 -3-83 5 -19-15 -9 89-72 - 12-83 7 -89-81 12-83 taking moments about the other end. In the column of Ws set this for the first force, and since all the loads act in the contrary direction, put negative signs opposite them, and in writing down the next column of IB add algebraically. We shall at the bottom of the column deter- mine the supporting force at the right-hand end. At the bottom of the column of M's, that is, at the point where the right-hand supporting force acts, we ought to get a zero moment. The obtaining of this will be a test of the accuracy of the work. In this example the small difference between 89-72 and 89 '81 is due to our having taken the supporting force only to two places of decimals. Observation of the process of calculation leads us to a very important proposition, viz., where the shearing force changes sign, the bending moment is at that point a maximum. This will be true for all important practical cases, but exceptional cases may be imagined in which, where the shearing force changes sign, the bending moment is a minimum. Since &M=Fa, then, so long as F is positive, M will be an increasing quantity as we pass from point to point. But where F changes to negative there M commences to diminish. We will now explain the construction of a diagram of bending moment for a system of loads : and first let us consider how the i + # 2 ) + W% a 2- ^ that a t this Pi n t also the bending moment is represented by the area of the polygon of shearing force, reckoned from the end up to the point 3. And so on for every point. This important deduction may be stated generally thus : The ordinate of the curve of bending moment at any paint is proportional to the area of the curve of shearing force reckoned from one end of the beam up to that point. 30. Application to the case of a Loaded Beam. We will next take the case of a beam supported at the two ends. PA i Fig - 36 - First, calculate the supporting force P, set it up at the end of the base line as an ordinate, and draw the stepped polygon by continually subtracting the W'a. At some point in the beam we shall cross the base line. At that point the shearing force changes sign, and there the bending moment is a maximum. The shearing force on the last interval will give the magnitude of the supporting force Q. The polygon thus drawn will be the polygon of shearing force. -CH. in. ART. 31.] STRAINING ACTIONS. 61 The polygon of bending moment may be drawn without previously determining the supporting force at either end thus : Commencing at (Fig. 36), the point of application of P, draw any sloping line 012' cutting W^ in 1, and W. 2 in 2'. Then set up 2' 2 to represent W^ join 1 2, produce it to cut W z in 3'. 3' 3 tf> 2 , 23, JF 4 in4'. 4' 4 ^3% , 3 4, ,, W b in 5', and so on. 7' 7 will represent 9^ . Now join 7 with the point 0, where 012' cuts the line of action of P. This is called the Closing Line of the polygon of moments. Any vertical intercept of this polygon will represent the bending moment at the corresponding point of the beam. The proof of this may be stated shortly thus : If we produce 1 to meet the line of action of Q in L, then LI will, from what has been said before, represent the sum of the moments of all the weights W about the end of the beam where Q acts. And from the conditions of equilibrium this must equal the moment of P about that end. Accordingly, if we take any point K t the vertical intercept M T below it will represent the moment of P about K. This is an upward moment. The four weights which lie to the left of K will together have a downward moment about K represented by MN. Therefore, the difference NT will represent the actual bending moment at the point K. It sometimes happens that we want the moment of the forces not about K, the section which separates the two parts of the structure, but about some other point, say JT, in the figure. We can obtain this moment also with equal facility ; for if we prolong the line 4 5 of the polygon to meet the vertical through X in the point S, we find, reasoning in the same way, that SZ, the intercept between the side so prolonged and the closing line, is the moment required. Polygons of moments and shearing forces may also be constructed by making use of the fundamental relations shown above to exist between them and the load, as will be seen presently, while a third purely graphical method is explained farther on, based on a most important property which they possess. 31. Application to the case of a Vessel floating in the Water. We some- times meet with cases in which the beam or structure is loaded not at intervals, but continuously, the distribution of the load not being uniform, but varied in some given way. In such a case the diagrams of shearing force and bending moment become continuous curves. The most convenient way of expressing how the load is distributed is by STATICS OF STEUCTURES. [PART Fig.36a. means of a curve, the ordinate of which at any point represents the intensity of the load at that point. Such a curve is called a curve of loads. It may be regarded as the profile of the upper surface of a mass of earth or other material resting on the beam. We will consider, first, the case of a beam fixed at one end and loaded continuously throughout, in a manner expressed by a curve of loads LL (Fig. 36a). The total area enclosed by the curve of loads will represent the total load on the beam, and between the two ordinates of any two points will be the load on the beam between the two points. Now, the area of the curve of loads, reckoned from the end A up to any point, K say, since it represents the total load to the left of K, will be the shearing force at K. If at K we erect an ordinate KF, to represent on some convenient scale the area ALK, and do this for many points of the beam, we shall obtain a second curve FF, the curve of shearing force. Having done this, we may repeat the process on the curve FF, and obtain the curve of bending moment. For we have previously proved that if the load on the beam is concentrated at given points, then the ordinate of the curve , of bending moments is at any point proportional to the area enclosed by the curve of shearing force for the portion of the beam between the end and that point. The truth of this is not affected by supposing the points of application of the load to be indefinitely close to one another, in which case the load becomes continuous. Accordingly, if we set up at K an ordinate, KM, to represent on some convenient scale the area AFK of the shearing force curve, and repeat this for many points, we obtain the curve of bending moment, MM. Thus the three curves form a series, each being the graphical integral of the one preceding. This process has an important application in the determination of the bending moment to which a ship is subjected on account of the unequal distribution of her weight and buoyancy along the length of the ship. On the whole, the upward pressure of the water, called the buoyancy, must be equal to the downward weight of the ship ; and the lines of action of these two equal and opposite forces must be in the same vertical. But for any portion of the length, the upward pressure and the downward weight will not, in general, balance one another ; so, on account of the difference, shearing and bending of the ship will be induced. In the case of a rectangular block of wood floating in water, CH. III. ART. 31.] STRAINING ACTIONS. the upward pressure of the water will, for every portion of its length,, equal the downward weight, and there will be no shearing and bending action on it. But in actual ships, the disposition of weight and buoyancy is not so simple. Taking any small portion of the length of the ship, the difference between the weight of that portion of the ship and the weight of the water displaced by that portion of the ship, will be a force which acts on the vessel sometimes upwards and sometimes downwards, according to which is the greater, just in the same way as forces act on a loaded beam producing shearing and bending. In the construction of the vessel, strength must be provided to resist these straining actions, and it is a matter of great practical importance to determine accurately the magnitude of them for all points of the length of the ship. We will select an example of very frequent occurrence, that in which at the ends of the ship the weight exceeds the buoyancy, whilst at the centre the buoyancy exceeds the weight. If the ship were very bluff ended, and carried a cargo of very heavy material in the centre hold, the distribution of weight and buoyancy would probably be the reverse of this. Fig.37. w In the example the ship is supposed to be divided into any number of equal parts, and the weight of water displaced by each of those 64 STATICS OF STRUCTURES. [PART i. parts determined ; ordinates are set up to represent those weights, -and so, what is called a curve of buoyancy, BBB (Fig. 37), is drawn. The whole area enclosed by the curve will represent the total buoyancy or displacement of the vessel, and is the same thing as the total weight of the vessel. Next we suppose that the weights of the different portions of the ship are estimated, and ordinates set up to represent these weights, then what is called a curve of weight, WWW, is obtained. In the figure it is set up from the same base line. The total area enclosed by this curve will also be the total weight of the ship, and must therefore equal the area enclosed by the curve of buoyancy. Thus the sum of the two areas marked 1 and 2 must equal the area marked 3. Not only must this be true, but also the centres of gravity must lie on the same ordinate. The difference at any point between the ordinates of the two curves will express by how much at the ends the weight exceeds the buoyancy, and in the middle portion by how much the buoyancy exceeds the weight, representing, in the first case, the intensity of the downward force, and, in the second, the intensity of the upward force. Wfyere the curves cross one another and the ordinates are the same height, as at K l ajid K, 2 , the sections are said to be water-borne. If now we set off from the base line ordinates equal to the difference between the ordinates of the two curves BBB and WWW^ we obtain the curve of loads LLL ; some portions where the weight is in excess will lie below the base line, and the rest, where the buoyancy exceeds the weight, will lie above the base line. From what has been said before, the area above the base line must equal the area below. Having obtained the curve of loads, the curve of shearing force is to be obtained from it in the manner previously described, by setting up, at any point, an ordinate to represent the area of the curve LLL between the end of the ship and that point. In performing the operation, due regard must be paid to the fact that the loads on different parts of the ship act in different directions, and for one direction they must be treated as negative, and the corresponding area of the curve as a negative area. Having thus determined the curve of shearing force FFF, the same operation must be repeated on that curve to determine the curve of bending moment. In drawing the curve of shearing force it will be found that at the further end of the ship we return again to the base line from which we started at first, for the shearing force at the end must be zero. Also the bending moment at the end must be zero. This gives us tests of the accuracy of our work. In this example the bending is wholly in one direction, tending to make the ends of the ship droop or the ship to " hog " in the technical CH. in. ART. 33.] STRAINING ACTIONS. 67 beam of span /, supported at the ends, and suppose a single concentrated load W to travel across it in the direction of the arrow. Let us consider any point K (Fig. 38) in the beam, distant a and b from the ends. As the load traverses the beam, each position of the load will produce a certain shearing force and bending moment at the point K. To find their greatest value let x = distance of W from A, then the supporting force at ~B = P = W-v So long as the weight lies between A and K the shearing force at K will be simply P. consequently the shearing force will increase as x increases, until the load reaches the point K. So long as the weight lies to the left of K, the tendency will be for the portion KB to slide upwards relatively to the portion AK. This we describe in accordance with our definition on p. 34 as a negative shearing force. Therefore, putting x = a, Max. negative shearing force at K= W-,- Now, supposing the weight to move onward, it will in the next instant have passed to the other side of K, and the shearing force will have undergone a sudden change. It will now be equal to the supporting force at the end A, ~ ^b But not only is the magnitude of the shearing force suddenly changed, but the tendency to slide is now in the other direction, and the shearing force is positive. As the weight moves further to the right of K the shearing force diminishes, thus Max. positive shearing force at K= W-j. Wherever we take the point K it will always be true that the maximum positive shearing force will occur when the weight lies immediately to the right of K, and the maximum negative when the weight lies immediately to the left. The maximum negative shearing force for every point in the beam may be represented by the ordinates of a sloping line AB 1 below the beam, the length BB' being taken to represent W. And similarly the maximum positive shearing force at any point by the ordinates of the sloping line A'B about AA' also being taken to represent W. Next as to the bending moment. When the weight lies to the left of K, and is at a distance from A equal to x, the bending moment at K -is given by Pb=W-x. 68 STATICS OF STRUCTURES. [PART i. This goes on increasing as x increases until the weight reaches the point K. After having passed K the bending moment at K must be differently expressed, being then which becomes smaller as x increases; so that the greatest bending moment at K occurs when the load is immediately over K, and then the Max. bending Moment at K= p- If the point K is taken in the centre of the beam, Max. Moment at centre = \Wl as before. If ordinates be set up at all points to represent the maximum bending moments at these points, a parabola (AGE) will be obtained. For the expression for the maximum bending moment is just twice that previously obtained for the same weight distributed uniformly. If there are more weights, W^, W^ etc., on the beam, and W l lie to the right of K, the shearing force at K= P - W l , where P is the right- hand supporting force. Now, suppose we shift W l to the left of K, we shall diminish the supporting force to P' say, and this will be the new shearing force at K. The difference between P and P' will be less than W^, and the shearing force will be increased by passing W 1 to the left of K, If we were to remove W^ altogether the diminution of P will be less than the whole of JF lt and so the shearing force at ^Twill be increased by so doing. We obtain the greatest positive shearing force at K when all the weights are to the right of K, but as near to K as possible. The greatest negative shearing force will occur when all the weights lie to the left of K, as near to K as possible. The maximum bending moment at K will occur when the weights are as near K as possible, whether to the right or left. Any addition to the load, on whichever side of K it is placed, will cause an addition to the bending moment. There is another important case, that in which we have a continuous load of uniform intensity passing over the beam, as when a long train passes on to a bridge. We observe that as the train, coming from A, approaches K, the supporting force at B, and therefore the shearing force at K, increases. When any portion of the weight lies to the right of K, the supporting force will be increased by a part of the weight lying to the right of K ; but when we have subtracted the whole of that weight, the difference, which will be the shearing force at K, will be less than before ; thus the maximum negative shearing force at K will occur when the portion AK is fully loaded, and no CH. in. ART. 34] STRAINING ACTIONS. 69 part of the load is on KB. To find its value we have only to determine the supporting force at B, by taking moments about A ; then that is, the magnitude is proportional to the square of the distance of the point from the end A. .It will be graphically represented by the ordinates of a parabola which has its vertex at A^ and axis vertical, cutting the vertical through B in a point B' such that BB' = ^wl, that is, half the weight on the beam when fully loaded. As the load travels onward the shearing force diminishes at last to zero, and then changes sign, becoming positive, the numerical magnitude increasing as the rear of the load approaches K. The maximum positive shearing force will occur when the portion KB only is loaded. The ordinates of a para- bola set below the line of the beam having its vertex at B and axis vertical, will represent the maximum negative shearing force. The question of maximum bending moment is more simple. It will occur at any point when the beam is fully loaded ; for at any point the bending moment is the sum of the bending moments due to all the small portions into which the load may be divided, and the removal of any one of them will cause a diminution of bending action throughout the whole length of the beam. A parabola, with its highest ordinate at the centre = %wl 2 , will represent it at any point. 34. Counter-bracing of Girders. In the design of a framework girder it is very important to take account of the maximum positive and negative shearing forces due to a travelling load. In such a structure the shearing force is resisted by the diagonal bars, and in general these bars are so placed as to be in tension, for the bar may then be made lighter than if subject to a compressive force of the same amount. Suppose the diagonal bars so arranged as to be all in tension when the girder is fully loaded, or when there is only the dead weight of the girder itself to be taken account of. There may be ample provision made for withstanding the tensile forces, and yet it will be important to examine if there may not be some disposition of the travelling load which would cause a thrust on some of the diagonals. If so, the maximum amount of this must be calculated, and the structure made capable of withstanding it. If the shearing force at any section of the girder is what we have called a positive shearing force, that in which the left-hand portion tends to slide upwards relatively to the right, then, in order that it may be withstood by the tension of a diagonal ^bar, the bar must slope upwards to the left. If the bar so slopes, and by the movement of the travelling load the 70 STATICS OF STRUCTURES. [PART i. shearing force becomes negative, then the bar will be subjected to com- pression. Now, it will frequently happen that in the central divisions of a girder the positive or negative shearing forces due to the dead load are less than the negative or positive shearing forces due to the travelling load, so that if those bars are arranged to be in tension under the dead load, then, on the passage of the travelling load, the stress will be changed to compression. In some cases the bars are slender and not suited to sustain compression; the shearing force is then provided for by the addition of a second diagonal, sloping in the opposite direction, which, by its tension, will perform the duty the first bar would otherwise have to perform by compression. Such a bar is called a counter-brace. We frequently see such additional bars fitted to the middle divisions of framework girders. Again, the powers of resistance of a piece of material to a given maximum load are greater the smaller the fluctuation in the stress to which it is exposed ; and therefore, in determining its dimensions, it is important to know not only the maximum but also the minimum stress to which it is exposed. This can be done on the principles which have just been explained. EXAMPLES. 1. A single load of 50 tons traverses a bridge of 100 feet span. Draw the curves of maximum shearing force and bending moment, and give the values of these quantities for the quarter and half span. 2. A train weighing one ton per foot-run, arid more than 100 feet long, traverses a bridge 100 feet span. Draw the curves of maximum shearing force and bending moment, and give the values of these quantities at the quarter and half span. 3. In the last question, suppose the permanent load fths ton per foot-run. Find within what limits counter-bracing will be required. Ans. 21 feet at the centre. 4. In Ex. 5, p. 49, the maximum rolling load is estimated at 1 ton per foot-run. Determine which of the diagonals will be in compression, and the amount of that com- pression, assuming a complete number of divisions to be loaded. The two centre diagonals are the only ones which can be in compression, the maximum amount of which will be=(3'2-2)V2=I'7. It will occur when the rolling load occupies four divisions only of the bridge. 5. In the last question, suppose a single load of 20 tons to traverse the bridge. Find the maximum stress, both tension and compression, on each part of the girder. Divisions. i 2 3 4 5 Max. tension, bottom boom, - 27 48 63 72 Max. compression, upper boom, 27 48 63 72 75 Max. tension of diagonals, 381 311 24 17 9-8 Max. compression of diagonals, - 2'8 6. In the two preceding questions, find the fluctuation of stress on each part of the girder. . III. ART. 35.] STRAINING ACTIONS. METHOD OF SECTIONS. 35. Method of Sections applied to Incomplete Frames. Culmann's Theorem. The straining actions due to a vertical load may either be wholly resisted by internal forces called into play within the structure itself, or also in part by the horizontal reaction of fixed abutments : the supporting forces being in the first case vertical, and in the second having a horizontal component. The distinction is one of the greatest importance in the theory of structures, which are thus divided into two classes, Girders and Arches, including under the last head also Chains. It is the first class alone which we consider in this chapter. The general consideration of internal forces is outside the limits of this part of our work, and we shall here merely consider some cases of framework structures, commencing with that of an incomplete frame. Incomplete frames are in general, as in Chapter I., structures of the arch and chain class, but by a slight modification we can readily convert such a frame into a girder and thus obtain very interesting results. Fig. 39a shows a funicular polygon such as that in Fig. 11, page 14, except that the supports are removed and replaced by a strut 06. By this addition the polygon becomes a closed figure, and 06 is therefore called its "closing line." The structure is carried by suspending rods at the joints 06, and loaded as shown. The construction of the diagram of forces, Fig. 39b, has been sufficiently explained on the page referred to, and it only remains to observe that the supporting 72 STATICS OF STRUCTURES. [PART i_ forces PQ are immediately derived from the diagram by drawing parallel to the closing line, which is not necessarily horizontal. The horizontal thrust of the strut and tension of the rope is found as before by drawing ON horizontal. The structure may now be regarded as a girder, the load on which, together with the vertical supporting forces, produce definite straining actions M and F on any section. Let the section be KK' in the figure,, cutting one of the parts of the rope and the strut as shown in the figure : let the intercept be y. Consider the forces acting at the section on the left-hand half of the girder, the horizontal components of these forces are equal and opposite, acting as shown in the figure, each being H or ON in the diagram of forces. The vertical components are balanced by the shearing force, and the horizontal components by the bending moment, which last fact we express by the equation Hy = M, that is to say, the funicular polygon corresponding to a given load is also a polygon of bending moments, the intercept between the poly- gon and its closing line multiplied by the horizontal force is equal to the bending moment due to the load. Hence, by a purely graphical process, we can construct a polygon of moments, for we have only to construct a funicular polygon corresponding to the load as shown in the article already cited, and complete it by drawing its closing line. This is one of the fundamental theorems of graphical statics, a subject which of late has been extensively studied. The construction is intimately connected with the process of Art. 29 as the reader should show for himself. In its complete form it is due to Culmann and is generally known by his name, having been given in his work on graphical statics. Fig.40. 36. Method of Sections in general. Bitter's Method. In frames which are complete the number of bars cut by the section, instead of being two only, as in the preceding case, is in general three at least. In Fig. 40 let KK' be the section cutting the three bars in three CH. in. ART. 36.] STRAINING ACTIONS. 7 points, which may be considered as the points of application of three forces PQR due to the reaction of the bars, which balance the shearing and bending actions to which the section is subject. Resolving horizontally and vertically, and taking moments, we should re- membering that the load being wholly vertical the sum of the horizontal components must be zero obtain three equations which would determine P, Q, Pi. It is, however, simpler to employ a method introduced by Ritter which enables us to obtain the value of each force at once. Let the lines of action of P, Q intersect in the point l r Q and R in 2, P and R in 3, and let the perpendicular dropped from each intersection on to the line of action of the third force be r, p r q respectively : by measurement on the drawing of the framework structure we are considering it is always easy to determine these perpendiculars. Then taking moments about the three points we get Rr = L l -, Pp = L. 2 ; Qq = L 3 , where L v L 2 , L 3 are the moments of the forces acting on the left-hand half of the structure about the points 1, 2, 3 respectively. On page 5& it was shown how to get these moments graphically from the polygon of moments, but they also may be obtained by direct calculation. We may write down a general formula for this method, thus Hh = L, where H is the stress on any bar, h its perpendicular distance from the intersection of the two others cut by a section, and L is the moment of the forces about that intersection. The special case in which the intersection lies on the section considered so that the moment L becomes the bending moment (M) on the section has already been considered in Chapter II. When the stress on a single bar is required as a verification of results obtained by graphical methods, or where the maximum stress due to a travelling load has to be determined, thi& method is often serviceable, but as a general method it is inconvenient from the amount of arithmetical labour involved. The shearing action on the section is resisted by the components parallel to the section of the stress on the several bars. In the case of the incomplete frame of Fig. 39, p. 71, these components are given at once by the diagram of forces. In general, however, three bars and only three, must be cut by the section if the frame be neither incomplete nor redundant ; when two of these are perpendicular ta the section the case is that considered in Chap. III. of a framework girder with booms parallel, in which the diagonal bars alone resist the shearing. When one bar only is perpendicular to the section, the other two collectively resist the shearing action : this case is common. Y4 STATICS OF STRUCTURES. [PARTI.] in bowstring and other girders of variable depth. The upper boom together with the web here resists the shearing. When more than three bars are cut by the section, the stress in each is generally indeterminate on account of the number of bars being redundant. On this question it will be sufficient for the present to refer to Chapter II., Section II. EXAMPLES. 1. In example 3, page 66, construct the polygon of bending moments by Culmann's method. 2. In example 6, page 32, find the stress on each part of the roof by Hitter's method. 3. In example 7, page 32, find the stress on each by Eitter's method. 4. If a parabolic bowstring girder be subject to a uniform travelling load, represented fcy the application of equal weights to some or all of the verticals, show that the horizontal component of the maximum stress on each diagonal is the same for all. 5. In the roof shown in Fig. 21, p. 28, emploj" Bitter's method to find the stress on the sloping struts and deduce the stress on each division of the tie rod. 6. The curve of shearing force for a vessel consists of two similar parabolas plotted with vertical axes on a base line representing the length of the vessel. The excess of weight over buoyancy of each end of the vessel up to the nearest waterbone section is T Vth her displacement ; find the maximum bending moment. Ans. YgWL 7. A uniform raft of rectangular section, which when floating freely is immersed to ^rds of its depth, has one end stranded so that the lower edge of that end is in the plane of flotation. Draw a diagram of shearing force, giving the value of some ordinates in terms of the whole weight of the raft, and show that the maximum bending moment is -^ 7 Wl. 8. A circular ring cut out of a piece of sheet metal is balanced in a horizontal plane upon knife edges placed in a central line. Find the shearing force and bending moment at any radial section. 9. In question 6, p. 54, draw curves of shearing force and bending moment for one of the stiffening pieces. 10. A flat-bottomed vessel of length L and beam B floats horizontally in the water The sides are vertical and the water lines curves of sines given by the equation B . x where x is measured from one end. The weight is uniformly distributed. Find the curves of shearing force and bending moment. Deduce their maximum values. -F =^ W ; M = WL nearly. REFERENCES. For further information on subjects connected with the present chapter the reader may refer to Naval Architecture. W. H. WHITE. Murray. Elements of Graphic Statics. L. M. HOSKINS. Macmillan. Graphical Statics. CREMONA. Clarendon Press. The last mentioned work is a translation by Professor Beare, revised by Professor remona, of two small treatises on graphical calculation and reciprocal figures. (See Note in Appendix.) CHAPTEK IV. FRAMEWORK IN GENERAL. 37. Straining Actions on the Bars of a Frame. General Method of Reduction. When the bars of a frame are not straight, or when they carry loads at some intermediate points, the straining action on them is not generally a simple thrust or pull, but includes a shearing and bending action. The present and two following articles will be devoted to some cases of this kind. First suppose the bars straight, but let one or more be loaded in any way, and in the first instance consider any one bar, AB (Fig. 41), apart from the rest of the frame, and suspended by strings in an inclined position. Let any weights act on it as shown in the figure, then the tensions of the vertical strings will be just the same as in a beam, AB, supported horizontally at the ends and loaded at the same points with the same weights. Resolve the forces into two sets, one along the bar, the other transverse to the bar. The second set produce shearing and .bending just as if applied to a beam in a horizontal position, while the first set produce a longitudinal stress, which will be different in each division of the bar. Let 6 be the inclination of the bar to the vertical, then the pulls on the successive divisions are P . cos : (P - W^ cos d : (P - W z - W^ cos : ... , the last being a thrust equal to Q . cos 0, so that the stress varies from Q cos to - P . cos 9. Now observe that we can apply to AB at its ends. 76 STATICS OF STRUCTURES. [PART i. in the direction of its length, a thrust, ZT , of any magnitude we please without altering P and Q, but that we cannot apply a force in any other direction, whence it follows that when AE forms one of the bars of a frame, its reaction on the joint A must be a downward force, P, and a force H , which must have the direction BA, while the reaction on B in like manner consists of a downward force, Q, and an equal force, H , in the direction AE. The downward forces P, Q are described as the part of the load on AE carried at- the joints A, E, and it is now clear that if these quantities be estimated for each bar and added to the load directly suspended there, we must be able to determine the forces H by exactly the same process as that by which we find the stress on each bar of a frame loaded at the joints. The actual thrust on AE evidently varies between H - P . cos 6 at the top, to H Q + Q . cos at the bottom, so that H may be described as the mean thrust on the bar, while the shearing and bending depend solely on the load on the bar itself, and not on the nature of the framework structure of which it forms part, or on the load on that structure. In the particular case where the load on the bar is uniformly distributed, the forces P, Q are each half the weight of the bar, and the thrust H Q is the actual thrust at the middle point of the bar. This question may also be treated by the graphical method of Art. 35 with great advantage. Through A and E draw a funicular polygon corresponding to the load on AB, the line OF in the diagram of forces will be parallel to AE and may be taken to represent H . This funicular polygon will be the curve of bending moment for the bar, and the other straining actions at every point are immediately de- ducible. It will be seen presently that the bar need not be straight. For simplicity it has been supposed that the forces acting on the bar are parallel : if they be not, the reduction is not quite so simple. It will then be necessary to resolve the forces into components along the bar and transverse to the bar, the second set can be treated as above, while the total amount of the first set must be considered as part of the force supplied to the joints either at A or B. Such cases, however, do riot often occur, and it is therefore unnecessary to dwell on them. The joints have been supposed simple pin joints or their equivalents, but the method used for frames loaded at the joints will apply even if the real or ideal centres of rotation of the bars are not coincident, provided only the centre lines prolonged pass through the point where the load is applied. The method of reduction just explained then requires modification. Such cases are of frequent occurrence, and the next article will be devoted to them. H. iv. ART. 38.] FRAMEWORK IN GENERAL. 77 38. Hinged Girders. Virtual Joints. The case of a loaded beam, the ends of which overhang the supports on which it rests, has already been considered in Art. 21, where it was shown that the straining actions at any point might be expressed in terms of the bending moments at the points of support, which of course will be determined by the load on the overhanging part. If the overhanging parts be supported, as in the case of a beam continuous over several spans, or with the ends fixed in a wall, the same formula will serve to express the straining actions at any point in terms of the bending moments at the points of support, but those bending moments will not be known unless the material of the beam and the mode of support are fully known. Hence the full consideration of such cases forms part of a later division of our work. Certain general conclusions can be drawn, however, which are of practical interest. The graphic construction for the bending moment at any point of a beam, CD, which is not free at the points of support, is given in Fig. 28, p. 40. The figure refers to the case where the bending action at C and D is in the opposite direction to the bending action near the centre, as it is easily seen must be the case in general. The points of intersection of the moment line with the curve of moments drawn, as explained in the article cited, on the supposition of the ends being free, show where the negative bending at the ends passes into the positive bending of the centre. Here, there is no bending at all, and the central part of the beam (EF in figure) is exactly in the position of a beam supported but otherwise free at its ends. We may therefore treat the case as if E and F were joints, the position of which will be known if the bending moments at the ends are known, and conversely. In some cases there may be actual joints in given positions, while in others there will be "virtual joints," the position of which may be supposed known for the purposes of the investigation. Fig.42. Fig. 42 shows a beam AB continuous over three spans, the moment curves for which will be known when the load resting on each span is known. It is evident from what has been said that the moment line must be the broken line AcdB, cutting the moment curve of the centre span in two points, and the moment curves of the end spans each in 78 STATICS OF STRUCTURES. [PARTI. one point, the others being the ends of the beam. Thus there are four virtual joints, of which two must be supposed known in order to find the straining actions at any point. Their position will depend (1) on whether the supports are on the same level or not, (2) on the material and mode of construction of the beam, (3) on the load. Such a beam is in a condition analogous to that of a frame with redundant bars, considered in Chapter II., Section III. ; the straining actions are indeterminate by purely statical considerations, for the same reason as before. We can, however, see that the bending action at each point is in general less than if the beam were not continuous. In one particular case the position of the virtual joints can be foreseen. Suppose a perfectly straight beam, of uniform transverse section, to be continuous over an indefinite number of equal spans : let the weight of the beam be negligible, and let equal weights be placed at the centre of each span. Then since the pressure on each support must be equal to the weight, the beam is acted on by equal forces at equal distances alternately upwards and downwards, and there being perfect symmetry in the action of the upward and downward forces, the virtual joints must be midway between the centre and the points of support of each span. In the special case where the beam is uniformly loaded we can further see that the load resting on the supports is not one half the weight of the parts of the beam resting there, as it would be if the beam were not continuous, but must in general be greater for the centre supports and less for the end supports. For if the virtual joints be LNML', as in the figure, it is easily seen that A carries half the weight of AL, not of AC, while C carries half the weight of AL and NM, together with the whole weight of CL and CN. This observ- ation shows that in trussed beams where, as is usually the case, the loaded beam is continuous through certain joints, the effect of the continuity is generally to transfer a part of the weight from the joints where the ends are free to the joints where the beam is con- tinuous. We shall return to this point hereafter. The principle of continuity is frequently taken advantage of in the construction of girders of uniform depth by making them con- tinuous over several spans. The virtual joints then vary in position for each position of the travelling load, rendering it a complicated matter to determine the maximum straining actions, while there is always an element of uncertainty about the results, for reasons already referred to and afterwards to be stated more fully. In some structures, however, the joints have a definite position. Fig. 43 shows a cantilever bowstring girder, consisting of a central CH. iv. ART. 39.] FKAMEWORK IN GENERAL. bowstring girder NM, the ends of which rest on parts ACN, BDM, projecting from the piers, technically described as " cantilevers." The joints here are at N and M. In structures of great span, in which .A B. Fig.43. the weight of the structure is the principal element, so that the variations in distribution are small, this type of girder is economical in weight. In the great bridge over the Forth, the central portion for each of two principal openings consists of a simple girder 350 feet span, while the cantilevers are each no less than 675 feet in length, making a total span of 1700 feet. These cantilevers are of great depth near the piers, and to provide against wind pressure, they are there likewise greatly increased in breadth, and solidly united to them. Full descriptions of this bridge, a structure which, from its gigantic dimensions and other unusual features, deserves- attentive study, appeared in the engineering journals for 1890, and several have since been re-published in a separate form. 39. Hinged Arches. In the second section of Chapter I. certain forms of arches were considered which are simply inverted chains, and require for equilibrium a load of a certain definite intensity at each point. We shall now take the case of an arch rib capable of sustaining a load distributed in any way. We shall suppose the load vertical, and, to take the thrust of the arch, we shall imagine a tie rod introduced so as to convert it into a bowstring girder. If the straining actions at each point of the rib are to be determinate without Fig.44. reference to the relative flexibility of the several parts of the rib, and other circumstances, we must have, as in the case of the continuous- beam, joints in some given position. The necessary joints are in this SO STATICS OF STRUCTURES. [PART i. instance three in number, and, we shall suppose, are at the crown (Fig. 44), and one at each springing A and B. Taking a vertical section KK' through the rib and tie, let the bending moment due to the vertical load and supporting forces be M. This bending moment is resisted, first, by the horizontal forces called into play ; that is to say, the pull of the tie rod H at K', and the equal and opposite horizontal thrust of the rib at K ; secondly, by the resist- ance to bending of the rib itself, the moment of which we will call /x. Hence if y be the ordinate of the point considered, we must have To determine H we have only to notice that at the crown where y = h there is a joint, that is, /* = 0, .-. M = Hh, where M is the bending moment due to the load for the central section. Thus, to determine /x we have the equation The graphic representation of /x is very simple. Let us imagine the curve of moments drawn for the given vertical load, and let it be so drawn as to pass through A, B, and C, which is evidently always possible. Then, if Y be the ordinate of the curve, M = H. Y. Therefore, by substitution, :So that the bending moment at each point of the rib is represented graphically by the vertical intercept between the rib and the curve of moments. In the figure, the curve AZCB is the curve of moments, and KZ is the intercept in question. Arched ribs in practice are rarely, if ever, hinged, and the straining actions on them occasioned by a distribution of the load not corre- sponding to their form depend, therefore, upon the relative flexibility of the several parts of the rib, and other complicated circumstances. If the position of the virtual joints be known, or the bending moments at any three points, the graphical construction just given can be applied. Instead of a rigid arch, from which a flexible platform is suspended, we may have a stiff platform suspended from a chain. This is the ase where a suspension bridge is adapted to a variable load by means of a stiffening girder. For this case it will be sufficient to refer to Ex. 3, page 87. 40. Structures of Uniform Strength. In any framework structure without redundant bars, the stress on each bar may be determined as CH. iv. ART. 40.] FRAMEWORK IN GENERAL. 81 in Chapter I., by drawing a diagram of forces for any given load, W, and expressed by the formula where k is a co-efficient depending on the distribution of the load. If A be the sectional area of the bar we find by division the stress per sq. inch, which must not exceed a certain limit, depending on the nature of the material as explained in Part IV. of this work. When the structure is completely adapted to the load which it has to carry, the stress per square inch is the same for all the bars, and it is then said to be of Uniform Strength. Uniformity of strength cannot be reached exactly in practice, but it is a theoretical condition which is carried out as far as possible in the design of the structure. Other things being equal, the weight of a structure of uniform strength is less than that of any other. Such a structure is therefore less costly, for weight is to a great extent a measure of cost. Whenever the load is known, the weight of a structure of given type, and of uniform strength can be calculated thus. Suppose A the sectional area of a piece, Zf, the stress on it, /, a co-efficient of strength, then TJfA Next let w be the weight of a unit of volume, usually a cubic inch, and assume /. A = A w a then A is a certain length, being in fact the length of a bar of the material which will just carry its own weight. Its value in feet for various materials is given in Chapter XVIII. Then assuming the piece prismatic and of length s, its weight is A HS w As = -^, and therefore the weight of the whole structure must be for the same value of A, the summation extending to all the pieces in the structure, and being performed by integration in a continuous arch or chain. It will be observed that s is the length of any line in the frame -diagram, and H that of the corresponding line in the diagram of forces ; we have only then to take the sum of the products of these lines and divide by A, the result will be the weight of the structure. It is, however, generally necessary to find the weights W v W^ of the parts in compression and in tension separately, because the value of A is generally different in the two cases. C.M. F 82 STATICS OF STKUCTUEES. [PART i. A remarkable connection was shown by the late Prof. Clerk Maxwell to exist between W l and TFo. Let us take a structure of the girder class and suppose the total load upon it G, and the height of the centre of gravity of that load above the points of support h. Imagine this structure to become gradually smaller without altering either its pro- portions or the magnitude and distribution of the load G, then O descends and does work during the descent in overcoming the resistance (T) of the bars in compression to diminution of length, while at the same time the bars in tension (P) do work during contraction. The values of T and P do not alter, for the diagram of forces remains .the same, and therefore if we conceive the process to continue till the structure has shrunk to a 'point, Gh='STs-2Ps= \ W l - \ 2 W 2 . In particular, if the centre of gravity of the load lies on the line of support, and if the co-efficients be the same, the weights of the parts in compression and tension will be equal. A corresponding formula may be obtained for structures of the arch-class by taking into account the thrust. The weight of an actual structure is always greater than that found by this method. First, an addition must be made to allow for joints and fastenings. Thus, for example, in ordinary pin joints the eye of the bar weighs more than the corresponding fraction of the length of the bar, and in addition there is the weight of the pin. Secondly, in all structures there is more or less redundant material necessary to provide against accidental strains not comprehended in the useful load. Thirdly, there are local straining actions in the pieces occasioned by their own weight and other causes. 41. Stress due to the PFeight of a Structure. The total load on any structure consists partly of external forces applied to it at various points and partly of its own weight : the total stress on any member is therefore the sum of that due to the external load and of that due to the weight of the structure itself. As that stress cannot exceed a certain limit, depending on the strength of the material, it necessarily follows that the stress due to the weight is so much deducted from the strength. Thus the consideration of the weight of a structure is an essential part of the subject, even if we disregard the question of cost. The weight of each member is, of course, distributed over its whole length, and so also may be a part or the whole of the external load. Applying the general method of reduction explained in Art. 37, we suppose an equivalent load applied at each joint, and drawing a diagram of forces, we determine the mean stress, H, on the member. If the unsupported length of the bars be not too great, a matter to be considered presently, this stress will be the principal part of the straining action on the bar, and the bending may be neglected as in the preceding article. Now, consider two structures similar in form and loaded with the same total weight, distributed in the same way, so that the only CH. iv. ART. 41.] FRAMEWORK IN GENERAL. 83 difference in the structures is in size : then the stress on corresponding bars must be the same, for the structures have the same diagram of forces. That is to say, in the formula H=kW, the co-efficient k depends on the type of structure and the distribution of the load upon it, but not on its dimensions. Dividing by the sectional area, the intensity of the stress is Next let WQ be the weight of the structure itself, and suppose the relative sectional areas of the several pieces the same, then W = w .cAl, where c is a co-efficient depending on the type of structure, arid / a length depending on the linear dimensions of the structure. For example, in roofs and bridges I may conveniently be taken as the span. Then if & be the value of k, which corresponds to the dis- tribution of the weight of the structure, which will be the same whether the structure be large or small, W will be the stress due to the weight of the structure. In other words, the stress due to the weight of similar structures varies as their linear dimensions. Since p cannot exceed /, it follows at once that there must be a limit to the size of each particular type of structure, beyond which it will not carry its own weight. If L be that limit given by the stress due to the weight of any similar structure of smaller dimensions will be simply is the strength which may be allowed in calculations made irrespect- ively of weight. If the structure be of uniform strength throughout under its own weight, the value of p will be the same for each member, but this is not necessarily the case, and there may be a different value of /' for each member. The actual limiting dimensions of the structure will, of course, be the least of the various values corresponding to the various members. The conclusion here arrived at is obviously of the greatest import- 84 STATICS OF STRUCTURES. [PART i. ance, for it immediately follows that in designing a roof, bridge, or other structure of great size, the weight of the structure is the principal thing to be considered in estimating the straining actions upon it, while a certain limiting span can never be exceeded. On the other hand, in small structures the straining actions due to the weight are unimportant; it is the magnitude and variations of the external load which have the greatest influence. This remark also applies to the local straining actions which produce bending in the pieces, their relative importance increases with the size of the structure, and it is necessary to provide against them by additional trussing. A large structure is therefore generally of more complex construction than a small one, as is illus- trated by the various types of roof-trusses considered in Chapter I. The difference of type of large structures and small ones, as well as the circumstances mentioned at the close of the last article, render tentative processes generally necessary in calculations respecting weight. If the type of structure and the distribution of the total load, py, be supposed known, the value of the co-efficients k and c will be known for some given member. By assuming the stress on that member equal to the co-efficient of strength /, we find W,= W.ck. 1 -, a formula which gives the weight of the structure in terms of the load, but the co-efficients will generally vary according to the span. Among the circumstances on which they depend, the ratio of the vertical to the horizontal dimensions of the structure is most important. For a given span k diminishes when the depth is increased, while on the other hand c generally increases, so that for a certain ratio of depth to span, the weight of the structure is least. In ideal cases c may remain the same (Ex. 10, p. 88), but in actual structures the redundant weight of material necessary to give stiffness and lateral stability increases, so that the most economical ratio of depth to span is generally much less than would be found by neglecting such considerations. These points are illustrated by examples at the end of this Chapter and Chapter XIL, where the question is again considered briefly ; but for detailed applications to actual structures the reader is referred to works on bridges, in the design of which it is .of the greatest importance. 42. Straining Actions on a Loaded Structure in General. The results obtained in the last chapter for the case of parallel forces acting on a structure possessing a plane of symmetry in which the forces lie, may be readily extended to structures which have an axis of symmetry acted on by any forces passing through that axis and perpendicular to CH. iv. ART. 43.] FRAMEWORK IN GENERAL. 85 it. This is the case, for example, of a beam acted on by a vertical load, arid also by some horizontal forces arising say from the thrust of a roof or from wind pressure. We have then only to consider the vertical and horizontal forces separately. Each will produce shearing and bending in its own plane, which may be represented by polygons as before. The total straining action will be simply shearing and bending, and will be as before independent of the particular structure on which the forces operate. The magnitude of the straining action, whether shearing or bending, will be the square root of the sum of the squares of its components, and may therefore be readily found by construction and exhibited graphically by curves. In shafts such cases are common, and some examples will be given hereafter. Another entirely different kind of straining action sometimes occurs in structures proper (roofs, bridges, etc.), and in machines is one of the principal things to be considered. Imagine a structure of any kind to be divided by an ideal plane section into parts A and B. and to be acted on by forces parallel to that plane. Let the forces acting on A reduce to a couple the axis of which is perpendicular to the section, the forces on B are equal and opposite, and the two equal and opposite couples tend to cause A and B to rotate relatively to each other. As already stated in Art. 16, this effect is called Twisting, and the magnitude of the twisting action is measured by the magnitude of either of the couples which form its elements. Simple twisting sometimes occurs in practice, for example, when a capstan is rotated by equal forces applied to all the bars, but it is generally combined with shearing and bending. It is then necessary to know about what axis the twisting moment should be reckoned, which will depend on the nature of the structure. In shafts and other cases to be considered hereafter the geometrical axis is an axis of symmetry which at once determines this. When twisting exists the shearing and bending are determined by the same method as before, for they are independent of the axis of refer- ence. Should, however, the structure be subject to a thrust or a pull (Art. 16), the axis about which the bending moment should be reckoned, must be known, for it will depend on the nature of the structure. These general observations will be illustrated hereafter, and are only introduced here to show how far straining actions can be regarded as depending solely on the external forces operating on the structure without reference to any other circumstances. 43. -Framework with Redundant Parts. In a complete frame, without redundant bars (pp. 11, 50), suppose a link applied to any two bars, 86 .STATICS OF STEUCTUEES. [PARTI. one end attached to each. Let the link be provided with a right and left-handed screw, or other means of altering its length at pleasure, then by screwing up the link a pull may be produced in the link of any magnitude we please, while a corresponding stress will be pro- duced in each bar of the frame which will bear a given ratio to the pull. Such a link may be called a straining link, and by its addition we obtain a frame with one redundant bar. The stress ratio on the parts of a frame of this kind is completely definite, but the magnitude of the stress may be anything we please. Instead of one straining link we may have any number, and if the stress on each of these links be given, the same thing will be true. Thus it appears that a frame with redundant parts may be in a state of stress even though no external forces act upon it. This is of practical importance on account of the effect of changes of temperature. If all the bars of a frame with redundant parts are equally heated or cooled, the frame expands or contracts as a whole, but no other effect is produced ; any inequality, however, causes a stress which may, under certain circumstances, be very great. This (at least theoretically) is one of the reasons why redundant parts are a source of weakness. The necessity of providing against expansion and contraction is well known in large structures resting on supports. The ground connecting the supports suffers little change of temperature, and the structure, therefore, cannot be attached to the supports, but must be enabled to move horizontally by the intervention of rollers. The magnitude of the stress produced when changes of length are forcibly prevented will be considered hereafter (Chapter X1L). There is no essential difference between a frame the stress on the parts of which is due to the action of straining links, and a frame acted on by external forces; for every force arises from the mutual action between two bodies, and may therefore be represented by a straining link connecting the bodies. Even gravity may be regarded as a number of such links connecting each particle of the heavy body with the earth. Accordingly, if we include in the structure we are considering, the supports and solid ground on which it rests, we may regard it as a frame under no external forces, but including a number of straining links screwed up to a given stress. If the original frame be incomplete, its parts will be capable of motion, and it becomes a machine, as will be explained in Part III. of this work. 44. Concluding Remarks. Various other questions relating to frame- work remain to be considered, especially with reference to the joints by which the parts are connected, but these, involving other than CH. iv. ART. 44.] FRAMEWORK IN GENERAL. 87 purely statical considerations, do not come within the present division of our work, but are referred to at a later period. EXAMPLES. 1. In Ex. 4, page 10, if the weight be supposed uniformly distributed, find the thrust, shearing force, and bending moment at each point of each rafter, and exhibit the results graphically by drawing curves. Diagrams of shearing force will be sloping lines crossing each rafter at the centre. Max. shearing for short rafter= 91 Ibs. long ,, =158-5 Diagrams of bending moment will be parabolas. Max. moment at centre of short rafter=117 ft. -Ibs. long =290 2. A triangular frame ABC, supported at A and (7, with AC horizontal, is constructed of uniform bars weighing 10 Ibs. per foot, the length being AB3 feet, BC=4 feet, and AC=o feet. Suppose, further, that AB and BC each carry 50 Ibs. in the centre. Draw curves of thrust, shearing force, and bending moment for each bar. 3. The platform of a suspension bridge is stiffened by girders hinged at the centre arid at the piers. The chains hang in a parabola, and the weight of the platform, chains, and suspending rods may be regarded as uniformly distributed. Find the bending moment, at any point of the stiffening girder, and exhibit it graphically by a curve when a single load TFis placed (1) at the centre of the bridge, and (2) at quarter span. First case. On account of W each half of the girder will tend to turn downwards about the ends, and will be supported by the uniform upward pull of the suspending rods. .". total upward pull for each \ girder = W, because the centre of action is at \ span. Thus each girder will be in the state of a beam loaded uniformly with W, and supported at the ends. Max. moment at middle of each half |TFxhalf span. Second case. The upward pull of the suspending rods will still be uniform, but for each half girder will now be only ^W, found by assuming an equal action and reaction at the centre joint, and taking moments of each half about the ends. For the half girder which carries the weight the bending moment will be the difference between that due to W concentrated in the centre and \W distributed uniformly. . ' . Max. = T \ W x half span. On the other half it will be due simply to a distributed load of ^ W. Max. =^Wx half span. 4. A timber beam 24 feet span is trussed by a pair of struts 8' apart, resting on iron tension rods forming a simple queen truss 3' deep without a diagonal brace. The beam is loaded with 5 tons placed immediately over one of the vertical struts. Find the shearing force and bending moment at any point of the beam, supposing it jointed at the centre, and the centre only. The thrust on each strut must be 2^ tons; therefore, curves of shearing force and bending moment for each half of the beam are the same as those for a beam 12 feet long loaded at a point 4 feet from one end with 2^ tons. The problem should also be treated by the method of sections. Results should also be obtained for the case where one half the beam is uniformly loaded. 5. A beam uniformly loaded is fixed horizontally at the two ends, and jointed at two given points. Draw the diagrams of shearing force and bending moment. Show that the beam will be strongest when the distance of each point from centre is rather less than | span. 6. The platform of a bowstring bridge of span 2a is suspended from parabolic arched ribs hinged at crown and springing. One half the platform only is loaded uniformly with w Ibs. per foot-run. Show that the greatest bending moment on the ribs is -r^wa 2 . 88 .STATICS OF STRUCTURES. [PART i.] 7. In the last question, if a weight of W tons travel over the bridge, how great will be the maximum bending moment produced? Wa AnS ' ^3 8. A girder is continuous over three equal spans, and is hinged at points in the centre span midway between centre and piers. Find the virtual joints in the end spans when uniformly loaded throughout. 10 Ans. ^ span from end. 9. The weight of the chains, platform, and suspension rods of a suspension bridge may be treated as a uniform load per foot-run which at the centre of the bridge is double the weight of the chain. The dip of the chain is T Vth the span. The weight of iron being 480 Ibs. per cubic foot, and the safe load per square inch of sectional area of chain being 5 tons, find the limiting span, and deduce the sectional area of chains for a load of \ ton per foot-run on a similar bridge, 300 feet span. If A= sectional area of chains at centre in sq. ins., then %--4 = weight of bridge per foot-run in Ibs. Horizontal tension =---AL= 5 x 2240 . A, .'. L= 1034 feet. If A' -area of one chain of the bridge 300 feet span, Whole load on chain=(-V-^' + 2 -V- a ) 300, Horizontal tension = & (*A' + ^-\ 4Q ) 300 x 13 = 5 x 224 2 re ~ present the ends of the stroke of the piston. If now we place the crank in any position OP we obtain the corresponding position of the piston by cutting the line of stroke with a circular arc of radius = PD and with centre P. DD^ DD^ will be the distances of the piston from the ends of its stroke. Since A 1 A 2 = D 1 D 2 , the length of the stroke, it will be convenient to find the point in A^A^ which corresponds to the position of the piston in its stroke. This may be readily done by striking a circular arc PN with centre D. N will be the point, for A 1 D l = PD = ND, therefore A 1 N=D 1 D, and the point N is the same distance from A l and A^ as the piston is from the ends of its stroke. C.M. G 98 KINEMATICS OF MACHINES. [PART IK We may just as easily solve the converse problem of finding the position of crank corresponding to any given position of the piston in its stroke. Let D be any position, cut the crank-pin circle by a circular arc of which D is the centre and DP the radius, then OP or OP' will be the corresponding position of the crank. Let the direction A v PA^ be the ahead direction of the crank, and let us call the motion D^D Z towards the crank the forward stroke, and D 2 D 1 the back or return stroke of the piston, then when the piston is at D in the forward stroke the crank will be at OP, and again when the piston is at D in the return stroke the crank will be at OP. Drop a perpendicular PM on to the line of stroke. Then the longer the connecting rod the smaller NM will be, and the more nearly the circular arc PN will coincide with the perpendicular PM. Hence in the limiting case of an indefinitely long connecting rod, M will be the position of the piston corresponding to the position OP of the crank. M being the position, neglecting the effect of the obliquity of the connecting rod, and N the true position, MN is what we may call the error, or deviation due to obliquity. In general the slide valve is worked by an eccentric, the radius of which is set at a particular angle on the shaft, so that the cut-off takes place when the crank occupies a certain angular position in its revolution, and it consequently follows that the fraction of stroke completed before cut-off takes place will be affected by the obliquity of the connecting rod, so that in the ordinary setting of the slide valve the rates of cut-off will be different in the two strokes. This is well illustrated by Ex. 4, page 103. "We may obtain a convenient approximate expression for MN, the error due to obliquity. Referring to Fig. 46, NM=DN-DM=DN(l-cos). Now the length of the connecting rod may be conveniently expressed as a multiple of the length of the crank radius a or stroke s. DN=na suppose =^ns. .. In the triangle POD, the sides being proportional to the sides of the opposite angles, OP 1 $inint at which steam will be cat off in the two strokes. Also when the obliquity of the connecting rod is neglected. A us. Fraction of stroke at which steam is cut off is '175 in forward stroke, '118 in backward stroke, '146 neglecting obliquity. 5. Obtain the results of the two last questions for the case of an oscillating engine, 6 feet stroke, the distance from the centre of the trunnions to the centre of the shaft being feet. An*. Angles 51 and 68 : Cut-off "2 and '115. 6. In. Ex. 3 construct both curves of piston velocity. If the revolutions be 70 per minute, find the absolute velocity of the piston in the positions given. Find also the maximum velocity of the piston. Ans. stroke forward, velocity 810 feet per 1'. i back, = 730 Maximum =900 Find also the points in the stroke at which the actual speed of piston is equal to the mean speed. Ans. 4f in. from commencement of forward stroke. 6|in. end 7. The travel of a slide valve is 6 in., outside lap 1 in. Find, in feet per second, the velocity with which the port commences to open when the revolutions are 70 per minute. Ans. Port commences to open when the valve is 1 in. from the centre of its stroke. Neglecting the obliquity of the eccentric rod, velocity of valve is then 1'72 feet per second. 8. Show that the maximum velocity of the piston occurs when the crank is nearly at right angles to the connecting rod, the difference being a small angle, the sine of which is 2 + 2) near ^' wnere n is the ratio of connecting rod to crank. 9. Referring to Fig. 48, p. 100, show that when the crank rotates uniformly the angular velocity of the connecting rod is proportional to PT. Draw a curve represent- ing it. With the notation of Art. 49, p. 102, show that approximately Angular velocity of rod= . cos 6. Angular acceleration = ^ sm & 10. At any point K of a linear curve of velocity (such as BKA in Fig. 48, p. 100) draw an ordinate KD to meet the base line in D and a normal KZ to meet this line in Z. Show that the acceleration of the piston or other moving piece is proportional to DZ. Note. This well-known construction is due to Proell. It is perfectly general, but difficult to apply with accuracy, because the exact direction of the normal is generally unknown. 11. Referring to Fig. 48, page 100, draw additional lines as follows. (Hoar's con- struction) (1.) TG horizontal to meet the crank OP produced in G. (2.) G8 vertical to meet the connecting rod PD in S (3. ) SZ at right angles to the rod to meet the line of centres in Z. Prove that when the crank turns uniformly Acceleration of Piston _OZ Acceleration of Crank Pin~ OQ' 104 KINEMATICS OF MACHINES. [PART n. SECTION II. EXAMPLES OF CHAINS OF LOWER PAIRS. 50. Mechanisms Derived from the Slider-Crank Chain. In the investi- gation just given it has been supposed, for simplicity, that the crank turns uniformly, but if this be not the case the curve constructed will show the ratio of the velocities of the piston and crank pin. In all cases it is the velocity-ratio of two parts, not the velocities themselves, which are determined by the nature of the mechanism. The velocities are of course reckoned relatively to the frame, but as both piston and crank pair with the frame, they are also the velocities of the piston-frame pair and the crank-frame pair (see p. 95), the crank being the radius of reference. The velocities of the other pairs will be determined presently, but in this mechanism are of less importance. We will now direct our attention to other examples of the simple chain of lower pairs, of which the direct-acting engine is only a particular case. In Fig. 49, D is a block capable of sliding in the slot of the piece A. By means of a pin this block is connected with one end of the link C. B is a crank capable of rotating about a pin attached to the piece A, and united to C by another pin. Each of the four pieces of which this mechanism is com- posed, together with either of the adjacent pieces, constitutes a "pair," of which there are four, viz., three turning pairs, AB, BC, CD, and a sliding pair, DA. This simple combination of pairs is known, in the modern theory of machines, as a Slider-Crank Chain. Since the relative motions of the parts depend solely on the form of the bearing surfaces of the pairs and the position of their centres, not on the size and shape of the pieces in other respects, we may vary these at pleasure, and thus adapt the same chain to a variety of purposes. Especially we may interchange the hollow and solid elements of the pairs, a process which occurs constantly in kinematic analysis, and is called " inversion of the pair." Again, any one of the four pieces may be fixed and the other move, so that we can obtain four distinct mechanisms from the same chain, simply by altering the link which we regard as fixed, a process called " inversion of the chain." (1.) Let A be fixed, then we obtain the mechanism of the direct-acting cu. v. ART. 50.] LOWER PAIRING. 105 engine already fully considered. In this, however, the connecting link C is much longer than the crank B ; by supposing them equal we obtain a mechanism well known in various forms. In Fig. 50, C is prolonged beyond the crank pin a to a point d, such that ad = ac, a circle struck with centre a then passes through c, d, and the centre of the block, thus cd is at right angles to the line c of stroke, so that d, when the crank turns, describes a straight line. This property renders the mechanism applicable to a parallel motion. It has also been used in air-compressing machinery. (See page 116.) The various forms of the well-known toggle joint, some of which will be referred to hereafter, are examples of the same mechanism with different proportions of C to B. (2.) Instead of A, let us suppose C to be the fixed link, so that A and the other pieces have to take a corresponding motion. With this, by a change in the shape of the pieces, we are able to derive a mechanism well known in two forms. C being fixed, and B caused to rotate, A will have given to it an oscillating motion about the block D, and, at the same time, will slide to and fro on the block, the block itself having a vibrating motion about the other end of the piece C. Now, the relative movement of the parts of this mechanism is identical with that of the oscillating steam engine, and by a suitable alteration in the shape of the pieces, that mechanism may be, derived. Thus, suppose, in the first place, the hollow element of A to become the solid one, in the shape of a piston rod and piston, whilst the block D is enlarged into a cylinder to surround the piston, and so becomes the hollow element of the pair. The cylinder D will oscillate on trunnions, in bearings in the fixed piece (7, which must be so con- structed as to be a suitable frame for carrying the engine, and have bearings in which the crank shaft and crank B can turn. The oscillating cylinder is in general mounted on bearings, the centre line of which coincides with the centre of the stroke of the piston, so that the distance apart of the shaft and trunnion bearings is equal to the length of the piston rod. An example is shown in Fig. 4, Plate I. Next let us consider the relative motions of the parts. Returning to Fig. 49 above, suppose a, b, c to be the centres of the turning pairs, and draw d, an perpendicular to the line of centres be, to meet C and A in t and TO, then it was shown above (page 1 00) that the velocity-ratio of the pairs DA, BA in the direct-acting mechanism was ct/ac-, and as fixing a link makes no difference in the relative motions, this must also be the 106 KINEMATICS OF MACHINES. [PART n. ratio of the speed of the piston of the oscillator in its cylinder, to the speed of the turning movement of the crank relatively to the piston rod. Again when C is fixed, as in the oscillator, the link A (Fig. 49) slides on the block D with a velocity the direction of which is perpendicular to an, while the point c in it moves perpendicular to ac. Hence it follows that the triangle of velocities is acn, and therefore the velocity-ratio of piston and crank pin is anfac. The curve of piston velocity can be drawn as before ; it differs little in form from that of the direct actor, but the maximum velocity of the piston is equal to that of the crank pin, instead of being somewhat greater. Once more, remembering that fixing a link does not alter the relative motions, it appears that, in all cases, the velocity-ratio of the pairs DA, BC must be an/ac, so that we have determined the ratio of the speed of piston in the direct actor to the speed of the turning movement of the crank relatively to the connecting rod. Comparing our results, we see that the velocity-ratio of the turning pairs BC, BA must be d : an, or what is the same thing, bt : ah. Since the three angles of the triangle abc are always together equal to 180, it is clear that the sum of the speeds of the three turning pairs must be zero, due regard being taken of the direction of rotation, and it follows, therefore, that in any slider-crank chain the speeds of the three turning pairs are as at : ab : bt. By the introduction of a suitable radius of reference, we may compare these velocities with that of the sliding pair. The most convenient radius to take is that of the crank, then assuming, as before, ab = n. ac, the velocities of the pairs are shown by the annexed table : VELOCITY-RATIOS IN A SLIDER-CRANK CHAIN. Pair, DA BA ac BC DC Velocity, ct bt n at n In the oscillator the angular velocity-ratio of the cylinder and crank is the velocity-ratio of the pairs CD, CB, and is therefore at : bt or en : be. This can readily be constructed by drawing nz parallel to ac, then, since ab is constant, the angular velocity of the cylinder is proportional to az, as may be verified by an independent investigation. The result may be exhibited by a polar curve similar to the curve of piston velocity already drawn. In Fig. 51, c is the crank pin, TST'S' the crank circle, az' is set off along the crank radius equal to az, then a curve with two CH. V. ART. 50.] LOWER PAIRING. 107 Fig.51. unequal loops is obtained, which shows the law of vibration of the cylinder. The motion of. the cylinder is such that, in the swing to the left, whilst the crank pin moves along the arc T'ST, the angular velocity is much greater than in the return swing to the right, whilst the crank pin moves along the arc TS'T'. Supposing the crank to revolve uniformly, the times occupied by the forward and return swings are as the arcs T'ST and TS'T', which are proportional to the angles subtended by them. By measuring or otherwise estimating these angles, the mean angular velocities in the for- ward and backward oscillation may be determined. This peculiar vibration, rapid one way and comparatively slow the other, has been made use of to obtain a quick return motion of a cutting tool in a shaping machine. The velocity with which a tool will make a smooth cut in metal is limited, and since in general the tool is made to cut in one direction only, time is saved by causing the return stroke to be made more quickly. One construction of such a quick return motion may be thus described. A slotted lever D vibrates on a fixed centre in the frame-piece C, its motion being derived from the revolution of a crank B on another fixed centre in the same frame-piece C. The crank pin of B turns in the block A, which slides in the slotted lever D. There is in addition a connecting rod, by means of which a to-and-fro motion of a headstock carrying the cutting tool is communi- cated from the oscillating lever, the headstock sliding in a guide. Omitting the connecting rod, we have the same kinematic chain, with the same fixed link C, as in the oscillating engine. There has been a change made only in the form of some of the pieces. What was the oscillating cylinder is now the slotted lever, and instead of a piston and rod, we have here the simple block A sliding in the slot. The crank B and frame-link C remain practically unaltered. The slotted lever will vibrate according to the same law which we have investigated for the oscillating cylinder, and thus with a uniform rotation of the crank, a quick return motion of the tool will be obtained. This mechanism 108 KINEMATICS OF MACHINES. [PART n, is shown in Fig. 5, Plate I., in a form employed for giving motion to the table of small planing machines. (3.) Let us next take an example in which B is the fixed link,, and becomes the frame, its form being of course modified to suit the new conditions. A crank arm C (Fig. 52) turns on a fixed centre in the frame-piece E\ so also does another arm A on a second fixed centre, D slides on A, being connected by a pin to the second end of C. Both A and C may make complete revolutions. If we suppose C to turn with uniform angular velocity, A will rotate with a very varying angular velocity, the movement of A in the upper part of its revolu- tion being much more rapid than in the lower. This device has been employed by Whitworth to get a quick return motion of a cutting tool in a shaping machine. When separated from the rest of the machine, the construction may be thus described : A spur wheel C which derives its motion through a smaller wheel from the engine shafting, revolves on a fixed journal B, of large dimension. Standing Fig.52. s> from the face of the journal is a fixed pin placed out of the centre of the journal. On this fixed pin a slotted k^y \ lever A rotates, in which a block D slides, a hole in the block receiving a pin which stands out from the face of the spur wheel. A second slot in A, on the other side of the pin, contains another block, which, by a screw, can be adjusted and secured at any required distance from the centre of rotation, so as to give any stroke at pleasure. This mechanism, omitting the adjustment by which the stroke is varied, is shown in Fig. 6, Plate I. The same mechanism in a somewhat different form is often employed in sewing machines to give a varying motion to the rotating hook. (4.) The fourth possible mechanism which can be derived from the slider crank chain is obtained by fixing the block D. This case is not so common as the three preceding, but in Stannah's pendulum pump, shown in Fig. 3, Plate I., we find an example. In a simple oscillating engine driving a crank shaft and fly-wheel, suppose the cylinder D fixed instead of the piece C which carries the cylinder and crank shaft. The crank and fly-wheel B has become the bob, and the link C the arm of the pendulum, from which the mechanism derives its name ; D is a fixed cylinder, and A is a piston and rod: Plate. To face page 109. CH. v. ART. 51.] LOWER PAIRING. 109 As the crank rotates the crank pin moves up and down, while its -centre vibrates in the arc of a circle. (5.) The four mechanisms here described are all which can be obtained from the simple slider-crank chain, but an additional set may be derived by supposing that the line of stroke of the slider does not pass through the centre of the crank. A common example is found in the chain communicating motion from the piston to the beam in a beam engine. Although the mechanisms derived by inversion from a given kine- matic chain may be described as distinct, it must be carefully observed that there is in reality no kinematic difference between them, the distinction consisting merely in a different link being chosen to reckon velocities from. If we consider the velocities of the pairs which consti- tute the chain, those velocities are always related to each other in the same way, and the same machine may be regarded sometimes as one mechanism and sometimes another. For example, suppose a direct- acting engine working on board ship ; the ship may be imagined to roll so that the connecting rod of the engine is at rest relatively to the arth, and the engine becomes an oscillator to an observer outside the ship. Dynamically and constructively, however, there is a great difference, for the fixed link is the frame, and is attached to the earth or other large body, the predominating mass of which controls the movements of all bodies connected with it. To illustrate and explain the inversion of a slider-crank chain, Plate I. has been drawn. The six examples which have just been described are here placed side by side with the same letters A B C D attached to corresponding links so that they may readily be recognized. It will be seen that each link assumes very various forms ; thus, for example, the link A is the frame and cylinder in Figs. 1 and 2, a piston and rod in Figs. 3 and 4, a block in Fig. 5, and a rotating arm in Fig. 6. The relative motions of corresponding parts are, however, always the same. 51. Double Slider-Crank Chains. We now pass on to the consideration of a kinematic chain consisting of two turning pairs and two sliding pairs. We will commence by showing how this chain may be derived from that previously described. Suppose the piece Z), instead of being simply a block, is a sector shaped as shown in Fig. 1, Plate II., having a slot curved to the arc of a circle of centre 0, while the piece (7, which was before the connecting rod, is compressed into a block sliding in the curved slot. The law of relative motion of the parts of this mechanism will be precisely the same as in the direct-acting engine, for the block C will move just as if it were attached by a link, shown by the dotted line, to a point 0, a fixed point in the piece D. The piece D will slide 110 KINEMATICS OF MACHINES. [PART n. in A, just as if there were a connecting link from C to and no sector that is, it will slide just as the piston does in the cylinder of a direct-acting engine. Moreover, there are in reality exactly the same pairs in this as in the mechanism of a direct-acting engine, for C and D together make a turning pair, although only portions of the surfaces of the cylindric elements are employed. This being so, let us now imagine the radius of the circular slot in the piece D to be indefinitely increased, so that the slot becomes straight, and is at right angles to the line of motion D. In such a case the pair CD would be transformed into a sliding pair, and the mechanism would consist of two turning pairs, and two sliding pairs, and is known as a double slider-crank chain. The most important example of this kinematic chain is that found in some small steam pumping engines. (Fig. 4, Plate II.) The pressure of the steam on the piston is transmitted directly to the pump plunger. The crank B and sliding block C serve only to define the stroke of the piston and plunger, and, by means of a fly-wheel, the shaft of which carries an eccentric for working the slide valve, to maintain a continual motion. The law of motion of piston and crank pin may be readily seen to be the same as that in a direct-acting engine, in which the con- necting rod is indefinitely long. P being the position of the crank pin, M will represent the position of the piston and reciprocating piece, and PM will represent the velocity of the piston at the* instant, OP being taken to represent the uniform velocity of crank pin. (See Fig. 46, p. 97.) In this case the polar curve of velocity would consist of a pair of circles. This motion, shown in dotted lines in Fig. 48, is called a simple Harmonic motion^ because the law is the same as that of the vibration of a musical string. By a change of the link which is fixed, we may now derive other well-known mechanisms from this kinematic chain. Instead of A, which forms part of a sliding and part of a turning pair, being fixed, let B be the fixed frame-link. B contains the elements of two turning pairs, so that the frame must contain two bearings or journals. An example of such a mechanism is that known as Oldham's coupling, Fig. 5, Plate II., used for connecting parallel shafts, which are nearly but not quite in the same straight line, and which are required to turn with uniform angular velocity-ratio. Each shaft terminates in a disc, in the face of which a straight groove is cut. The two discs, A and C in the figure, with the grooves, face each other, and are placed a little distance apart, with the grooves at right angles to each other. Filling up the space between them is placed a disc D, on the two faces of which are straight projections at right angles to one Plate.ll. FIG.I. FIG. 4. FIG.2. FIG.5. FIG. 3. FIG.6. To face page 111. CH. v. ART. 5lA.] LOWER PAIRING. Ill another, which fit into the grooves in the shaft discs. In the revolution of the shafts each of these projections slides in the groove in which it lies, and rotates with it. The two grooves are, therefore, maintained always at right angles to one another, and the two shafts rotate one exactly with the other. Next, let the fixed link of the chain contain the elements of two sliding pairs, which would be obtained if we made D the frame-piece. An interesting example of this is the instrument sometimes employed in drawing ellipses. (Fig. 2, Plate II.) Two blocks slide in a pair of right-angled grooves. By means of clamp-screws a rod unites them at a constant distance from one another. Pins fitting in holes in the block allow the rod to rotate relatively to the blocks. Any point in the rod will describe an ellipse, as indicated in the figure. If the link C be fixed, the resulting mechanism does not differ from that derived by fixing A, and the three mechanisms just described are therefore all which can be obtained by inversion of a double-slider chain. In Figs. 2, 4, 5 of the plate referred to they are shown side by side with the same letters attached to corresponding links, as in Plate I. The directions of motion of the two sliding pairs have been supposed at right angles, but any other angle may be assumed, and mechanisms obtained which we need not stop to examine. A more important change is to suppose that the sliding pairs and turning pairs alternate, so that each link forms an element of one sliding and one turning pair. A mechanism known as " Rapson's Slide," employed as a steering gear in large ships, will furnish an example. Fig. 3, Plate II., shows one way in which it is applied. A is an enlarged pin made in two pieces between which the tiller B slides while turning about an axis fixed in the ship D. A' is carried by the piece (7, which slides in a groove fixed transversely to the ship, being drawn to port or starboard by the tiller chains passing round pulleys mounted on (7, as shown in the figure. The further the tiller is put over the slower it moves (Ex. 8, p. 124), and therefore the greater the turning moment (Ch. VIII.), a property of considerable practical value. Another example occurs in the motion of the compensating air cylinders employed in the Worthington direct-acting pumping engine. In this kinematic chain the same mechanism is obtained whichever link is fixed. The mechanism shown in Fig. 6 of this Plate is a compound chain, to be referred to hereafter. 51a. Wedge Chain. A chain also may be found which consists of sliding pairs alone : the number of pairs being 3, and the directions of sliding parallel to the same plane. 112 KINEMATICS OF MACHINES. [PART n. This chain consists of two sliding pairs, AB and AC, having a common element A. A block attached to B slides in an oblique slot cut in (7, thus forming a third sliding pair, BC. The effect of this arrangement is that the pairs A, AC are connected with uniform velocity-ratio. It is employed when it is desired to alter the direction and magnitude of a sliding motion. An incomplete form occurs in the strap and cotter employed to tighten the brasses of a bearing as they wear. The action of a wedge or the raising of a weight by drawing it up an incline furnishes another example of the same chain, here reduced by omission of one link which, as in various other instances, is replaced by force-closure (p. 123). We may describe it as a Wedge Chain ; only one mechanism can be derived from it. 52. Crank Chains in General Instead of having a chain of sliding pairs or of turning pairs, connected by one or two sliding pairs, we may have turning pairs alone. The number will be four, and their axes must meet in a point or be parallel. Taking the second case, the chain in its most elementary form consists of four bars united by pin joints at their extremities, as in Fig. 53. It is called a crank or four- bar chain, and from it may be derived the slider-crank chain already Fig.53. considered, in the same way as from that chain we derived the double- slider chain. All the mechanisms hitherto considered may therefore be regarded as particular cases of it. In its present form, however, many new A mechanisms are included, some of which will be briefly indicated, referring for descriptions and figures to works specially devoted to mechanism. Assuming A the fixed link, B and D which pair with it are called for distinction cranks or levers, according as they are or are not capable of continuous rotation, while (7, the connecting link, is called for shortness the coupler. (1.) Let B be a crank and D a lever, then the mechanism is a "lever-crank," an example of which occurs in the common beam engine, D being the beam, B the crank, C the connecting rod, and A the entablature, foundation, and all other parts connected therewith. (2.) The links B and D may be equal, and C may be equal to A. This may be called " parallel cranks " when B and D are set parallel, as in the coupled outside cranks found in locomotives, or "anti-parallel cranks " when they are set crosswise, a 'case to be hereafter referred to (page 163). CH. v. ART. 54] LOWER PAIRING. 113 (3.) The links D and B may still both be cranks if C be greater than A, provided that the difference between B and D be not too great. The mechanism is called "double cranks," and occurs in the common draglink coupling, and also in the mechanism of feathering paddles. (4.) If the coupling link be too short, neither B nor D will be capable of a complete rotation. The mechanism is then a "double lever," and an example occurs in the common parallel motion to be considered hereafter. (5.) A number of additional mechanisms may be derived by sup- posing the axes of the four turning pairs to meet in a point, instead of being parallel; we thus obtain a "conic crank chain." Hooke's joint is a particular case of this, but in general these mechanisms are of less importance. 53. Screw Chains. We have hitherto considered only chains of turning pairs and sliding pairs, but screw pairs also occur in a great variety of mechanisms which we can only briefly indicate. (].) In the Differential Screw, there are two screw pairs with the same axes but of different pitch, combined with a sliding pair, forming a three-link chain. The connection between the common velocity of rotation of the screws and the velocity of translation of the sliding pair is the same as that between the rotation and translation of a screw, the pitch of which is the difference between the pitches of the actual screws. The arrangement has often been proposed for screw presses, a mechanical advantage being obtained, at least theo- retically, with screws of coarse pitch, which would otherwise require a thread so fine as to be of insufficient strength. The right and left- handed screw is an example in common use. (2.) In the Slide Rests of lathes and other machine tools, the traversing motion of planing machines, and many other cases, we find a three-link chain, consisting of a screw pair, a turning pair, and a ^sliding pair. This may be regarded as a particular case of the preceding, the pitch of one of the screws being zero. (3.) In presses, steering gear, and many other kinds of machinery, we find a simple screw chain employed to work a slider-crank chain. Some examples will be given hereafter. 54. Parallel Motions Derived from Crank Chains. In beam engines the connecting rod by which the reciprocating motion of the piston is communicated to the vibrating beam is necessarily short, in order to diminish the height of the machine, and therefore, if guides are employed to retain the end of the piston rod in a straight line, there C.M. H 114 KINEMATICS OF MACHINES. [PART n. will be considerable lateral pressure which is difficult to provide against, and which involves a large amount of friction. The guides may then be replaced with advantage by some linkwork or other mechanism. Such a mechanism is called a Parallel Motion, and in the early days of engineering was employed more extensively than at the present time. In its most simple form it consists of two levers capable of turning about the fixed centres a and b (Fig. 54). The ends of the levers are connected by a coupling link pq, then, so long as the angular movement of the levers is not too great, there is a point in the link pq which will describe very approximately a straight line. In the first instance let us suppose the -link so set that 'when ap and bp are parallel, p q Q is at right angles to them. Let apqb be the extreme downward movement of the levers, then p lying to the left and q to the right, there will be some point P in pq which in this extreme position lies in the straight line p$ . In the upward extreme position the same point of pq will, approximately, also lie in this line. If, then, p q be the line of stroke, and the point P be selected for the point of attachment of the piston-rod head, then this point will be exactly in the line at the middle and bottom of the stroke, and at other points will deviate but little from it. To find the point where pq intersects p q Q , we must first obtain expressions for the amount that the point p deviates to the left of p Q and q to the right of q Q ; these amounts being the versines of the arcs in which the points move, and shown by dp and eq ot where pd and qe are drawn perpendicular to ap and bq . By supposing the CH. v. ART. 54.] LOWER PAIRING. 115 circle of which a is the centre to be completed, it is easy to see that d> P d * If the angle p ap is not greater than 20, we may write the error not being greater than 1 per cent. Now, neglecting the small effect due to the obliquity of the connecting link when in the extreme positions, pd-% stroke : therefore, supposing ap = r a and if-ni , (stroke) 2 Now P being the point where pq intersects p Q q , we have similar triangles in which ^J m and .-.*!* qP qn r a Thus the point P, which has most correctly the straight-line motion, is such that it divides the coupling link into segments which are inversely proportional to the lengths of the levers. If the levers be placed into all possible positions, then in the motion the connect- ing link will be inverted and the point P will trace a closed curve resembling a figure of 8. There are two limited portions of this curve which deviate very little from a straight line. We may approximate still more nearly to a straight line by a little alteration in the setting of 'the levers. Suppose the centres of vibration, a, b, are brought a little nearer together so that the line of stroke bisects the two versines, dp Q and eq . Then when the levers are parallel, the link slopes to the left upwards, whereas at the ends of the stroke the link will slope to the right upwards. At two intermediate positions about quarter stroke from the ends, the link will be vertical. If we choose the point P as previously described, the maximum deviation will be only about one-fifth of its former amount. In practice, the final adjustment of the centres of motion is performed by trial. In steam engines the use of parallel motions is almost exclusively confined to beam engines. In that case bq will be the half length of the beam, and in order that the angle through which the beam vibrates should not exceed 20 above and below the horizontal, the length of the beam should not be less than three times the stroke. 116 KINEMATICS OF MACHINES. [PART n. The radius rod may be somewhat shorter than the half beam, but should not be less than the stroke, or the error in the motion of P will be too great. This mechanism will, therefore, occupy a con- siderable space. To economize space, and also to provide a second straight-line path to guide the air-pump rod, a modification of the mechanism is made use of. In Fig. 54a, be being the half length of beam, a point q is chosen so that bq stroke of air pump be ~ stroke of piston ' and a parallelogram of bars qeQp provided, united by pins. The Fig.54a. point p is jointed to the end of the radius rod ap vibrating on the fixed centre a. Consequently there will be some point P in the back link qp which will describe very nearly a straight line. This point is such that Pq ap Now, if the proportions of the links are such that bPQ is a straight line, bQ/bP will be constant, and therefore the path described by Q will be an enlarged copy of the path described by P. That is to say, if P moves approximately in a straight line, then Q will do so also. If then the radius rod is of suitable length we provide a point Q for the attachment of the piston rod, and also a point P for the attachment of the air-pump rod. To find this length we have pQ-pP' whence multiplying by the preceding equation bf =pQ x ap, or Length of radius rod CH. v. ART. 54.] LOWER PAIRING. 117 The parallel motion just described which was introduced by Watt is that chiefly used in practice, but there is another form which possesses great theoretical interest because it is exact and yet in volves only turning pairs. Scott Russell's parallel motion (Fig. 50, page 105), modified by attaching D to the end of a long vibrating lever, is known as a "grasshopper" parallel motion, but then is only ap- proximate. In its original form it is exact, but as it involves a sliding pair its accuracy depends on the exactness with which the guides of the slides \* are constructed. Now, a straight edge or a plane surface can only be constructed by a process of copying from some given plain surface or by trial and error, whereas a circle can be described, by a pair of com- passes independently of the existence of any other circle. Hence an exact parallel motion, with turning pairs only, enables us theoretically to trace a straight line in the same way that a circle is traced with compasses. It has long been known that this could be done by a circle rolling within another twice its diameter, but this method does not satisfy the necessary conditions, and it \*as not till 1872 that Col. Peaucellier invented a linkwork mechanism for the purpose. This mechanism consists of two equal bars, OA, OB, jointed to each other at 0, and at A, B to a parallelogram of equal bars, APBQ, so that OQP are in a straight line (Fig. 55). This being so then, how- ever the bars are placed, there will always be some fixed relation existing between OQ and OP. Thus drop a perpendicular AN on OP, then OQ = ON-QN'&nd OP=ON+NP. Also, since AQ = AP, QN=NP, .'. OQ. OP=ON*-QN*. But ON=OA*-AN 2 and QN* = QA*-AN\ therefore OQ.OP = OA 2 - QA' 2 , and is a constant quantity for all positions ; that is to say, if we cause Q to move over any curve, then P will describe its reciprocal. We can now show how this mechanism may be employed to draw a straight line. Let be a fixed centre and PL be the straight line which it is required to describe. Draw the perpendicular OL on PL. Then the mechanism being placed in any position with P at any point on the line to be drawn, draw QZ at right angles to OQ. Bisect OZ in C and attach Q to by means of a jointed rod which can turn on the fixed centre C. The circle which Q describes during the motion 118 KINEMATICS OF MACHINES. [PART n. of the bars will have OZ as a diameter, for OQZ is a right angle, and therefore the angle in a semicircle. We observe now that we have similar triangles OQZ and OLP. but OZ=2 . OC is a constant quantity and so is the product OP, OQ. .'. OL is constant. That is to say, wherever P is, the length of the projection of OP on the perpendicular OL is a constant quantity. This can be true only so long as P lies in the perpendicular line PL. Thus, by the constrained motion of Q in a circle passing through 0, P is caused to move perfectly in a straight line. This mechanism has been applied to a small engine used for ven- tilating the House of Commons. SECTION III KINEMATICS OF LINKWORK MECHANISMS. 55. Combination of a Sliding Pair and a Turning Pair. The motion of the connecting rod in the mechanism of the direct-acting engine Fig 56a. (Arts. 48, 49) may be considered as a combination of a motion of rotation due to a turning pair, with a motion of translation due to a sliding pair, and we now propose to consider the effect of such a combination more generally. In Fig. 56a, D is a block sliding in guides attached to the frame- piece A, C is a wheel turning in bearings attached to D, the whole forming a chain of two pairs DA, CD. The block slides with velocity CH. v. ART. 55.] LOWER PAIRING. 119 F", while the wheel turns with angular velocity o> about an axis perpendicular to the plane of the paper. In consequence of the rotation, any point P at a distance r from the axis has a velocity . KP. Thus the velocity of P is the same as if C were rotating with its actual angular velocity oj about an axis through K, and this will be true for any point in C or rigidly attached to it, the effect of combining a motion of translation with a motion of rotation being simply to shift the axis of rotation through the distance R = F/o>. The point K does not remain fixed, but moves so as to be always in the perpendicular, and the axis through it is therefore described as the Instantaneous Axis of the moving piece C. Its position is completely represented by the point K which is often spoken of as an "instantaneous centre." If, as in the figure, the rotation be in the opposite direction to the hands of a watch and the translation be from right to left, the point K lies below D, but if either motion be reversed it will lie above, as in Fig. 56b, p. 120. Further, there is nothing in the demonstration just given which renders it necessary that the direction of the sliding motion should remain unaltered, and the construction will therefore be the same if the block D slide in a slot which is circular instead of straight, or be attached to a piece turning in bearings on A. That is, the point K can be found in the same way for a combination of two turning pairs, and the effect of the combination of two rotations about parallel axes is to produce a rotation about an instantaneous axis parallel to the 120 KINEMATICS OF MACHINES. [PART n. former and in the same plane. A particular case is when the rotations are equal and opposite, the instantaneous axis is then at an infinite distance, and the effect of the combination is a motion of translation, the direction of which continually changes. A locomotive coupling rod (p. 112) is a common example which should be carefully con- sidered, as a useful illustration of the meaning of this theoretical proposition. Fig. 56b. Fig. 56b shows in skeleton the mechanism of the direct-acting engine already considered at length. In this case, C is a rotating piece connected as just described with both the sliding block D and the rotating crank B. In consequence of the first connection it has. an instantaneous centre K in the perpendicular through Z>, and in consequence of the second an instantaneous centre in the prolongation of the crank OP. Hence K, the intersection of these two lines, must be that centre, and with the same notation as before If all points in C lying in a plane perpendicular to the axis be joined to K, the corresponding instantaneous centre, the set of radiating lines may be considered as a diagram of velocities, but, as in the case of stress diagrams, it is generally far preferable to draw a separate diagram on some suitable scale. This may be done, as previously described on page 100, by selecting a pole and drawing Oa, Ob perpendicular and proportional to the velocities of a and b, two given points in (7. Two figures may thus be drawn, corresponding to the two positions into which the triangle Oab (Fig. 47a) may be turned by a rotation through 90. In the first, ab is parallel to the corresponding line in C, and points in the same direction ; the diagram CH. V. ART. 56.] LOWER PAIRING. 121 is now similar and similarly situated to the set of lines radiating from K. In the second, ab is also parallel to the corresponding line in C, but it points in the opposite direction, and the diagram may be described as "reversed." In plotting a point p in a reversed diagram which corresponds to a given point in C we have only to draw ap, f>p parallel to the corresponding lines in C. The pole 0, of course, always corresponds to the instantaneous centre K. 56. Diagram of Velocities in LinkworL A simple construction has already been given, by means of which the velocity-ratios of the parts of a slider-crank chain are . determined, and we will now consider this question for any case of linkwork in which the axes of the pairs are parallel. Fig. 57a represents a chain of links, zOabcd ..., united by pins so as to form a succession of turning pairs. The first link, Oz, is fixed, so Fig. 57a. that the second turns about a fixed point, 0, as centre, and therefore a moves perpendicularly to Oa, with a velocity V a , which we may suppose known. The other points, b, c, d ..., move in directions which we suppose given, and with these data it is required to find the magnitudes of the velocities. In Fig. 56b from a pole draw radiating lines perpendicular to the given directions, and set off on the first Oa to represent V a , then draw ab, be, cd ... parallel to the links of the chain to meet the corresponding rays, then the lengths of those rays represent the velocities, For drop a perpendicular ON from on to ab, or ab produced, then ON represents the component of V a in the direction of the second link, but this must also be the component of V b in that direction, since ab is of invariable length ; that is, Ob must represent V b . 122 KINEMATICS OF MACHINES. [PART n. Similarly all the other rays must represent the velocities of the corresponding points. The figure thus drawn may be called the Diagram of Velocities of the chain. It may be constructed equally well, if the magnitudes of the velocities be given, instead of their directions, also any of the turning pairs may be changed into sliding pairs. If both ends of the chain be attached to fixed points, the diagram will evidently be a closed polygon. Its sides, when divided by the lengths of the corresponding links of the chain, represent their angular velocities, for each side is the algebraical difference of the velocities of the ends of the link perpendicular to the link. In the four-link chain (Fig. 58a), consisting of two links turning about o fixed centres, a, d, coupled by a link be, the diagram of velocities is a simple triangle, Obc (Fig. 58b), the sides of which when divided by the lengths of the links to which they are parallel, represent the angular velocities of the links. Through a draw aZ parallel to cd, and prolong be to meet it in Z, and the line of centres in T 9 then, since the triangle Zab is similar to the triangle of velocities, the angular velocities of the levers cd, ab will be proportional to Za/cd and ab/ab. The last fraction is unity, and therefore we have .. Za aT angular velocity-ratio = y = -ry,, showing that the ratio in question is the inverse of the ratio of the distance of T from the centres. If, instead of the link ad being fixed, the chain of four bars be imagined to turn about one joint such as d, the diagram of velocities would be a quadrilateral Oab'c, with sides parallel to abed. Returning to the general case, let p be any point rigidly connected CH. v. ART. 56A.] LOWER PAIRING. 123 with one of the links of the chain, say cd, in the figure ; then if we lay down on the diagram of velocities a point p, similarly situated with respect to the corresponding line cd of that diagram, it follows at once, by the same reasoning, that the ray, Op, drawn from the pole 0, must represent the velocity of p in the same way that the other rays represent the velocities of the point a, b . . . . Thus it appears that for any linkwork mechanism, consisting of pieces of any size and shape connected by pin joints, the axes of which are parallel, a diagram may be constructed which will show the velocities of all points of the mechanism. By constructing the mechanism and its diagram of velocities for a number of different positions, curves of position and velocity may be drawn, such as those described in preceding articles for special cases. 56 A. Closure of Kinematic Chains. Dead Points in Linkwork. A kine- matic chain, like a pair (p. 95), may be "incomplete," that is, the relative moments of its links may not be completely defined. It then cannot be used as a mechanism without employing some addi- tional constraint, a process called "closing" the chain. In order that a chain may be closed it must be endless, and the number of links must not be too great ; for example, in a simple chain of turning pairs with parallel axes we cannot have more than four links. If there be five the motion of any one link relatively to the rest will not be definite, but may be varied at pleasure. So also a chain may be "locked" either by locking one of the pairs of which it is constructed; or by rigidly connecting two links not forming a pair ; it then becomes a frame, such as was considered in a previous part of this book. As an example of an incomplete chain may be taken the combina- tion of a sliding pair and a turning pair considered in Art. 55, and shown in Fig. 56a (p. 118). The relation between the sliding and the turning is here undefined until the chain is closed by the addition of another pair as in Fig. 56b, or in some other way. A chain is often incomplete or locked for special positions of its links, though closed and free to move in all other positions ; this, for example, is the case at the dead points which occur in most linkwork mechanisms. A well-known instance is that of the mechanism of the steam engine, in which the chain is locked and the direction of motion of the crank indeterminate when the connecting rod and crank are in the same straight line. This instance further shows that it is necessary to distinguish between the two directions in which motion may be transmitted through the mechanism, for the dead points in question 124 KINEMATICS OF MACHINES. [PART n. would not occur if the crank moved the piston instead of the piston the crank. A piece, then, which transmits motion is called a "driver," in relation to the piece in which motion is transmitted, which is called a "follower," terms which will be frequently used hereafter in cases where both pair with the fixed link. (Compare Arts. 47, 86.) The dead points in a mechanism may be passed either by the union of two similar mechanisms, with dead points in different positions, as in a steam engine with a pair of cranks at right angles, or by aid of the inertia of the moving parts. This last method involves what is called "force-closure," a term which will be explained presently. EXAMPLES. 1. The stroke of an oscillating engine is 6 feet, and the distance between the centre line of the trunnions and the centre of shaft is 9 feet. Find the maximum and mean angular velocity of the cylinder in each of its two oscillations as compared with that of the crank. Find also the velocity of the piston at half stroke as compared with the speed of piston. Ans. Maximum angular velocity-ratios of cylinder and crank, \ and J. Mean ,, '277 and *178. Velocity of piston at \ stroke _1'54 Mean speed of piston 1 2. The travel of the tool of a shaping machine is to be 9 inches, and the maximum return, three times the maximum cutting, velocity. The connecting link is horizontal in the extreme positions of the lever and is attached to a point in it which is on a level with the crank pin when the crank is upright, find the proper proportions of the quick return motion (Fig. 5, Plate I.). Find also the revolutions per minute for a maximum cutting velocity of 6 inches per second, and compare the times of cutting and return. Ans. The length of slotted lever =9 inches. Distance apart of centres = twice length of crank. Time of cutting_2 Revs, per minute of crank =19'1. Time of return 1 3. In Whitworth's quick return motion find the proportions that the maximum return may be three times the maximum cutting velocity, and compare the times of cutting and return. 4. In Example 1, draw curves showing the angular velocity and position of the piston for any position of the crank. 5. A reciprocating movement is given to the table of a small planing machine by a uniformly rotating crank below connected by a rod to a projecting arm so that the rod is horizontal when the crank is upright. Find by graphic construction the ratio of the times of a forward and a backward movement. Also find the velocity of the table in any position. 6. In question 1, p. 102, supposing two pairs of driving wheels coupled, the lengths of cranks 1 foot, find the velocity of the coupling rod in any position. First, relatively to the locomotive ; second, relatively to the earth. 7. In Ex. 6, p. 103, find in feet per second the maximum and minimum velocity of rubbing of the crank pin, assuming its diameter 12 in., and the revolutions 30 per 1'. Draw a curve showing this velocity in any position of the crank. 8. In Rapson's Slide (p. Ill), if the tiller be put over through an angle 6, show that the velocity -ratio of tiller and slide varies as cos 2 0, and draw a curve of velocity. 9. In a draglink-coupling the shafts are 6 in. apart, the draglink 1 foot long, and the cranks each 3 feet long. By construction, determine the four positions of the following OH. v. ART. 56A.] LOWER PAIRING. 125 crank when the leading crank is on the line of centres, and at right angles to the line of centres. 10. The length of the beam of an engine is three times the stroke. Supposing the end of the beam when horizontal is vertically over the centre of the crank shaft at a height equal twice the stroke, and the crank also is then horizontal, find the length of connect- ing rod and the extreme angles through which the beam will sway. Adjust the crank centre so that the beam may sway through 20 above and below the horizontal. Length of rod =2 '06 stroke. The beam sways 22$ above the horizontal, and 17 below. 11. The depth of the floats of a feathering paddle wheel is Jth the diameter of the wheel, and the immersion of the upper edge in the lowest position th the depth of the float. Assuming the stem levers -fths the depth of the floats, find the position of the centre of the collar to which the guide rods are attached. Determine the length of the rods, and draw the float in its highest position. If O be centre of wheel, K centre of collar, OK= "054 of diameter of wheel, and is horizontal (very approximately). Length of guide rods = 1 '01 radius of wheel. 12. In Ex. 9, find the angular velocity-ratio of the shafts when the cranks are in the positions mentioned. 13. In Oldham's coupling, show that the centre of the coupling disc revolves twice as fast as the shafts, and hence show how to give two strokes of a sliding piece for one revolution of a shaft. 14. In a simple parallel motion the length of the levers are 3 feet and 4 feet respect- ively, and the length of the connecting link is 2^ feet. Find the point in the link which most nearly moves in a straight line, and trace the complete curve described by this point as the levers move into all possible positions, the motion being set so that, when the levers are horizontal, the link is vertical. Ans. The required point in link is 17f in. from the 3-feet lever. 12fin. 4-feet 15. In a beam engine the stroke of piston is 8 feet, of air-pump 4^ feet, length of beam 24 feet, the front and back links of the parallel motion being 4 feet. Find the proper length of radius rod, and the point in the back link where the air-pump rod should be attached. Ans. Length of radius rod =8 feet 8g inches. Point of attachment of air-pump rod =2 ,, 3 ,, below beam. 16. Suppose in last question the parallel motion set for least deviation from a straight line, find the correct positions of the centre lines of air-pump and piston, and the position of the centre of motion, of the radius rod. A ns. Horizontal distances from centre of beam Line of stroke of piston, - 11 feet 8 inches. ,, air-pump, - 6 ,, 6| ,, Centre of motion of radius rods, 15 ,, 1^ 17. Draw diagrams of velocity (Art. 56) for any position of the mechanism (1) In the beam engine of Ex. 10 ; (2) In the quick return movement of Ex. 2 ; (3) In the Peaucellier parallel motion (Fig. 55). 18. In the last question, draw curves showing in case 1 the velocity of the piston for any position of the crank, and in case 2 the velocity of the tool at any point of the cutting and return strokes, assuming in each case that the crank rotates uniformly. REFERENCE. A good collection of linkwork and other mechanisms, some of which do not occur in the larger works cited on page 92, will be found in the later editions of Professor Goodeve's Elements of Mechanism. Much valuable information on the details of machine design is contained in a treatise on Machine Design by Prof essor W. C. Unwin, M.I.C.E. (Longman). CHAPTER VI. CONNECTION OF TWO LOWER PAIRS BY HIGHER PAIRING. SECTION I. TENSION AND PRESSURE ELEMENTS. 57. Preliminary Remarks. Tension Elements. If one of the elements of a pair be not rigid, or if the contact be not of the simple kind con- sidered in the preceding chapter, the pairing is said to be " higher," because the relative motion of the elements is more complex. Higher pairing is seldom employed alone ; it is generally found in combination with lower pairs, the elements of which it serves to connect. The most important case is that where a chain of two lower pairs is completed by contact between their elements or by means of a link which is flexible or fluid. Motions may thus be produced in a simple way which are impossible or difficult to obtain by the use of lower pairing alone. The present chapter will be devoted to mechanisms derived from chains of this kind, the fixed link being generally a frame common to the two lower pairs. The velocities of each of the pairs are thus the same as those of their moving elements. We commence with the case of non-rigid elements. A body which was incapable of resistance to any kind of change of form and size would of course be incapable of being used as part of a machine, for it could not furnish any constraining force whereby the motion of other pieces could be affected, but if it resists any particular kind of change it will supply a corresponding partial constraint which may be supplemented by other means. The first case we take is that of a flexible inextensible body, such as is furnished approximately by a rope, belt, or chain. This is called a Tension Element, being capable of resisting tension only, and it is plain that when any two points are connected by it, their distance apart, measured along the element itself, must be invariable so long as the rope remains tight. If the rope be straight, it may be replaced by a link, and we obtain the mechanisms already considered, but we now suppose it to pass over a surface of any form. [CH. vi. ART. 58.] CONNECTION BY HIGHER PAIRING. 127 In Fig. 59a, let A be a fixed body of any shape, round which an inextensible rope PQ passes, the ends hanging down. If P moves .2V+v Fig.59a. Fig.59c. P,-v-9v-< f.lq downwards with velocity V, Q moves upwards with the same velocity, the rope slipping over A at all points with velocity V. In practice A is generally circular, and is mounted on an axis, upon which it re- volves. We have then a " pulley block," of which A is the " pulley " or "sheave," and the rope causes it to rotate instead of slipping over it, but this makes no difference in the motion, and the only object of the arrangement is to diminish friction and wear. Next suppose the pulley movable (Fig. 59b), and imagine P attached to a fixed point, while Q moves upwards with the same velocity V relatively to A as before. Then A must move upwards with velocity V, because its motion relatively to the fixed point P is unaltered, and hence Q moves with velocity "IV. More generally, if P, instead of being fixed, moves downward with velocity v, Q must move upwards with a velocity 2V +v, or to express the same thing otherwise the difference of velocities of the two sides of the rope is twice the velocity of lifting a principle applicable to all questions relating to pulleys. The velocity of rotation cf the pulley is V+v, its radius being the "radius of reference " (Art. 46). The motion of rope and pulley may be repre- sented by a diagram of velocity. Thus, in Fig. 59c, describe a semi circle with radius equal to V+ v, then the radius of that circle represents the velocity of rotation or the velocity of any point in the rope relatively to the centre of the pulley. The actual velocity of any point K in the rope is found by compounding this with V, the velocity of the centre of the pulley. The pole of the diagram is therefore a point 0, distant V from the centre of the circle, so that if k be the point in the diagram corresponding to the point K of the rope, Ok represents the velocity of K. 58. Simple Pulley Chain. Blocks and Tackle. We have now a simple means of solving one of the most important problems in mechanism namely, to connect two sliding pieces with a constant velocity-ratio. 128 KINEMATICS OF MACHINES. [PART u. In Fig. 60a, B, C are pieces sliding in guides attached to a frame-piece A, thus forming two sliding pairs with one link common. In B a number of pins are fixed, and in A an equal number placed as in the figure, so that a rope passing round them as shown may form a number of plies parallel to B's motion.* The rope is attached at one end to C, and led to the nearest fixed pin, over a guide pin placed so that this part of the rope may be parallel * sa jj r B I .* V to C"s motion, while the other end is attached to a fixed point K. The effect of this arrangement is that when C moves in the direc- tion of the arrow, B also must move with a velocity which is readily found by the principle just explained, for the difference of velocities of the two parts of each ply must be the same, being twice the velocity of B. Thus reckoning from the fixed end, if B's velocity be V t the velocities of the several parts of the rope must be 0, 2P, 2F, 4P, 4F, 6F, 6F..., so that if there are n pins in B, the velocity of the other end of the rope must be 2nV, and the velocity-ratio 2n:l. The diagram of velocities consists of a number of semi-circles (Fig. 60b), the lower set struck with centre a and the upper with centre 0, where is the pole and Oa the velocity of lifting. The simple kinematic chain here de- scribed may be inverted, by fixing B or C instead of A. In the blocks and tackle so common in practice, the pins are replaced by movable sheaves, usually, but not always, of equal diameters, and placed side by side so as to rotate on the same axis. Some of the various forms they assume will be illustrated hereafter. The diagram of velocities shows that, if the diameters of the sheaves are proportional to the diameters of the circles shown in the diagram, they will have the same angular * This figure is taken, with some modifications, from the second edition (1870) of Willis's Mechanism. CH. vi. ART. 59.] CONNECTION BY HIGHEE PAIRING. 129 velocity, and may therefore be united into one, an idea carried out in White's Pulleys. In all cases the mechanism which we have been considering (Fig. 60a) is a closed kinematic chain only so long as the rope remains tight. One method of securing this would be to supply a second rope passing under another set of pins below B (not shown in the figure) and led to the other side of C by a suitably placed guiding pulley; we should then, by tightening up the ropes, have a self-closed chain similar to those considered in the preceding chapter. In practice, however, forces are applied to B and C which produce tension in the rope ; thus, for example, when employed for hoisting purposes, the weight which is being lifted keeps the rope tight. This is the simplest example of what is called force-closure, where a kinematic chain, which is not in all respects closed, is made so by external forces applied during the action of the mechanism. In practical applications the principle of force-closure is carried still further, for the guides which compel the pieces B and C to move in straight lines are usually omitted. In the case of B the weight and inertia of the load which is being raised or lowered supply sufficiently the necessary closure, while in the case of C the end of the rope may be guided by the hand. 59. Wheel and Axle. When mechanical power is employed for hoisting purposes, the end of a rope is frequently wound round an axle the rotation of which raises or lowers the weight, and this leads us at once to a different and equally important method of employing tension elements, namely, by attaching one end to a fixed point in the cylindrical surface of an element of a turning pair. The rope in this case passes over the surface and is guided by it, but does not slip over it as it does over the pins of the previous arrangement. The most useful case is that where the transverse section of the surface is a circle, and the direction of the rope always at right angles to the axis of rotation ; then it is clear that the motion of the surface is the same as the motion of the rope. The well known Wheel and Axle is a combination of two chains of this kind. In its complete ideal form it consists of two sliding pairs AB, AD, with planes parallel and one link A (Fig. 61) common. A rope is attached to D and, passing partty round a wheel, is attached to it at a fixed point K in its circumference ; a second rope is attached to B, and passing partly round an axle, is attached to a fixed point k in its circumference, the two ropes lying in parallel planes. The wheel .and axle are fixed together, and form with A the turning pair AC. We have thus a second means of connecting two sliding pieces so that C.M. I 130 KINEMATICS OF MACHINES. [PART 11. Fig.61. their velocity-ratio may be uniform, for the velocities of B and D* must be inversely as the radii of the wheel and the axle. As before,. the ropes must be kept tight, also the guides of the pieces B and D may be omitted and replaced by force-closure, and this will be necessary if the wheel is to make more than one re- volution, for then a lateral movement is required to enable the rope to coil itself on the surfaces. In practical applications the second rope is generally omitted and the wheel turned by other means ; the lateral movement is sometimes provided for by permitting the axle to move endways in its bearings, but more often, in cases where the load is not free to move laterally, the effect of a moderate inclination of the rope to the axis is disregarded. We may, however, escape this difficulty by the use of force-enclosure of a different kind. Instead of attaching the rope to a fixed point in the surface, let it be stretched over it by a force at each end, there will then be friction between the rope and the surface, which will be sufficient to prevent slipping if the tendency to slip be not too great. The Differential Pulley is a good example of the application of these principles. As is shown in Fig. 62, there are two blocks, of which the upper, which is fixed, carries a compound sheave, consisting of two pulleys A and C, of somewhat different diameters, fixed to one another. The lower block carries a single sheaf B, the diameter of which should theoretically be a mean between those of A and (7, in order that the chain may be vertical. The chain is endless, and passes round the pulleys in the manner shown, so that when the side P is hauled downwards with a given velocity V, it will raise the lower block B with a velocity which we will now determine. In passing around A and C the chain is not capable of slipping. To ensure its non-slipping the periphery may be recessed to fit the links of the chain. In passing around B the slipping is immaterial - Fig.62. CH. vi. ART. 60. J CONNECTION BY HIGHER PAIKING. 131 the raising of B would take place with the same velocity, whether there were an actual slipping of the chain round the circumference, or whether B were a rotating pulley. When the point P is hauled downwards with velocity F", it neces- sitates the rotation of A, and with it of C. Thus the left-hand portion of the chain passing round B will be hauled upwards with the same velocity as the point P downwards, and the right hand will descend with a velocity which is less in the ratio of the radii, c, a, of the united pulleys, and thus on the whole there will be an ascending motion given to B. Now, since the upward velocity of B is half the difference between the velocities of the two portions of the chain, Thus, by making the difference between a and c small, the relative velocity of B to P may be made as small as we please. This apparatus, in a somewhat modified form, is much employed. It is called Weston's Differential Pulley Block, and possesses the valuable property that the weight will not descend when the hauling force is removed, for reasons which will be explained hereafter (Ch. X.). 60. Pulley Chains with Friction Closure. Belts. A tension element may also be employed to connect the elements of two turning pairs. The most important case is that where two shafts are connected by an endless belt passing over a pair of pulleys and stretched so tightly that the friction between belt and pulley is sufficient to prevent slipping. If the belt were absolutely inextensible the speed of centre line of the belt would be the same at all points, and therefore the angular velocities of the pulleys would be inversely as their radii each increased by half the thickness of the belt. This mode of connection is unsuitable where an exact angular velocity-ratio is required, for even though the belt may not slip as a whole, yet it will be seen hereafter (Ch. X.) that its extensibility causes a virtual slipping to a greater or less extent. In the case of leather belts, the error in the angular velocity-ratio due to this cause is said to be about 2 per cent. There are two ways in which the belt may be wrapped around the pulleys, being either crossed or open. If the belt is crossed, the pulleys will revolve in opposite directions. The crossed belt embraces a larger portion of the circumference of the pulleys than the open belt, and there is thus less liability to slip. There is a proposition of some importance connected with the length of a crossed belt,. which it will be useful to give here. 132 KINEMATICS OF MACHINES. [PART n. A C and BD (Fig. 63) being radii, each drawn at right angles to the straight portion of the belt CD, will each make the same angle 6 with the line of centres. Hence the por- tion of the belt in contact with the pulley A = (2ir -26) r A and that in contact with the pulley B=(2ir - 26) ?> The length not in contact=2 . CD=2(r A + r B ) tan 0. Thus whole length of belt = 2(-rr - 6 + tan 6)(r A + r B ). But cos 6 = (r A + r B )lA B, and consequently, if the distance AB between the centres is a constant quantity, and if, further, the sum of the radii r A + t* B is constant, then the angle 6 will be constant. That being so, the total length of the belt will be a constant quantity. This property is made use of when it is desired to connect two parallel shafts with an angular velocity-ratio, which may be altered at pleasure. A set of stepped pulleys, such as are shown in Fig. 1, Plate III., are keyed to each shaft, and the belt being shifted from one pair to another of the pulleys, the angular velocity-ratio is altered at will. If the belt is crossed, then the same belt will be tight on any pair of pulleys, if the sum of the radii is the same for each pair. This does not hold good for open belts. The actual length o , belt required in any given example is best found by construction. The tightness of the belt necessary to effect closure by friction of this kinematic chain may be produced simply by stretching the belt over the pulleys so as to call into play its elasticity, but the axis of rotation of one pulley is sometimes made movable, so that the belt may be tightened by increasing the distance apart of the shafts, while in other cases an additional straining pulley is provided. The belt may then be tightened and slackened at pleasure, a method frequently used in starting and stopping machines. In order that the belt may remain on the pulleys they must be pro- vided with flanges, or, as is more common in practice, they must be slightly swelled in the middle, for when the shafts are properly in line, a belt always tends to shift towards the. greater diameter. Great care, however, is necessary in lining the shafts that each side of the belt lies exactly in the plane of the pulley on to which it is advancing. Thus, for example, if the shafts be in the same plane,, they must be exactly parallel, otherwise the belt will shift towards the point of intersection. This remark, however, does not apply to the receding side of the belt, and the shafts may make a considerable angle with each other, subject to the above restriction. Friction-closure is always imperfect, because the magnitude of the CH. vi. ART. 62.] CONNECTION BY HIGHER PAIRING. 133 friction is limited, but this is often a great advantage, since it permits the chain to open when the machine encounters some unusual resist- ance, which would otherwise produce fracture. By the use of grooved pulleys provided with clips the friction may be increased to any extent, so that great forces may be transmitted, but these devices are only suitable for low speeds, as in steam-ploughing machinery. Slipping may be avoided altogether by the employment of gearing chains, the links of which fit on to projections on the pulleys ; force- closure is here replaced by chain-closure, and the action is in other respects analogous to toothed gearing. The speed is limited, as will be seen hereafter. 61. Shifting of Belts. Fusee Chain. By the use of drums of considerable length as pulleys, the belt may be shifted laterally at pleasure. This principle is much employed in practice, as for example (1) To stop and set in motion a machine. The drum on one of the shafts is divided into two pulleys, one fast and the other loose on the shaft. (2) To reverse the direction of motion. The drum is divided into three pulleys, the centre one fast, the two end ones loose on the shaft. Two belts, one crossed and the other open, are placed side by side. By shifting the belt either is made to work on the fast pulley at pleasure. (3) To produce a varying angular velocity-ratio. The drums are made conical instead of cylindrical. The fusee employed in watches to equalize the force of the main spring is a common example. The kinematic character of these devices will be considered in the next chapter. 62. Simple Hydraulic Chain. Employment of Springs. Incompres- sible fluids may be employed to connect together two or more rigid pieces forming a class of elements which may be called "pressure elements," since they are capable of resisting pressure only. The pressure must be applied in all directions, and the fluid must therefore be enclosed in a chamber which pairs with the different pieces to be connected. For constructive reasons lower pairing must generally be adopted, and almost all cases are included in the following investigation. Suppose two cylinders, each fitted with a piston (A and B in 134 KINEMATICS OF MACHINES. [PART n. Fig. 64), to be connected by a pipe, the space intervening between the pistons being filled with fluid. Then when the piston B moves downwards with velocity v, the piston A will rise with velocity V, which is easily found by considering the spaces traversed by the two pistons in a given time. Let A, B be the areas of the pistons, a, b the spaces traversed, then, since the volume of the fluid remains the same, we must have Aa = Bb, and therefore, 7 a 3 ~v~b~A' The chain here considered, in which the elements of two sliding pairs are connected by a fluid, is kinematically identical with the arrangement of Fig. 60a, p. 128, the replacement of a tension-element by a pressure-element constituting merely a constructive difference between the mechanisms. In the hydraulic press, in pumps, in water- pressure engines driven from an accumulator, and in other cases this kinematic chain is of constant occurrence, and will be frequently referred to hereafter. Combinations of an hydraulic chain with blocks and tackle are common in hydraulic machinery. Some examples will be found in Chapter XX. Springs, compressible fluids, and even living agents, are employed in mechanism, not only in a manner to be explained hereafter as a source of energy, by means of which the machine does work, but also in force- closure, and especially for the purpose of supplying the force necessary to shift pieces which open and close, or lock and unlock kinematic chains, and so produce changes in the laws of motion of the mechanism. The force of gravity, which, as has already been shown, frequently produces closure, should be regarded as the tension of a link of in- definite length connecting the frame-link of the mechanism with the link we are considering. The inertia of moving parts likewise gives rise to forces which are not unfrequently applied to similar purposes. Examples will be given in a later section. EXAMPLES. 1. A shaft making 90 revolutions per minute carries a driving pulley 3 feet in diameter, communicating motion by means of a belt to a parallel shaft, 6 feet off, carry- ing a pulley 13 inches diameter. Find the speed of belt and its length 1st, when crossed, and 2nd, when open. Find also the revolutions of the driven shaft, allowing a slip of two per cent. Speed of belt =847 '8 feet per minute. Length when crossed =19 feet 2 inches. open =18 8 Revolutions of the follower =244|. 2. Construct a pair of speed pulleys to give two extreme velocity -ratios of 7 to 1 and CH. vi. ART. 63.] CONNECTION BY HIGHER PAIRING. 135 3 to 1, and two intermediate values. The belt is to be crossed and the least admissible diameter is 5 inches. Velocity-ratios - - |- l l ^ |, Diameter of pulleys {J ^| J 3. The diameters of the compound sheave of a differential pulley block are 8 inches and 7 inches respectively ; compare the velocities of hauling and lifting. Velocity-ratio = 16 to 1. 4. In a pair of ordinary three-sheaved blocks compare the velocity of each part of the rope with the velocity of lifting. 5. In a hydraulic press the diameter of the pump plunger is 2 inches and that of the ram 12 inches, determine the velocity-ratio. Ans. 36. SECTION II. WHEELS IN GENERAL. 63. Higher Pairing of Rigid Elements. We next consider pairs of rigid elements in which the relative motion is not consistent with continuous contact over an area. The elements then touch each other at a point or along a line which is not fixed in either surface, but continually shifts its position. The form of the surfaces is not then limited as in lower pairing, but may be infinitely varied, with a corresponding variety in the motion produced. This kind of pairing occurs when a chain of two lower pairs is com- pleted by simple contact between their elements. In the double slider- crank chain shown in Fig. 4, Plate II., of the last chapter, let us omit the block C arid enlarge the crank-pin so as just to fill the slot. By so doing the relative motions of the remaining parts will be unaltered, but we shall have three pairs instead of four, the turning pair EG and sliding pair CD being replaced by a single higher pair ED. This process is called Reduction of the chain, and when higher pairing is admissible the reduced chain serves the same purpose as the original, but with fewer pieces. The crank-pin and slot are in contact along a line only which during the motion continually shifts its position. In practice, the elements not being perfectly rigid, the contact extends over an area, but this area is of very small breadth, and consequently, if heavy pressures are to be transmitted at high velocities, the wear is excessive. If we trace the development of pieces of mechanism we observe that in the earlier stages higher pairing is much employed for the sake of simplicity of construction, but is gradually replaced by lower pairing. Nevertheless, where the object of the machine is mainly to transmit and convert motion rather than to do work, or where the velocity of rubbing is low, higher pairing may be employed. In many cases it is necessary, because the required motion cannot be produced by any simple combination of lower pairs. 136 KINEMATICS OF MACHINES. [PART n. Higher pairing of rigid elements may be divided into two classes according as the surfaces in contact do or do not slip over one another, just as in the case of tension elements considered in the last section. In the first case the contact is spoken of as Sliding Contact and in the second as Rolling Contact. In rolling contact the difficulty of wear does not occur, and friction is greatly reduced, so that it is always used when possible. When a roller rests on a hard plane surface the points in contact lie on a line which, if there be no slipping, remains for an instant at rest as the roller moves. On reference to Art. 55, page 118, it will be seen that the motion of the roller is completely represented by a turning about this instantaneous axis, the point K (Fig. 56a) being in this case on the periphery of the wheel of which R is now the radius. The same is true when one circle B rolls within or without another fixed circle A, a case to be considered further on : the motion at B at the instant is a simple rotation about the point of contact. We first however consider the simple and important case in which both surfaces move, the line of contact being fixed. 64. Polling Contact. Rolling contact may be employed for the communication of motion between two shafts, the centre lines of which are either parallel or intersect, by means of surfaces rigidly attached to the shafts. In the first case the surfaces are cylindrical and in the Fig-.64b. Fig.64a. b e second conical, the apex of the cone being the intersection of the shafts. By far the most important case, and the only one we shall CH. vi. ART. 64.] CONNECTION BY HIGHER PAIRING. 137 here consider, is that in which the transverse sections of the surfaces, are circular. Portions of the surfaces are used, as in Figs. 64a, 64b, and are pressed together by external forces, so that sufficient friction is produced to prevent the slipping of the surfaces. In other words, force-closure is necessary, as in the case of connection by a belt. This being supposed, it will immediately follow that the velocity of the two surfaces at the points of contact is the same, and hence, as before, the angular velocity-ratio of the shafts is inversely proportional to the radii of the wheels. In the case of intersecting shafts, the surfaces are frustra of cones called " bevel," or, if the semi-angle of the cone be 45, " mitre-wheels," and their radii may be reckoned as the mean of that at the inner and outer periphery. The shafts revolve in opposite directions, unless one of the surfaces be hollow so that the other may be inside it, in which case the corresponding wheel is said to be " annular." When it is inconvenient to use an annular wheel, the same result may be obtained by transmitting the motion through an inter- mediate or " idle " wheel. If the radius of a wheel be infinite, it becomes a " rack," and the surface a plane. In the case of bevel wheels the corresponding cones may be found r when the centre lines of the shafts and the angular velocity-ratio are given, by a simple construction. In Fig. 64b, let OA, OB be the centre lines of the shafts, and let distances Oa, Ob be marked off* upon them in the ratio of the required angular velocities. Complete the parallelo- gram Oacb, then OC must be the line of contact of the required cones. For drop perpendiculars cm, en, on OA, OB, then cm sin aOc Ob en sin bOc Oa 1 so that the radii of any frustra of the cones employed for wheels will be inversely as the angular velocities of the shafts. The particular case may be mentioned in which one of the cones becomes a plane ; the corresponding wheel is then a " crown " or " face " wheel. The shaft of a wheel which is to work correctly with a crown wheel must be inclined to the plane of that wheel at an angle depend- ing on the angular velocity-ratio required, a restriction not generally attended to, especially in the earlier stages of machinery in which face wheels were of common occurrence. If, as generally happens, it is required to transmit a working force of a considerable amount, then the friction between the two circumferences will be found not to be sufficient to prevent slipping taking place, unless a considerable pressure to force the shafts together is employed, which involves an excessive friction on the bearings. In what is 138 KINEMATICS OF MACHINES. [PART n. known as " frictional gearing," this is partially avoided by the use of wheels with triangular grooves fitting each other as the thread of a screw fits into its nut ; but, in general, to prevent slipping, teeth are , I ~z = aT =A '^T' CH. vii. ART. 83.] MECHANISM IN GENEEAL. 167 Similarly supposing A fixed, v velocity of d _ ~z = aT ' from which it appears that A dT agree with those obtained in the article cited by a Fig.81. results which different method. The centrodes in this case, as well as in that of the four-link chain from which it was derived by reduction, may be traced graphically by plotting the position of T for a number of positions of the pieces, but they are known curves only in exceptional cases such as those of Art. 80, and generally have infinite branches which render their use inconvenient. When the point P lies on the line of centres it coincides with T 7 , and the velocity of rubbing is zero ; the centrodes are then no other than the profiles themselves of A and B. The curves are then said to roll together : a particular example is that of the equal ellipses of Art. 80 which are not unfrequently used to connect two revolving shafts with A r ariable angular velocity-ratio. In this case the velocity-ratio is the ratio of the focal distances of the point of contact, but by properly determining the profiles it is theo- retically possible to give any velo- oity-ratio to the shafts at pleasure. The question, however, is not one of much practical interest. 83. Construction of Centres of Cur- vature of Profiles Willis's Method. In the four-link chain A BCD shown in Fig. 81, D is the fixed link and B the coupling link : a, b, c, d are sec- tions of the axes of the pairs which are supposed parallel. If the coupling link be be pro- longed to meet the line of centres ad in the point t, and ab to meet cd in 0, it appears as in previous cases that must be the instantaneous centre of B, and that the angular velocity-ratio of A and C is dt : at. Join Ot, and imagine bt an actual 168 KINEMATICS OF MACHINES. [PART n. prolongation of the bar be, so that t is rigidly connected with it, then f s motion will be perpendicular to Ot. Suppose now that the propor- tions of the links are taken so that Ot is perpendicular to bt, then t moves in the direction of the length of the rod, and the rod therefore may be imagined to slide through a fixed swivel at t. This reasoning shows that the levers A and (7, when in this position, will move for a short interval with uniform angular velocity-ratio, and the movement of a pair of wheels in gear is thus imitated by a linkwork mechanism. Let us now form a reduced chain by omission of the coupling-link, and we shall be able to solve the important problem of finding a pair of circular arcs which will serve for the profiles of a pair of teeth in contact. For this purpose, with centres b and c, strike arcs through any point p on cbt produced, and let these arcs be rigidly connected with A and C respectively ; the coupling-link may now be removed and A imagined to drive C by direct contact of the arcs. Evidently wherever p is, the pieces will move for the moment with uniform angular velocity-ratio and pitch point t. The uniformity, however, is only momentary, because the position of changes, and to trace the profiles with accuracy it would be necessary to perform the construction for a succession of positions of cbt, hence the face and flank of a pair of teeth in contact cannot be exactly represented by a pair of circular arcs. When it is sufficiently approximate to do so, the arcs are found by assuming a mean position for the point p, and the mean value for the obliquity i, found by experience to give good results. The method here described was invented by the late Professor Willis, and the value of i recommended by him was sin" 1 '25, or about 14J, being about the actual mean value of the obliquity in cycloidal teeth of good proportions. Also the value of pt was taken by him as half the pitch, p being then about midway between the pitch point t and the point of the tooth. Having made these assumptions, it still remains to fix the position of the point 0, which may be taken anywhere on a line through t inclined at 14J to the line of centres. This is done by observing that must be the same for all wheels D intended to work with a given wheel A, and that teeth never should be undercut (Art. 70); that is, c and b must lie on the same side of t. Hence in the smallest wheel intended to work with A, c is at infinity, so that if d is its centre, d is parallel to pt, and therefore perpendicular to Ot. The flank of the tooth in this case becomes a radius d p. The position of is thus completely determined for all the wheels of a set when the pitch is given. CH. vii. ART. 85.] MECHANISM IN GENERAL. Willis's method is of great theoretical interest, and has consequently been given here, but the form of teeth obtained is not always suffi- ciently approximate. It may, therefore, with advantage be replaced by other methods, as to which the reader is referred to a work by Professor W. C. Unwin on Machine Design. 84. Sphere Motion. When a body moves about a fixed point it& motion is completly represented by that of a portion of a spherical shell of any radius which fits on to a corresponding sphere, and moves on it just as in the case of plane motion. Everything which has been said respecting plane motion also applies to sphere motion, but the axoids are conical instead of cylindrical surfaces, the centrodes spherical instead of plane curves, and all straight lines are replaced by great circles of the sphere on which the motion is imagined to take place. The corresponding crank chains are called " conic n crank chains, the axes of the pairs lying on a cone instead of a cylinder. 85. Screw Motion. In the plane motion of two pieces, endways motion of the cylindrical axoids is supposed to be prevented by some suitable means. Let us now remove this restriction and imagine the axoids to slide endways, while continuing to roll together, the relative movement will now not be completely defined, but additional constraint will be required. In the first place take the case of a lower pair in which the axoids are coincident straight lines ; if endways sliding be permitted we obtain an incomplete pair, unless the nature of the surfaces in contact define the relation between the endways motion and the rolling motion. In the simple screw pair the two are in a fixed ratio, in the screw cams of Art. 78 they have a varying ratio. In every case of non-plane motion with cylindrical axoids, not only must the axoids be given, but also a connection between the endways sliding and the motion of rotation. In the most general case possible the instantaneous axis changes its direction as in spherical motion, its position as in plane motion, and in addition there may be an endways sliding. This is expressed by the rolling and sliding of certain surfaces on one another, which are now neither cylindrical nor conical. These surfaces are in all cases of the kind known as "ruled" surfaces, being generated by the motion of a straight line, along which they touch each other. The surfaces are still called Axoids, and the line is in the Instantaneous Axis. The hyperboloidal pitch surfaces for wheels connecting two shafts which do not intersect are examples of this kind; but for the discussion 170 KINEMATICS OF MACHINES. [PART n. of this question, which is not of very common occurrence, the reader is referred to the works already cited. 86. Classification of Simple Kinematic Chains. On observing the action of any mechanism, several of the pieces of which it is con- structed may be readily distinguished as having functions different from the rest. These pieces, like the rest, occur in pairs, and may be described as such, though the pairing is not necessarily kinematic. First, one or more perform the operations which are the object of the mechanism ; these may be called the Working Pairs, as, for example, the tool and the work in machine tools, the weight raised and the earth in the hoisting machines. Second, one or more form the source from which the motion is transmitted, as, for example, the crank handle and frame of a windlass, the piston and cylinder of a steam engine. These may be called the Driving Pairs. Thirdly, various sudsidiary working pairs carry out various operations incidental to the working of the machine. The object of the mechanism is always to convert the motion of the driving pairs into that of the working pairs. The simplest case is that in which the motion has only to be trans- mitted without alteration; a single pair will then suffice. Thus by means of a long rod sliding in guides or turning in bearings, a motion of translation or rotation may be transmitted to a distance only limited by n on-kin ematical considerations. By use of flexible elements among which should be included the flexible shafts recently intro- duced the direction may be altered at pleasure and any desired position reached. If, however, the magnitude of the motion is to be altered, a mechanism must be employed in which at least one element of the driving and working pairs is different. The driving pairs are usually kinematic ower pairs, and the working pairs are so very frequently, and this is why so many of the simplest and most important mechanisms are examples of the connection of lower pairs. The peculiar motions of lower pairs being translation and rotation, a number of mechanisms may be classed as examples of the conversion of rotation into translation or rotation and conversely, with uniform or varying directional relation or velocity- ratio. This is especially the case when, as so frequently happens, the driving and working pairs have a common link which is fixed. It has been shown, however, that many apparently different me- chanisms are in reality closely connected, being derived from the same kinematic chain. Mechanisms are therefore to be classed according to the kinematic chains to which they belong. The number of simple chains actually employed in mechanism is limited by the preceding ) = & Thus = __ P~~ cos< ~ smOTB~~OB' That is, the crank effort is to the steam pressure as the intercept OT is- to the crank arm OB. But we have previously shown (see p. 101) that this fraction expresses the velocity-ratio of piston to crank pin ; hence we have again found in this case that the force-ratio is the reciprocal of the velocity ratio, and the curve which we previously drew to represent the varying velocity of the piston, the crank pin moving uniformly, will represent also the varying crank effort, the pressure of the steam on the piston being uniform throughout the stroke. It is therefore described as the Curve of Crank Effort. (3) The same thing may be proved to be true for every mechanism,. the forces acting on which balance one another. In some cases it may be easier to determine the force-ratio than the velocity-ratio or vice versa* In any case, either may be inferred by taking the reciprocal of tte other. As an additional example take the case of two pieces driving one another by simple contact (Fig. 92). We have already found the velocity -ratio by a direct process (p. 152), but we may also determine it in the following way. When A presses on B there is a resistance R equal and opposite to the pressure, and normal to the portions of the surfaces in contact, if we suppose no friction to exist. Drop 192 DYNAMICS OF MACHINES. [PART m. perpendiculars p A and p B on the common normal. Then the moment R -^^ of the driving pressure R which A exerts on E or the turning moment due to A=M A = Rv.. Similarly the moment of the resisting force which B exerts on A or the moment of re- sistance to turning which B opposes to A = M B = Rp B . Hence it appears that Driving moment __M A p A Resisting moment M B ~~ p B But we have previously proved that this fraction is the angular velocity-ratio of the piece B to the piece A, and thus we show that the moment-ratio is the reciprocal of the angular velocity ratio. 96. Periodic Motion of Machines. One of the most essential char- acteristics of a machine is the periodic character of its motion. Each part goes through a cycle of changes of position and velocity and returns periodically to its original place. When moving steadily the periods are equal and the velocity of each piece is the same at the beginning arid end of each period. That this may be the case it is not necessary that the driving effort should balance the working resist- ance in every position ; on the contrary, this seldom happens ; it is sufficient if the mean effort be equivalent to the mean resistance, or as we may otherwise express it, Energy exerted during a period = Work done in the period ; a condition which always governs the action of a machine in steady motion. In reckoning the energy and work the action of gravity on any piece of the machine may be omitted, for, if the piece rise through any height during one part of the period, it will fall through an equal height during another part. The work done consists partly of the work which the machine is designed to do, and partly of frictional resistance to the relative motion of the parts of the machine, or, in other words, of Useful Work and Waste Work. The ratio of the useful work to the energy exertecL is called the Efficiency of the machine and its reciprocal the Counter-Efficiency. The efficiency of a machine depends partly on the kind of machine and partly on the speed, as will be explained in the chapter devoted to frictional resistances (Chap. X.). In estimating the power required to drive a machine a value is assumed for the efficiency derived from experience of machines of the same or nearly the same type. Examples will be given hereafter. CH. viii. ART. 97.] PRINCIPLE OF WORK. 193 97. Power. Sources of Energy. The sources of energy are (1) Living agents; (2) Gravity acting usually by means of falling water ; (3) Springs and elastic fluids ; (4) Gunpowder and other explosive agents. The energy thus derived may be traced further back to the action of heat and chemical affinity, and we may add to the list electric and magnetic forces, but the foregoing is a sufficient statement for our pre- sent purpose. In general, the motion and effort which are proper to the source of energy, and to which the driving pair must be adapted, are entirely different from the motion and the resistance necessary in the working pair. Besides which the work will generally be required to be done at various places more or less distant from the source of energy. To connect the source and the work mechanism is therefore necessary, which (1) receives energy from the source and converts into a form suitable for transmission and distribution ; and (2) receives the trans- mitted energy and adapts it to the work to be done. The same machine may serve both these purposes, especially when a living agent is the source of energy, as in a crane worked by hand, a sewing machine or a lathe driven by the foot. But in most cases distinct machines are employed, one of which receives energy directly from the source, and is described as [a Prime Mover, or more briefly a Motor, while the rest receive energy from the motor, either directly or by a train of connecting mechanism, and adapt it to the work. A machine then effects something more than mere transmission of energy ; it is directly connected with the source or the work, and converts the energy it receives into a form in which it can be utilized. Thus in a factory the engine is a machine which adapts the energy of the steam to the purpose of driving a shaft ; the loom or the mule are machines which adapt the energy transmitted to them to the purposes of weaving or spinning, but the train of belt or wheel gearing distributing the energy through the factory is not a machine, for it is employed solely for transmission purposes. Theoretically the connection between the source of energy and the work might be effected by a single machine ; the separation into distinct machines connected by a transmitting train is simply an augmentation (p. 139) adopted for constructive reasons. The variety of movements of which a living agent is capable renders the separation less necessary. The rate at which energy is exerted is called Power; it is this which measures the value of a source of energy and the expense of the work which is being done. The ordinary unit of measurement is C.M. N 194 DYNAMICS OF MACHINES. [PART in. the conventional horse-power of 33,000 foot-pounds per minute, or 550 per second, a quantity greater than the working power of an ordinary draught horse on the average of a day's work, except under the most favourable conditions (see Appendix). The unit of power employed universally on the Continent is somewhat less, being 75 kilogrammetres per second or 32,550 foot-pounds per minute. In measurements of electrical power the "watt" is often used; 1000 watts, a quantity also known as a "kilowatt," being 1'34 horse- power. For small powers the watt is a convenient unit. In prime movers the effort may generally be regarded as applied at a point which moves with a known mean velocity ; then the horse-power is given by the equation PV TT p _ r y ~ 33,000' where P is the mean value of the effort in Ibs. and V the mean velocity in feet per minute. In machines driven from a prime mover the effort is generally a moment M which exerts the energy MSir in every revolution of a driving shaft. We then have HP =- 33,000' where M is the mean moment and n the revolutions per minute. 98. Reversibility. Conservation and Storage of Energy. The resistance overcome at the working point may be either frictional as in machine tools or reversible as in machines for raising weights. In the second case, if the machine were stopped and set in motion in the reverse direction it would, if friction could be neglected, work equally well, the driving effort and working resistance would be interchanged, and constructive modifications might be required, but otherwise the action is unaltered. This may be described by saying that the machine is Reversible. Many machines actually occur in both their direct and their reversed forms ; thus a pump is a reversed hydraulic motor. Hence it appears that in reversible machines the power of doing work, that is to say, energy, is not lost after being exerted, for by reversing the machine it may be employed a second time. Thus it is that we describe the action of reversible machines as a transfer of energy, and are led to conceive of energy as indestructible, and speak of it as if it were independent of the bodies through which it is manifested. No machine, indeed, is completely reversible, for in all cases frictional resistances occur to a greater or less extent, while many machines are CH. vin. ART. 98.] PKINCIPLE OF WORK. 195 completely non-reversible ; but we shall see as we proceed that even then energy is not lost but only converted into another form, so that we have in reversible machines the first and most simple example of the great natural law called the Conservation of Energy. The import- ance of reversibility as a test of maximum efficiency will be seen more fully hereafter. Again, we can store up energy and use it as required when it is inconvenient to resort to any of the usual sources. For example, by a few turns of the watch key we store energy in the mainspring which is supplied at a regular rate to the watch throughout the day. So the hydraulic accumulator (Part V.) receives energy from the pumping engines and supplies it at irregular intervals to the hydraulic machines which lift weights and move gates in a dockyard or work the guns in a ship of war. A large part of what follows in the present work is merely a development of what has been said here : in the succeeding chapters of the present division we consider machines comprising solid elements only, while in a future division we shall consider the transmission and conversion of energy by means of fluids. The simpler machines are treated in much greatea detail with numerous additional examples in a smaller treatise by the author of this work and Mr. J. H. Slade.* EXAMPLES. 1. A waggon weighs 2 tons and its draught is ^th of its weight. Find the work done iu drawing it up a hill 1 in 20, half a mile long. Find also how long three horses will take to do it, supposing each horse to work at the rate of 16,000 foot-pounds per minute. Work done = 370 ft. -tons. Time occupied = 17' 15". 2. A force of 10 Ibs. stretches a spiral spring 2", find the work done in stretching it successively 1", 2", 3", etc!, up to 6". Ans. 2J, 10, 22, 40, 62J, and 90 inch-lbs. 3. Find the H.P. required to draw a train weighing 200 tons at the speed of 40 miles an hour on a level, the resistance being estimated at 20 Ibs. per ton. Find also the speed of the train up a gradient of 1 in 100, the engine exerting the same power. Ans. H.P. required =426. Speed up the iodine =18*87 miles per hour. 4. The resistance of H.M.S. "Iris " at 17 knots is estimated at 40,000 Ibs., what will be the H.P. required simply to propel the ship? Find also in inch-tons the moment, on each of the twin screw shafts, equivalent to this power, the revolutions being 80 per minute. Ans. H.P. required =2088. Moment on each shaft =367 inch-tons. 5. The curve of stability of a vessel is a common parabola, the angle of vanishing stability 70, and the maximum moment of stability 4,000 ft. -tons. Find the statical and dynamical stabilities at 30. Ans. Statical stability = 3 '918 ft. -tons. Dynamical stability = 1 '283 ft. -tons. 6. Verify the principle of work, neglecting friction, in : (a) The differential pulley (Art. 59). (6) A pair of 3-sheaved blocks, (c) The hydraulic press (Art. 62). * Lessons in Applied Mechanics. Macmillan. 1891. 196 DYNAMICS OF MACHINES. [PART in. 7- From the results in question 6, p. 103, deduce the crank efforts for the given posi- tions of the piston and the mean crank effort, supposing the effective steam pressure on the piston 20 tons and neglecting friction. Crank effort at (forward stroke =18 '4 tons. Mean = 1274 tons, quarter stroke in the \ backward ,, =16 '6 tons. 8. Show that the efficiency of a machine is equal to the velocity-ratio multiplied by the force-ratio. SECTION II. UNBALANCED FORCES (KINETICS). 99. Kinetic Energy of Translation. Sliding Pair. We now proceed to consider the cases in which efforts or resistances arise from the changes of velocity of the parts of a system, which changes thus become a source of energy or require energy in order to produce them. The commonest observation is sufficient to show the importance of such cases : a cannon ball possesses a great power of doing work, and a railway train requires energy to be exerted by the steam to obtain the requisite speed, quite irrespectively of that necessary to maintain the speed when once produced. First, suppose a weight under the action of gravity only. Unless it be supported by a vertical force exactly equal to the weight it will fall with a gradually increasing velocity. Let it be wholly unresisted by external bodies, let it start from rest and fall through a height h, then, whatever the material, we know that it will acquire a velocity v given by the formula v* = 2gh, where g is a number measuring the acceleration of the weight which, for velocities in feet per second, ranges from 32-1 at the equator to 32 '25 at the pole, and having intermediate values at other points on the earth's surface according to the intensity of gravity at the point. The average value 32*2 is usually adopted for this important constant, and the height h is called the " height due to the velocity." During the whole fall, the weight W of the body has been exerting an effort upon it which overcomes an equal resistance occasioned by the change of velocity which is taking place ; thus an amount of energy has been exerted, and an amount of work done equal to Wh. Resistance of this kind is of the reversible kind, for if we imagine the weight, after reaching the ground, projected up again with the same velocity, it will, if not otherwise resisted, attain the height from which it originally fell. Hence we describe the weight as possessing energy, and the amount it possesses when moving with velocity v is CH. vin. ART. 99.] PRINCIPLE OF WORK. 197 Energy due to motion is called Kinetic Energy, to distinguish it from that kind of energy considered previously, which is a consequence of the relative position of the parts of a system, and which is called Potential Energy. The kinetic energy of a body depends only on the velocity of each of the particles of which it is made up, not on the direction of its motion nor on the way in which its motion has been produced ; and the energy exerted in changing the motion of a body is always represented by an exactly equivalent increase of kinetic energy, whether the effort be uniform or variable, or whether its direction coincide with the direction of motion or not. The fall of a weight under the action of gravity is a particular case of the motion of a sliding piece under the action of a known force P in the direction of motion, the other element of the sliding pair being fixed. The piece here has a simple motion of translation, each particle traversing the same space with the same velocity. Let the velocity change from V to v as the piece moves through the space x, then equating the change of kinetic energy to the energy (Px) exerted by the force P _ 1) ~^g"W" an equation which is true whatever be the size, shape, or material of a sliding piece of weight W. The equation may be written *-r*-*jpx, showing that the piece moves with uniform acceleration as in the case of a falling weight, the magnitude of the acceleration being If the sliding piece be under the action of a force 8 which is not in the direction of motion, then we know (p. 180) that the energy exerted by S is the same as if its resolved part P in the direction of motion existed alone. The acceleration of the sliding piece therefore is independent of the component of 8 perpendicular to the direction of motion. These results are, of course, in direct accordance with the laws of motion. If P be a resistance instead of an effort, then work is done at the expense of the kinetic energy which is now diminished. If P be variable we must represent it graphically by a curve as in Art. 90, and it should be especially remarked that the ordinate of the curve of areas deduced in Art. 31 will, on affixing a suitable scale, and measuring the ordinates from a suitable base line, represent the height 198 DYNAMICS OF MACHINES. [PART in. due to the velocity, or, as it may otherwise be described, the "height equivalent to the kinetic energy " of the body. 100. Partially Unbalanced Forces. Principle of Work. Again, the effort which is changing the motion of the body may be partly balanced by an external resistance to which the body is subject. If this be the case we can imagine it separated into two parts, a part which is, and a part which is not, balanced. The energy exerted by the first is employed in overcoming the external resistance, while that exerted by the second is employed in increasing the kinetic energy of the body. Or the resistance may be greater than the effort, then the excess is overcome at the expense of the kinetic energy of the body, the velocity of which now diminishes. In the present treatise we shall use the phrases " energy exerted " and "work done" only in reference to efforts and resistances other than those due to inertia, subject to which convention, we may state, the principle of work as applied to cases where the forces are partially unbalanced, as follows Energy exerted = Work done + Change of Kinetic Energy. In this statement the work done may be greater or less than the energy exerted. In the first case the change of kinetic energy is a decrease, in the second an increase. Not only does this principle apply to a single body, but subject to the observations of the preceding section to a set of bodies mechanically connected in any way, provided that one of them be fixed to the earth ; or, in other words, that a body of great mass like the earth be one of the set. A single body is in reality one of a set of two bodies, the other being the earth. When no one of the set predominates over the rest it is necessary to consider further how the kinetic energy should be reckoned : for the present, however, we shall suppose this condition satisfied. A simple case is that of Atwood's machine. Let the descending weight P be greater than the rising one Q. Neglecting friction, the excess sets the two weights in motion. Let P descend through a distance y, then Q rises through the same distance, and therefore Energy exerted = Py. Work done = Qy. Let v be the velocity of the two weights ; then supposing them to start from rest, Kinetic energy acquired = (P + Q). CH. vni. ART. 101.] PRINCIPLE OF WORK. 199 From principle of work The law of increase of velocity is, therefore, the same as that of a body falling freely, but the rate of increase is less. This formula is the same as that obtained by other methods, and we have therefore here a verification of the principle of work. In applying the principle any pair of elements may be a driving or a working pair, whether or not one of them be the fixed link attached to the earth. Thus, for example, in a locomotive the steam exerts an amount of energy measured by its pressure and by the motion of the cylinder piston pair which it drives. This energy is employed in drawing the train while overcoming frictional and other resistances which oppose the motion of the various pairs making up the whole mechanism. Any excess or defect is represented by a change of kinetic energy in the whole train, inclusive of the mechanism of the locomotive estimated relatively to the earth as fixed. A rotating cylinder engine, in which the steam cylinders, instead of being fixed to the frame, are attached to a rotating fly-wheel, furnishes another instructive example. 101. Kinetic Energy of Rotation. Turning Pair. Instead of a single body, every point of which moves with the same velocity, suppose we have a system of bodies, and we require to know the total kinetic energy of the system. The direct method is to find the energy of each separate particle of the system and add the results. In the particular case of a rotating rigid body we are able to express the result of the summation in a convenient and simple form. First consider a ring of small section rotating about an axis in the centre perpendicular to its plane. Every portion of the ring will move with the same velocity, v say, and the kinetic energy of the ring may, as before, be written WiPfig. We may express this another way, as follows : If n be the revolu- tions per second, and a the radius, v = 2 If the ring is not complete, but W is the weight of a portion which has the same centre of rotation, the expression will still hold. Now, suppose we have a body consisting of a number of particles rigidly connected together, rotating about a centre 0, at n revolutions per second. 200 DYNAMICS OF MACHINES. [PART in. Let the weights of the particles be w v w. 2 , w. B , w^ etc. rotating about at distances y v y 2 , y s , ?/ 4 , etc. By adding together the results for each particle, we obtain for the kinetic energy of the system, w $ and therefore that is, the velocity of the weight is proportional to the ordinates of a semicircle. The curve of areas corresponding to the curve of effort which, as we have before found in a different problem (Ex. 5, p. 66), is a parabola, gives the kinetic energy, but it is not shown in the diagram, not being required for our present purpose. Let V Q be the velocity with which Q moves as N returns with velocity V towards C y then since F" when resolved parallel to AA must be equal to F, from which it appears that CQ rotates with uniform angular velocity, describing a complete circle in the Period = -= which gives the time of a complete oscillation to and fro. As the formula shows, the period does not depend on the extent of the oscillation, but only on the intensity of the force as measured by the magnitude of the coefficient p. If we call c the distance from the centre at which the force is equal to the weight of the vibrating mass, then c = I/p and the formula becomes Period = 2: CH. viii. ART. 103.] PRINCIPLE OF WORK. 205 being as will be seen presently the same as that of a pendulum of length c. (2) Next take the case of a rotating piece vibrating backwards and forwards about a mean position under the action of a couple of magni- tude proportional to the angle turned through. For example, the balance of a watch vibrating under the action of the balance spring which exerts a moment M, proportional to the angle turned through from the position of rest. Here the moment is given by the equation H as before being a coefficient measuring the intensity of the moment. Writing now r for the radius of gyration of the wheel and referring to page 201 the equation of energy will be exactly as in the case just considered of a sliding piece, The motion is now represented by the same diagram as before in which A A is now 20 19 the whole angle through which the wheel oscillates in the 27T7- Period = :== In this as in the preceding case the time does not depend on the extent of the oscillation, and the oscillations are therefore described as "isochronous." In a wheel, however, the period also depends on the radius of gyration r : the coefficient /x is here a certain length, being the leverage at which W must act to balance the moment M at unit angle in circular measure, and the length of the corresponding pendulum is r 2 //*. (3) In the two preceding cases the motion is of the kind called "harmonic," let us next consider a pendulum vibrating to and fro under the action of gravity. We have now a rotating piece, the radius of gyration of which is r (suppose), oscillating about a horizontal axis at a distance L from the centre of gravity g. Referring to page 184 it will be seen that in any position inclined at an angle to the vertical, the potential energy reckoned from the lowest position is Applying once more the principle of the conservation of energy we have as the equation of energy W V 1A1 ^~- + W. L(l - cos 0) = W. L(l - cos ^), #! being the extreme angle reached, that is to say, half the total angle of swing. The equation may be written W . 2L sin* =W.'2L. sin*. 206 DYNAMICS OF MACHINES. [PART in. When the angle # x is the equation reduces to f) When the angle 6 l is not too great sin - may be replaced by 6/2 and This is the same equation as in the preceding case, and indicates that the vibrations are isochronous, an oscillation to and fro taking place in the * Period = 2ir . -=. In a " simple " pendulum consisting of a heavy particle suspended by a string of length I from a fixed point, and vibrating in a vertical plane ?' = Z = , and ,- Period = 2ir\ -. M g The length of such a simple pendulum is often adopted as a measure of the time of a vibration. In a so-called " compound " pendulum let the radius of gyration about a horizontal axis through the centre of gravity be r , then (p. 201) arid consequently the length of the simple equivalent pendulum is This is least when L = r and the quickest time of vibration of a body of radius r is consequently that of a pendulum of length 2r Q ; but the period may be made as long as we please by taking the axis near the centre of gravity, as for example in the beam of a pair of scales which is balanced on knife edges slightly above the centre of gravity. f\ Returning to the original equation observe that sin r is always less n than 6/2 and that therefore the potential energy is always less than if the motion were harmonic. The difference is greater the greater the value of 6, it is therefore greater for 6 } than for 6, and the kinetic energy is consequently always less than in harmonic motion. When 6 l is not small the diminution is perceptible and the vibrations are then not isochronous, but the period is less the greater the angle of swing. If T be the time of small oscillations and T the actual time for the half angle of swing 6 V then it is shown in treatises on the kinetics of a particle that ^ 2 ^ T T Q \ 1 + ~ [ approximately. (4) When a vessel rolls in still water a part of her kinetic energy corresponds to the movement of her centre of gravity : this, however, is usually a small fraction of the whole and may be neglected. If we also CH. vin. ART. 103.] PRINCIPLE OF WORK. 207 neglect the resistance to rolling due to friction and disturbance of the water the equation of energy will be where r is the radius of gyration about a horizontal longitudinal axis through the centre of gravity, U the potential energy at the angle of heel 6, and U^ the value of U at the extreme angle through which she rolls. The potential energy is here the same quantity as that already described as the " dynamical stability " and in the typical case con- sidered on page 185 is given by the equation hence by substitution ancl multiplication by 9 + Wm(\ - cos kO) =Wm(l- cos kOJ. Referring now to the equation of energy of a simple pendulum just obtained, suppose it to swing through k times the angle of heel of the vessel it will be seen that the angular velocity of the pendulum will be kd., and that therefore the motion of the rolling vessel will follow the motion of such a pendulum if j.!l m Hence the period of the small isochronous oscillations of a vessel when unresisted is Period = 2^-, where r is the radius of gyration and m the metacentric height. Being independent of k the formula applies to any case whatever the stability curve so long as the oscillations are small, not exceeding 15 probably on each side of the vertical. For larger oscillations the deviation from isochronism is much greater than in a simple pendulum swinging through the same angle, being proportional to k 2 in the case just considered. It should be observed that throughout this article the periods given refer to a complete oscillation to and fro. By many writers the time of a single oscillation is described as the period. In the case of a pendu- lum the "time of vibration" generally means the time of a single oscillation. The number of vibrations per second is known as the "frequency." EXAMPLES. 1. The energy of 1 Ib. of pebble powder is 70 foot-tons. Find the weight of charge necessary to produce an initial velocity of 1300 feet per second in a projectile weighing 700 Ibs., neglecting the recoil of the gun and the rotation of the shot. "VVt. of powder required =117 Ibs. 208 DYNAMICS OF MACHINES. [PART m. 2. In Example 1 suppose the gun fired at an elevation of 30, and resistance of the atmosphere neglected, find the kinetic and potential energies of the shot at its greatest elevation. Also deduce the greatest elevation. Horizontal velocity = velocity at highest point = 1300-^-- Kinetic energy at highest point=6150 ft. -tons, Potential ,, =2050 Potential energy , , "rFZ~""f i, . =oc>60'6 feet= maximum elevation. 3. A train is running at 40 miles an hour, find the resistance in pounds per ton necessary to stop the train in 1000 yards on a level. Also find the distance in which the train would be brought up by the same brake power on a gradient of 1 in 100, both when going up and when going down. Resistance = 39 '9 Ibs. per ton. Distance required to bring up the train when ascending the gradient ............ M . ... = 640 yards. When descending .................. =2280 4. The reciprocating parts of an engine running at 75 revolutions per minute weigh 25 tons, of which parts weighing 20 tons have a stroke of 4 feet, and parts weighing 5 tons a stroke of 2 feet. Find the energy stored in the parts, assuming a pair of cranks, OP, OQ at right angles and neglecting obliquity of connecting rod. Here if V is the velocity of the crank pin and PN, QM are perpendiculars on the line of centres, y Velocity of parts attached to crank P-PN- Further assuming weights attached to these cranks each equal W. WVi -' 1 WV Z Energy stored in these weights = ^-(P^ 2 +QM2)^L_=-^-. In example, total kinetic energy =40 '7 ft. -tons. 5. One weight draws up another by means of a common wheel and axle. The force- ratio is 1 to 8 and the velocity-ratio is 9 to 1. Find the revolutions per minute after 10 complete revolutions have been performed, neglecting frictional resistances and the inertia of the wheel and axle. Diameter of axle 6 inches. Revolutions per second=2'14. 6. In Ex. 1 suppose the gun rifled so that the projectile makes 1 turn in 40 diameters, find the additional powder charge required to provide for the rotation of the shot, the diameter of shot being 12 inches and the radius of gyration 4J inches. Additional powder required ='407 lb. 7. A disc of iron rolls along a horizontal plane with velocity 15 feet per second, and comes to an incline of 1 in 40 on to which it passes without shock. Find how far it will ascend the incline, neglecting friction. Distance along incline it will run =209 '6 feet. 8. In Ex. 5 suppose the weight of wheel = weight of axle, and the two together = sum of weights, obtain the result, taking account of the inertia of the wheel and axle. After 10 revs, it will rotate at 1*22 revs, per second. 9. A fly-wheel, the radius of gyration of which may be taken as 8 feet, rotates at 40 revolutions per minute ; find the height due to the revolutions and also the height equivalent to the energy of rotation. Ans. &=1'835; J?=17'45. 10. The beam of a pair of scales is 2 feet long, radius of gyration 6 inches ; the scale, pans, and weights are equivalent to a weight of 3 Ibs. placed at each end of the beam, H. viii. ART. 103.] PRINCIPLE OF WORK. 209 which itself weighs 3 Ibs. If the beam rest on knife edges placed J inch above the centre of gravity, find the time of vibration. A ns. 3 '3 seconds. 11. The centre of gravity of a connecting rod 5 feet long has been found by the method of suspension to be 3 feet from the crosshead end. To determine the radius of gyration it is made to oscillate as a pendulum on knife edges fixed at the crosshead end. It is then found that 53 vibrations are made in a minute ; find the radius. Ans, 3 feet 6 inches. 12. From a curve of "tons per inch immersion" it is found that a vessel sinks one inch in the water by the addition to the weight on board of a small fraction e of her original displacement ; show that the period of small unresisted dipping oscillations is . v/386e REFERENCES. Numerous elementary examples on the application of the Principle of Work will be found in Cotterill and Slade's Lessons on Applied Mechanics. C.M. CHAPTER IX. DYNAMICS OF THE STEAM ENGINE. 104. Construction of Polar Curves of Crank Effort. One of the most common and important applications of the principles of the preceding chapter is to the working of steam engines, and we shall investigate this question, chiefly with reference to fluctuations of stress energy and speed. Throughout, frictional resistances are neglected. In Chapter V. a curve was constructed which shows the velocity- ratio of piston and crank pin, and it has been proved (p. 191) that this curve must also give the ratio of the effort tending to turn the crank to the pressure of the steam on the piston, so that it may also be called a Curve of Crank Effort. If there are two or more cranks, the crank effort can be obtained by suitably combining the results for each taken separately, and a curve may then be drawn representing the combination. There are two kinds of such curves, the Polar and the Linear. First suppose two cranks at right angles, steam pressure uniform, and the same on both pistons. Let us commence with the polar curve. Suppose OT^By OT 2 (Fig. 93) to represent the polar curve of crank effort for an engine constructed as in Art. 49, and let the two cranks be in the positions OQ^ OQ 2 , each pointing towards the cylinder. Add together the corresponding crank efforts OT^ OT%, which are given by the curve, and set off their sum along OQ^ we thus obtain a radius OT", which represents the total crank effort for the two engines taken together. It may also be considered as the leverage at which the pressure on one piston must act to produce the same turning moment. Performing this construction for a number of positions of the cranks, we obtain a polar curve showing the crank effort in every position. If the connecting rod is indefinitely long the single curve of crank effort consists of the pair of circles on OB^ OB^ shown dotted in the diagram. If we add together radii of these circles, the combined curve of crank effort will consist of four portions of circles passing the points- THE STEAM ENGINE. 211 (PART in. CH. ix.] A^B^AvB^ \ each of the circular arcs if produced would pass through the point 0. These arcs are also dotted in the diagram. When the crank is in a quadrant lying towards the engine, the actual crank effort Fig.93. is in excess of that due to a long connecting rod. So for the positions OQ V OQ 2 , shown, for each the crank effort is in excess, and thus the curve of combined effort will for the quadrant A l B l lie outside the circular arc. When the cranks are in the two upper quadrants the effort for the leading crank is less than when the connecting rod is long, whereas for the following crank it is greater ; and the diminution of one is very approximately equal to the excess of the other ; that is, the sum is the same as that obtained by neglecting the shortness of the rod. The true combined effort is then for the quadrant B^ 2 represented by the circle. In the next quadrant both are diminished ; and the true curve will lie inside the circle A^B^ while for the fourth quadrant it will again coincide with the circular arc. We may, if we please, lay off the sum of the radii on the following crank instead of the leading ; the same series of curves would be obtained, but would be turned backwards through an angle of 90. 212 DYNAMICS OF MACHINES. [PART in. To add to this the circle of mean crank effort we equate the work done on the two pistons in the double strokes to the work due to the mean effort R m exerted through a complete revolution. P x 2 x 4a = R x 2ira. In these curves the steam pressure P is represented by the radius of the crank-pin circle, so the mean crank effort will be represented on the diagram by drawing a circle, shown dotted, with centre and radius = 4 OQ/ir. If there are three or more cranks inclined at any angles, the com- bined crank effort diagram can be constructed by adding together three or more radii vectores of the curve of single crank effort, and laying the sum off on either of the cranks. 105. Construction of Linear Curves of Crank Effort. The linear curve of crank effort, which is more useful for most purposes, is constructed as follows : Take a base line, A t A 2 = semi-circumference of the crank-pin circle, and let the circle and this base line be divided into the same number of equal parts, and at the points of division of the base line set off ordinates such as SN, VM both above and below the base equal to lengths of the common ordinates of the single crank effort diagram such as OT^, OT%, and so we construct the linear crank effort diagram for a single crank. Neglecting the obliquity of the connecting rod, the diagram will consist of two curves of sines shown dotted, one above, the other below (Fig. 94). To get the combined crank effort diagram we have only to add together proper ordinates according to the angle between the cranks, just as we did in drawing the polar diagram. When the cranks are at right angles it will be seen that when the leading crank is, for example, at Q l or N the following crank is at Q% or M ; and if the ordinate MF is laid off on the top of ordinate NS we obtain a point W on the curve of combined 2 1-31 1-077 Four Cranks. Minimum. 785 794 The great influence which the length of the connecting rod has on the results should be especially noticed ; we shall return to this here- after, but now go on to consider the motion of the engine under the action of the varying crank effort. 107- Fluctuation of Energy. We have already referred to the periodic character of the motion of a machine, and explained that when the mean motion is uniform we have for a complete period Energy exerted = Work done. It .will seldom happen however that this equation holds good for a por- tion of the period. In general, during some part of the period the work -done will be greater, and in some part less, than the energy exerted. In the first case some part of the kinetic energy of the moving parts is absorbed in doing a part of the work, and the speed of the machine diminishes ; while in the second, a part of the energy exerted is employed in increasing the kinetic energy of the moving parts and the speed of the machine increases. Thus the kinetic energy of the moving parts alternately increases and diminishes, the increase exactly balanc- ing the decrease. At some instant in its motion, the energy of the moving parts will be a minimum, and at some other point a maximum. The difference between the maximum and minimum energies is -described as a " fluctuation " of energy of the machine. In general a number of these fluctuations occur in the course of a period, and the CH. IX. ART. 107.] THE STEAM ENGINE. 215 greatest of them is called the Fluctuation of Energy. It is most con- veniently expressed as a fraction of the whole energy exerted during a complete period of the machine, and this fraction is called the Co- efficient of Fluctuation of Energy. All this will apply to any machine taken as a whole, or to any part of that machine; for every piece of the machine has a driving point and a working point, and the equation of energy may be applied to it. Take now the case of the mechanism of a direct-acting engine. Suppose the pressure P on the piston to be uniform. This through the connecting rod will produce a crank effort S, the magnitude of which for each position of the crank may be found as just now shown. To the crank and shaft S is the driving force and furnishes the energy exerted. At every point of the revolution of the shaft a certain resistance will be overcome, which resistance will tend to prevent the shaft from turning ; it will not depend on the steam pressure, but on the sort of work that is being done. As the most simple ordinary case we will suppose the Fig.95. resistance overcome to be uniform, and we will neglect the inertia of the reciprocating parts (Art. 110). We may represent this constant resistance by a constant force E applied to the crank pin Q (Fig. 95), at right angles to the crank arm, resisting its motion. The magnitude of R is immediately determined by the application of the principle of work to a complete period, say one revolution. We have This constant resisting force is the same as the mean crank effort. 216 DYNAMICS OF MACHINES. [PART m. Then so long as S>E the speed of the crank shaft will increase, and when S), Also considering a complete revolution of the screw, Energy exerted = M . '2rr = Rird . sin (a + $), Useful work done = H. p = Pip . cos (a CH.x.ART.116.] FRICTIONAL RESISTANCES. 243 from which it follows that the efficiency of the screw is tan a It is not difficult to show that this fraction is greatest when a = 45 - J< r and its value is then (1 - f\ 2 f) approximately. 1 * 27' For ordinary values of/ then, the best pitch angle is approximately 45 and the efficiency is considerable. In practice, however, the pitch angle is much smaller, its value in bolts and the screws used in presses ranging from '035 in large screws- to -07 in smaller ones ; the efficiency is then less, often much less, than one-third, the object aimed at being not efficiency but a great mechanical advantage. If the pitch be sufficiently coarse, it will be possible to reverse the action, the driving force being then H and the resistance a moment opposing the rotation of the screw. In a well-known kind of hand drill and a few other cases this occurs in practice ; the force E is then inclined on the other side of the normal, and the efficiency is in the same way as before found to be _ . Efficiency = tana In most cases, however, a is less than , and the screw is then incapable of being reversed. Non-reversibility is often a most valuable property in practical applications, the friction then serving to hold together parts which require to be united or to lock a machine in any given position. In estimating the efficiency of screw mechanisms the friction of the end of the screw acting like a pivot or of the nut upon its seat must be included ; in screw bolts this item is generally as great as the friction of the threads. The friction due to lateral pressure of the screw on its nut may usually be neglected, but when necessary it may be estimated by the same formula as is used for shafts. The above investigation strictly speaking, applies only to square-threaded screws; it has, however, been shown that the efficiency is only slightly diminished by the triangular or other form of thread usually adopted for the sake of strength.* The formulae here given for screws may be applied to any case of a sliding pair in which the driving effort is at right angles to the useful resistance. A simpler case is that in which the driving effort is parallel to the direction of sliding. This is given in Example l r * Cours de Mtcanique Appliquee aux Machines, par J. V. Poncelet, p. 386. Paris,. 1874. 244 DYNAMICS OF MACHINES. [PART in. page 260. In all cases observe that the efficiency diminishes rapidly when the velocity-ratio is increased. This, which is common to most mechanisms, limits the mechanical advantage practically attainable. The hydraulic press is an exception, as will be seen hereafter. 117. Efficiency of Mechanism by Exact Method. In the preceding cases the efficiency is the same for any motion of the mechanism whether large or small. Generally, however, it will be different in each position of the mechanism, and by the " efficiency of the mechanism " is then to be understood the ratio of the useful work done in a period to the energy exerted in the period. The exact calculation of the loss of work by frictional resistances in mechanism is generally very complicated, so that it is best to proceed by approximations the nature of which will be understood on consider- ing an example with some degree of thoroughness. The case we select is that of the mechanism of the direct-acting vertical steam engine such as is represented in Plate I., page 108. The losses by friction are (1) the loss by piston friction, (2) friction of guide bars, (3) friction of crosshead pin, (4) friction of crank pin, (5) friction of crank shaft-bearings. Of these, the first two are considered separately (Ex. 2, p. 261), and for the present neglected, whilst the last three are treated by a graphical method as follows. In Fig. 106 CQA are the friction circles of the three parts in question, Fig.106. vW which for the sake of clearness are drawn on a very exaggerated scale while the bearings themselves are omitted. We will neglect the weight of the connecting rod and its inertia; of these the first is generally relatively inconsiderable, but in high speed engines the last is often very large and makes the friction very different at high speeds and low speeds (see Chap. XL). The weight of the crank shaft and all the parts connected with it is supposed to act through the centre of the shaft ; for simplicity we will call it W. The pressure on the piston after correction for piston anpl guide-bar friction is denoted by P. Then, in the absence of friction, the line of action of the thrust on the connecting rod is the line joining the centres of the friction circles, and the moment CH.X.ART. 118.] FRICTIONAL EESISTANCES. 245 of crank effort is P . CT , where T is the intersection of that line with the vertical through C. But the line of action in question must now touch the friction circles (Art. 114), and the true moment of crank effort on the same principle must be P . CT, where T is the intersection of this common tangent with the vertical CT. Thus P . TT is the cor- rection for friction of the crosshead and crank pins. Next observe that the forces acting on the crank shaft are W the weight and S the thrust of the connecting rod ; these may be compounded into one force R passing through T as shown in the diagram. The reaction of the crank-shaft bearing is an equal and opposite force R which must touch the friction circle and cut CT in a certain point K. Now the horizontal component of R is the same as that of S, namely P ; therefore the true moment of crank effort after allowing for friction is P . TK. By performing this construction for a number of positions, as in the last chapter, we obtain a diagram of crank effort corrected for friction. The area of this curve will give us the useful work done in a revolution, the ratio of which to the energy exerted is the efficiency of the mechanism : and its intersections with the line of mean resistance will give the points of maximum and minimum energy and the fluctuation of energy as corrected for friction. .When the crank makes a certain angle with the line of centres TK vanishes. Within this angle no steam pressure, however great, will move the crank, as is well known in practice. It may be called the " dead angle," all points within it being dead points. 118. Efficiency of Mechanism by Approximate Method. The process just described is not too complicated for actual use in the foregoing example, but in many cases it would be otherwise, and it may there- fore be frequently replaced with advantage by a calculation of the efficiency of each of the several pairs of which the mechanism is made up taken by itself. Each pair consists of two elements, one of which transmits energy to the other, with a certain deduction caused by the friction between the elements. The ratio of the energy transmitted to the energy received may be called the efficiency of the pair. If c lt c 2> c 3J ... be the efficiencies of all the pairs in the mechanism it is evident from the definition that the efficiency of the whole mechanism must be c = c 1 .c 2 .c s .... In some cases the efficiency of each pair will be independent of the frictional resistances of all the other pairs, and may be found separately. In general this is approximately, but not exactly, true, a point which will be best understood by a consideration of the foregoing diagram. 246 DYNAMICS OF MACHINES. [PART in. For example, the friction of the guide bars is diminished in consequence of the friction of the crank pin, because the obliquity of the connect- ing rod is virtually diminished. The supposition is, however, often sufficiently nearly true to enable a rough estimate to be made of the efficiency of the mechanism by finding the efficiencies of the several pairs taken alone, all the others being supposed smooth. In doing this mean values are taken for variable forces, if the amount of variation be not considerable. The uncertainty and variability of the co-efficients on which frictional efficiency depends are such as to render refined calculations of little practical value. 119. Experiments on Sliding Friction (Morin). The ordinary laws of friction, which may be comprised in the single statement that the co-efficient of friction depends on the nature of the surfaces alone, and not on the intensity of the pressure or on the velocity of rubbing, were originally given by Coulomb in a memoir published in 1785, although some facts of a similar kind were previously known. They are there- fore often called Coulomb's laws. Yet Coulomb's experiments were scarcely sufficient to establish them, and the subject was reinvestigated by others, especially by the late General Morin, whose memoirs were presented to the French Academy in 1831-4. Morin's experiments were so elaborate and exact that they may be considered as conclusively proving the truth of Coulomb's laws within certain limits of pressure and velocity, and under the circumstances in which they were made : it will therefore be advisable to explain them briefly. A sledge loaded with a given weight was caused to slide along a horizontal bed AB more than 12 feet long (Fig. 107), the rubbii Fig. 107. T W surfaces being formed of the materials to be experimented on. Th< necessary force was supplied by a cord passing over a pulley at B to descending weight Q. The tension of the cord T was measured by spring dynamometer, and could likewise be inferred from the magni- CH. X. ART. 120.] FRICTIONAL RESISTANCES. 247 tude of the weight after correction for the stiffness of the cord and the friction of the pulley. In one form of experiment the weights were so arranged that the sledge moved nearly uniformly : the corresponding friction was measured and found to be constant. In a second form, the times occupied by the sledge in reaching given points were automatically measured and compared with the spaces traversed, by setting them up as ordinates of the curve CZ shown below. The curve proved to be a parabola, showing that the space varied as the square of the time, from which it was inferred that the acceleration of the sledge was constant. From both methods it appeared that the co-efficient of friction was exactly the same, whatever the pressure and whatever the velocity, provided the nature and condition of the surfaces were the same. A few important results are given in the annexed table; they are taken from Morin's latest memoir,* containing, besides many new experi- ments, tables of the results of the whole series. The limits to their application will be considered presently. NATURE OF SURFACES. CONDITION OF SURFACES. CO-EFFICIENT OF FRICTION. Wood on Wood, /Perfect!}' dry and\ ( clean, J 25 to -5 Metal or Wood on\ Metal or Wood, / Slightly oily, 15 Do., do., Well lubricated, 07 to '08 Do., do., / Lubricant con- \ \ stantly renewed, / 05 Full tables of Morin's results will be found in Moseley's work cited on page 257. The friction between surfaces at rest is often greater than when they are in motion, especially when the surfaces have been some time in contact : the excess, however, cannot be relied on, as it is liable to be overcome by any slight vibration. In the case of metal on metal whether dry or oily, or in wood on wood when dry and clean, Morin, however, found that there is no such difference. 120. Limits of the Ordinary Laws. From the exactitude with which Coulomb's laws were verified by Morin's experiments the inference was naturally drawn that they were universally true, but this is probably erroneous, even for dry surfaces without sensible adhesion. Although no complete and thorough investigation has been made, it can hardly *Nouvelles Experiences . . . faites a Metz en 1834. Page 99. 248 DYNAMICS OF MACHINES. [PART in. now be a matter of doubt that there are cases in which the laws of friction are widely different. The known cases of exception for plane surfaces may be grouped as follows : (1) At low pressures the co-efficient of friction increases when the pressure diminishes. This has been shown by various experimentalists, as, for example, by Dr. Ball.* The lowest pressure employed by Morin was about three-fourths of a Ib. per square inch, and this is about the pressure at which the deviation noticed by Ball becomes insensible. This eifect may be ascribed to a slight adhesion between the surfaces independent of friction proper. (2) At high pressures, according to certain experiments by Eennie,t the co-efficient increases greatly with the pressure. The upper limit of pressure in Morin's experiments was from 114 to 128 Ibs. per square inch. At 32*5 Ibs. per square inch Rennie found for metallic surfaces at rest '14 to '17, nearly agreeing with Morin; but on increasing the pressure the co-efficient became gradually greater, ranging from 35 to *4 at pressures exceeding 500 Ibs. per square inch. The metals tried were wrought iron on wrought and cast iron, and steel on cast iron. Tin on cast iron showed only a slight increase in the co-efficient. This increased friction at high pressures may be ascribed to abrasion: of the surfaces. (3) At high velocities the co-efficient of friction, instead of being independent of the velocity, diminishes greatly as the velocity in- creases. This was shown by M. Bochet in 1858. Similar results have been obtained by others, especially by Captain (now Sir Douglas) Galton in some important experiments on railway brakes. J The limit of velocity in Morin's experiments was 10 feet per 1", and at some- what greater velocities than this the diminution becomes perceptible. Morin's results have been shown to be applicable at the very lowest velocities by the late Professor F. Jenkin and Mr. (now Professor) Ewing, the friction increasing at excessively low velocities in those cases only in which there is a difference between the friction of rest and the friction of motion. It appears difficult to explain the diminution at high speeds merely by a change in the condition of the surfaces ; is should, probably, be regarded as part of the law of friction. Professor Franke in the Civil Ingenieur for May, 1882, has proposed the formula * Experimental Mechanics, by R. S. Ball, p. 78. Macmillan, 1871. t Phil. Trans, for 1829. Engineering, vol. 25, pages 469-472. Phil. Trans., vol. 167, part II. CH. x. ART. 121.] FRICTIONAL RESISTANCES. 249 where / is about *29, and a (for velocities in metres per 1") ranges, from '02 to -04, according to the nature and state of the surfaces. 121. Axle Friction. It has already been pointed out that the co- efficient of axle friction is not necessarily the same as that for plane surfaces sliding on one another, and, besides, the continuous contact of a shaft and its bearing is very different from the brief contact occurring in sledge experiments. Morin, however, made special ex- periments on the friction of axles, and showed that the co-efficients were constant and nearly the same in the two cases. The diameters employed, however, were 4 inches and under, while the revolutions did not exceed 30 per minute, so that the rubbing velocity was not more than 30 feet per minute. The pressures were not great, the value of pv not exceeding 5,000. Much greater values of pv than this are common in modern practice, and then it is certain that the value of the co-efficient is much less and diminishes with the pressure. Already in 1855 M. Hirn had made a long series of experiments on friction, especially of lubricated surfaces. The following summary of his results is given by M. Kretz, editor of the third edition of the Mdcanique Industrielle* (a) That a lubricant may give a regular and minimum value to the friction it must be " triturated " for some time between the rubbing surfaces. (b) The friction of lubricated surfaces diminishes when the tempera- ture is raised, other things being equal. (c) With abundant lubrication and uniform temperature friction varies directly as the velocity. When the temperature is not main- tained uniform, the relation between friction and velocity depends on the law of cooling of the special machine considered. In ordinary machinery friction varies as the square root of velocity. (d) The friction of lubricated surfaces is nearly proportional to the square root of the area and the pressure. Experiments made in 1883-4 by Mr. Tower, f and subsequently by others, have however conclusively shown that, with thoroughly efficient lubrication, the friction of a bearing under pressures exceeding 100 Ibs. per sq. inch is independent of the pressure. This is the well-known law of friction between a fluid and a surface in contact with it, and indicates that the bearing surfaces are not in actual contact but are separated by a thin film of lubricant, a fact which has also been proved * Introduction a la Mecanique Industrielle, par J. V. Poncelet. Troisteme edition,. Paris, 1870. . Page 516. "^Proceedings of the Institution of Mechanical Engineers. 250 DYNAMICS OF MACHINES. [PART in. by direct experiment. The co-efficient of friction in such cases varies inversely as the pressure and under heavy loads may be less than one- tenth of the value (-05) given by Morin. Thus Hirn's law of pressure, though verified by other experiments, is only true in special cases ; in other respects his conclusions appear on the whole correct. With thorough lubrication very heavy pressures and high speeds may be employed, the value of pv reaching, or even exceeding, 100,000, but any inaccuracy of fit in consequence of which parts of the bearing surfaces come into actual contact will at once cause heating. On the causes which produce heating the reader is referred to a paper by Professor Denton.* SECTION II. EFFICIENCY OF HIGHER PAIRING. 122. Rolling Friction. We now proceed to consider higher pairing, ^commencing with the case of rolling contact. The friction is then described as " rolling friction." When a wheel rolls on soft ground the resistance to rolling is due to the fact that the wheel makes a rut arid depresses the ground as it .advances over it. Thus the resistance to motion is proportioned to the product of the weight moved into the depth of the depression. The depth of the rut depends on the radius as well as the breadth of the wheel. It is found that the resistance may be expressed by bJT B => where W= weight, r = radius of wheel, and b is approximately a onstant length. This might have been anticipated, since the depth of the rut is of the versed sine of the arc of contact, and therefore for a given small arc is inversely as the radius. If the wheel roll on hard ground over a succession of obstacles of small height the law of resistance will be expressed by the same formula. When the surface rolled over is elastic, and the pressure on it is not -sufficient to produce a permanent rut, the resistance to rolling is not so easily explained. If we consider an extreme case, as for instance a heavy roller rolling on india-rubber, we shall be able to see to what .action the resistance is due. The wheel will sink into the rubber, which will close up around it both in advance and behind, as shown in Fig. 108. At C the rubber will be most compressed. As the wheel advances and commences to crush the rubber in advance of it the rubber moves away to avoid the compression, heaping itself up con- * ' ' Special Experiments on Lubricants, " by Prof. J. E. Denton. American Society of Mechanical Engineers. November, 1890. CH. X. ART. 122.] FRICTIONAL RESISTANCES. 251 tinually in advance of the wheel. In this movement it rubs itself over the surface Ca of the wheel, exerting on it a frictional force in the -direction shown by the arrow F, which opposes the onward motion of the wheel. Again, the rubber in the rear is continually tending to recover its normal position and form of flatness, and in doing so rubs itself over the surface bC of the wheel in the direction shown by the arrow F', which also tends to oppose the onward motion of the wheel. The effect of this creeping action of the rubber over the surface of the wheel is to cause the onward advance of the centre of the wheel to be different from that due to the circumference rolled out.* Moreover Fig. 108, the vertical component of the reaction of the surface no longer passes through the centre of the wheel as it must do in the absence of friction, but is in advance by a small quantity b such that Wb is the moment of resistance to rolling. Experiments on rolling resistance present considerable discrepancies, but within the limits of dimension of rollers which have been tried it appears that b is independent of the radius ; this leads to a formula of the same form as before for the force necessary to draw the roller, namely where b is a constant which for dimensions in inches is from -02 to -09 according to the nature of the surfaces. With very hard and smooth surfaces of wood or metal, the lower value '02 may be employed. Rolling friction is not sensibly diminished by lubricants, but depends mainly on smoothness and hardness of the surfaces. It is probably influenced by the speed of rolling, but this does not appear to have been proved by experiment unless in cases where the resistance of the atmosphere and other causes make the question more complicated. In many cases of rolling the surfaces are partly elastic and partly *See a Paper by Professor Osborne Reynolds, Phil. Trans., vol. 166, to whom the true explanation of resistance to rolling in perfectly elastic bodies is due. 252 DYNAMICS OF MACHINES. [PART in. soft, so that the resistance to rolling is partly due to surface friction and partly to permanent deformation. The value of the constant b is then much increased. For wagon wheels on macadamized roads in good condition the value of b is about -5", and on soft ground four to six times greater. The draught of carts is said to be increased by the absence of springs. 123. Friction of Ropes and Belts. Fractional resistances are also produced by the changes of form and dimension of the parts of a machine occasioned either by the stresses necessarily accompanying transmission of energy or by shock. In the present chapter we con- sider tension elements only, that is to say, chiefly ropes and belts. In Fig. 109 AB is a pulley, the centre of which is 0, over which a rope passes embracing the arc A KB and acted on by forces T 1 T 2 at its ends. If their be sufficient difference between 1\ and T 2 the rope will slip over the pulley not- Fig ' 109 - withstanding the friction which tends to prevent it. Let the rope be just on the point of slipping, then its tension will gradually diminish from 1\ at A to 7' 2 at B. Let T, T' be the tensions at the intermediate points K, L, then the portion KL of the rope is kept in equilibrium by the forces T, T' at its ends, and a third force S due to the reaction of the pulley, the three forces meeting in a point E. On OL set off to 01 to represent T, and draw Ik perpendicular to $ to meet OK in k, then the sides and the triangle Okl will be proportioned by the three forces, so that Ok represents T' and Ik S. The angle S makes with the radius will be the same for all arcs of the same length, and if KL be taken small enough will be the angle of friction (Art. 113). This construction can, if we please, be commenced at A and re- peated for a number of small portions of the rope till we arrive at E\ we shall obtain a spiral curve alkb, the last radius Ob of which represents T 2 on the same scale as the first Oa represents T v It is convenient, however, to have an algebraical formula to calculate T 2 . Let the angle KOL be i and the angle S makes the radius <, then T Ol_sir\0kl_cos(i-) cos cos i + sin i tan . H. X. ART. 124.] FRICTIONAL RESISTANCES. 253 If now the angle i be diminished indefinitely we may write cos i 1 and sin i = i, so that T-T , 4 , p, -- = . tan. Replacing i by A0, T - T by A7 T , and proceeding to the limit \dT which being integrated gives where / is the co-efficient of friction, 6 the angle subtended by the part of the pulleys embraced by the rope, and e the number 2-718 being the base of the Naperian system of logarithms. The formula is applicable even if the pulley be not circular. For a circular pulley the spiral curve, representing graphically the tension at every point, is the equiangular or logarithmic spiral of which the formula may be regarded as the equation. In constructing it graphically, the value of <, for a small yet finite angle i, is found by replacing T/T by e fi and expanding the exponential : we thus get approximately 1 +fi = cos i + sin i . tan = 1 - \i 2 + i . tan , .. tan =f+^i. With small values of the co-efficient 2f may be a sufficiently small Fig.110. angular interval, but in general it will be advisable to take the angular in- terval equal to the angle of friction, then the value of is 1J times that angle. The construction being one in which errors accumulate, the formula is preferable when great accuracy is desired. 12-4. Driving Belts. When a belt is stretched over a pulley by equal weights, the tension of the belts is not necessarily the same everywhere fwl in the first instance ; but if the pulley move steadily and the stiffness of the belt be disregarded, it must be so. Assuming this, let one of the weights be increased by a certain quantity Q*.and the pulley be held fast, then the tension of that side of the belt will be increased by an amount equal to Q at A, but diminishing to zero at L v a point determined by the intersection of the friction spiral a v ^ (Fig. 110) with the circle alb, the radius of which represents the weight W. 254 DYNAMICS OF MACHINES. [PART m. Similarly if the other weight be diminished by Q', the tension will be diminished by an amount equal to Q' at B, but diminishing to zero at L 2 . The portion L^ 2 will remain at the original tension W. If QQ' be increased sufficiently, L lt L 2 will coincide in one point L, the position of which will depend on the proportion between Q and Q'. While these changes take place in tension, corresponding changes of length must occur in the parts of the belt exposed to them, AL^ increases and BL 2 diminishes in length. Hence both these parts slip over the pulley and work is lost by friction, while L^L^ remains fixed. If now, instead of altering the weights W, we imagine these weights held fast, and the pulley forcibly rotated so as to increase A's tension by Q, and diminish B's tension by Q', L^L^ will rotate with the pulley, and the total increase of length of the one side must be equal to the total diminution on the other, from which con- sideration it is possible to calculate the ratio Q bears to Q'. In practical cases, however, the difference between Q and Q' is so small that it may be neglected without sensible error, and therefore, in all questions relating to the working of belts, it may be assumed that the mean tension of the two sides of the belt is independent of the power which is being transmitted. The difference of tensions, however, is directly proportional to the power, and may at once be calculated if the speed be known, while the ratio of tensions may be determined, so that the belt shall just not slip, by means of the formula above obtained. The value of the co-efficient of friction of leather on iron ranges from '15 to '46 according to the degree of lubrication : under ordinary circumstances '25 may be considered an average value. This, however, is often greatly exceeded in practice,, and one reason why large values are admissible is said by some to be the effect of atmospheric pressure. The sectional area of belts is fixed by consideration of strengths, and as their thickness varies little, this is equivalent to saying that a certain breadth of belt is required for each horse-power transmitted. (See Ex. 11, p. 261.) 125. Slip of Belts. When a belt is stretched over a pair of pulleys r one of which drives the other, notwithstanding a resistance not so great as to cause slipping of the belt as a whole, it appears from what has been said that a certain arc exists on each pulley on which the belt does not slip. The length of these arcs has already been found, but in the present cases the movement of the pulleys causes them to place themselves where the belt winds on to the pulleys, so that the driving pulley has the speed of the tight side of the belt and the driven pulley that of the slack side. The two sides have different speeds, because the same CH. x. ART. 126.] FRICTIONAL EESISTANCES. 255 weight of belt must pass a given point in a unit of time, wherever that point be situated, and therefore the speed must be greater the greater the elongation, that is to say, the greater the tension. Hence the driving pulley moves quicker than the driven pulley by an amount which can be calculated when the tensions and the elasticity of the leather are known, and the " slip " measures the loss of work due to the creeping of the belt over the pulleys described above. In ordinary belting the slip does not exceed 2 per cent., and is believed to be often insensible. The length of belts, however, must not be too great, or its extensibility will be inconvenient, especially if the motion of the machine be not sufficiently uniform.* Within moderate limits extensi- bility is favourable to smooth working. 126. Stiffness of Ropes. When a rope is bent it is found that a cer- tain moment is required to do it depending on the dimensions of the rope and, besides, on its tension. The reason of this is best understood by referring to the corresponding case in a chain with flat links united by pin joints. If d be the diameter of the pin, T the tension of the chain, there will be a certain moment of friction resisting bending which, if the pin be any easy fit, will be simply J/7W, but if it be tight will be where T is a constant depending on the tightness. If the chain pass- over a rotating pulley without slipping, this frictional moment has to be overcome both when bending on and when bending off the pulley. The effect shows itself by a shaft outwards on the advancing and inwards on the retiring side of the chain, so as to increase the leverage of the resistance and diminish that of the effort. In the present case the two shifts are equal, being each given by the formula The case of a rope differs from this only in being more complex : in the act of bending, the fibres move over each other, and the relative motion is resisted by friction due to pressures which are partly constant and partly proportional to the tension. The shift of the centre line of the rope is visible on the side of the resistance, but hardly perceptible on the side of the hauling force, showing that most of the loss of work is- due to the bending on the pulley. The magnitude of the shift varies so- much according to the mode of manufacture and the condition of the rope that it is useless to attempt more than a very rough estimate. *See a footnote by M. Kretz, Covrs de Mtcanique Appliquee aux Machines, par Poncelet, p. 264. 256 DYNAMICS OF MACHINES. [PART in. According to a formula given by Eytelwein, if d be the diameter of the rope, where c is a constant, which for dimensions in inches is taken as '47 for hemp ropes ; but this value is too large, except for light loads, and small diameters of pulley. The loss of work per revolution is T. 2-n-x, and if D be the effective diameter of the pulley, D There is a loss of work by the stiffness of belts of a similar kind, but of uncertain amount. By most authorities it is considered so small as to be negligible. The shift of the line of action of the tension of a rope due to its stiffness has the effect of diminishing its strength. 127. Friction of Toothed Wheels and Cams. The friction of toothed wheels is partly rolling and partly sliding, but the first is relatively small and may be neglected. To determine the sliding friction, let PT=z (see Fig. 71, page 149), then (page 153) the velocity of rubbing is given by the formula, which may be written, if V be the speed of periphery of the pitch circles, R, R the radii, -44}* If, therefore, the wheels be supposed to turn through a small space 8x measured on the pitch circles, the pair of teeth will slide on one another through the small space &y, given by the formula 1 This enables us to find the work done in overcoming friction, for if P be the pressure between the pairs of teeth, Work done= [/. Pdy= (4.*Jp) [/ P^x. The pressure between the teeth will vary as the wheels turn according to some unknown law, depending on the way the teeth wear and the co-efficient / probably varies. Assuming fP constant, and further, supposing that the chord PT (Fig. 71) is equal to the arc PT, and there- fore to x the arc turned through by the wheels after the teeth pass the line of centres, CH. x. ART. 127.] FittCTIONAL EESISTANCES. 257 The same formula applies before the line of centres, and if we assume the arcs of approach and recess each equal to the pitch p, we shall have for the whole work lost by the friction of a pair of teeth, Whole Work lost=/p(4 + 4 \/i K The energy transmitted during the action of a pair of teeth is therefore the counter efficiency is where n, ri are the numbers of teeth in the wheels. A smaller arc of action is sometimes employed in practice, and the friction will then be less. This is also the case in bevel gear. The formula shows that the friction is diminished by increasing the number of teeth. A more exact solution of this question* can be obtained on the assumption that P varies as it would do if there were only one pair of teeth ; but as this is uncertain it is not practically useful. In all cam and wheel mechanisms the efficiency for a small movement in any position can be determined exactly by a graphical or other pro- cess. For the velocity-ratio can be found, as shown in Part II., and the force-ratio is determinate by the principles of statics, therefore the quotient which gives the efficiency can also be found. In the case of toothed wheels this method shows at oncef that the friction of the teeth before the line of centres is greater than the friction after the line of centres. The difference appears insufficient to account for the injurious effects generally ascribed to friction before the line of centres, which however may be due to other causes. In cam mechanisms the efficiency in one position is little guide to the efficiency in a complete period, which can only be found by a process too intricate to be useful, or by making some supposition as the mean value of the pressure between the rubbing surfaces. The counter efficiency of a train of m equal pairs of wheels is = I + m , /I 1 \ if IT ( - + ) . J \n ri / Assume now that a given velocity-ratio is to be provided by the train, and that the number of teeth in one wheel is given, then it is possible to find the value of m that the friction may be least. The solution of this problem is the same as that of finding the least possible number of teeth, and it was shown by Young that, for this, we ought to take m, so that the velocity-ratio for each pair of wheels is, as nearly as possible, * See Moseley's Mechanical Principles of Engineering. tlbid., page 286. C.M. R 258 DYNAMICS OF MACHINES. [PART in. 3-59. For example, if the train is to give a total velocity-ratio of 46, there should be three pairs of wheels. The gain over a single pair in this case is one-third, but will be much greater for higher velocity-ratios. The solution (first given by Mr. Gilbert) takes no account of axle friction, a circumstance which would greatly modify the result. Some experiments on the friction of toothed and worm gearing have been made by Mr. W. Lewis, a brief account of which, with a table of results, will be found in a treatise on the Mechanics of Machinery by Prof. A. B. W. Kennedy. The table shows that the efficiency of spur gearing increases from '94 at a speed of periphery of 10 feet per minute to -986 at a speed of 200 feet per minute, but is nearly independent of the pressure. In a worm wheel there is a similar increase with the speed, but the efficiency is much smaller, diminishing with the angle of the worm. The experiments by Sir Douglas Gal ton on railway brakes already referred to (p. 248), show that the friction of a wheel when " skidding," that is when sliding on the rail without rotation, is much less than the friction of the brake blocks on a rotating wheel. It diminishes rapidly as the speed increases. In cases of higher pairing by contact we may therefore probably say generally that the co-efficient of friction is relatively small except at the lowest speeds, the difference being greater the higher the speed. SECTION III. FRICTION AL RESISTANCES IN GENERAL. 128. Efficiency of Mechanism in General. It appears from what has been said that an exact calculation of the frictional resistances is im- practicable, partly because the process is too complex to be useful, but chiefly because the co-efficients to be employed are variable according to circumstances, and within limits which are not precisely known. Hence when possible the efficiency of a machine is estimated, not by considering each particular element, but by direct experiment on the machine as a whole, and we conclude this chapter with some general principles which bear on this question. The effort employed to drive a machine may be greater or less, according to the resistance which is being overcome, and therefore the stress between each element will also vary according to this effort. As, however, these stresses depend also on other forces, such as weight and elasticity, which have no connection with the effort, but are always the same, they will not increase so fast, and the frictional resistances will accordingly be proportionally less the greater the effort. Some resist- ances are absolutely constant, for example, the friction of bearings, the load on which is simply the weight of a fly-wheel or other moving part L CH. x. ART. 128.] FRICTIONAL RESISTANCES. 259 or the friction of a piston rod in its stuffing box. Others are sensibly proportional to the driving effort or the useful resistance, in which case, when the ordinary laws of friction apply, the loss of work increases in direct proportion to these quantities. The greater number depend on both variable and constant forces, but these may be in great measure separated into two parts, one of which is approximately constant and the other approximately proportional either to the driving effort or to the useful resistance. Hence, if U be the useful work done and E the energy exerted in a period of the machine, where k, k' are numerical co-efficients and B the work done in over- coming the constant resistances. In hydraulic and other machines, where fluid resistances occur, terms depending on the speed of the machine must be added, indeed this is so in all machines when driven at a high speed ; because forces due to inertia increase the friction, and besides, shocks and the resistance of the atmosphere have to be considered. Such cases, however, are not considered here. If we transfer the term k'E to the other side of the equation and divide by 1 - k', we get where e, E Q are two new constants derived from the former ones, of which E Q is the work done in driving the machine when unloaded, and \+e the counter-efficiency when the load is very great. The same formula may also be written in a way which is some- times more convenient. Let P be the mean value of the driving effort and R that of the useful resistance during a complete period, r the mean value of the velocity-ratio of the working and driving pairs, then P = (l+e)Er+P Q , where P is now the effort required to drive the machine when unloaded. In hoisting machines R is the weight lifted and P the hauling force usually called the power, RjP is the mechanical advantage or purchase. In the steam engine, if p m be the actual mean effective pressure, p' m the part of that pressure employed in overcoming the useful resistance, p Q the pressure necessary to drive the engine when unloaded, p m =(l+e)p' m +p . The value of e may be taken as '15 or in large engines somewhat less. The constant^, often called the "friction pressure," is from 1 to 1 J Ibs. or in marine engines 2 Ibs. or more per square inch. At high speeds and pressures the ordinary laws of friction fail and e is diminished, the constant friction is then relatively of more importance. 260 DYNAMICS OF MACHINES. [PART in. Experiments recently made by Professor Thurston on the friction of a horizontal engine driving a crank shaft and fly-wheel showed that the loss of work by friction was nearly independent of the power transmitted, 47 J per cent, of the whole being due to friction of the crank shaft. The question is one on which little is definitely known, but it seems clear that the "constant" friction must be the most important element, and that it must be to a great extent uncertain, varying from time to time even in the same engine. If the direction of motion of the machine be reversed so that the original resistance becomes the driving effort and the effort the resistance, the same general formula is approximately true, but the constants k, k' are interchanged. Unless under special conditions the efficiency is not the same in the two cases, and in fact is generally very different. Let us suppose that in a machine working against a known reversible resistance, the driving effort is gradually diminished until the machine reverses, and let E' be the work done when reversing, we have the equations from which by subtraction and dividing by U we find ^__2__1^' E_ U~l + k' l+k'' U' a formula which gives the efficiency when reversing. If the original efficiency be less than |(1 - k'), the machine will not reverse even when the driving force is entirely removed. In most forms of hoisting machines k' is small enough to be neglected, and we have the important principle that a machine will not reverse if its efficiency is less than '5. It will not reverse under any circumstances if k > 1 . As previously explained in the case of a screw, non-reversibility is a property so valu- able in practical applications as to be worth obtaining at the sacrifice of efficiency. The differential pulley block is a common example. Frictional resistances, though a source of waste of energy, are usefully employed in machines for various purposes. In screws and driving belts we have already found them used for the purpose of locking a pair or closing a kinematic chain, and many instances of the same kind might be referred to. Another application of equal importance will be considered in the next chapter. EXAMPLES. 1. A weight is moved up a plane inclined at 1 vertical to n horizontal by an effort parallel to the plane : show that the counter-efficiency is 1 + r?/, where /is the co-efficient of friction. Find the value of n for a mechanical advantage of 10 : 1 and a co-efficient -05. Ans. n=2Q. cH.x.ART.128.] FRICTIONAL RESISTANCES. 261 2. Show that the pressure on the guide bars of a direct-acting engine is approximately proportional to the ordinates of an ellipse, and deduce the work lost per stroke. Referring to Fig. 91 let X be that pressure, then p X=S . sin 0=P . tan 0= sin 6 approximately. If the radius of the crank circle represent P, and an ellipse be drawn with the same major axis, and minor axis=P/n, X will be the ordinate of the ellipse at a point repre- senting position of piston. Loss of work per stroke =/x Area of semi-ellipse P TrfsP = * f -- a n=-^' where s is the stroke and / the co-efficient of friction. 3. A bearing 16" diameter is acted on by a horizontal force of 50 tons and a vertical force of 10 tons. Find the work lost by friction per revolution, using a co-efficient of one-eighteenth. Find also the horse-power lost by friction at 70 revolutions per minute. Ans. Loss of work=ll'87 foot-tons. H.P. =56 '4. 4. The thrust of a screw propeller is 20 tons, the pitch 28 feet. The thrust block is 18" diameter at the centre of the rings. Find the efficiency with a co-efficient of friction of "06. A ns. Efficiency ='986. 5. Find the efficiency of a common screw and nut with pitch angle 45 and co-efficient 16. Ans. Efficiency =72. 6. A screw bolt is " diameter outside and "393" at the base of the thread. The effective diameter of the nut is 3", the pitch angle '07, and the co-efficient of friction '16 ;. supposing it screwed up by a spanner two feet long, find the mechanical advantage. Tension of bolt =218 x pull on the spanner. 7. Find the efficiency of a pair of wheels, the number of teeth being 10 and 7o, and the co-efficient of friction '15. Ans. '954. 8. The stroke of a direct-acting engine is 4 feet, piston load 58 tons, load on crank- shaft bearings 10 tons, connecting rod 4 cranks : trace the curve of crank effort when friction is taken into account, assuming all bearings 16" diameter and co-efficient one- eighteenth. Find the "dead angle." 9. In the last question, if the engine drive the screw propeller of question 4, find the efficiency of the mechanism, including thrust block, by the approximate method. The connecting rod may be supposed indefinitely long except for the purpose of estimating the efficiency of the guide bars. Efficiency = "989 x ( "97) 2 x "986= '92. 10. A rope is wound i/hrice round a post, and one end is held tight by a force not exceeding 10 Ibs. What pull at the other end would be necessary to make the rope slip, the co-efficient of friction being supposed '366? Ans. 10,000 Ibs. 11. Find the necessary width of belt three-sixteenths inch thick to transmit 1 H.P., the belt embracing 40 per cent, of the circumference of the smaller pulley and running at 300 feet per 1'. Co-efficient= "25. Strength 285 Ibs. per sq. inch. Ans. Breadth = 4". 12. In question 10 construct the friction spiral showing the tension of the rope at every point. 13. The axles of a tramway car are 2^" diameter, and the wheel 2' 6": find the resistance, being given, that the co-efficient of axle friction is '08 and that for rolling '09. Ans. Resistance =28 Ibs. per ton. 14. Find the efficiency of a pulley 6" diameter, over which a rope ^" diameter passes, the axis of the pulley being " diameter, and the load on it twice the tension of the rope. Co-efficient of axle friction '08. Co-efficient for stiffness of rope "47. Ans. Efficiency =94 per cent. 15. From the result of the preceding question deduce the efficiency of a pair of three- sheaved blocks. Ans. Efficiency =71 per cent. 262 DYNAMICS OF MACHINES. [PART in.] 16. A wheel weighing 20 Ibs., radius of gyration 1', is revolving at 1 revolution per second on axles 1" diameter. It is observed to make 40 revolutions before stopping : find the co-efficient of axle friction. -4ns. Co-efficient= '059. 17. In a pair of three-sheaved blocks it is found by experiment that a weight of 40 Ibs. can be raised by a force of 10 Ibs., and a weight of 200 Ibs. by a force of 40 Ibs. Find the general relation between P and W, and the efficiency when raising 100 Ibs. P=^s W+$. Efficiency = 784 when raising 100 Ibs. e=i- 18. Find the distance to which power can be transmitted by shafting of uniform diameter, with a loss bj r friction due to its weight of n per cent., assuming that the angle of torsion is immaterial, and co-efficient for strength 9,000 Ibs. per square inch. If /be co-efficient of friction, then the length of shafting is 13^ ~e in feet. REFERENCES. For further information on subjects connected with the present chapter the reader is referred to KENNEDY, Mechanics of Machinery. Macmillan. CHAPTER XL MACHINES IN GENERAL. 129. Preliminary Remarks. In the motion of a machine the relative movements of the several parts are completely denned by the nature of the machine, and the principal action consists in a transmission and conversion of energy. Hence it is that the principle of work is of such importance in all mechanical operations that it is desirable to consider it as an independent fundamental law verified by daily experience. Even in applied mechanics, however, we have sometimes to do with sets of bodies, the relative movements of which are not completely defined by the constraint to which they are subject, but partly depend on given mutual actions between them. When this is the case, the principle of work, though still of great importance, is not by itself sufficient to determine the motions. Again, if we wish to study the forces which arise when the direction of a body's motion is changed, the principle of work does not help us, for no work is done by such forces. For example, the position of the arms of a governor, revolving at a given speed, cannot be found, except, perhaps indirectly, by the methods hereto employed. We then resort to the ordinary laws connecting matter and motion, which form the base of the science of mechanics, and of which the principle of work itself is often considered as simply a consequence. The present chapter will be devoted in the first place to a brief sum- mary of elementary dynamical principles, and afterwards to various questions relating to machines and the forces to which they are subject. SECTION I. ELEMENTARY PRINCIPLES OF DYNAMICS.* 130. Quantity of Matter. Mass. The effect of an unbalanced force P, acting during a certain time t, on a piece of matter, is to generate a velocity v, which is proportional to P and t directly and the quantity of *'The brief statement here made of principles assumed in subsequent articles of the treatise is not intended as a substitute for a treatise on elementary dynamics. 264 DYNAMICS OF MACHINES. [PART in. matter inversely. When the force P is equal to the weight fF, as in the case of a body falling freely, the velocity generated in 1" is known to be g, where g is a number which varies slightly for different positions on the earth's surface (Art. 99), but is precisely the same for all sorts of matter. We may express this by the equation ft-Hk 9 Since we use gravitation measures exclusively, the symbol W in this formula must be understood to mean the weight of the piece of matter as compared with that of a standard piece at some definite place, as, for example, Greenwich Observatory. The weight PFiheu varies according to the actual position of the piece of matter upon, above, or below the earth's surface ; but these variations are in exact proportion to cor- responding changes in the value of g, so that the quotient W/g, commonly known (p. 200) as the Mass, furnishes a definite measure of the inertia and therefore of the quantity of matter in the piece. The quotient thus described as the " mass," however, is not numeri- cally equal to the quantity of matter because the unit of measurement is different. The unit of mass is here derived from the unit of force, being necessarily a quantity of matter such that W/g is unity, that is, the weight of the unit mass must be g units of force. But the weight of the standard piece at Greenwich is one unit of force, and therefore the unit mass is the quantity of matter in the standard piece multiplied by g Q , the value of g at Greenwich. Now quantities of matter are practically determined by the process of weighing them against the same standard pieces as are employed in measuring forces ; the quantity of matter in the standard piece is therefore the unit of measurement. So much is this the case that in ordinary language the terms "pound" or "kilogramme" are used indiscriminately for force, or the matter on which force acts. The unit of mass then in gravitation measure, as usually defined, is the unit quantity of matter multiplied by g Q , and therefore, if /* be the quantity of matter, and m the mass, It is obvious that the value of p is an absolute measure of the quantity of matter, being independent of time, space, and the place where the weighing takes place ; it is numerically the same as W , the force with which the quantity of matter //, is drawn to the ground at Greenwich, for which reason the term "weight" in ordinary language is used in the sense of quantity quite as often as in that of force. On the other hand, the value of m is a measure which is on. xi. ART. 131.] MACHINES IN GENERAL. only relative to the numerical value of g . Hence in gravitation measure, the word " mass " means the quantity of matter measured on a special scale, dependent on the units of time, space, and force adopted. Some remarks on the " absolute " system of measurement employed by physicists, in which the mass and the quantity of matter are identical, will be found in the Appendix, but as this system has not as yet been introduced into practice, either at home or abroad, it is unnecessary for the purposes of this work to dwell on the subject here. 131. Equation of Momentum. Centrifugal Force. Denoting, then, the mass by m, the equation connecting P, t, and v, becomes Pt^mv. The products Pt, mv are called IMPULSE and MOMENTUM respectively, and the equation may be written Impulse exerted = Momentum generated. A unit of impulse is unit force exerted for unit time, usually 1 Ib. for 1", a quantity for which the expression " second-pound " may conveniently be used. If P be variable, then impulse is calculated in the same way as the energy exerted by a variable force (Art. 90), the abscissa of the diagram now representing time instead of space. The body we are considering may have a velocity at the commence- ment of the time t, and the force may be partially balanced ; if so, v must be understood to be the change of velocity, and P the unbalanced, part of the force. So far, the equation] of momentum is analogous to the equation of work, impulse representing the time effect of force as energy represents its space-effect. There are, however, two important differences, which we consider in the present and next succeeding article. Change of kinetic energy arises from a change in the magnitude of the velocity irrespectively of direction, whereas change of momentum must be estimated in the direction of the force producing it, and includes change of direction. Hence the equation is applicable when the direction of the force is perpendicular to the direction of motion, so that the only effect produced is change of direction. The rate of change of velocity taken in the most general sense, is called Acceleration, and the equation of momentum may also be written P~mf t where /' is the acceleration estimated in the direction of the force. Bjr taking the force perpendicular to the direction of motion we get the 266 DYNAMICS OF MACHINES. [PART HI. equation which connects the curvature of the path of a moving body with the force ft, which compels it to deviate from the straight line, namely, where v is the velocity and r the radius of the circle in which it is moving at the instant considered. Since v/r is the angular velocity of the line in which the body is moving the formula shows that the deviating force is equal to the product of the momentum and the rate of deviation. Like other forces this arises from the mutual action between two bodies : one of these is the moving body ; the other, the fixed body which furnishes the necessary constraint. If we are thinking of the fixed body instead of the moving bod}'', we call the force R the Centrifugal Force, being the equal and opposite force with which the moving body acts on the body which constrains it. The two forces together con- stitute what we have already called a Stress (Art. 1). To determine a stress of this kind it is necessary to refer the direction of motion to some body which we know may be regarded as fixed, and we are not at liberty to choose any body we please for this purpose, as in kinematical questions. What constitutes a fixed body is a question of abstract dynamics, into which we need not enter. For practical purposes the earth is taken as fixed. If a body rotate about a fixed axis the centrifugal forces, arising from the motion of each particle, will not balance one another unless the axis be one of three lines, passing through the centre of gravity, which are called the " principal axes of inertia " at that point. In most cases occurring in practical applications the position of these lines can be at once foreseen as being axes of symmetry. This is the case, for example, in homogeneous ellipsoids and parallelepipeds. In the com- mon case of a homogeneous solid of revolution, the axis of revolution, and any line at right angles to it through the centre of gravity, are principal axes. If the axes of rotation be parallel to one of these axes, hut does not pass through the centre of gravity, the centrifugal forces reduce to a single force, which is the same as if the whole mass were concentrated at the centre of gravity. In all other cases there is a couple depending on the direction of the axis of rotation, as well as the force just mentioned. (Ex. 15, p. 291.) 132. Principle of Momentum. Again, every force arises from the mutual action between two bodies, consisting in an action on one accom- panied by an equal and opposite reaction on the other. Hence, if we -CH. xi. ART. 133.] MACHINES IN GENERAL. 267 understand by the total momentum of two bodies in any direction, the sum or the difference of the momenta of each, according as the bodies move in the same or in the opposite direction, it appears that the total momentum will not be affected by the mutual action between the two. And more generally, if there be any number of bodies we shall have Total impulse exerted = Change of total momentum, where, in reckoning the impulse, we are to take into account external forces alone, and not the internal forces arising from the mutual action of the parts of the set of bodies we are considering. This equation expresses one form of what we may call the Principle of Momentum ; other forms will be explained hereafter in connection with questions relating to fluid motion (Part V.). The total momentum of a number of bodies may be reckoned by direct summation, with due regard to sign, but it may also be expressed in terms of the velocity of the centre of gravity : for, let m be the mass of any particle of the system, the ordinate of which, reckoned from a given origin parallel to a given line, is x ; also, let ^mx denote the sum of all the separate products mx, for all the particles of the system, and let M be the total mass, then we know that the ordinate of the centre of gravity * is given by the formula ^mx x = -=-=-. M Let the velocity of a particle parallel to the given line be u, then if jc v x. 2 , be the ordinates at the beginning and end of 1" we shall have u = x 2 - x r Hence if u be the velocity of the centre of gravity parallel to the same line ' which equation may be written Mu = 2m%, showing that the total momentum of the system is the same as if its total mass were concentrated in its centre of gravity. We conclude from this that the motion of the centre of gravity can only be influenced by external forces and not by any action between the parts of the system. 133. Internal and External Kinetic Energy. If we multiply the equation just obtained by 2u and remember that u being constant may be placed within the sign of summation, we obtain * Called more correctly by Young "the centre of inertia" and by modern writers on mechanics the "centre of mass," or more briefly the "ceutroid." 268 DYNAMICS OF MACHINES. [PART in.. which, adding 2wiw- 2 to each side and re-arranging the terms, may be written M~u? + ?m (u -i*) 2 = SWIM*. This is true in whatever direction the velocities are estimated, and we can therefore write down two similar equations for the velocities in two directions at right angles to the first. Now the resultant of three velocities at right angles is the square root of the sum of the squares of the components, also u - u is the velocity parallel to x relatively to the centre of gravity ; hence if U be the resultant velocity of the centre of gravity, v, v the velocities of any particle relatively to the body regarded as fixed and relatively to the centre of gravity respectively, we have, adding the three equations together, and dividing by 2, The first term on the left-hand side of this equation is what the energy would be, if the whole mass were concentrated at its centre of gravity, a quantity which may be described as the External Energy, or otherwise as the Energy of Translation of the system. The second term is the energy relatively to the centre of gravity considered as fixed, which may be called the Internal Energy. The right-hand side is the total energy of motion, and we see therefore that this is the sum of the internal and external energies. In the case of a single rigid body the motion relatively to the centre of gravity is always a rotation about some axis, and therefore Energy of Motion = Energy of Translation + Energy of Rotation, a principle already employed in a preceding chapter (p. 202). In the case of a set of rigid bodies the internal energy is the sum of the energies of rotation of each together with the internal energy of a set of particles of the same mass occupying the centres of gravity of the bodies and moving in the same way. 134. Examples of Incomplete Constraint. In the cases which occur in applications to machines and structures we usually have to consider two bodies moving in straight lines without rotation. CASE I. Recoil of a Gun. When a cannon is fired the shot is pro- jected and the cannon recoils with velocities dependent on the relative weights of the shot, the cannon, and the charge of powder. Here, the motion is due to the pressure of the gases generated by the combustion of the powder one way on the shot, the other way on the cannon. If the inertia of these gases could be neglected these pressures would be exactly equal at each instant and would cease as soon as the shot left the bore. The impulse exerted on shot and cannon would then be equal. In fact, the inertia of the powder gases CH. xi. ART. 134.] MACHINES IN GENERAL. 269 causes the pressure to be greater and to last longer on the cannon than on the shot, so that the impulses on the two are not nearly equal. For the present we shall neglect this, and shall further suppose that the material of both shot and gun is sensibly rigid. In general, recoil is checked by an apparatus called a " compressor," which supplies a gradually increasing resistance to the backward move- ment of the gun, while friction and the resistance to rotation of the shot resist the forward movement of the shot. In the first instance suppose there are no such resistances, let V be the velocity of recoil and M the mass of the gun, v the velocity and m the mass of the shot ; then, since the impulse exerted is the same for both, MF=mv. Further, if the weight of the charge and the amount of work 1 Ib. of it is capable of doing be known, the explosion will develop a definite amount of energy (E) which will be all spent in giving motion to the shot and the cannon. Energy of Explosion = \MV- + %mv 2 . Here E is the sum of two parts Energy of Shot = ^ - E, Energy of Recoil = -^ E. M + m The energy of recoil has to be absorbed by the compressor, usually an hydraulic brake, which will be considered hereafter (see Part V.). CASE II. Collision of Fessels. When two vessels come into collision an amount of damage is done depending on the size and velocities of the vessels. Here we may suppose the vessels moving in given directions with given velocities ; let the velocities parallel to a given line be u^ u v and the masses m v m , then, as in Art. 133, the velocity of the centre of gravity parallel to the same line is m 1 and therefore the velocities of the vessels relatively to their common centre of gravity must be . _m l ('u 2 -u l ) . tto II - . % + m. 2 Two similar equations may be written down for the velocities in a -direction at right angles to the first. Square and add corresponding 270 DYNAMICS OF MACHINES. .[PART m.. equations, multiply by ^m v Jra.,, and add the pair of products, then (Art. 133) Internal Energy = 1 . m where V is the velocity of either vessel relatively to the other, a quantity found immediately from the given velocities of the vessels by means of a triangle of velocities. The total kinetic energy of the vessels is found by adding the energy of translation. As, however, this quantity cannot be altered by the collision, it is clear that the amount of work done must depend on the internal energy alone : we may properly call it there- fore the "energy of collision." If the displacements in tons of the vessels be W v W^ we shall have, in foot-tons, W W V' 1 Energy of Collision = ^ ^ . . It is not, however, to be supposed that the whole of this is neces- sarily expended in damage to the vessels ; if the circumstances of the collision be such that the vessels, even though completely devoid of elasticity, would have a motion of rotation or a velocity of separation of their centres of gravity, then the corresponding internal energy must be deducted. Also the influence of the water surrounding the vessels has been left out of account ; this somewhat augments the effect by increasing the virtual mass of the vessels. The same formula may be used for other cases of impact, but the effects of impact depend so much on the strength and stiffness of the colliding bodies that the subject cannot be further considered here (Ch. XVI.). CASE III. Free Rotation. If the axis of rotation of a solid be free to move, it will shift its position as already stated unless the axis be one of the principal axes of inertia : but if it be a principal axis it will remain fixed in direction unless external forces act upon it. When the solid rotates rapidly it offers a considerable resistance to any change of direction of its axis which can only be overcome by the action of forces which have a moment about an axis inclined to the axis of rotation. In consequence a body in rapid rotation may possess considerable stability in circumstances where in the absence of rotation equilibrium would be impossible. The principle is important and has many applications, the well-known gyroscope being a common example. The question, however, requires a con- siderable amount of explanation to render it intelligible, and the limits of this work render it impossible to do more than mention it here. The theory of the gyroscope is given in a clear and simple CH. XI. ART. 135.] MACHINES IN GENERAL. 271 form by Professor Worthington, in a small treatise referred to at the end of this chapter. SECTION II. REGULATORS AND METERS. 135. Preliminary Remarks. Revoking Pendulum. Centrifugal forces may be employed in machines to do work by energy transmitted from a source, or derived from the kinetic energy of the moving parts. Sometimes the work thus done is the object of the machine, as in certain drying machines where the substance to be dried is caused to rotate with great rapidity so that the fluid is expelled at the outer circumference : or, partially, in centrifugal pumps. Mor frequently they serve to move a kinematic chain connected with a shifting piece which regulates the speed of the machine. Such mechanisms are called Centrifugal Regulators or, more briefly, Governors. In Fig. 112 Q is a heavy particle attached by a string to a fixed point and revolving in a horizontal circle the centre of which is ^vertically below 0. This will be possible if the centrifugal force due to the motion of the particle is equal to the horizontal component of the tension of the string. Let S be that tension, W the weight of the particle, and let the string make an angle 6 with the vertical, then the horizontal and vertical components of S are Fig.112 Let A be the angular velocity of the revolving particle, then it is shown in works on elementary dynamics that the centrifugal force is X= .A*.QN. Equating these values of X and eliminating S, W.texie = .A*.QN. Since QN= ON. tan 0, this reduces to the simple formula which shows that the vertical distance of Q below the point of sus- pension depends on the speed, not on the length of the string or the magnitude of the weight. 272 DYNAMICS OF MACHINES. [PART in. This distance is called the " height " of the revolving pendulum, and will be denoted by h. If t be the period, that is the time of a complete revolution, we find, since At = '2-!r, showing that the period is the same as that of a double oscillation of a simple pendulum of length h (see Art. 103). The height of a simple revolving pendulum may, as already explained in Art. 101 (p. 201), often be conveniently adopted as a measure of a speed of revolution. It is inversely proportional to the square of the speed being given in inches at n revolutions per minute by the equation 35,232 li = , n- Instead of supposing the string attached to a point in the axis of revolution, we may suppose it attached to a point K, rigidly connected by a cross-piece KE, with a revolving spindle ON. The same reasoning applies, being now an ideal point, found by prolonging the string to meet the axis. The height of the pendulum is still ON, and is found !>y the same formula. 136. Speed of a Governor to overcome given Fridional Resistances. Loaded Governors. In the simplest centrifugal governors two heavy balls are attached to arms, which are jointed either directly to a revolving spindle, or to the ends of a cross-piece attached to a spindle. Motion is communicated by links from the arms to a piece sliding on the spindle, the movement of which is communicated by a train of linkwork, either to a throttle valve directly controlling the supply of steam, or to an expansion valve which regulates the cut-off. In either case an upward movement of the arms has the effect of diminishing the mean effective pressure, and a downward movement of increasing it. Two forms of this mechanism are shown in the figures of Plate VI.: in one of these (Fig. 1) the weight of the sliding piece is increased by a large additional weight, the governor is then said to be loaded ; while in the other (Fig. 2) the arms cross each other, the spindle being slotted, or the arms bent to permit this. The object of these arrangements we shall see presently. If now the speed of revolution be increased or diminished, the arms move outwards or inwards, arid so adapt the mean effective pressure to the work which is being done. If there were no frictional resistances the smallest variation of speed would produce a corresponding motion in the arms ; but, as the linkwork mechanism necessarily offers a certain resistance, motion cannot take place until the change of speed has To face page 2 7 2. CH. xi. ART. 136.] MACHINES IN GENERAL. 273 reached a certain magnitude, which is smaller the more sensitive the governor is. These frictional resistances are measured by a certain addition to, or subtraction from, the weight of the sliding-piece, which might be determined experiment- ally, and therefore will be supposed a known quan- tity F. We first investigate what change of speed will be necessary to overcome them. In Fig. 113 AQB is a triangle revolving about AB which is vertical, a heavy particle is placed at Q, and T the weights of the bars AQ, BQ are small enough to be neglected. If the triangle revolve at a speed cor- responding to the height AN of a simple revolving pendulum AQ, there will be no stress on BQ, but if it be greater or less there will be a pull or thrust, the magnitude of which is determined thus : Set up NO equal to the height due to the revolutions, and join QO. Then it appears from what was said in the last article that if NO be taken to represent the weight W of the particle, NQ will represent X the centrifugal force, and therefore the resultant force on Q must be represented by QO. Through draw OZ parallel to BQ, then QOZ is a triangle of forces for the joint Q of the triangular frame AQB, so that QZ, OZ must represent the stresses on AQ, BQ respectively. For our purposes we require the vertical component of the stress on the link BQ, which is obtained by drawing ZL horizontal : OL must be the force in question which we call T. In the Figure T is an upward force, being below A, and the speed of revolution therefore great. In this construction the links need not be actually jointed to the spindle AB; they may, as in the simple pendulum, be attached to the extremities of cross-pieces fixed to AB. A and B are then ideal points of intersection of the links with the axis of rotation. In general AQ and BQ are equal; we may then obtain a simple formula for T. Let N0 = h, a quantity given by the same formula as before for a given speed, and let AN, the actual height of the governor, be denoted by H, then OA = H-h; but in the case supposed, OA '20 L, therefore h formulae which give the pull for any speed, and conversely the speed for which the pull will have a given value. In practical applications there are always two balls, so that if W be the weight of one, '2T will be the pull due to both. C.M. s 274 DYNAMICS OF MACHINES. [PART in. We can now find within what limits of speed the mechanism can be in equilibrium. Let w be the weight of the sliding-piece B, inclusive of any load which may be added to it, h the height due to the speed at which there is no tendency to move the arms, h l9 A 2 , the heights due to the speeds at which they are on the point of moving upwards or down- wards respectively, then W W W In general F will be small compared with W+w, and then we have very approximately, F fVt) ll = IV //'-I == IV TTr W + w These results show that loading a governor gives it a power of over- coming frictional resistances which would otherwise require a weight of ball equal to the sum of the load and the actual weight. Light balls may therefore be used as in the figure (Plate VI.) without sacrificing power, as the load may be made great with- out inconvenience. The speed of a loaded governor is greater than that of a simple governor of the same actual height, as appears from the formula for h. It may be altered at pleasure by altering the load. This arrangement is known as Porter's governor, from the name of the in- ventor. 137- Variation of Height of a Pendu- lum Governor by a Change of Position of the Arms. Next suppose the speed to alter so much that the arms actually change their position, then if H re- mained the same, the tendency to move would also be the same, and the movement must therefore continue until the speed is brought back within the limits for which rest is possible. In the ordinary pendulum CH. xi. ART. 138.] MACHINES IN GENERAL. 275 governor, however, H alters in a way which depends on the mode of attachment and arrangement of the arms, as will appear from the annexed diagram (Fig. 114) which shows three cases. In the centre figure the arms are jointed to the spindle so that their centres of rotation are in the axis, in the two others they are jointed to a cross-piece KK, but differently arranged in the two cases. In all three, as explained in the preceding article, the height H is measured to A, the real or ideal intersection of the arms with the axis of rotation. Suppose the arms to move from position 1 to position 2 in the figure ; H diminishes to H', but the amount of diminution is different in the three cases : in the right-hand figure it is greatest, and in the left-hand least. Indeed in the latter case where the arms are crossed it is possible by making KK long enough, to change the diminution into an increase. (Ex. 4, p. 289.) If then, by an increase in the speed, the arms move into a new position, the speed required for equilibrium does not remain the same but increases, so that, when the adjustment has been effected between the energy and the work, the speed is increased, instead of being the same as before. Conversely, after adjustment to suit a diminished speed, the speed actually attained is diminished. Thus the effect of the variation in H is to widen the limits within which the speed can vary. 138. Parabolic Governors. A governor may be constructed in which H does not vary at all. In Fig. 115 Q is a ball resting on a curve CC attached to a vertical spindle. The curve lies in a vertical plane, and D is the lowest point. When at rest Fig. 115. the ball can only be in equilibrium at 7), but, if the spindle revolve, it may rest at another point, the position of which depends on the speed of revolution. If the curve be a circle we have only the pendulum gover- nor in a different form, for, drawing the normal QA and the perpendicular QN, A will be a point to which Q might be attached by a string and the curve removed. Hence, AN must be equal to h, the height due to the speed of revolution. But if the curve be not a circle the same thing must be true, only A is now not a fixed point; hence in every case the sub-normal AN of the curve at the point of equilibrium must be equal to h. In general this geometrical condition determines one, and only one, position for a given speed ; but if the curve be a 276 DYNAMICS OF MACHINES. [PART m. parabola with vertex at Z), AN will be constant, and therefore Q will rest in any position for one particular speed, but for lower speeds will roll down to D, and for higher speeds will move upwards inde- finitely. We have here a governor for which, neglecting frictional resistances, only one speed is possible. Such a governor is said to be "isochronous." The curve arrangement is inconvenient for constructive reasons, but if it be replaced by a linkwork mechanism the ball still moves in a parabola. An isochronous governor is therefore often said to be "parabolic." The term is preferable, for no governor is actually isochronous on account of frictional resistances. A pendulum governor is much more nearly parabolic when the arms are crossed, and by properly taking the length of the cross-piece (Ex. 4, p. 289) it may be made exactly parabolic for small displacements. This arrangement is called Farcot's governor from the name of the inventor. 139. Stability of Governors. If the curve CC be not a parabola JET, which in this case is the sub-normal, will diminish or increase as the ball Q moves outwards. Take the first case and suppose Q in equi- librium at a certain point when the speed of revolution has a given value. Let Q now be moved up or down, then, if released, it will not remain at rest, but will return towards its original position and oscillate about it, or in other words the equilibrium of Q is stable. A governor possessing this property is described as "stable," and its stability is greater the quicker H diminishes. Similarly when H increases for an outward movement of the balls the governor is " unstable," and a parabolic governor may properly be described as " neutral." A certain degree of stability is necessary for the proper working of a governor, and the amount required is greater the greater the frictional resistances. For assuming the revolutions at which the machine is intended to work to be n, the balls commence to work outward at the speed n + x, where x is a small quantity depending on the frictional resistance. After starting, the frictional resistances are not increased, but on the contrary will somewhat diminish ; and, in a neutral governor, the balls therefore move outwards with increasing speed until by alteration of the regulating valve the supply of energy is diminished and the speed of the machine lessened. This change however requires time, and besides the balls are in motion and have to be stopped. The consequence is that they move outwards too far, and the supply of energy being too small the revolutions diminish to n-x, the speed necessary to move the balls CH. xi. ART. 140.] .MACHINES IN GENERAL. 277 inwards, notwithstanding the frictional resistance. Thus the motion is unsteady, the balls oscillating, and the speed fluctuating, between limits wider than nx without ever settling down to a permanent regime, an action known as "hunting." The oscillation of the balls may be checked by a suitable brake, but it is preferable to employ a governor possessing a moderate degree of stability ; the tendency to move the balls then diminishes as soon as the balls move, and they stop before moving far. The greater the frictional resistances the greater is the change required to enable the balls to return at once if they have moved too far for equilibrium. An important characteristic therefore of a good centrifugal governor is that the stability be capable of adjustment to suit the frictional resistances. Certain forms of compound governors, as for example that known as the "cosine," fulfil this condition and can, probably, be made more perfect than the simple pendulum governor. It should also be remarked that a governor should not be so sensitive as to be called into action by the changes of speed in the course of a revolution consequent on the fluctuation of energy of the moving parts. These changes are regulated by the fly-wheel as already fully described in Ch. IX. All such mechanisms are however imperfect in principle, for they cannot come into operation till a certain change of speed has actually existed for a perceptible length of time. Where the changes of resistance are sudden and violent the best governor will scarcely prevent violent fluctuations in speed. In screw vessels, where this difficulty is much felt, it has been proposed to employ an auxiliary engine rotating against a uniform resistance ; any difference of speed of which and the screw shaft immediately shifts the regulating valve. 140. Brakes. In order that a machine may be under complete control when the changes of resistance are sudden and violent, and especially when it is required to stop it, it is not sufficient to cut off the supply of energy, but it is necessary in addition to have some means of absorbing the energy stored in the moving parts. An apparatus for this purpose is called a Brake. The surplus energy may in some cases be stored by springs or an elastic fluid, and subsequently applied to useful purposes ; the brake is then combined with an accumulator. In general, however, this cannot conveniently be accomplished arid frictional resistances are then employed to convert the energy into heat, which is dissipated by radiation and conduction. When the amount to be disposed of is not too great the friction of two solids pressing against one another may be used 278 DYNAMICS OF MACHINES. [PART in. for the purpose, but care must be taken to provide sufficient surface to prevent temperature from rising too high during the process. A brake of this class is generally applied to a rotatory wheel or drum, and consists either of a solid block of wood or metal pressed against the wheel by some suitable mechanism, or else of a strap of metal often lined with small blocks of wood embracing the drum and tightened by a lever or otherwise. Three common forms are shown in Plate VII., two of these are used as dynamometers, and will be referred to again presently. The most powerful brakes however are those in which hydraulic resistances are employed, some examples of which will be found in a later chapter. In the "cup governor," invented by Dr. Siemens,'* a regulator and an hydraulic brake are combined. A cup containing water rotates within a cylindrical casing; at low speeds the water remains within the cup, but as soon as the speed exceeds a certain limit centrifugal action causes it to pour over the edge of the cup into the space between the cup and the casing. A set of vanes attached to the cup rotate between fixed vanes attached to the casing, and break up the descending water, which re-enters the cup by an orifice in the bottom. There is then a great resistance to the motion of the cup which absorbs surplus energy. Some other forms of governor will be considered hereafter. 141. Dynamometers.- Mechanisms employed for the purpose of measuring physical quantities, such as time, speed, etc., are called generally Meters. The subject is very extensive, and would require a complete chapter to deal with even in outline. We can only notice here very briefly the apparatus used for the measurement of Power, a class of instruments known as Dynamometers. They are of very various construction, the most common being those in which the instru- ment measures the driving effort while the speed is independently determined and the power thence obtained as in Art. 97, page 193. (1) In Fig. 4, Plate III., page 141, a common form of " transmission " dynamometer is represented. A shaft transmitting power is divided into parts and bevel wheels ED attached to each. A lever A turning about an axis concentric with the shaft in a plane perpendicular to it carries bevel wheels (7, gearing with BD, through which the power is transmitted. If A be held fast a couple will be required to prevent it turning, which is just twice the driving couple being transmitted, and hence if a weight sliding on A, as shown in the figure, be so placed by *Phil. Trans., 1866. PLATE VII Fig- I Fig. 3 Fig. 4 To face page 279. CH. xi. ART. 141.] MACHINES IN GENERAL. 279 trial that A just remains horizontal the driving couple in question will be determined. Hence the revolutions of the shaft being known the power can be found. (2) Two shafts being connected by a belt some arrangement may be adopted by which the difference of tensions of the two sides of the belt is measured, and thus the driving effort being transmitted may be determined. For example, in the apparatus employed by Froude and Thorneycroft to measure the power required to drive a model screw propeller the two sides of the belt pass round pulleys mounted at opposite ends of a lever turning about a fulcrum at the centre. The force required to prevent the turning furnishes a measure of the difference of tension. (3) In Fig. 1, Plate VII., a "friction dynamometer" is represented in one of the various forms in which it is applied. A is a lever from which a weight is suspended, B is a block fixed to A, which rest on a revolving drum. A strap passes below the drum and is tightened by the nuts JVTVtill the friction just balances the weight, which in its turn is adjusted by trial till it just balances the driving couple tending to turn the shaft. Stops are provided to prevent the lever from moving except within narrow limits, and when the adjustment is perfect the lever remains horizontal without resting against either stop. Here the driving couple and consequently the power are determined as in the pre- ceding example, from which it only differs in the way in which the power is employed. Instead of being transmitted to a machine which is being driven it is all absorbed by a friction brake which replaces the machine for the time being. A modification is shown in Fig. 2, in which the strap passes over a wheel and is tightened by a suspended weight, the difference between which and the tension of a spring balance, to which the other end of the strap is attached, measures the driving effort. In both these forms of friction dynamometer any variation in the driving effort requires a corresponding adjustment. The more complex form shown in Fig. 4 is provided with a compensating lever DEC, which tightens the friction strap embracing the wheel when the driving effort is great and loosens it when the effort diminishes. A self- acting adjustment is thus obtained, but the pressure of the fixed pin D fitting into a slot in the end of the lever renders the indications inaccurate, and the error may be serious unless special care is taken. (4) In both the preceding cases the driving effort and the speed of the driving pair are constant, but in the indicator universally employed to measure the power of steam and other heat engines we find an example in which both vary. The driving effort is now measured for each 280 DYNAMICS OF MACHINES. [PART in. position of the piston and a curve drawn which represents it ; the area of this curve will be the work done per stroke, and divided by the length of the stroke will give the mean driving effort. This will be further explained in Part V. SECTION III. STRAINING ACTIONS ON THE PARTS OF A MACHINE. 142. Transmission of Stress in Machines. We have seen (Art. 94, p. 189) that a mechanism becomes a machine if certain links are added capable of changing their form or size, and so producing forces which tend to move the mechanism combined with other forces which resist the motion. Each link so added exerts equal and opposite forces on the elements it connects, and for a pair of forces the general word "Stress" may be used, which has already been employed in Article 1 in the case of the bars of a framework structure. When the machine is at rest the forces, being all in pairs, balance each other, and have no tendency to move the machine as a whole. For example, in the direct-acting vertical engines represented in Fig. l r Plate I., page 109, the driving link is the steam, pressing with equal force, one way on the cylinder cover, and the other way on the piston ; the working, link is the resistance to turning of the crank shaft, which exerts equal and opposite forces, one way on the crank, the other way on the frame which carries the crank-shaft bearings. The steam pressure and the working resistance may each be described as a " Stress." The forces which make up the stress are transmitted from the piston through the connecting rod to the crank, and, in the opposite direction, from the cylinder cover through the frame to the crank shaft. The horizontal pressure of the cross-head on the guide-bars is in like manner balanced by the equal horizontal thrust of the connecting rod on the crank pin> combined with the moment of the working resistance. So in every machine, when at rest, or moving slowly and steadily, the stress is transmitted from the driving pair to the working pair, not only through the movable parts of the machine, but in the opposite direc- tion, through the framing ; and this is a circumstance which must be always borne in mind in designing the framing. Thus, in our example, the steam cylinder and crank-shaft bearing must be rigidly connected by a frame strong enough to withstand the total steam pressure, and, in addition, the bending due to the lateral pressure on the guide bars. We have here one of the simplest examples of the transmission of stress; whether in a machine or in a structure it always takes place in a closed circuit. If the driving pair and the working pair are the same, and acted on CH. xi. ART. 143.] MACHINES IN GENERAL. 281 by the same stress, the whole state of stress is the same for all the mechanisms which are derived by inversion from the same kinematic chain. All such mechanisms are therefore statically as well as kine- matically identical ; it is only when we consider machines in motion, or the straining actions due to gravity, that it is necessary to consider which link (if any) is fixed to the earth. For example, the only differ- ence between the direct-acting engine of Fig 1, and the oscillating engine of Fig. 4, Plate I., is that the working pair is BA in the first and EG in the second. So again, in Plate III., the only difference between the water wheel of Fig. 2 and the horse gear of Fig. 3 is in the nature of the driving link, which in the first case is gravity acting on the falling water, and in the second a living agent. A striking example, of the balance of forces in a machine occurs in the hydraulic riveting machines. Here the working pair is a small hydraulic cylinder and its ram, between which the rivet is compressed. This cylinder is suspended from a crane by chains, and can be moved into any position, as it communicates with the accumulator (Part V.) by a flexible pipe. . Any portable machine, however, is an example of the same kind : machines which require foundations have no complete frame apart from the solid ground which connects their parts together. 143. Reversal of Stress. In many machines the direction in which stress is transmitted through one or more of the moving parts is reversed in the course of the period. For example, in a double-acting engine of the ordinary type the piston rod and the connecting rod are alternately in compression and tension as the crank turns through a revolution. Such a reversal of stress is a cause of shocks which, though they may individually be small, yet from the rapidity with which they recur at high speeds are ultimately destructive, and require in any case to be carefully considered in the design. Suppose a crank which is rotating uniformly to be connected by a rod with a reciprocating piece such as a piston, but in the first instance let there be no steam admitted to the cylinder. When the piston is at the end of its stroke it is at rest, and has to be set in motion ; it con- sequently drags on the crank with a force which we have already investigated in Art. 109, p. 224. As the piston moves onwards the drag diminishes and becomes zero near the middle of the stroke at the point where the velocity of the piston is greatest. In the second half of the stroke the piston is being gradually reduced to rest, and conse- quently presses against the crank pin and drives the crank, thus reversing the stress on the rods. A small amount of play is necessary for the purposes of lubrication between the crank pin and the brasses 282 DYNAMICS OF MACHINES. [PART in. into which it fits, and consequently at the instant of reversal a " knock " occurs, thus damaging the bearing surfaces and wasting energy. The intensity of a knock of this kind depends on the acceleration of the moving piece, and would be small in the case here supposed where there is no steam admitted to the cylinder, so, that reversal occurs in the middle of the stroke. Next imagine steam admitted to the cylinder in the usual way, then, as already described fully in the article cited, the pressure on the crank pin is due to the difference between the steam pressure and the force called into play by inertia, and the effect is that reversal occurs at or near the ends of the stroke. If the speed Tje moderate and the moving parts light, the knock will occur at the ends of the stroke, and if the steam be suddenly admitted and there be no compression, will be of considerable intensity. It may, however, be much diminished by " cushioning," that is, by closing the exhaust port before the end of the return stroke and thus enclosing in the cylinder a mass of steam, the compression of which behind the returning piston furnishes a force which, by its increase, gradually diminishes the stress and renders the reversal at admission less violent. At very high speeds or with heavy moving parts reversal occurs after the stroke has begun ; as shown by the point K on the dotted line L'CL' shown in Fig. 100, p. 225, the effect of reversal in the absence of cushioning is then not so great as if it occurred in the absence of cushioning at the ends of the stroke. Heavy reciprocating parts may therefore, under certain circumstances, be advantageous. When the speed is excessive the forces called into play by inertia are so great that reversal must be avoided altogether. For driving a fan or some similar purpose a small engine of three inches stroke is sometimes run at 1000 or even 2000 revolutions per minute ; on making the calculation by the formula of page 224 we find that the force P necessary to start the piston is now 150 times its weight, and the shock at reversal necessarily great. If the engine is made single acting, reversal can be prevented entirely by cushioning. In the Willans high speed engine the piston rod prolonged moves as a plunger in an independent cylinder containing air, which serves as the cushion, an arrangement which admits of any compression being used in the steam cylinder, which may for other reasons be convenient or economical. The speed in the foregoing case is limited by the amount of cushioning employed, and this is also the case in cam mechanisms with force closure, such as have already been discussed in Ch. VI. 144. Stability of Machines. Balancing. In a machine with recipro- CH. xi. ART. 144.] MACHINES IN GENERAL. 283 eating parts the balance of forces (Art. 142) is destroyed by their inertia when the machine is in motion, and, in consequence, the machine must be attached to the earth or some massive structure by fastenings of sufficient strength. The straining actions on these fasten- ings will now be briefly considered. Taking the case of a direct-acting horizontal steam engine, let P be the total pressure of the steam on the cylinder cover, then the pressure (P r ) transmitted to the crank pin is not equal to P, but there is a difference (S), given by the formula (Art. 109, p. 224) : neglecting obliquity, S=P-P'=W* h This difference will be a force acting on the engine as a whole, and straining the fastenings. The direction of this force is reversed twice every revolution, and its maximum value is obtained by putting x = a in the above formula. In slow-moving engines the value of S is small, but at high piston speeds it becomes very great, and must be carefully provided against, especially when, as in locomotives, the engine cannot be attached to the ground. Fig.116. In most cases there are two cranks at right angles, and therefore two forces S t S' given by the equations S=IT.~.cos6', S' = ~ .sintf, fi h where 6 is the angle the first crank makes with the line of centres. These two forces are equivalent to a single force (Fig. 116), acting midway between them, and a couple L = (S-S')c= W. ^ . c(cos - sin 0), where 2c is the distance apart of the centre lines of the cylinders. The total effect therefore is the same as that of a single alternating force 284 DYNAMICS OF MACHINES. [PART in. combined with an alternating couple, which tends to turn the engine as a whole about a vertical axis. The maximum values are , a , T Wac and they are each reversed twice in every revolution. In locomotives this action produces dangerous oscillations at high and must therefore be counteracted by the introduction of Fig.117. ' ?Ft to the bisector be the weight, gravity lies, being an suitably placed balance weights, so as to neutralize both the force and the couple. Fig. 117 shows a projection on a ver- tical plane of the two driving wheels and their cranks. On each wheel a balance weight is placed, occupying a segment between two or more spokes. The centre of gravity of each weight is in a radius nearly, but not exactly, opposite the nearer crank, the angle of inclination angle i somewhat less than 45. If B the radius of the circle in which its centre of is its centrifugal force ; and by rightly taking the values of B and i the horizontal components of these forces derived from the two balance weights may be made to counteract both the force and the couple (Ex. 10, p. 290). In practice the weights are fixed approximately by a formula derived in this way, and the final adjustment is performed by trial. The engine is suspended by chains, and its oscillations, when perfectly adjusted, are very small even at very high speeds. In high speed marine engines similar forces arise, of great magni- tude, which must add considerably to the strain upon the fastenings, and which are now known to be the principal cause of vibration of the vessel. The question of balancing these forces has therefore become of great importance, and we shall recur to it hereafter. When the speed of a machine is excessive, we have already seen that reversal of stress must be avoided, and besides this the greatest care is necessary that the axis of rotation of each rotating piece passes through its centre of gravity, and coincides with one of th< axes of inertia of the piece (Art. 132). The magnitude of the forces which arise, in case of any error, may be judged of from the results of Exs. 13, 16, pages 290, 291. The vibrations due to these forces will, CH. XL ART. 144A.] MACHINES IN GENERAL. 285 however, in some cases be greatest at some particular speed depend- ing on the natural period of vibration of the frame of the machine which could only be determined by trial. (Ch. XVI.) In similar machines the forces due to inertia will be in a fixed proportion to the weight of the pieces, when the revolutions vary inversely as the square root of the linear dimensions of the machine. 144A. Stability of Machines (continued). The pressure on a piston necessary to overcome the inertia of a connecting rod has already been investigated, but to complete the subject it is now desirable to study the relation which exists between the pressure thus found and the horizontal momentum of the reciprocating parts upon the change of which the disturbing forces considered in the last article depend. Referring to page 229 the horizontal velocity of a point on the rod distant fj-l from the crank pin is /^{sin + p . tan ^> . cos 0}, which may be written Horizontal velocity = (1 - p) V^ . sin + pY. Taking therefore as before / as the acceleration of the piston, we have at the poifit in question F 2 Horizontal acceleration = (1 - /x) - . cos 6 + ft/, as may be easily proved independently. Hence, by summation, the rate of change of the total horizontal momentum of the rod is WV* f X=( 1 -o-F-^- . cos + vW. 1, i/ / where, as before, crl is the distance of the centre of gravity of the whole rod from the crank pin. Writing for /its approximate value given on page 228 we find -. cos 201 n ) X=- Jcoe0+-.coa ga \ If a- = I we have the case of weights actually or virtually attached to the piston, when the inertia-pressure P already found is equal to X. But in the case of the rod P is a smaller quantity, the reason of which is that the angular swing of the rod requires a force at the crank pin in order to produce it. As the rod swings outwards it is gradually stopped, as it swings inwards it is gradually accelerated, by a pull of the crank arm towards the centre. Now if T be this pull, T cos 6 will be a horizontal force at the crank pin which assists in producing the horizontal acceleration, so that 286 DYNAMICS OF MACHINES. [PART in. Taking for X the value just given and for P the value already found, we find after dividing by cos 6, The force T thus found is a tension of the crank arm which furnishes an unbalanced force on the crank centre. The total horizontal dis- turbing force is therefore not P but the larger quantity X given by the formula above. In balancing the reciprocating parts of an engine the formula shows that we have two separate sets of horizontal periodic disturbing forces to consider. The first are given by the formula X-i = . cos 0, ga where 2/F is the total weight of all the parts, connecting rod included. These are independent of the length of the rod, and they go through their variations in one revolution of the crank shaft. The second ai given by the formula X 9 = -. - . cos 20, n ga where W is the weight of the piston and all parts attached to it, whil< W is the weight of the connecting rod, and they go through theii variations in half the time. In the production of vibration the effect of the two sets is entirely different, as will be fully explained hereafter. It is the first set alone which are considered in the preceding article the second set are a consequence of the obliquity of the connecting rod, and though of much less intensity, for a complete balance requii consideration as well as the first. Besides the horizontal forces there are also periodic forces perpendi- cular to the line of centres which may here be briefly noticed. The force T found above has a vertical component T sin 0, and ii addition the pressure of the cross-head on the guide is altered by th( inertia of the connecting rod. The effect of these two vertical force taken together is best seen by dividing the weight W of the rod im two parts q.ti =pl . BB . cos 6 =pl . NN', NN' being the projection of BB' on the plane of section. Summing up the pressures on all the small arcs BB', composing the semicircle, we obtain the total separating force. P=pl.?NN'=p.l.d, pel or ? = ^; thus the tensile stress is directly proportional to the diameter, and inversely proportional to the thickness of the cylindrical shell. For greatest accuracy d should be taken as the mean of the internal and external diameters. The formula just obtained is true only when the thickness is small compared with the diameter. If t is large, the stress is not uniform over the section ; the formula will then give the me* stress if d be understood to mean the internal diameter. CH. xii. ART. 151.] TENSION AND COMPRESSION. 303 We next consider the tendency for the cylinder to tear across a transverse section when there are no longitudinal stays to take the pressure on the ends. The total pressure on each end of the cylindrical shell is the separating force, and in the absence of stays the resistance to separation is due to the tensile stress, q' suppose, called into action over the annular area ird . t of the transverse section. .-. irdt . q' = ^d*p ; or ?'=^- This is just half the stress on the longitudinal section. If the vessel is spherical in form, the stress produced on all sections of the sphere through the centre is the same as at the transverse section of the cylinder. The formula just obtained is used to estimate the strength of a boiler which is more or less cylindrical ; but since the boiler is made up of plates overlapping each other, connected together at the edges by rivets, and since also a line of rivets in a longitudinal section is generally found only for a portion of the length of the boiler, the question of strength is complicated. But a longitudinal section through the greatest number of rivet holes is the weakest section, and if for q we write /, where / is a co-efficient of strength to be determined from experience, the value of it depending, among other things, on the form of joint, then the formula 2# . pd P = ~d> Tt lf may be used as a semi-empirical formula to determine the greatest pressure which can be employed in a given boiler, or the thickness of metal required to sustain a given pressure. The value of the co- efficient for iron boilers with single rivetted joints is about 4,000 Ibs. per square inch, or, when double rivetted, as is usual in large boilers, 5,500. With steel the value is about one-third greater. In large boilers at high pressure these values, however, have of late been very greatly exceeded, for reasons which will be considered in a subsequent chapter. 151. Remarks on Tension. The results obtained in the present sec- tion are, strictly speaking, only applicable when the piece of material considered is of uniform transverse section, but they nevertheless may be used when the transverse section is variable, provided the rate of variation be not too great and the other conditions mentioned are strictly fulfilled. The intensity of the stress is then different at different parts of the bar, varying inversely as the transverse section and in determining the elongation this must be taken into account. 304 STIFFNESS AND STRENGTH. [PART iv. In many cases of tension the effect of the weight of the tie and other circumstances introduces an additional stress, the amount of which is often imperfectly known. This is allowed for, either by making a certain addition to the theoretical diameter, or by the use of a factor of safety adapted to the particular case. On the other hand it also often happens, as in the case of ropes for example, that the strength of the material is greater in small sizes than large ones for reasons connected with the mode of manufacture. 152. Simple Compression. When the forces applied to the ends of a bar act in a direction towards one another the bar is in a state of compression. If the bar is long compared with its transverse dimen- sions, then any slight disturbance from uniformity will cause it to bend sideways under the compressive force, and we have then, not simple compression, but compression compounded with bending, an important case to be considered hereafter. To obtain simple com- pression the ratio of length to smallest breadth should not exceed certain limits which depend on the nature of the material, viz., cast iron 5 to 1, wrought iron 10 to 1, steel 7 to 1. These values, however, depend to some extent on the type of section. Further, it is necessary that the material be perfectly homogeneous, and that the line of action of the load should be in the axis of the bar. Then the results we have obtained for simple tension apply to this case of simple compression P * = A* and the strength of the column is given by P = Af, where / is the co-efficient of strength. The compression x which the column under- goes is connected with the stress by the equation p = E*. The modulus of elasticity E would, in a perfectly elastic body, be the same as for tension. In actual materials it sometimes appears to be less ; but within the elastic limit only slightly less. EXAMPLES. 1. A rod of iron 1 inch in diameter and 6 feet long is found to stretch one-sixteenth inch under a load of 7^ tons. Find the intensity of stress on the transverse section and the modulus of elasticity in Ibs. and tons per square inch. Stress =21, 382 Ibs. =9 '55 tons. Modulus of elasticity = 24, 632,000 Ibs. =11, 000 tons. 2. "What should be the diameter of the stays of a boiler in which the pressure is 30 Ibs. per square inch, allowing one stay to each 1J square feet of flat surface and stress of 3,500 Ibs. per square inch of section of the iron? Ans. 1^ inch. 3. In Example 1 find the work stored up in the rod in foot-pounds. Ans. 43|. CH. xii. ART. 153.] BENDING. 305 4. If in the last question the rod were originally 2" diameter and half its length were turned down to a diameter of 1", compare the work stored in the rod with the result of the previous question. Ans. Ratio = |. 5. In Example 1 assume the given load of 1\ tons to be the proof load ; find the modulus of resilience. Ans. 18 '56 in inch-lb. units. 6. Find the thickness of plates of a cylindrical boiler 4' 2" diameter to sustain a pressure of 50 Ibs. per square inch, taking the co-efficient of strength of plate at 4,000 Ibs. Ans. yV'. 7- A spherical shell 4' diameter " thick is under internal fluid pressure of 1,000 Ibs. per square inch. Find the intensity of stress on a section of the sphere taken through the centre. Ans. 48,000 Ibs. per square inch. 8. Find the necessary thickness of a copper steam pipe 4" diameter for a steam pressure of 100 Ibs. above the atmosphere, the safe stress for copper being taken as 1,000 Ibs. per square inch. Ans. "2". 9. A circular iron tank, diameter 16 feet, with vertical sides 5" thick, is filled with water to a depth of 12 feet : find the stress on the sides at the bottom. How should the thickness vary for uniform strength throughout? Ans. 1,024 Ibs. per square inch. 10. What length of iron suspension rod will just carry its own weight, the stress being limited to 4 tons per square inch, and what will be the extension under this load ? Ans. 2,700 feet. Extension=5". 11. The end of a beam 10" broad rests on a wall of masonry ; if it be loaded with 10 tons what length of bearing surface is necessary, the safe crushing stress for stone being 150 Ibs. per square inch? Ans. 15". 12. Find the diameter of bearing surface at the base for a column carrying 20 tons, the stress allowed being as in the last question. Ans. 20" nearly. 13. Compare the weight of the shell of a cylindrical boiler with the weight of water it contains when full. Ans. Ratio =15 "5 p/f. SECTION II. SIMPLE BENDING. 153. Proof that the Stress at each Point varies as its Distance from the Neutral Axis. The nature of the straining action producing bend- ing has been sufficiently explained in the third section of Chapter II., and we shall now consider the kind of stress which results on the ultimate particles of a solid bar of uniform transverse section and of perfectly elastic material. The bar is supposed symmetrical about a plane through its geometrical axis, and the bending is supposed to take place in this plane, which may be called the Plane of Bending. In the first instance the bending is supposed to be " simple," that is, it is not combined with shearing as is most often the case in practice, but is due to a uniform bending moment (see Art. 21). The curvature of the beam is then uniform, that is to say, it is bent into a circular arc. The investigation consists of three parts. Fig. 122 shows a longitudinal section AE and a transverse section LL through the centre of the beam ; by symmetry it follows that if the bending moment be applied to both ends in exactly the same way, that transverse section, if plane before bending, will be still plane C.M. u 306 STIFFNESS AND STRENGTH. [PART iv. after bending, for there is no reason for deviation in one direction rather than another. It will be seen presently that if the bending moment be applied to the ends of the beam in a particular way all transverse sections will be in the same condition, and we may there- fore assume that not only the central section, but any other sections KK we please to take, will remain plane notwithstanding the bending of the beam. All such sections, if produced, will meet in a line the intersection of which by the plane of bending will be a point 0, which is the common centre of the circular arcs KL, PP, NN, etc., formed by the intersection of the same plane with originally plane longitudinal layers. These layers after bending have a double curvature, one in the plane of bending, the other in the transverse plane ; the transverse bending however need not be considered at present, and the transverse section of the layers may be treated as straight lines. Before bending, the layers were all of the same length, being cut off by parallel planes, but now they will vary in length since they lie between planes radiating from an axis 0. We shall find presently that some layers must be lengthened and some shortened, an intermediate layer, NN in the figure, being unaltered in length. This layer is called the Neutral Surface, and the transverse section of that layer SS is called CH. xii. ART. 153.] BENDING. 307 the Neutral Axis, the last expression being always used in reference to a transverse section, not a longitudinal section. Let the radius of the neutral surface be R. The more the beam is bent, that is the less R is, the greater will be the stress produced by the bending action ; and the first step in the investigation is to obtain the relation between the stress produced at any point of a transverse section and the radius of curvature R. If we bisect SS in N and draw LNL at right angles to SNS, it is necessary that the section of the beam should be symmetrical on each side of LNL ; with this restriction the section may be any shape we please. Now consider any layer PP of the beam between the planes LL and KK which is at the distance y from the neutral surface NN or neutral axis SNS. This layer will be curved to a circle whose 1 radius is R + v, and it must undergo an alteration of length from NN which it had before bending, to PP which it now has. Thus the alteration of pp _ length per unit of length, that is, the strain e = - j^= , but since D D JJ arcs are proportional to radii -j = t^, .A iv 41 PP-NN If the layer we are considering is taken below the neutral surface, the strain, which will then be compression, will be given by the same expression e = y/R, e and y both being negative. Accompanying the longitudinal strain just estimated there must be a longitudinal stress proportional to the strain. Let p be the intensity of that stress, then p = Ee, where E is a modulus of elasticity. If we imagine the beam divided into elementary longitudinal bars, and if we imagine each of those bars independent of the others, it will follow that E is the same modulus of elasticity as we have previously employed in Section I. of this chapter. This, however, implies that the bar can freely contract and expand laterally when stretched and compressed, and we therefore could not be sure a priori that the union of the bars into a solid mass would not cause the value of E to be different from that for simple stretching, and to vary for different layers of the beam. It will be seen hereafter, however, that there are good reasons for the assumption. Accordingly we write where E is the ordinary (also called Young's) modulus of elasticity. 308 STIFFNESS AND STRENGTH. [PART iv. If y is taken below the neutral axis then p is negative, signifying that the stress is now compressive. Inj perfectly elastic material the value of E is the same for compression as for tension, and so, within the limits of elasticity, the same equation will apply for all parts of the transverse section. Thus the stress at any point of the transverse section of the bar is proportional to its distance from the neutral axis. 154. Determination of Position of Neutral Axis. The second step in the investigation is to find the position of the neutral axis, which may be done by dividing the beam into two portions, A and B, by a section LL, and considering the horizontal equilibrium of either portion, say B. The external forces, being vertical, have no horizontal component, and we have therefore only to take account of the internal molecular forces which act at the section LL. Above the neutral axis the action of LA is a tendency to pull B to the left ; but below the neutral axis, the tendency is to thrust B to the right. In order that it may remain in equilibrium, and not move horizontally, it is necessary that the total pull should equal the total thrust ; or the total horizontal force at the section must be zero. To estimate the horizontal force, consider the force acting on a thin strip of the transverse section, of breadth b, and thickness t, distant y from the neutral axis. The thrust or pull on this elementary strip =p .b.t. Summing the forces on all the strips composing the sectional area, we must have but p = EyjR, where E and R are the same for all strips of the section. That is to say, the sum of the products of each elementary area into its distance from the neutral axis must be zero. This can be true only if the axis passes through the centre of gravity of the section ; for it is the same thing as saying that the moment of the area about the neutral axis is to be zero. 155. Determination of the Moment of Resistance. The third and \i step in the investigation is to obtain the connection between the bending moment applied, and the stress which is produced by it Again, considering either portion, AL or BL, of the beam, say A. the external forces on A produce a . bending moment or couple, which has to be resisted by the internal stresses called into action the section K; so that the total moment of these stresses must be eqm CH. xii. ART. 155.] BENDING. 309 to M. The moment of the resisting stresses, being a couple, may be estimated about any axis with the same result. For convenience we will estimate it about the neutral axis of the section. Let us again consider the elementary strip of area bt, distant y from neutral axis, on which the intensity of stress is p, the force, pull, or thrust, on this strip being pbt. The moment of the force =p.U. y. Seeing that forces on all elementary strips, whether pull or thrust, all tend to turn the piece AL the same way, the total moment of the stresses will be found by summing all terms, p . bty, for the whole area of the section. .-. M=2p.bty. Since p = Ey/JK, substitute, and remember that EjR is the same for all strips, then In this formula the area of each strip has to be multiplied by the square of its distance from the neutral axis and the sum of the pro- ducts taken. This, or an analogous sum, is of constant occurrence in mechanics, and has a name assigned to it. 26/y is the simple moment of an area about an axis, 2bty 2 may be called the moment of the second degree, but the common name is the Moment of Inertia; be- cause a similar sum (differing only from this in involving the mass) occurs in dynamics under that name. To distinguish the two cases area-moment and mass-moment, the former is sometimes called the geometrical moment of inertia. Let / denote the moment of inertia, so that / = 2% 2 , the value of which for any form of section can be obtained by geometry, then M E T M E M = rf> or 7=5* thus connecting the curvature of the beam with the moment producing it. Having previously found p/y = E/B, we can now connect the moment with the stress by writing p_M y~ i' This equation may be employed to determine the strength of a beam to resist bending. The limit of strength is reached when either the greatest safe tensile stress on one side of the neutral axis, or the greatest safe compressive stress on the other side of the neutral axis is called into action. Thus in the equation pfy = M! I we must put p=f v the co-efficient of strength under tension, or pf v the co-efficient of strength under compression; and for y, either y lt the distance of the most remote point on the stretched 310 STIFFNESS AND STRENGTH. [PART iv. side, or y 2 , the distance of the most remote point on the compressed side, so that M=&I, or &I. vi y* The strength of the beam, or maximum moment of resistance to lending, is measured by the least of these quantities. y l or y 2 is readily determined from geometry, the form of the section of the beam being given. It may be most conveniently expressed as a fraction of the depth of the beam. Thus y^ or y 2 may be put = qh, where the co-efficient q has different values. In a rectangular section q J, in a triangular section q = ^ or f , and so on. Next to express the value of /. It will be found that whatever be the form of the section, / may always be written = nAh?, A being the area of the section of the beam, h the depth in the direction of bending, and n a numerical co-efficient, the value of which depends on the form of the section. For a rectangular section, n = T ^, so that 7 = For an elliptical or circular section, ?i = T V, so that I= For a triangular section, w = T V, so that I= and so on. Therefore assuming q and n known, we can write --.> qh J q & formula which shows that for sections in which n/q is the same, the moment of resistance to bending is proportional to the product of the area and depth of the beam. Sections with the same n and q are said to be of the same type. They are often, but not correctly, said to be similar. In estimating the numerical value of M, care must be taken with the units. It is generally advisable to use the inch unit throughout. 156. Remarks on Theory of Bending. In the foregoing theory of simple bending it is supposed (1) That the bar is homogeneous and of uniform transverse section and perfectly elastic ; (2) That sections plane before bending are plane after bending, for which it is theoretically necessary that the bending moment should be uniform, and applied at the ends of the bar in a particular way ; en. xn. ART. 157.] BENDING. 311 (3) That longitudinal layers of the beam expand and contract later- ally in the same way, as if they were disconnected from each other. These assumptions are not obvious a priori, and require justification, which at the present stage of the subject we are not in a position to give ; for the present it may be stated that if the material be homo- geneous and perfectly elastic, the equations hold good with certain qualifications to be considered hereafter (Chap. XVII.), even though the transverse sections and the curvature vary and however the bending moment is applied. The strength of the material, however, is not generally the same, as if the layers were disconnected, and co-efficients of strength require therefore to be (determined by special experiment on transverse strength (Chap. XVIII.). 157. Calculation of Moments of Inertia. We have frequently to deal with beams of complex section, in which case to determine 7 it is convenient to divide the section up into simple areas, the eye of each of which is known, and the total moment of inertia of the section will be the sum of these 7's. In employing this process we require to know the relation between the moments of inertia of an area about two axes parallel to one another, one being the neutral axis. We make use of a general theorem which may be thus proved. Let A be an area of which we know the moment of inertia about the neutral axis, SS (Fig. 123), and we require to know the moment of inertia about any parallel axis, XX, distant y from SS. Dividing the area into strips of breadth b, and thickness t, s Moment of Inertia required 7= 26 . t . Now *2bty 2 = moment of inertia about neutral axis, Fig.123. ~bt . y = 0, because the neutral axis passes through the centre of gravity of the section, and 26/ = Area A ; The moment of inertia of an area about any axis is, therefore, determined by adding to the moment of inertia of the area about a parallel axis through the centre of gravity the product of the area into the square of the distance between the two axes. This theorem, together with previously quoted values of / , will enable us to determine the following results, which will be useful in application to beams Rectangle of height y about its base, ... I Triangle ... ... I Triangle about a parallel to its base through vertex, /= 312 STIFFNESS AND STRENGTH. [PART iv. Many other forms will divide up into rectangles or triangles, or both ; for example, the moment of inertia of a trapezoid about the neutral axis may be readily determined by taking, for the area above the neutral axis, the / for a rectangle about one end, and triangles about the base. For the area below, a rectangle about one end and triangles about the vertex, and add the results. 158. Seams of I Section with equal Flanges. The case of a beam of I section is very important. First, suppose the flanges of equal breadth and thickness, and the web of uniform thickness b', the depth being h', b being the breadth of the flange, and h the whole depth of the beam. The moment of Fig.124. fy| - u , K inertia of the section may be taken as the difference of the moments of inertia of two rectangles (see Fig. 124). This is the accurate value of /, and when the flanges are thick this expression for / must be used; but if the flanges are thin compared with the depth, an approximation can be obtained by supposing each flange to be concentrated in its centre line, and taking for the depth of the beam the distance h Q to the centre of flanges. If A = area of each flange and C = area of web, then /= + A + fv\ l^T Putting p=f and y = -|A , in the formula - = -y, Since the total area of the flanges is 1A it appears that, area for area,, the web has only one-third the resistance to bending of the flanges. The result given by this formula is. too large, the excess being greater the thicker the flanges, partly because a part of the web is reckonec twice over, and partly for the reason mentioned below. We previously deduced an approximate expression for the strerij of an I beam, viz., M=Hh=fhA (see Art. 27), in which the effect of the web in resisting bending was neglected, th< whole of the bending action being supposed to be taken by the flanges. The present formula shows the amount of the error involved in th{ assumption. In using this approximation when h the effective depth it CH. xii. ART. 160.] BENDING. 313 reckoned from centre to centre of the flanges, two errors are made, one in supposing the resistance to bending of the web neglected, and the other, in supposing the mean stress on the flange equal to the maximum. When the web is very thin the first of these errors is the least, and the effective depth is more nearly h^/h', where h' is the outside depth and ^ the depth from centre to centre of flanges. This is little greater than the inside depth. On the other hand, in beams rolled in one piece the web 'is thick. The first error is then the greater, and Prof. Philbrick has pointed out that the approximation gives fairly accurate results if h be taken as the outside depth.* Such approximate rules are useful in rough preliminary calculations of dimensions, but always require verification. 159. Ratio of Depth to Span in I Beams. The formula just obtained for the moment of resistance of a beam of I section shows that the greater the depth of the beam and the thinner the web the stronger will the beam be for the same weight of material, or in other words that the best distribution of material is as far away from the neutral axis as possible. The practical limitation to this is that a certain thickness of web is necessary to hold the flanges together and give sufficient power of resistance to lateral forces and to the direct action of any part of the load which may rest on the upper flange. Hence the weight of web rapidly increases as the depth increases, and a certain ratio of depth to span is best as regards economy of material (see Ex. 17, page 319). This is especially important in large girders in which economy of material is the primary consideration. In smaller beams the proper ratio of depth to span is generally in great measure a question of stiffness, a part of the subject to be considered in Chapter XIII. The moment of resistance of I sections of practical proportions is generally nearly double that of a rectangular section of equal area and mean depth. The straining actions on the web will be considered in Chapter XV. 160. Proportions of I Beams for Equal Strength. Materials in general are not equally strong under tension and compression, so that a beam whose section is symmetrical above and below the neutral axis will yield on one side before the material on the other side of the neutral axis has reached its limiting stress. Accordingly we might obtain a more economical distribution of material if we were to take some from the stronger side and put it on the weaker, so that the limiting tensile on one side and the limiting compressive stress on the other * Van Nostrand's Magazine. Nov., 1886. 314 STIFFNESS AND STRENGTH. [PART iv. may be produced simultaneously. The section of the beam will be different above and below the neutral axis, which will not now be at the centre of depth of the beam, but in such a position that the distance to the top and bottom of the beam are in the proportion of the greatest allowed stresses to one another. The neutral axis in all oases must pass through the centre of gravity of the section. Let /,,' fg be the co-efficients of strength under compression and tension respectively, y A , y B distances of the most strained layer from the neutral axis, then the beam will be strongest when For simplicity of calculation we will consider a beam (Fig. 125) in which the web is of uniform thickness through- out the depth, and so of rectangular section, and each flange also of rectangular section, H : V and determine the relation which should hold ...-N-^i. between the areas of flanges and web for YB I \ maximum strength of beam, and the moment of resistance to bending where this condition is satisfied. We will further suppose each flange to be concentrated in its centre line. Let A = area of compressed flange, B = area of stretched flange, (7 = area of web. Since the neutral axis is at the centre of gravity of the section, we obtain, by taking moments about the axis, or, substituting the previously given values of y A and Supposing f A and / B known, A, JB, and C must be such as to satisfy this relation. We have some liberty of choice between these quantities, and frequently find one of the flanges omitted, so producing a beam of T or _L section. In a cast-iron beam, where the resistance to compression is greater than for tension, the compressed flange A may be omitted. Putting A = we get <7=-%^, and supposing = 4 ; <7 = , or B=\\G. 4/fi JB In a wrought-iron beam on the other hand, if we take f A jf B to be f , the stretched flange B is to be omitted. Putting JB 0, we find A JB ~JA ri \ri ^._-c->a CH. xn. ART. 160.] BENDING. 315 Otherwise we may assume the depth and thickness of the web to be given (Art. 159), then the equation furnishes a relation between the areas of the flanges. For example, in cast iron, if we assume f A = 4/ fl , we find B = 4A+%C. Having decided on the proportions between the parts of the section we can now calculate the moments of inertia and resistance. Still considering the flanges concentrated in their centre lines, I=Ay* + By* + C . . yj + JC . . y* a result which admits of ready calculation. Further whence we obtain M = (f A +/B)T- The calculation just now made is one which has been frequently given in dealing with beams of I section,* but in applying it to actual examples it should be remembered that the results are obtained on the supposition that the flanges are concentrated in their centre lines, and are consequently only approximate when the co-efficients /!, f B mean the intensities of the stress at those centre lines, not at the surface of the beam where the stress is greatest. If, for example, F A be the maximum stress on the flange A, where t A is the thickness of the flange. The difference is especially great in the case of the larger flange of cast-iron beams, and the true ratio of maximum compressive and tensile stress is much less than it appears in the preceding article. On the other hand, in extreme cases, such as we are now considering, the stress may not be uniformly distributed along a line parallel to the neutral axis. Extensive experiments were made on cast-iron beams by Hodgkinson with the object of determining the best proportions between the flanges, with the result that rupture always took place by tearing asunder of the lower flange, unless it was at least six times the size of the compressed flange. This proportion is rarely adopted in practice, from the difficulties of obtaining a sound casting, and the necessity * See Rankine's Civil Engineering, page 257. 316 STIFFNESS AND STEENGTH. [PART iv. of having sufficient lateral strength. Nor is it certain that the pro- portions which are best for resisting the ultimate load are also best in the case of the working load ; it is, in fact, probable that a smaller proportion is better even on the score of strength. If we take f A = 2J/ , instead of 4f B we find Bm-fyA' + tG, which agrees moie closely with practice. Ihe ratio of maximum com- pressive and tensile strength is in this case about 2, which, according to some authorities, is the ratio of elastic strengths in the two cases. In wrought-iron beams the areas of the flanges are usually equal, and this is correct if the elastic strength, and not the ultimate strength, is regarded as fixing the proper proportions, and if there be sufficient piovision against the yielding of the top flange by lateral flexure. Small-sized beams of this kind are rolled in one piece, while large girders are constructed of iron or steel plates and angle iron?, rivetttd together. Some of the forms they assume are shown in Plate VIII., Chapter XVIII. In making calculations respecting girders, approximate methods may be used for preliminary tentative calculations, but should be checked by a subsequent accurate determination of the neutral axis and moment of inertia. A previous reduction of the section to an equivalent solid section is required when, as is often the case, all parts of the section do not offer the same elastic resistance to the stress applied to them, either because they are not sufficiently rigidly connected or from the material being different. This is especially the case in determining the resistance to the longitudinal bending of a vessel occasioned by the unequal distribution of weight and buoyancy already considered in Chapter III. On this important question the reader is referred to a treatise on Naval Architecture by Mr (now Sir) W. H White. In many cases of built-up girders the shearing action which generally exists has considerable influence, a matter for subsequent consideration (Ch. XV.). The effect of the weight of the girder itself has been con- sidered in Chapter IV. (See also Ex. 13, p. 319, and Art. 192.) 161. Beams of Uniform Strength. A beam of uniform strength is one in which the maximum stress is the same on all sections. For beams of the same transverse section throughout, this can only be the case when the bending moment is uniform, but, by properly varying the section, it is possible to satisfy the condition however the bending moment vary. For this purpose we have only to consider the equation M-fl.Ak, CH. XII. ART. 162.] BENDING. 317 which must now be satisfied at all sections. Suppose A=Jcbh, where k is a numerical factor depending on the type of section, then All sections of the beam being supposed of the same type we have only to make Ah or bh 2 vary as M, that is, as the ordinates of the curve of bending moments. The principal cases are (1) Depth uniform. Here the breadth must vary as the bending moment, whence it is clear that the curve of moments may be taken as representing the half plan of the beam. (2) Sectional area uniform. Here the depth must vary as the bend- ing moment, that is, the curve of moments may be taken to represent the elevation or half elevation of the beam. (3) Breadth uniform. Here the elevation or half elevation of the beam must be a curve, the co-ordinates of which are the square roots of the co-ordinates of the curve of moments. (4) TCatio of breadth to depth constant. Here the half plan and half elevation are each a curve, the ordinates of which are the cube roots of the ordinates of the curve of moments. The first, third, and fourth of these cases are common in practice with some modifications occasioned by the necessity of providing additional erial at sections of the beam where the bending moment vanishes, as it usually does at one or both ends. 162. Unsymmetrical Bending. It occasionally happens that the plane of the bending moment is not a principal plane of the beam, as for Fig.126. M example when a vessel heels over, the plane of longitudinal bending will not coincide with the plane of symmetry of the vessel which is 318 STIFFNESS AND STRENGTH. [PART iv. obviously the plane of the masts. The neutral axis does not now coincide with the axis of the bending couple, though in other respects the theory of binding still holds good. In Fig. 126 let MM be the axis of the bending moment M, inclined at an angle to the principal axes of inertia GX, GY of the plane section. Then the couple M may be resolved into two components M cos 6 and M sin 6, each of which will produce stress at any point P as if the other did not exist. Let p be the stress, x, y the co-ordinates of P referred to the axes GX, GY, the moments of inertia about which are /!, / 2 , then M . cos 6 . y M . sin . x p = - j -- Z+~ -- . 1 l 1 2 The position of the neutral axis NN is found by putting p = 0, then the angle which it makes with GX is given by This equation shows that the neutral axis is parallel to a line joining the centres of the circles into which the beam would be bent by the component couples supposed each to act alone. The neutral axis being thus determined and laid down on the diagram the points can be found which lie at the greatest distance from that axis. At these points the stress will be greatest, and if X, Y be their co-ordinates, still referred to the axes GX, GY, the moment of resistance will be determined by the equation F.cos0 .X. sin (9) -/- -+ 7 *! 2 2 J For a different method of expressing the moment of resistance see Rankine's Applied Mechanics, p. 314. EXAMPLES. 1. A bar of iron 2" diameter is bent into the arc of a circle 372' diameter. Find in tons. per square inch, 1st, the greatest stress at any point of the transverse section ; 2nd, the stress on a line parallel to the neutral axis half an inch from the centre, E being taken 29,000,000. Ans. Maximum stress =5 '8. Stress at " from centre = 2'9. 2. Find the diameter of the smallest circle into which the bar of the last question can be bent ; the stress being limited to 4 tons per square inch. Ans. Diameter =540 feet. 3. Find the position of the neutral axis of a trapezoidal section ; the top side being 3", bottom 6", and depth 8". Also find the ratio of maximum tensile and compressive stresses. Ans. Neutral axis 3 '56 inches from bottom. Ratio of stresses 5 to 4. 4. A cast-iron beam is of I section with top flange 3" broad and 1" thick and bottom flange 8" broad and 2" thick ; the web is trapezoidal in section %" thick at top and V at bottom ; total outside depth of beam 16". Find the position of the neutral axis and the ratio of maximum tensile and compressive stresses. Ans. Neutral axis 4 '81 inches from bottom. Ratio of stresses 3 to 7. 5. A wrought-iron beam of rectangular section is 9" deep, 3" broad, and 10 feet long. CH. xii. ART. 162.] BENDING. 31D Find ho\v much it will carry loaded in the centre, allowing a co-efficient of 3 tons per square inch. Also deduce tlie load the same beam will bear when set flatways. Ans. When upright load =4 '05 tons. "When set flatways load = 1 '35 tons. 6. A piece of oak of uniform circular section is 16" diameter and 12 feet long. It is supported at the two ends and loaded at a point 5 feet from one end. How great may the load be, allowing a stress of \ ton per square inch? Ans. Load may be 5 '7 4 tons. 7. In Example 5 suppose half the weight of metal formed into a beam of I section, of the same depth, each flange being equal to the web ; what load will the beam carry ? Ans. Load may then be 4 tons. 8. Find the moment of resistance to bending of the section given in Example 4, the co-efficient for tension being 1 ton per square inch. Ans. 1=798 inch units. Moment of resistance to bending=166'4 inch- tons. 9. Suppose the skin and plate deck of an iron vessel to have the following dimensions at the midship section, measured at the middle of the thickness of the plates. Find the position of the neutral axis and moment of resistance to bending. Breadth 48' and total depth 24', the bilges being quadrants of 12' radius. Thickness of plate " all round and co-efficient of strength 4 tons in compression. Arts. Neutral axis 13" above centre of depth. Moment of resistance to hogging =32, 500 ft.-tons, and to sagging 39,000. 10. What should be the sectional area of a T beam of wrought iron to carry 4 ton* uniformly distributed? Span 20', depth of beam 10". Co-efficient for compression 3 tons, and for tension 5 tons. Ans. Area=13'7 square inches. 11. If, in the last question, the flange is made equal to the web instead of being pro- portioned for equal strength, show that to carry the same load the beam must be about one quarter heavier. 12. In Example 8 find the moments of inertia and resistance on the supposition that the flanges are concentrated at the centre lines, and thus by comparison with previous results show the amount of the error involved in the assumption. Ans. Moment of inertia =861 '5 inch units. Moment of resistance = 227 inch-tons. 13. Show that the limiting span (Art. 41) of a beam of uniform transverse section is Sn Li \ . -zf=- , Nq where N is the ratio of span to depth, and the rest of the notation is the same as on pages 81 and 310. Obtain the numerical result for a wrought-iron beam of rectangular section, taking X from Table I., Ch. XVIII., and supposing aV=12. Ans. =336 ft. ; in an ordinary I section the result would be doubled. For the case of large girders see Art. 192. 14. If I be the length of an iron rod in feet, d its diameter in inches, just to carry its own weight when supported at the ends, show that when the stress allowed is 4 tons per square inch I=*j224d. 15. If /!, / 2 be the moments of inertia of two plane areas, A lt A. 2 , about their neutral axes which are supported parallel at distance apart z, show that the moment of inertia of their sum or difference about their common neutral axis is I=I l I + z*. . \_ * AiA z Apply this formula to the trapezoidal section of Question 3. Ans. 7=185 inch units nearly. 16. Find the moment of resistance to bending of a beam of I section, each flange con- sisting of a pair of angle iions 3i"x4" rivetted to a web '37" thick and 16" deep between them. Assuming it 24 feet span, find the load it would carry in the middle, using a co- efficient of 3 tons per square inch. Ans. M =288 inch-tons. W=4 tons. 17. If it be assumed that for constructive reasons the thickness of web of an I beam with equal flanges must be a given fraction of the depth, show that for greatest economy of material the sectional area of the web should be equal to the joint sectional area of the flanges. Prove that in this case M=&f. Sh. 320 STIFFNESS AND STRENGTH. [PART iv. 18. In a cast-iron beam of I section of equal strength for which fA=2^/B', if it be assumed that for constructive reasons the thickness of the web should be a given fraction of the depth, show that for greatest economy of material the large flange, the web, and the small flange should be in the proportion 25, 20, 4. Prove also that the moment of resistance is given by the same formula as in Question 17, supposing 2//= 1//U + I//B. 19. A beam of rectangular section of breaith one-half the depth is bent by a couple the plane of which is inclined at 45 to the axes of the section. Find the neutral axis, and compare the moment of resistance to bending with that about either axis. Ans. Ratio =V2/3 and j2/3. 20. If a beam be originally curved in the form of a circular arc of radius R instead of being straight, show that the neutral axis does not pass through the centre of gravity of the section. In a rectangular section of depth h show that the deviation is, approxi- mately, h? 21. In the preceding question, if JR is large show that the equations of bending are * */I. CHAPTER XIII. DEFLECTION AND SLOPE OF BEAMS. 163. Deflection due to the Maximum Bending Moment. It is not only necessary that a beam should be strong enough to support the load to which it is subjected, it is also necessary that its changes of form should not be too great, or in other words, that it should be sufficiently stiff, and we next proceed to deter- mine under what conditions this will be the case. The question is simplest when the beam is bent into an arc of a circle; we have then p M E - = -f = -^ = constant. // / R Two cases may be especially mentioned (1) Depth uniform. We then have p constant, that the beam is of uniform strength. (See Case 1 of Art. 161.) (2) Sectional area uniform. s Fig.127. We then have, since E M= T>I= n . -j . H A the depth of the beam varying as the square root of the bending moment, as in case 3 of the same article. Let I be the length of the beam, i the angle its two ends make with one another, then since i is also the angle subtended by the beam at the centre ._ I _Ml ~ R~ El' If the beam be supported at the ends i is twice the angle which the ends make with the horizontal, an angle called the Slope at the ends. Let AB be the beam (Fig. 127), the centre of the circle into which it is bent KL, the diameter of the circle through K the middle point of C.M. x 322 STIFFNESS AND STEENGTH. [PART iv. the beam. Then KN is the deflection which is given by a known proposition of Euclid, Hence remembering that the diameter of the circle is very large* we have, if 8 be the deflection, This formula gives the deflection in any case where the curvature is uniform. When the transverse section is uniform the curvature varies. Unless the bending moment be likewise uniform, the deflection curve is not then a circle AKB, but for the same maximum bending moment a flatter curve A'KB'. Thus the deflection is less than that calculated by the above formula, which may be described as the " deflection due to the maximum moment." The actual deflection may conveniently be expressed as a fraction of that due to the maximum moment. It is possible to construct the deflection curve graphically by observing that the curvature at every point is proportional to the bending moment. We] have then only to strike a succession of arcs with radii inversely proportional to the ordinates of the curve of bending moment. It is however more convenient to proceed by an analytical method, f The fraction is least when the beam is least curved, which is evidently the case when it is loaded in the middle, and we shall show presently that it is then two-thirds, while, when uniformly loaded, it is five-sixths. 164. General Equation of Deflection Curve. It was shown above that M 1 l = ET 1 ' If the bending moment vary, then we must replace I by an element of the length ds and i by the corresponding element of the angle ; we shall then have an equation <& Jf ds~j~r which by integration will furnish i. It will generally be convenient to reckon i from a horizontal tangent and it then means the slope of the beam at the point considered. To perform the integration it is in most cases necessary to suppose the slope of the beam small, as it actually is in most important cases in practice, and we may then replace ds the * For clearness it is made small in the figure. t Readers who have no knowledge of the Calculus may pass over the next four articles. CH. xni. ART. 165.] DEFLECTION OF BEAMS. 323 element of arc by dx, the corresponding element of a horizontal tangent AN (Fig. 128) taken as axis of z, whence an equation which can generally be integrated because M is usually a function of x. The deviation y of any point Q of the beam from the straight line AN can now be found since dy/dx = i, from which we further obtain the fundamental equation <5r Jf dx*~ Ef which applies to all cases where the bending of the beam is occasioned by a transverse load. We shall first give some elementary examples of the determination of the deflection and slope of a beam and then consider the question more generally. Fig.128. 165. Elementary Cases of Deflection and Slope. Case L Suppose a beam supported at the ends and loaded in the middle. In Fig. 128 CD is the beam resting on supports at C, D, and loaded in the middle with a weight W. Take the centre A as origin and the horizontal tangent at A as axis of x, then if / be the whole length Wfl _ dx 2 El" El ' ' dx El is the slope of the beam at Q, no constant being required since i is zero when x = 0. If x = 1/2 we get the slope at the ends of the beam Integrating a second time As before no constant is required because y = when x = 0. 324 STIFFNESS AND STEENGTH. [PART iv. If now we put x = 1/2 we get the elevation of D above A N or, what is the same thing, the depression Ag of A below the level of the supports. This is called the Deflection of the beam ; if we denote it by 8, El a result which we may also write 8 2 Mf 2 d-^.-^-g ' <" where M Q is the maximum moment and 8 Q the deflection due to it. Case II. Let the beam be supported at the ends and loaded uniformly with w pounds per foot-run. It will be sufficient to give the results, which are obtained in precisely the same way, remembering that the bending moment is now \w(a 2 - x-) where a is the half span. We have was Wl* wa? The value of 8 may be expressed as in the previous case in terms of the deflection due to the maximum moment. We have 8 = | . 8 . 166. Beam propped in the middle. When a beam is acted on by several loads the deflection and slope due to the whole is the sum of those due to each load taken separately. An important example is Case HI. Beam supported at the ends and propped in the middle, uniformly loaded. (Fig. 129.) Here the deflection of the beam is the difference between the down- ward deflection due to the uniform load and the upward deflection due Fig.129. ;g A Z Hence we write down at once for the Wl* to the thrust Q of the prop, deflection at the centre, 8 = 384'^T~48E~r an equation which may be used to determine the load carried by the prop when its length is given, and conversely. CH. xiii. ART. 167.] DEFLECTION OF BEAMS. 325 First suppose the centre of the beam propped at the same level as the supports, then 8 = 0, and 5 so that the prop in this case carries five-eighths of the weight of the beam, the supports C, D only carrying three-eighths. Each supporting force is ^wl, I being as before the whole length of the beam ; hence the bending moment at a point distant x from C is given by the formula M = -^wlx - ^ivx 2 = im^(f I - x), from which it appears that the beam is bent downwards until a point Z is reached, such that CZ&1 &AC Here the bending moment is zero, that is Z is a " point of contrary flexure" or "virtual joint." (Compare Art. 38.) Beyond Z the beam is bent upwards, and at the centre A we get, by putting x = U, -M Q = ^wl*. The case here discussed is also that of a beam, one end of which is fixed horizontally and the other supported at exactly the same level. Let us next inquire what will be the effect of supposing the centre of the beam propped somewhat out of the horizontal line through the supports at the ends. Let us suppose 8 to be l/n ih the deflection of the beam when the prop is removed, then 1 5 #T5 ___ n ' 384 ' "El ~ 384 ' El 48^7 ' that is = a formula which gives the load on the prop. If, for example, ?i = 5, Q-^i^ or if n = - 5, Q = f W ; thus if the centre of the beam be out of level, by as much as one-fifth the deflection when the prop is wholly removed, the load on the prop will vary between \W and f W y a result which shows the care necessary in adjustment to obtain a definite result. 167. Beams fix^d at the Ends. Case IV. Uniformly loaded beam, with ends fixed at a given slope. Fig.130. In Fig. 130 AB is a uniformly loaded beam, with the ends A, P> fixed not horizontally but for greater generality at a slope i. Here 32(5 STIFFNESS AND STRENGTH. [PART iv. the central part of the beam will be bent downwards and the end parts upwards; at Z, Z there will be virtual joints; let OZ=r, then taking as origin the bending moment at any point between and Zis M~W*-&)> a formula which will also hold for points beyond Z, as can be seen from Art. 38, or proved independently. We have then dx ~ El El No constant is required, because i is zero at 0. Let a be the half span OA, or OB, then putting x = a, we get for the slope at the ends _ 1 El a formula from which r can be determined if i^ be given. If r = a, we get the case where the ends are free ; let the slope then be i Q , we have Now, assume the actual slope to be l/7i th of this, we get El that is, r' 2 = a?(\+ If the ends are fixed exactly horizontal, then r 2 = Ja 2 , and by substitution we find for the bending moment at the centre and the ends M Q = %wa 2 ; M A = M B = \wo?. If the ends were free, the bending moment at the centre would have been ^wa 2 , so that the beam will be strengthened in the proportion 3:2. The formula obtained above, however, shows that a small error in adjustment of the ends will make a great difference in the results. It is theoretically possible so to adjust the ends that the bending moments at the centre and the ends shall be equal, in which case the beam will be strongest. For this we have only to put wr 2 = ^w(a~ - f 2 ), that is, r 2 = Ja 2 , whence by substitution we get >i = 4; that is, the ends should be fixed at one-fourth the slope which en. xin. ART. 168.] DEFLECTION OF BEAMS. 327 they have when free, and the strength of the beam will then be doubled. By proceeding to a second integration the deflection of the beam can be found. In particular when the ends of the beam are horizontal it can be shown that the deflection is only one-fifth of its value when the ends are free. On the effect of shearing see page 332. The graphical representation of the bending moments in Cases III., IV., is easily affected, as in Fig. 42, page 77. 168. Stiffness of a Beam. The stiffness of a beam is measured by the ratio of the deflection to the span. In practice, the deflection is limited to 1 or 2 inches per 100 feet of span when under the working load; that is, the ratio in question is ^-^ to T^bV*. It appears from what has been said that if M Q be the maximum moment the deflection is .MJ? where k is a fraction, which in beams of uniform section, varies from two-thirds to unity, depending on the way in which the beam is loaded.* Hence the greatest moment which the beam will bear consistently with its being sufficiently stiff is xu ~ kl 'I' If we express / as usual in terms of the sectional area and depth, Mt~9gfji where s is a co-efficient depending on the material and on the admissible deflection which may be called the " Co-efficient of Stiffness." We thus obtain a value for the moment of resistance of a beam which depends on its stiffness, not on its strength, and if that value be less than that previously obtained for strength (p. 310), we must evidently employ the new formula in calculating dimensions. On comparing the two, we find that they will give the same result if sh_f m ^ h_fk. that is to say, for a certain definite ratio of depth to span, and if there is no other reason for fixing on this ratio, it will be best to choose the value thus determined. The two formulae then give the same result. In large girders a greater depth is generally desirable, then the strength formula must be used ; while in small beams it may often be convenient * When the transverse section is not uniform the co-efficient k may be greater than unity. 328 STIFFNESS AND STEENGTH. [PART iv. or necessary to have a smaller depth, and then the stiffness formula must be employed. 169. General Graphical Method. The foregoing simple examples of the determination of the deflection and slope of a beam are perhaps those of most practical use, but, by the aid of graphical processes, there is no difficulty in generalizing the results which are of considerable theo- retical interest. We can, however, afford space only for a hasty sketch. The general equations given in Art. 164 show that the angle (i) between two tangents to the deflection curve of a beam is proportional to the area of the curve of bending moments intercepted between two ordinates at the points considered. Starting from the lowest point of the deflection curve, let us now imagine a curve drawn, the ordinate of which represents that area reckoned from the starting point, then that curve will represent the slope of the beam at every point, and may therefore properly be called the " Curve of Slope." But referring again to the general equations we see that the ordinate of the deflection curve reckoned upwards from the horizontal tangent at the lowest point, is connected with the slope in the same way as the slope with the bending moment, and is consequently proportional to the area of the curve of slope. Thus it appears, on reference to Chapter III., that the curves of Deflection, Slope, and Bending Moment are related to each other in the same way as the curves of Bending Moment, Shearing Force, and Load. The five curves, in fact, form a continuous series each derived from the next succeeding by a process of graphical integration. We now see that any property connecting together the second three quantities must also be true for the first three. For example, we know, from the properties of the funicular polygon, that two tangents in the curve of moments intersect in a point vertically below the centre of gravity of the area of the corresponding curve of loads. (See Arts. 31, 35.) It must therefore be true that two tangents to the deflection curve intersect vertically below the centre of gravity of the corresponding area of the curve of moments, a useful property, which can be proved directly without much difficulty. The deflection curve of a beam may therefore be constructed in the same way that the funicular polygon is constructed in Art. 35, the perpendicular distance (H) of the pole from the load line in the diagram of forces being made equal to EL To do this we have only to divide the moment curve into convenient vertical strips and regard each as representing a weight. Set down these ideal weights as a vertical line and choose a pole at a distance from the line equal to El, measured (on account of the largeness of E) on a scale less in a given ratio. Now, CH. xin. ART. 170.] DEFLECTION OF BEAMS. 329 construct the polygon and draw its closing line, the intercept multiplied by the scale ratio is the deflection of the beam. A parallel to the closing line in the diagram of forces gives the slopes at the extremities of the beam which correspond to the supporting forces of the loaded beam in the original case. We have hitherto supposed the beam to be of uniform stiffness throughout ; if not, let the quantity El, which is now variable, be E Q I , at some datum section. Keduce the ordinates of the curve of moments in the proportion E^I^ to El, then the reduced curve is to be employed in the way just described for the original curve. 170. Examples of Graphical Method. Theorem of Three Moments. Let us now take some examples. Case I. Symmetrically loaded beam, of flexibility also symmetrical about the centre. Let ACE Fig 131 (Fig. 131) be the curve of moments, reduced if neces- sary, AOB the deflection curve ; both curves, of course, will be symmetrical about the centre vertical, then from what has been said, tangents at A, B to the deflection curve intersect the tangent at in points T vertically below the centres of gravity of the two equal areas AGO, SCO. Hence if S be the area of the whole curve of moments, 1: the horizontal distance of either point T from the nearer end, s .- * must be the slope of the ends of the beam and its deflection. Case II. Beam continuous over several spans loaded in any way. (Fig. 132.) Let AGO', EDO' be the moment curves due to the load on two spans AO', BO' of a beam AOB, continuous over three supports A, 0, B, of which the centre is somewhat below the level of A, B. Being continuous, there will be bending moments at A, 0, B, which are represented in the diagram by AE, O'L, BF. Joining EL, FL, the actual bending moment at each point of the beam will be repre- sented by the intercept between the line ELF and the curves of moments due to the load and corresponding supporting forces. (See Art. 38.) The curve AOB is the deflection curve, AT, BT are the tangents at A, B and TOT is the tangent at 0, intersecting AT, BT in the points T. 330 STIFFNESS AND STRENGTH [PART iv. Now, let i A be the angle between the tangents at and A, then, as before, <-, ^ = ir where S is the area of a curve representing the actual bending moment at each point. In the present case S is the difference of two areas, c Fig.132. one the moment curve for the load, the other the trapezoid EO' for the moments M A9 M Q . where A is the area of the moment curve AGO' and 1 A is the span AO'. Let the horizontal distance from A of the common centre of gravity of the two curves be x ; then, as before, x is also the horizontal distance of T from A and Sx , , = * asbefore - To find x let Z A be the horizontal distance of the centre of gravity of ACS from A, then We have thus found y A the distance of A from the tangent through ; and y B the corresponding distance of B, is written down by change of letters. Assuming now the depression of 0, the centre of the beam, below the level of the two other supports to be 8, it appears from the geometry of the diagram that _ g _ ^ + 3 or hence dividing the values of y AJ y B , by 1 A , 1 B respectively, and adding CH. xili. ART. 171.] DEFLECTION OF BEAMS. 331 This equation connects the bending moments at three points of support of a continuous beam, the centre support being below the end supports by the small quantity 8. It can readily be extended to the case where the flexibility of the beam is variable by reducing the moment curves as previously explained, then the moments M, which are the results of the calculation, will, in the first instance, be reduced, and can afterwards be increased to their true values. The above equation is the most general form of the famous Theorem of Three Moments, originally discovered by Clapeyron, which is much employed in questions relating to continuous beams a somewhat large subject, on which we have not space to enter. The general method of Art. 169 can, however, be applied directly without using the Theorem of Three Moments. Further information on this point will be found in Mr. R. H. Graham's work on the Geometry of Posi- tion. (Macmillan, 1891.) 171. Elastic Energy of a Bent Beam. The work done in bending a beam by a uniform bending moment M is evidently \Mi, where i is the angle which the two ends of the beam make with each other, as in Art. 163; hence by substitution for i we find for the elastic energy U, M , = and if the bending moment vary, TT f^ 2 ,7 U =\2EI' dx ' An important case is when the beam is of uniform strength, then we have **., where the suffix refers to a datum section. Then Mf (I y m>k'f-' Assuming now the section (A), though varying, to remain of the same type f If, therefore, we call V the volume of the beam, M^ r i - With the notation of Art. 155 this gives 2 n 332 STIFFNESS AND STRENGTH. [PART iv. For the resilience we have only to change p into /, the proof strength. It thus appears that in beams of uniform strength with transverse sections of the same type the resilience is proportional to the volume, and less than that of a stretched or compressed bar, as might have been foreseen from general considerations. The ratio of reduction is q 2 : n, being 3 : 1 in rectangular sections, 4 : 1 in elliptic sections. When the beam is not of uniform strength the ratio of reduction must be greater for the same type of section. The reduc- tion is of course least in / sections of uniform strength. The elastic energy U is a function of great importance in the theory of continuous beams and other similar structures, the relative yielding of the several parts of the structure being always such that this func- tion is less than it would be for any other distribution of stress and strain. It may also be called the Elastic Potential, and when known all the equations necessary to determine the distribution of stress may be found by simple differentiation. (See Appendix.) In the case of a beam supported at the ends and loaded at a given point, the elastic energy may also be expressed in the form where W is the load and 3 the deflection of the loaded point. Taking the load in the middle and substituting by the formula on page 3^4, we find ~ results which we shall have occasion to use hereafter. 172. Concluding Remarks. Throughout this chapter it has been supposed that the deflection and slope of a beam are exclusively due to the bending action of the load, and this supposition is sufficiently accurate when the object is solely to estimate the stiffness of a beam in practical cases. The effect of the shear, which nearly always accompanies bending, will be briefly noticed in a later chapter (Art. 190, Ch. XV.), and it need only here be added that in some of the examples discussed in this chapter, where the results depend on a nice adjustment of the slope of the ends of a beam or the level of the supports on which it rests, the effect of shearing may be very considerable. Structures, the straining actions on which depem on a delicate adjustment should, like frames with redundant part (Art. 26), be avoided when possible, but when employed the effect of shearing should be carefully examined. CH. xin. ART. 172.] DEFLECTION OF BEAMS. 333 EXAMPLES. 1. If I be the length of an iron rod in feet, d its diameter in inches, just to carry its own weight with a deflection of 1 inch per 100 feet of span, show that 1= Compare this result with that of Ex. 14, p. 319, and state what formula is to be used when both stiffness and strength are required. 2. Find the ratio of depth to span in a beam of rectangular section loaded in the middle, assuming stress=8000, #=28,000,000, deflection = . Ans. J..ZUU L i O 3. A beam is supported at the ends and loaded at a point distant a, b from the sup- ports with a weight W. Show that the depression of the weight below the points of support is 4. In the last question deduce the work done in bending the beam, and verify the result by direct calculation. (See Art. 20.) 5. A dam is supported by a row of uprights which take the whole horizontal pressure of the water. The uprights may be regarded as fixed at their base at the bottom of the water while their upper ends at the water level are retained in the vertical by suitable struts sloping at 45, the intermediate part remaining unsupported. Find the bending moment at any point of the upright, and show that the thrust on the struts is about two-sevenths the horizontal pressure of the water. 6. A timber balk 20 feet long of square section supports 160 square feet of a floor, find the dimensions that the deflection of the floor, when loaded with 60 Ibs. per square foot, may not exceed inch. Ans. 12f". 7. A shaft carries a load equal to m times its weight (1) distributed uniformly, (2) con- centrated in the middle. Considering it as a beam fixed at the ends, find the distance apart of bearings for a stiffness of T *W-. Ans. If I be the distance apart in feet, d diameter in inches, then for a wrought-iron or steel shaft Vd 2 STi' (1) I 8. A beam originally curved, as in Ex. 21, p. 320, is fixed at one end and loaded in any way. If i be the change of slope at any point and X, Y the displacements parallel to axes of x, y of the point consequent on any load, prove that di_M f dX = . m dY_. ds El ' dy ' dx Apply these formula? to find the straining actions at any point of one of the rings of a chain of circular links. 9. A weight W is fixed to the centre of a vertical rotating shaft, and, by its centri- fugal force when the shaft is slightly bent, tends to increase its lateral deflection. Show that the number of revolutions of the shaft per minute must not approach that given by the equation all dimensions being in inches. Note. This is the simplest case of what is known as "centrifugal whirling," a question considered in Art. 203A, Ch. XVI. CHAPTER XIY. TENSION OR COMPRESSION COMPOUNDED WITH BENDING. CRUSHING BY BENDING. 173. General Formula for the Stress due to a Thrust or Putt in combina- tion ivith a Bending Moment. The bars of a frame and the parts of other structures are often exposed, not only to a pull or thrust alone, or to a bending action alone, but to the two together ; and the total stress at any point of a transverse section is then the sum of that due to each taken separately. That is to say, if H be the thrust, reckoned negative if a pull, M the bending moment, the stress at any point distant y from the neutral axis of the bending (see Art. 155), reckoned positive on the compressed side, must be given by H My H f _ q M } P = -j + r = -jl 1 + -" rff f A I A \ n Hh } the notation being as in the article cited. This formula shows how the effect of a thrust or pull is increased by a bending action : it has many important applications, some of which we shall now briefly indicate. 174. Strut or Tie under the Action of a Force parallel to its Axis in cases where Lateral Flexure may be neglected. Case I. Bar under the action of a force in a principal plane parallel to its axis. Let z be the distance from the axis of the line of action of the force, then For example, let the section be circular, then n = T \, q ^, and we find from whence it appears that a deviation from the axis of T y h the diameter of a rod increases the effect of a thrust or pull 50 per cent. Similarly it can be shown that if the line of action of the force lie outside the [CH. xiv. ART. 174.] COMPRESSION AND BENDING. 335 middle fourth of the diameter of a circular section, or the middle third of a rectangular section, the maximum stress will be more than double the mean, and at certain points the stress will be reversed. In designing a structure, then, the greatest care must be exercised that the line of action of a thrust or pull lies in the axis of the piece which is subjected to it ; to effect which, the joints, through which such straining actions are exerted, must be so designed that the resultant stress at the joint is applied at the centre of gravity of the section of the piece. This is a condition which cannot always be satisfied, and allowance in any case must be made for errors in workmanship. In practical construction it is the joints which require most attention, being most often the cause of failure. In frames which are incompletely braced the friction of pin joints causes the line of action of the stress to deviate from the axis. The effect is increased in the case of a thrust and diminished in the case of a pull by the curvature of the piece, which increases or diminishes z. Fig. 133 shows the axis of a column, under the action of a weight W y suspended from a short cross piece of length a. The column bends laterally, as shown in an exaggerated way in the figure. The inclination of AB to the horizontal is so small that the difference between the actual and the projected length of AB may be disregarded ; the bending moment at is therefore W '(a + 8), where 8 is the lateral deviation AN of the top of the pillar. This deviation we will in the first instance suppose small compared with a, and then determine the condition that this may actually be the case. Neglecting it, the axis of the pillar is bent by the uniform bending moment Wa into a circular arc of radius R, and as in Art. 163, S.2 = Z 2 ; substituting for R its value (Art. 155) we get ' Jfl* JPofl 1EI~ 2EI ' 8 WV whence we find a 2E1' The condition, then, that the lateral deviation should be small is that W should be much less than 2 El '/I 2 , and if this condition be satisfied the stress will not be much increased beyond that indicated by the formula given above. The very important cases in which W is large will be treated presently. In the case of a pull this restriction on the use of the formula need not be attended to, the deviation diminishing the stress. 336 STIFFNESS AND STRENGTH. [PART iv. Case II. Uniformly loaded beam supported at the ends and subject to compression. Let the load be W and the thrust H, then n' Hh Y For example, let the section be rectangular, then q = \> w= iV, and we find Let us further suppose the ratio of depth to span one-sixteenth, then which shows how greatly the effect of a thrust is increased by a moderate bending moment. If the deflection be supposed 1 inch in 100 feet then H will in con- sequence produce an additional bending action at the centre equal to Bl/1200, which will be equivalent to an addition to W of H/150. For safety H ought not to exceed 3JF, and the stress due to the bending action of the uniform load on the beam will then be increased about 25 per cent. This calculation shows why it is often necessary to support a beam at points not too far apart by suitable trussing even when support is not required to give sufficient stiffness. Theoretically a proper " camber " given to the beam will counteract the bending action, and, conversely, a small accidental deflection will increase it. 175. Remarks on the Application of the General Formula. The formula given in Art. 173 is much used in questions relating to the stability of chimneys, piers, and other structures in masonry and brickwork. The stress on horizontal sections of such structures varies uniformly or nearly so, and the formula then shows where the stress is greatest and also where it becomes zero, tension usually not being permissible. It must be borne in mind however that the bending is frequently unsymmetrical, so that the axis of the bending moment will not coincide with the neutral axis of the bending stress on the section (Art. 162). The stability of blockwork and earthwork structures is a large subject which will not be considered in this treatise. The use of the term " neutral axis " to denote the line of zero stress, a line which varies in position according to the proportion between the thrust and the bending, though common, is better avoided. TH. xiv. ART. 177.] COMPRESSION AND BENDING. 337 176. Straining Actions due to Forces Normal to the Section. The reasoning of this section shows that when a structure is acted on by forces some or all of which have components normal to a given section, the straining actions due to the normal components will in general depend on the relative yielding of the several parts of the section (Art. 42). These normal components however can always be reduced to a single force, acting through any proposed point in the section, and a couple, and if the point be properly chosen according to the nature of the structure at the section that single force will be a simple thrust or pull ; thus in the cases we have mentioned the point is the centre of gravity of the section. Having done this the couple will be so much addition to the bending action. An important example of this is the ase of a vessel floating in the water in which the horizontal longi- tudinal component of the fluid pressure generally produces bending, the arm of the bending couple being the distance of the intersection of the line of action of the resultant with the section considered, from the neutral axis of the " equivalent girder." 177. Maximum Crushing Load of a Pillar. When the compressing force is sufficiently great it produces a strong tendency to bend the pillar even though there be no lateral force. We have already seen that the condition that this shall not be the case is that W shall be small compared with the quantity 2J/// 2 , and we now proceed to inquire the effect produced when W has a larger value. All these cases come under the head of what is called Crushing by Bending, and are very common and important in practice. As in the case of the deflection of a beam the question is much more simple when the pillar bends into an arc of a circle, which it will do in various cases explained in Art. 163. The case which we select is that in which the sectional area remains constant and the thickness varies. Such a pillar is of uniform strength when very slightly bent, and when more bent the weakest point is at the base. When the load is applied exactly at the centre the elevation of such a pillar is a semi-ellipse with vertex at the summit; when not exactly at the centre the ellipse is truncated. As in other cases of uniform strength the section is ideal, requiring modification at the summit when applied in practice. Assuming then the form of the bent pillar to be a circular arc we liave as before s = but we have now, since we cannot neglect M C.M. 338 STIFFNESS AND STBENGTH. [PART iv. Hence by substitution we find . 1EI ' where / is the moment of inertia at the base, from which we find a This result shows that the pillar bends laterally more and more as W increases, and breaks with some value of W which we will find presently by substitution in the formula of Art. 172. First, however, observe that if a = 0, that is, if the line of action of the load pass through the centre of the pillar at its summit, then 8 = unless the denominator of the fraction be also zero, that is, unless W=*> The interpretation of this is. that if W be less than the value just given the pillar will not bend at all, but if disturbed laterally will return to the upright position when the disturbing force is removed. If W have exactly that value then, when put over into any inclined position the pillar will remain there in a state of neutral equilibrium, while the smallest increase of W above this limit will cause the pillar to bend over indefinitely and so break. Thus the foregoing equation may be regarded as giving the crushing load of the pillar under certain conditions to be defined more exactly presently. If the form of the bent pillar be not a circular arc but some other given curve, the corresponding type of section can be found by use of the general equation given on page 323. A formula of the same form is then obtained, but the co-eificient 2 is replaced by some not very different number depending on the form assumed. In Fig. 134 let y be the deviation from the vertical BB of any point in the pillar BAB at a distance x from the summit, then Wy is the bending moment M at that point, and the equation may be written Wy=EL*y dx 2 The case of most importance is that in which the curve BAB is a curve of sines given by the equation 7T 'T* 21 being the height of the pillar and 8 the deviation from the vertical at the centre. Differentiating, substituting and dividing by y, CH. xiv. ART. 177.] COMPRESSION AND BENDING. 339 an equation which shows that a pillar of uniform transverse section when bent into a curve of sines will be in equilibrium for this value of IV and no other ; a result the interpretation of which is the same as in the preceding case, from which it only differs in the number 2 being replaced by ?r 2 /4 or 2 -47. In Fig. 134 both ends of the pillar are rounded so as to be free to change their direction while remaining in the same vertical, the whole height L of the pillar is then 21 and In Fig. 1346 the pillar is fixed in direction at both ends and con- sequently there are two points of contrary flexure or " virtual joints n BB. The position of these joints is easily foreseen, for the four pieces CB, BA, ABj BC are all acted on by the same compressing force applied virtually in the same way and are therefore all of equal length. The whole height L of the pillar must consequently now be taken as 4 instead of 21 and ?r 2 replaced by 4?r 2 . Figf.134. Fig. 1340. ' In Fig. 134& we have an intermediate case, the summit being rounded and the base fixed in direction. The two ends are still supposed in the same vertical, so that the pillar now bends into the form BABC, having only one point of contrary flexure near the base while the upper portion BAB is in the condition of a pillar with rounded ends. To find the length 21 of this upper portion in terms of 340 STIFFNESS AND STRENGTH. [PART iv. the whole height L of the pillar, we must observe that the point of contrary flexure is not in the same vertical as the ends but deviates from it by a small quantity which we will call y . So that the deviation of any point distant x from the summit is now s . TT x x y= m 2'7 + 2/0 ' 2/' a formula which applies to points below B as well as above. To determine the position of B we have only to observe that when x = L both y and dyjdx must be zero ; hence differentiating and eliminating y , irL L tan-^TT.^, a transcendental equation which when solved by trial gives from which we find that the upper portion BAB of the pillar is about 70 per cent, of the whole height and that, in the formula for W, ?r 2 should be replaced by 2*04 7w 2 . For the purposes to which this formula is applied 2?r 2 is sufficiently accurate; Rankine employed in Gordon's Formula (p. 344) a coefficient obtained by supposing BB instead of BC vertical, which corresponds to the value 2J?r 2 . We thus obtain the three formulae known as Euler's Formulae, for the three cases in question with a uniform section. If the pillar be bent into a circle as above, then -rr 2 is to be replaced by 8. 178. Manner in which a Pillar crushes. Formula for Lateral Dematic The value of W here found is the maximum load, consistent with stability, which a pillar, free to deflect laterally, can sustain under any \ circumstances ; but, in order that it may actually be sustained, the pillar must be perfectly straight, the material must be perfectly homogeneous, and the line of action of the load must be exactly in the axis. These conditions cannot be accurately satisfied, and consequently a lateral deflection is produced, which increases indefinitely as the load approaches the theoretical maximum. This may be expressed by supposing that a is not zero, but some known quantity depending on the degree of accuracy with which the conditions are satisfied, and which may called the " effective " deviation ; since, when the pillar is straight am homogeneous, it will be the actual deviation of the line of action of the load from the axis. Let W Q be the theoretical maximum load as CH. xiv. ART. 178.] COMPRESSION AND BENDING. 341 calculated from the preceding formulae, and W the actual load, then w~ thus we see that a load of J, f , f the theoretical maximum produces a lateral deflection of la, 2a, 3a, increasing the deviation of the load from the axis of the column to 2a, 3&, 4a. These numbers are only exact when the pillar is so formed as to bend into the arc of a circle ; when this is not the case they follow a more complicated law of the same general character, depending on the type of pillar and the nature of the deviation. For our purpose the simple case is sufficient. It is convenient to express the load in pounds per square inch of the area (A) of the pillar at its base, then we may write with the notation of Art. 155 w , 2 W for the case where the pillar is rounded at both ends, the number ?r 2 being replaced by 2?r 2 or 4;r 2 in the two other cases of the last article. Similarly writing p = W\A for the actual load on the pillar, we get by substitution o = a . ^ , or a + 8 = a . -* Po-P Po-P The deviation is accompanied by an increase in the maximum stress (/) on the transverse section, which is given by the formula /=?K1) <>*> ' : from which we get, replacing H by W and M by W(a + 8), a result which shows that / increases indefinitely as p approaches p ot so that the pillar must break before the theoretical maximum is reached, however small the original deviation is. The greatest value of / must be the elastic strength, for as soon as this is past an additional lateral deviation at the most compressed part will occur, sooner or later accompanied by rupture. The formula may be written in the more convenient form (/-iYi-lp-1 \p A pj nh in which it is worth while to observe that the right-hand side is unity for the deviation necessary to produce double stress when the pillar is BO short that no sensible augmentation of the deviation is produced by lateral bending. In materials like cast iron which have a low 342 STIFFNESS AND STEENGTH. [PART iv. tenacity, very long pillars give way by tension on the convex side : the formula then becomes ;j where /' is the tensile stress at the elastic limit. The two formulae give the same result if For loads greater than this the first formula applies, and for small loads the second. In pillars flat, but not fixed at the ends, without capitals /' may be zero. 179. Actual Crushing Load. We thus see that if a pillar were absolutely straight and homogeneous it would crush, by direct com- pression if p Q were greater than /, and by lateral bending if p Q were less than /, the crushing load being the least of these two quantities ; but that the smallest deviation will be augmented by lateral bending, so that the actual crushing load will be less than the least of these quantities. Experience confirms this conclusion. When a long pillar is loaded we do not find that it remains straight till a certain definite load p is reached, and then suddenly bends laterally. We find, on the contrary, that a perceptible lateral deflection is produced by a small load, which gradually increases as the load is increased, till rupture takes place, showing, as we might anticipate, that some small deviation existed originally. And as that deviation evidently depends upon accidental circumstances it is impossible, from imperfection of data, to find the actual crushing load of a pillar for those proportions of height to thickness, for which its effect is greatly augmented by a small deviation. The augmentation is on the whole greatest when that is, when . This gives, by taking the values of E and / from Table II. , Chaptei XVIIL, Wrought Iron, L = SGx/A . h = '28h (Circular Section). Mild Steel, L = 29x/A .h = 23h Hard Steel, L=23j^n. h=l8h Cast Iron, L = 20 V A .h=l6h In the case of cast iron there is a difficulty in determining the value of/, but if we suppose that the elasticity of the material is not greatly CH. xiv. ART. 180.] COMPRESSION AND BENDING. 343 impaired at half the ultimate crushing load, we get the value given The case of timber is exceptional, and will be referred to further on. For pillars fixed or half-fixed at the ends the number ir' 2 is to be replaced by 4?r 2 or 2?r 2 as before. Let us assume this condition satisfied, and let us imagine the pillar loaded with three-fourths the theoretical maximum crushing load, then by substitution we find, qa/nh = ., or since n/p = for a circular section, from which it will be seen how small a deviation will cause the pillar to crush under three-fourths the theoretical maximum load, when the proportion of height to thickness is that just given. With a pillar of double this height the magnitude of the original deviation (a), always supposing it small, has little influence, and with a pillar of one-third this height lateral flexure has little influence, on the resistance to crushing. On the whole, then, it would seem that the most rational way of designing pillars would be to calculate the theoretical maximum load, and then adopt a factor of safety depending on the value of the devia- tion found from the above formula ; it is obvious that in some cases a much larger deviation may be considered likely than in others. For example the probable deviation from straightness may easily be imagined to be proportional to the length of the pillar. The Gordon- Rankine formula given in the next article may be regarded as a formula for the average factor of safety necessary on account of the exaggerated influence of errors of workmanship on the strength of pillars in cases where the deviation is not greatly influenced by the length. For the case of thin tubes see Chapter XVIII. 180. Gordon's Formula. A considerable part of our experimental knowledge respecting the strength of pillars is due to Hodgkinson.* His results show that in cast-iron pillars with flat ends, the length of which exceeds 100 diameters, the theoretical maximum is closely approached, while with shorter lengths the strength falls off consider- ably, as might be expected. In other respects the theoretical laws are approximately fulfilled, the principal difference being that columns with one or both ends rounded are somewhat stronger relatively to columns with flat ends than theory would indicate, an effect which may be partly due to imperfect fixing of the ends. Various empirical formulae have * Phil. Trans., 1840, Part II. An abridgment is given in Hodgkinson's work on Cast Iron. Weale, 1846. 344 STIFFNESS AND STRENGTH. [PART iv, been given to express the results of experiment on the crushing of pillars. That which has been most used was originally devised by Navier, but is commonly known as Gordon's. It is so constructed as to agree in form with the theoretical formulae in the extreme cases in which those formulae give correct results. As employed by Rankine, only replacing r 2 , the square of the radius of gyration, by nh in the notation of this work the formula is W f A which becomes, when l/h is small, and when l/h is large, while for intermediate values it gives smaller results. If we compare this last with Euler's formula for a column with flat ends, we get and this may be called the " theoretical " value of the constant c. The values actually used for c are somewhat different, being deduced from such experiments as have been made, and the results for different forms of section are not always consistent. Rankine gives in his Useful Rules and Tables, VALUE OF CONSTANTS. Value of /. Value of c. Wrought Iron, . . . . 36,000 36,000 Cast Iron, ..... 80,000 6,400 Dry Timber, .... 7,200 3,000 These values refer to struts fixed at the ends and to the crushing load. If one end be rounded, the value of c must be divided by 2, and if both ends are rounded, by 4. Rankine's formula has been very extensively tested for the case of wrought-iron columns of large size of various transverse sections, con- structed of ri vetted plates, and has been found to give good results.* In the case of timber Hodgkinson found, from a limited number of experiments on struts of oak and red pine of small dimensions, a formula which agrees with the formula for the theoretical maximum * Minutes of Proceedings of the Institution of Civil Engineers for May, 1878, vol. liv., page 200. CH. xiv. ART. 180A.] COMPRESSION AND BENDING. crushing load when the value of E in that formula is taken as about 900,000 Ibs. per square inch. It is possible that the low lateral tenacity of this material increases its flexibility under a heavy crushing load. The values just given of the constants for timber in Gordon's formula appear rather low. Recent good authorities give 9400 for / and 6700 for c. In the case of steel the value of / may be expected to be increased and the value of c diminished in the ratio of the direct resistance to> crushing of steel and wrought iron respectively, conclusions on the whole borne out by experience.* Calculations made by Gordon's formula may be tested by calculating the deviation a by the formula on p. 341 ; the magnitude of this will be to some extent a measure of the safety of the proposed load. In all cases of struts of large size subject to a heavy load, special care is necessary in considering all the circumstances if a deflection be- occasioned by the unsupported weight of the strut itself, or if, as is often the case, it be constructed of rivetted plates, a large margin of safety is desirable. So also in pieces forming part of a machine in which a bending action may be produced by inertia and friction, or which are subject to shocks, the simple thrust alone is often a very imperfect measure of the stress to which they are subject. Returning to the case of a long slender column we observe that the resistance to crushing depends solely on the stiffness and not on the strength being proportional to the modulus of elasticity. Hence a long column is stronger when made of wrought iron than when made of cast iron, although with short columns the reverse is true. It appears from Gordon's formula that for a ratio of length to diameter of about 26J the two materials are equally strong. In very long columns steel is not stronger than iron, for its modulus of elasticity is not very different ; in shorter lengths, however, the greater resistance to direct crushing of steel gives it an advantage. 180A. Partial Fixture of Ends. The condition of the ends of a pillar has great influence on its resistance to crushing; thus by EulerV formula the crushing load of a pillar fixed at the ends is four times that of a pillar with both ends rounded. The ends of a pillar in practical cases can hardly ever be regarded as either rounded or fixed in a mathematical sense, and the influence which different methods of fixing may have is a matter of much importance. (1) The most effectual method of obtaining, for experimental pur- * A Practical Treatise on Bridge Construction, by T. C. FIDLER, page 180. Griffin, 1887. 346 STIFFNESS AND STRENGTH. [PART iv. poses, a pillar, the ends of which are freely movable in direction while remaining exactly in the same vertical, is to make its ends wedge-shaped, or, still better, conical. Experiments on pillars with conical ends were carried out in 1887 by the late Professor Bausch- inger, a well-known authority on strength of materials. The test- pieces were of rolled iron of various sections, among which may be especially mentioned some pieces of I section of sectional areas ranging from 10| to 63 J square centimetres and of lengths from 1 to 4 metres. The results show irregularities arising partly from causes already mentioned and probably partly from the difficulty of obtaining the moments of inertia with sufficient accuracy, but on the whole show a, crushing load of more than 85 per cent, of that given by Euler's formula for a pillar with rounded ends. In 1887-88 similar experi- ments on pillars with conical ends were made by Herr Tetmajer, the test-pieces being iron bars of circular section about 2 inches diameter and also pieces of wood. On comparison with Euler's formula similar results were obtained. These experiments point to the conclusion that the deviation a in- stead of increasing slowly with the length, as it would do (Ex. 9, p. 349) if the Rankine formula were satisfied, increases much more rapidly, so that the "constant" c in that formula diminishes rapidly with the length when the ends of the pillar are rounded. (2) At the same time Bauschinger also made experiments on pillars with flat ends simply butting against the compressing pieces without any attachment. The test-pieces were similar to those in the preceding case, but the results of the experiments now showed a comparatively constant value of c instead of the rapid diminution previously found. In this case also, however, c is not constant, but diminishes with the length. The same diminution of the constant c as the length increases has been found in many experiments on columns of larger size, as shown by the formulae proposed by Mr. Cooper.* (3) The ends of a pillar are very frequently pin-jointed, the load being then transmitted by the pressure of the pin upon its circular- bearing surface. The crushing load now depends on the diameter of the pin, because the total diviation cannot exceed the radius of the friction circle (p. 240) of the pins ; as soon as this is over- passed, the pillar instantly crushes in consequence of the release of the ends. Very instructive experiments were carried out at Watertown Arsenal, U.S.A., in 1883, by means of the well-known testing machine * Engineering Construction, by W. H. Warren, p. 196. Longmans, 1894. CH. xiv. ART. 181.] COMPRESSION AND BENDING. 347 there stationed.* The test-pieces were bars of iron about 3 inches square, of lengths ranging from 10 to 60 diameters, some with pin ends, the diameters of the pins ranging from | inch to 2| inches, and some with flat ends. The results of these experiments, tabulated on page 118 of the report cited, show that pins 2J inches diameter are nearly equivalent to flat ends, but that inch pins give a much reduced crushing load. The Rankine formula appears to agree fairly well with these experiments, small pin ends being treated as rounded and large ones as fixed. (4) The facts described in this article show clearly the empirical character of the Rankine formula ; the approximate truth of which, under the complicated conditions in which most experiments have been made, being due to the effects of increasing initial deviation as the length of the column increases, being partially compensated by the increasing influence of the partial fixture of the ends. How far the conditions of the experiment resemble the conditions of practice must always be carefully considered in each individual case. 181. Collapse of Flues. There are other cases of crushing by bend- ing, some of which will be considered in a later chapter ; but it will be convenient to mention the Fi 13g ,,-'' important practical problem of the yielding of a thin tube under external fluid pressure. The strength of a tube under external fluid pressure is as different from that of a tube \ under internal pressure as the strength of a bar under compression is different to its P^ Fig.136. strength under tension. ^N A tube perfectly uniform in thickness made of perfectly homogeneous hard material, and subject to perfectly uniform normal pressure ex- ternally would theoretically maintain its form until it yielded by the direct crushing of the material. But when the pressure exceeds a certain limit the tube is in a state of unstable equilibrium, and any deviation from perfect accuracy in the above conditions will cause the tube to yield by collapsing, the collapsing being accompanied by bulging. If the tube is very long it will collapse in the manner shown in Fig. 135, the circumference dividing itself up into four arcs, two * Report of Tests on Structural Material made at the Watertown Arsenal. 1883. 348 STIFFNESS AND STRENGTH. [PART iv. of which are concave outwards and the other two convex. A want of exactness in the construction will in practice generally prevent the collapsing from being symmetrical. Each portion of tube between the points A is under the action of forces applied at the ends towards one another, which crush it by lateral bending just as a long column is crushed. Just before collapsing, each segment A A (Fig. 136), of length s say, will be under the action of a thrust P suppose, applied at the ends tangentially. Equilibrium is maintained by fluid pressure of intensity p on the convex side. When the pressure exceeds a certain limit the equilibrium is unstable, some accidental circumstance determining the position of the point A of contrary flexure, and the consequent length 5 of any arc. As shown on page 302 the thrust per inch length of the tube may be taken as approximately proportional to pel. Thus if t = thickness of tube, we may expect that the collapsing pressure would be given by a formula like that which expresses the crushing load of a long slender rod of rectangular section, namely, pd = k'fi/s 2 where k' is an elastic co-efficient. All other things being equal, the diameter alone varying, the length s of an arc A A would be proportional to the diameter of the tube d, and, under those circumstances, the collapsing pressure of a thin tube (see Appendix), would probably vary with t z /'d s . But the length of the tube, as well as the diameter, influences the value of s. In all practical cases, as in all those on which experiments were made, the ends of the tube are rigidly constructed, and very much support the tube in the neighbourhood from collapsing ; thus the proximity of the ends has an important effect in determining the length of the arcs into which the circumference divides itself. If the length of the tube is decreased a limit will be reached below which the tube lff ' 137 on collapsing divides itself up into six arcs, three concave and three convex, as shown in Fig. 137. Then the length of each arc will bear a smaller proportion to the diameter than in the long tube. A still shorter tube will, when it collapses, divide it into eight arcs, and so on. Thus the length s is in some way dependent on the length of the tube. The correctness of this reasoning is borne out by experiments made by Fairbairn and others. In Fairbairn's experiments the tubes were made of rivetted wrought-iron plates. The ends were made rigid by a strong stay placed within the tube, keeping the ends apart. The tube thus constructed was placed in a larger cylinder of wrought iron and external pressure was applied by forcing water in. The pressure being gradually increased the tube will at last suddenly collapse, making a noise which indicates the instant of the occurrence. The results of CH. xiv. ART. 181.] COMPEESSION AND BENDING. 349 the experiments showed that the collapsing pressure may be approxi- mately expressed by the formula the dimensions being all in inches, the co-efficient k = 9,672,000. This formula must not be used for extreme cases nor for tubes of thickness less than f inch. Since a short tube is so much stronger than a long one, we have an explanation of the advantage of rivetting a T-iron ring around a boiler- furnace tube, which amounts to a virtual shortening of the length of the tube. Other formulae have been proposed, some of which represent the results of experiment more closely, but the materials at present avail- able do not admit of the construction of a satisfactory formula. Some further remarks on the subject will be found in the Appendix. EXAMPLES. 1. Find the thickness of metal of a cast-iron column fixed at the ends, 1 foot mean diameter, 20 feet high, to carry 100 tons. Factor of safety, 8. Ans. Thickness I". 2. Find the crushing load of a wrought-iron pillar 3" diameter, 10 feet high, rounded at the ends. .4ns. Crushing load =66, 218 Ibs. =30 tons nearly. 3. If in last question the pillar were of rectangular section of breadth double the thickness, what sectional area would be required for equal strength? Ans. Sectional area =9 '4 square inches instead of 7 square inches as before. 4. Assuming the crushing resistance of steel to be 1^ times that of wrought iron, and its modules of elasticity 10 per cent, greater, find the probable values of /and c in the Gordon-Rankin* formula. Ans. /=54,000, c=26,400. 5. Find the crushing load in tons of a timber pile 12 inches square, 30 feet long, fixed at one end, rounded at the other. Ans. 56J tons. 6. Find the collapsing pressure, according to Fairburn's formula, of a cylindrical boiler flue T 7 ^" thick, 48" diameter, and 30 feet long. Ans. Collapsing pressure = 107 Ibs. 7. In Ex. 1 calculate the deviation of the line of action of the load from the axis to produce a maximum stress of 10,000 Ibs. per square inch. Ans. 1'8". 8. In Ex. 2 calculate the deviation to produce a maximum stress of 9,000 Ibs. per square inch with a load of 11,000 Ibs. or of 22,000 Ibs. Ans. 1'5 or '5. 9. Assuming the crushing load of a pillar to be given by the Gordon-Rankine formula with the theoretical values of the constants, show that the deviation is given by the formula CHAPTER XV. SHEARING AND TORSION OF ELASTIC MATERIAL. SECTION I. ELEMENTARY PRINCIPLES. 182. Distinction between Tangential and Normal Stress, Equality of Tangential Stress on Planes at Pdght Angles. In the cases we have hitherto considered of simple tension, compression, and bending, the stress on the section under consideration has been at all points norma to the section. But we may take our section inclined at any angle to the stress, and the mutual action is then not normal to the section. The particles on each side of the section partly act on one another in the direction of the section itself, and so constitute a stress analogous Fig. 138. to friction, resisting the sliding of one portion relatively to the other. Such a stress is called tangential or shearing stress, being the stress called into action by shearing. Let us return to the case of the stretched bar carrying a load P (Fig. 138). On a transverse section of the bar only a normal stress is produced. Now suppose we take an oblique section, whose |p normal makes an angle 6 with the axis of the bar, and let us resolve the force P into two com- ponents, one perpendicular and the other parallel to the section. The normal component P cos tends to produce a direct separation at the section, producing a tensile stress similar in character to that on a transverse section, but of less intensity. If A = area of transverse section of bar, then A sec = area of oblique section ; the intensity of the normal stress ^ P. cos 6 Pcos(9 P pn = . -X = T COS-P = p cos^ 1 A sec 6 A where p = ', A [CH. xv. ART. 183.] SHEARING AND TORSION. 351 the other component P sin 6 produces a tangential or shearing stress of intensity P sin B , = --, = p sm cos u. *' A sec Similarly if the bar is subjected to a compressive instead of a tensile load. Many materials which offer great resistance to direct compression yield by sliding across an oblique plane. Now p t is a maximum when = 45, this is therefore approximately the angle of separation. The same maximum stress, the value of which is p/2, occurs on another plane sloping the other way at an angle of 45. We sometimes find fracture to occur across two oblique planes ; sometimes across one only. If in p t =p sin 6 cos 6 we change 6 into 90 + 6, p t has the same value ; so that the intensity of the tangential stresses on two planes at right angles to one another is the same. This is true generally in all cases of stress, as will be seen presently. 183. Tangential Stress equivalent to a Pair of Equal and Opposite Normal Stresses. Distorting Stress. In the example we have just considered we have both shearing and normal stress ; but there are cases in which there is only a shearing stress. Let A BCD f p Fig.139 (Fig. 139) be a rectangular plate of thickness t. Over the surfaces BO and A D suppose a tangential stress to be applied of intensity p t . Calling b and a the length of the sides of the plate, the total amount of the tangential stress on each side is P=p t .bt. To prevent the turning of the plate, suppose the forces P balanced by the application of an - ^ - -* i * uniform stress over the surfaces BA and DC P * | p of intensity p' t . The amount of the force on each of these sides Q=p' t .a.t. Since equilibrium is produced, the moment of the couple P must be equal to the moment of the couple Q. .'. p t .bt . a=p' t . at. b, or p t =p' t ; that is, the intensity of the stress is the same on BA as on AD. Shearing therefore cannot exist along one plane only. It must b& accompanied by a shearing stress of equal intensity along a plane at right angles. Such a pair of stresses unaccompanied by normal stress 352 STIFFNESS AND STRENGTH. [PART iv. constitute a Simple Distorting Stress, so called because it distorts the elements of the body. Let us now assume, for simplicity, the plate to be square (Fig. 140). The effect of the forces is to produce a change of form, which, in Fig. 140 perfectly elastic bodies, is exactly pro- portioned to the shearing force which produces it. The square ABCD becomes a rhombus AB'C'D, the angle of distortion being proportional to the stress p t . We may write where the co-efficient C is a kind of * Modulus of Elasticity, but of a different nature from that previously employed. 'The volume of the elastic body A is in general practically unaltered. Under the action of the forces it has simply undergone a change of form or figure, and the co-efficient C which connects the change of form with the stress producing it, is a co-efficient of elasticity of figure. It is sometimes called the modulus of transverse elasticity, but preferably the co-efficient of rigidity. The ordinary (Young's) modulus of elasticity E connects the stress and strain in a bar when it undergoes changes both of volume and figure. The co-efficient of rigidity C for metallic bodies is generally less than |^, and for wrought-iron bars may be taken as 10 to 10i millions, or in torsion somewhat greater. Let us now take a section of the square plate (Fig. 140) along one of the diagonals and consider the forces which act on the two sides of the triangular upper portion. Resolve these forces parallel and perpendicular to the diagonal. The components of the two P's along the diagonal balance one another, and there will be no tendency for this triangular portion to slide relatively to the other ; that is to say, there is no shearing stress on the diagonal section. But the other -components, perpendicular to the diagonal, cause the upper triangular portion to press on the lower with a force If we divide this force by the area of the diagonal section over which it is distributed, we obtain the intensity of this normal stress, P = ^^=p t On the diagonal section AC which we have been considering, this stress is compressive, but if we take the section along BD, the other CH. xv. ART. 184.] SHEARING AND TORSION. 353 diagonal, we find by the same reasoning a stress of the same magni- tude, but tensile. Thus it appears that a shearing stress on any plane necessarily involves tensile and compressive stresses of equal in- tensity on planes at 45 to this plane, so that a simple distorting stress, which was defined above as a pair of shearing stresses on planes at right angles, may P - also be defined as a pair of normal stresses of equal intensity and of opposite sign, as shown in Fig. 141. SECTION II. TORSION OF SHAFTS. SPRINGS. 184. Torsion of a Tube. Round Shafts. We now proceed with various examples of this kind of stress, commencing with the case of torsion. Torsion was mentioned as one of the five simple straining actions to which the bar as a whole may be exposed. It is produced by a pair of equal couples applied at the ends of the bar, the axis of the couples being the axis of the bar. When we consider the nature of the elastic forces called into action amongst the particles of the bar, Torsion reduces to a case of Shearing. To understand this, we will begin with a simple case. Imagine a thin tube (Fig. 142) with one end fixed, and the other acted on by an uniform tangential stress of intensity q. Let t be the thickness and d the mean diameter of the tube, then Sectional area of tube = irdt approximately ; Total shearing force = q-n-dt ; and since the force on each unit of area of the section acts approxi- mately at the same distance from the centre of the tube, the total twisting moment = qirdt x \d = \qndH. This twisting moment is balanced by the resistance to turning offered at the fixed end. At any transverse section KK of the tube there will be produced an uniform stress of intensity q. Let us now consider a small square traced on the surface of the tube, with two sides on two transverse sections. If we take the square small enough we may treat it as a plane square. To balance the shearing 354 STIFFNESS AND STRENGTH. [PART jv. stress q, which acts on the sides of the square lying in the transverse planes, a shearing stress of equal intensity is, as explained above, called into action on the other two sides of the square, in the direction of the length of the tube, so that, if the tube were cut by longitudinal slits, the power of resistance to torsion would be as effectually destroyed as if it were cut by transverse slits. But if we made spiral slits at an angle of 45, as shown at SS in Fig. 142 ; supposing the slits indefinitely fine,. and no material removed, the strength of the tube to resist torsion in the direction shown would not be impaired. The material of the tube would then be divided into spirally-bent ribands, which would be in tension along their length, and in compression laterally, the ribands being caused to press against one another. Along a second set of spirals- such as S'S'j longitudinal compression and lateral tension exist ; the lateral forces are indicated in both cases by arrows in the figure. So much for the state of stress induced in the tube by the torsion. Next as to the change of form which accompanies the stress. The square will be distorted into a rhombus. A straight line AD, drawn on the surface parallel to the axis of the tube passing through the centre of the square, will be twisted into a spiral AD', the angle of the spiral being the angle of distortion of the square. Let 6 be that angle, then q = CO, where C is the co-efficient of rigidity. The effect of this is that, relatively to the end A, the end D is twisted round through an angle DOD' = i suppose, called the angle of torsion. arc DD' In circular measure i = (r = radius of tube). Also arc DD' = W y 8 being a small angle. Therefore i = lO/r. Since also = q/C, we have the angle of torsion i = ql/Cr, in terms of the stress. From this we may express the angle of torsion in terms of the twisting moment producing the torsion. We now pass on to the consideration of the torsion of a solid or hollow cylindrical shaft. First, let us imagine the shaft to be made up of a number of concentric tubes exactly fitting one another, and let us further imagine that at the end of each tube a suitable twisting moment is applied, so that each tube is twisted round through exactly the same angle. This effect will be produced by applying over the section at the end of each elementary tube a tangential stress, which is propor- tional to the radius of the tube. If we make q/r = qjr^ where q l and i\ refer to the outside tube, then the angle of torsion will be the same for all the tubes, and they will not tend to turn relatively to one another, but altogether. We may then suppose them united together again in a solid mass. If the stress applied be proportional to the distance CH. xv. ART. 184.] SHEARING AND TORSION. 355 from the centre, the shaft will twist just as if it were a set of tubes, each being subjected to the same stress and strain as if it were an independent tube. Now in the actual case of the twisting of a solid shaft, all portions from the outside inwards to the centre must turn through the same angle, and hence the shearing stress at any point of the section of the shaft must be proportional to its distance from the centre. This is true except very near the point of application of the twisting moment. Suppose, for example, the twisting moment is applied by means of a wheel keyed on the shaft, then in the immediate neighbourhood of the key-way, the stress will not be as stated, but at a short distance along the shaft the stress distributes itself in the manner described. This is another instance of the general principle already employed in the case of stretching and bending. The total resistance to torsion of the solid shaft is the sum of the twisting moments of all the concentric tubes into which it may be imagined to be divided. Thus in which = r.&. that is, the product of the sectional area of each tube multiplied by the distance squared of the area from the axis of the shaft must be taken and summed. The result is called the Polar Moment of Inertia and will be denoted by 7, so that T-&L r i The same formula applies to hollow shafts, the summation now extend- ing from the internal radius ?' 2 to the external radius r^ and the value of / is then being double the corresponding value in the case of bending. Since i = ql/Cr we can eliminate q and thus obtain r-e/.l i a formula which gives the twisting moment in terms of the torsion per unit of length. Dropping the suffixes, taking r to be the outside radius, we can write the moment of resistance to torsion of a solid shaft, T=t*ff*, or T Vr/^; where / is the co-efficient of strength of the material to resist shearing. 356 [STIFFNESS AND STEENGTH. [PART iv. Thus the strength under torsion is proportional to the cube of the diameter. The formula shows that, assuming / to be the same in each case, the strength of a shaft to resist a twisting moment is double its strength to resist a bending moment. Having determined the diameter of shaft required to take a given twisting moment we are now able to obtain a solution of the practical question, What diameter of shaft is required to transmit a given horse- power at a given number of revolutions per minute 1 Let T Q = mean twisting moment transmitted in inch-tons, then T x 2irN= work transmitted per minute in inch-tons, where N= revolutions per minute of shaft. Let H.P. denote the horse-power to be transmitted, then 33000 x 12 TT T x2irN= 224Q H.P. 33000 x 1 2 H.P. '* ~ ~ Now the shaft must be strong enough to take not only the mean but the maximum twisting moment. We may express the maximum in terms of the mean by writing T=KT Q , where K is a co-efficient whose value is different in different cases and T= maximum twisting moment, but TTftfd* or d s = 16x33000x12 27r 2 x 2240 3 rj7 and d = 5-233^ /* The value of/ depends in some measure on the fluctuation to which the twisting moment is subject, but under ordinary circumstances should not exceed 3J tons per square inch (Art. 221) for wrought iron, 4J tons for steel, and (see Art. 229) 1J tons for cast iron. The value of K, the ratio of maximum to mean twisting moment, depem on the circumstances discussed in Chapter X. We may assume it equal to 1 J when the number of cranks is 2, allowing a small additior for the bending due to the weight of the shaft. On substitution w< obtain for wrought iron This formula agrees closely with the best practice in screw-propelh shafting. When the amount of bending to which the shaft is subject is con CH. xv. ART. 185.] SHEARING AND TORSION. 357 siderable, as in the case of crank shafts, the diameter determined by this formula is too small. It will be seen hereafter that when all the forces acting on the shaft are known, a value of K can be calculated which gives the effect of bending. If we assume K= 2, the co-efficient 4 in the above formula will be replaced by 4*5, and this agrees closely with practice in the crank shafts of marine screw engines when made of iron, the number of cranks being 2. In the formula for the angle of torsion, i- ql - -Or' if we replace q by its working value for wrought iron (7200 Ibs.), C by 5000 tons, and i by the circular measure of 1, we find /=13-6d, showing that under the working stress the shaft twists through 1 for each 13 \ diameters in its length. For many purposes this is much too small, and the dimensions of a shaft then depend on stiffness, not on strength, as in the case of beams (Art. 168). The itest angle of torsion permissible depends in great measure on the irregularity of the resistance, and no general rule can therefore be laid down for it. If the angle of torsion be given and the length, the diameter will depend on the fourth root of the twisting moment as shown by the formula already given which connects the two. In this, as in other cases where dimensions depend on stiffness, not on strength, steel has no advantage over iron, because the co-efficients of elasticity of the two materials are the same or nearly so. A hollow shaft is both stronger and stiffer than a solid shaft of the same length and weight, the central portion of a solid shaft not being twisted sufficiently to develop its full strength. The distance apart of the bearings of a shaft depends on the stiff- ness necessary to resist the bending due to the weight of the shaft itself, and of any pulleys or wheels upon it, together with the ten- sion of belts and other similar forces. If the total load be equivalent to m times the weight of the shaft itself uniformly distributed, the length between bearings for a wrought-iron or steel shaft d inches diameter will be given approximately for a stiffness of ^-^^ by Ex. 7, p. 333. When, as in screw propeller shafting, the bearings are liable to get out of line, too great stiffness in a shaft will produce great straining actions upon it. 185. Elliptic and other Sections. In the cases hitherto considered the stress called into play at each point of the transverse section is pro- 358 STIFFNESS AND STRENGTH. [PART iv. portional to the distance of the point from the centre, and its direction is tangential to a circle drawn through the point, but in non-circular sections this is no longer the case. In particular the direction of the stress at points near the circumference is necessarily tangential to the contour of the section, for by the principle of Art. 183 (p. 351), in the absence of shearing stress on the outer surface, there can be none on the transverse section in a direction perpendicular to the contour. Commencing as before with the case of a tube the thickness (t] of which is small compared with the radius of curvature, but which now may vary according to any law; the shear (q) at any point is along the circumference. Taking As a small element of the circumference, t . As will be the area of the element, and qt . As will be the whole shear upon it, while qt will be the shear (S) per unit of length of the circumference. Next take two cross sections of the tube distant A# apart and con- sider the rectangular portion of thickness t the sides of which are As and Aa. Reasoning in the same way as in Art. 183 already cited, the shear S on the face As of the cross section is necessarily accom- panied by an equal shear (S') on the longitudinal face Arc. Now if the cross sections are free from constraint caused by the neighbourhood of rigid ends, the state of stress of all cross sections may be taken as the same, and the normal stress (if any) on the two faces forming part of the two cross sections cannot be different : from which it follows that S' is constant, since any change in S' can only be balanced by a corresponding change in the normal stress on the faces at right angles to it Hence S, that is qt, is also constant, and we infer that the intensity of the shear at any point varies inversely as the thickness of the tube at that point. Now take a point inside the tube in the plane of the section and join to the extremities P, P' of the element As at the middle of the thickness, thus forming a triangle OPP on the base A.s. Then if p be the perpendicular dropped from on the tangent to the circumference drawn through PP' j?As=2xArea OPP', and if A be the whole area of the cross section of the tube measured to the middle of the thickness But S . 2/'3) may be taken as 42. Since the elastic energy U is necessarily equal to \Ti the foregoing formula gives TT C A * . 2 _2ir 2 // T 2 = 8^-/7- ~-^^'~c' in which as before 4?r 2 should be replaced by 42 in rectangular sections. (4) The maximum stress q lt due to a twisting couple T, may be found from the value of the moment of resistance already given in (1), (2), whence by substitution for T the elastic energy, or its limiting value the resilience, per unit of volume may be found. It is, however, also important to know the stress q 2 at a point 2 situated at the middle of the shorter side of a rectangular section. for when the section is exposed to bending as well as torsion the combined effect of the two is frequently greatest at this point. On examination of St. Venant's results, given on page 39 of the work already cited, it is found that q z is never less than 74 per cent, of q v and that it is given with fair approximation by the formula In an elliptic section the point 2 is at either extremity of the major axis and the stress there is pq v diminishing to zero as the ratio of axes is reduced. This shows that the distribution of stress for small values of /3 is quite different from that in a rectangular section; a point further illustrated by comparing the strength of the two for the same 362 STIFFNESS AND STEENGTH. [PART iv. area and ratio of axes ; when it will be found that for small values of fi the rectangle is the stronger form. Further remarks on torsion will be found in the Appendix. 186. Crank Shafts. The twisting of a shaft is due to the action of transverse forces which have a moment about its axis. The common crank shaft is a case which may here conveniently be considered as an example of the way in which such forces strain the shaft. In Fig. 143 ACB is a shaft turning in bearings A and B and acted on by twisting moments T 19 T 2 at its ends. The sides of the crank __T Fig-. 143 are generally at right angles to the shaft, but in the figure are shown inclined at an angle 0, a case which sometimes occurs. The crank pin is acted on by the thrust of .a connecting rod not shown in the figure, which together with other forces (if any) passing through the axis of the shaft and corresponding reactions of the bearings form a system of forces the straining actions due to which are now to be studied : the graphical methods explained in Part I. being employed as most suitable for the purpose. The first step is to resolve the forces into two sets, one set in the plane of the crank, the other perpendicular to that plane. The first set produce shearing and bending only, which actions may be repre- sented by polygons in the usual way and need not for the present be further considered ; the second set alone produce twisting. As regards the straight part of the shaft : if S be the force on the crank pin perpendicular to the plane of the crank and a the crank-radius, then the difference of the twisting moments T lt T 2 is determined, but the actual magnitudes depend on the twisting transmitted from the parts of the shaft lying beyond the bearings. If one end B of the shaft be free the corresponding moment T 2 will of course be zero. If the turning moment Sa supplied by the connecting rod furnish energy at both ends of the shaft, as is often the case, T 2 will be negative. CH. xv. ART. 187.] SHEARING AND TORSION. 363 Taking any point K (Fig. 143), on the crank arm at a distance z from the axis, let a polygon of moments be drawn, the force S for this purpose being taken as passing through the axis. The result at K is a bending moment m, the axis of which is shown in the figure by a dotted line perpendicular to the axis. In like manner a polygon of shearing force may be drawn giving the shear at K which we will call F. Taking a transverse section of the crank arm at K the shear on this section will be F while the bending moment M and the twisting moment T will be determined by the equations from which we obtain the values of T and M, namely, T=(T l -Fz)cos6 + m. sin 6, M= (Fz - 7\) sin + m . cos 0. For the crank pin we have only to put = 0, z = a, and we find T^Ti-Fa, M=m, and in the common case where = 90 we have for any point of the crank arm, T=m- M=Fz-T r These results refer to the side next the bearing A on the other side T l must be changed into T%. It must further be remembered that they refer exclusively to the set of forces perpendicular to the plane of the crank; the set of forces in that plane produce a shear F' and a moment M' perpendicular to those just considered, so that the resultant bending moment is *JM 2 + M' 2 . The crank arm, however, is usually of rectangular section and the components M, M' must then be considered separately. The method of compounding a twisting moment with a bending moment will be explained in Chapter XVII. 187- Spiral Springs may be flat or conical, but the simplest and most important case is that in which the spring consists of a strip of metal, usually of rectangular or circular section, coiled into a cylinder of radius r, the pitch angle (0) of the spiral being uniform. The length of the spring (a; ) measured along the axis of the cylinder is given by the formula X Q = I sin 0, and therefore can only vary sensibly by variation of the pitch angle. The ends of the strip are bent to meet the axis and are inclined to each other at an angle given by the equation r<{> = I . cos 0, 364 STIFFNESS AND STRENGTH. [PART iv. each complete convolution of the spring increasing the angle by 2?r. The angle 6 will for the present be supposed small. Such a spring may be used in two distinct ways. (1) One end being held fast the other may be attached to a spindle occupying the axis of the spring, and by a couple (M) applied to it the spring is turned through an angle - < . The action here is one of simple bending, the bending moment being M and the elastic energy ( U) being given by equations which give for the angle turned through 2U Ml *-*F"ir An example of this kind occurs in the spring of the balance of a, chronometer. (2) Much more important, however, is the case so common in practice in which the spring is altered in length from x to x by the action of a force P applied along the axis. Each section of the strip is now subject to the action of a twisting moment T=Pr, while the corresponding elastic energy is The value of U is found from the formula given in Art. 185 (3), and the radius between x-x and P is thus determined. (3) The action on a spiral spring is not exactly one of pure bending or pure torsion as just supposed unless the pitch angle be exceedingly small. In the first case the bending moment is M.cosO and there is in addition a twisting moment M.sinO; while in the second case the twisting moment is Pr . cos 6 which is accompanied by a bending moment Pr sin 6. It will be shown hereafter that the elastic energy due to the combination of bending and twisting is the sum of the values of U due to each taken alone. The total value of U can therefore be readily obtained by sum- mation : by use of which the preceding formulae for < - and X XQ will still apply. If be less than 15, however, the correc- tion is of little importance. For values of co-efficients, see Ch. XVIII. When the spiral is flat as in the main spring or balance spring of a watch the action is one of simple bending as in (1), and the same formulae apply with slight modification. The conical springs employed in the buffers of railway carriages and for other purposes act by torsion as in (2), but the calculation is somewhat more complex. CH. xv. ART. 188.] SHEARING AND TORSION. 365 SECTION III. SHEARING IN GIRDERS. JOINTS. 188. Web of a Beam of I Section. Torsion is one of the few cases in practice where a simple distorting stress occurs alone and not in combination with other kinds of stress. It generally happens that a normal stress is combined with it ; such, for example, is the case in the web of a beam of I section, to which we next proceed to direct our attention. Taking a transverse section, the normal stress at a point distant y from the neutral axis is given by the formula p M r~r and is therefore the same for the same values of M and /, whether the web be thin or thick, while it will be shown presently that the tangential stress is greater the thinner the web, and becomes the most important element when the web is thin. Let us suppose, for simplicity, the flanges equal, and also that the beam is supported at the ends and loaded in the centre with a weight W. As we have previously seen, the flanges will sustain the greater 1 ! K * K, ciB.J'i^t !H 2 ;T q 4 H 2 ^ qj H, H, CO III 1 ^ 1 < X, * v/ w /////< portion of the bending moment, the web carrying only a small portion of it, l/7 th , if the area of the web equals the area of each flange. For simplicity, let us imagine the flanges to take the whole of the bending. Let jfiTj and K 2 (Fig. 144) be two transverse sections of the beam at distances x l and x 2 from the centre of the beam, 2a being the span of the beam the bending moment at the first section, M l = \ W(a -x } ) and at the second M 2 = \ W(a - x 2 ). Now, supposing the flanges to take the whole of the bending, the stress H produced on the flanges is given by the formula Hh - M. Thus at /^ we have H, and at K we have #= Ztl and similar forces on the bottom flange only reversed in direction. There will thus be a resultant force H^ - H 2 tending to push the portion K 1 K 2 of the flange to the left, 366 STIFFNESS AND STRENGTH. [PART iv. W(V -*L\ HI~H Z = % -. This force is balanced by the resistance of the web to shearing along the line of junction with the flange. Since H l - U 2 is proportional to the length of K^K^ the shearing force per unit of length of web = W/2h. If we suppose t to be the thickness of the web, the intensity of the shearing stress will be K Thus, considering the portion of the web between the sections K^ and K 2 apart by itself, we see that on the upper and lower horizontal edges of it, where it joins the flanges, it is subject to a shearing stress of intensity q t to balance which there must act on the vertical sides KK a shearing stress of equal intensity. Now, the shearing force for the vertical sections KKis \W. Supposing the web to be of rectangular section and of height h, then, assuming the whole of the shearing force to be borne by the web, the mean intensity of the shearing stress on the vertical sections is Wfiht. Therefore the assumption that the flanges take the whole of the bending moment is equivalent to sup- posing the web to take all the shearing. Assuming this, we see that the shearing stress, taken as uniformly distributed over the vertical section, will be accompanied by an equal shearing stress on any hori- zontal section. When considered alone, the effect of these shearing stresses on planes at right angles to one another is to produce tensile and compressive stresses on the web in directions making an angle of 45 with the horizontal and vertical planes ; and thus the web may be superseded by an indefinite number of diagonal bars inclined at an angle of 45, thus forming a lattice girder. If the web is designed so as to be strong enough only to withstand the shearing stress, replacing q by / the co-efficient of strength against shearing /, we find pp The influence of the normal stress due to bending will be considered in a subsequent chapter. Its effect is greatly to increase the strain on the web (see Art 207), which besides will in most cases exhibit weakness on account of the compressive stress in one of the diagonal directions. If the distance between the flanges is great, the web will be liable to yield by buckling or lateral flexure (see page 340). To prevent this, the web must be stiffened by angle irons rivetted on it. But the girder would then be made heavy, and it is therefore more economical to make large girders with openwork diagonal bracing. W r e have in this investigation supposed the beam loaded in the CH. xv. ART. 189.] SHEARING AND TORSION. 367 middle, so that the shearing force is uniform throughout the length of each half, and the problem was thus simplified. But the same prin- ciples apply if the load be distributed in any manner. The shearing force will then vary from section to section along the beam. 189. Distribution of Shearing Stress on the Section of a Beam. The foregoing preliminary investigation will give some idea of the effect of a shear on the web of a flanged beam ; let us now consider the question more generally. Taking a section of any type, let a line be traced cutting off from the whole area A any given portion. The line may be curved, but in the first instance assume it straight and parallel to the Fig.i45 L neutral axis SS (Fig. 145). Divide the area into strips of breadth b and thickness Ay, as in Art. 154, then the normal pressure on the portion cut off is the second form being obtained by substituting for p from the bending moment formula and writing for the moment of the portion cut off about the neutral axis SS, a quantity which can be directly calculated by summation or deduced when the position of the centre of gravity of the portion is known. Assuming the transverse sections K lt K 2 as in the preceding article at a distance x 2 - x 1} which, however, we will now suppose to be the unity, let AM be the difference of the corresponding bending moment, then St-.ff,- Air.il But referring to Art. 29, p. 55, it will be seen that if F be the shearing force &M=F(x 2 -x 1 ), and, as before, H l - H 2 is balanced by a corresponding shearing stress called into play over the horizontal base of the prismatic portion intercepted between the sections. If then S be the total shear on the base, a formula which is equally true if the base be curved, or even if the portion is wholly enclosed in the solid mass of the beam. If the mean shearing stress FjA on the transverse section be q Q and on the base be q, the formula may be written, replacing 7 by nAh' 2 , 2 -*sr- where s is the periphery of the base whether straight or curved. By the principle of Art. 183 the shear at any point of the base is also 368 STIFFNESS AND STRENGTH. [PART iv. the shear on the transverse section in a direction normal to the base. Let us now consider various cases. (1) Returning to the case of the I section, let y l be the distance of the base of the flange (Fig. 144) from the neutral axis and y the ordinate of some other point in the transverse section of the web, then s = t and n o hence by substitution q = q l + q Q . ^ l ~ ^ , a formula which gives the shear at any point of the transverse section distant y from the neutral axis in terms of q t the shear immediately below the flange. The value of q 1 can be found from the formula, /^ being a given quantity. When the web is very thin, q Q is relatively very small, and the shear on the web is approximately the same at all points ; but otherwise, in addition to the shear on the web as a whole, there is a local shear represented by the ordinates of a parabolic arc, the chord of which is the depth of the web. The extreme case is that of a rectangular section when q l = 0, h = 2y v n = y 1 ^ , then -W'-&> At the neutral axis where y = the stress which is then a maximum is 1 J times the mean. (2) Consider a tube of circular section, mean radius a, thickness of metal tf, under the action of a shear F, producing on the section a shearing stress the mean intensity of which is F In Fig. 146 draw the radii OP, OP' inclined at an angle to the vertical cutting off the arc PP. Then in the general formula given above -P, Jo = 2<7 n . sin 6. Fig. 146 This gives the resultant shearing stress at any point P, which, as explained in Art. 185, is necessarily in the direction of the tangent. The maximum value occurs at the neutral axis SS and is double the mean. (3) Taking a section of any type, consider the portion cut off by th< CH. xv. ART. 189.] SHEARING AND TORSION. 369 neutral axis SS in Fig. 145, and let SS = b be the breadth of the beam there, then the mean shearing stress on the transverse section at points lying on the neutral axis is where /* is the moment of the area SLS about the neutral axis. For example, in a rectangular section -hh - and consequently q = f g as already found. Similarly in an elliptic or circular section q = |^ , This formula is commonly accepted as giving the ratio of maximum to mean stress on the section ; but this statement must be understood with very considerable qualifications, for all that is actually determined is the mean stress at points along the neutral axis; the maximum is generally greater, and sometimes very much greater, as will be seen from our next example. (4) Take a square bar and imagine it bent and sheared by forces parallel to a diagonal. Through the centre of the section draw two straight lines ON, OM perpendicular to the sides, cutting off one-fourth of the whole area. Then if h be the side is the mean stress along ON perpendicular to ON, and along OM per- pendicular to OM. If we suppose the stress at equal to the mean in each direction it must be the resultant of two forces at right angles, each equal to q and will therefore be f and the deflection due to it The corresponding deflection (d) due to bending is found by the formula on page 324, which, on writing as usual nAh 2 for /, becomes, Wl* ~AnEAh* The ratio of these two is therefore 8 19* E h * -d = Un 'C'F*' To obtain the actual value of the ratio of the deflection due to shearing and bending, the result here found requires multiplication by a factor the value of which can be calculated approximately as explained in Chapter XVII. This factor for a rectangular section is 1'2, whence taking E/C equal to 2*5 and % = T V we ^ n( ^ 8 3A 2 a"T' which agrees with Professor Pearson's estimate of the average value of this fraction for sections of various types. Thus if the ratio of depth to span be one-tenth, the correction due to shearing is 3 per cent. In the case of a tube considered on page 368 the factor is 2 and 71 = i> giving a ratio 2i times as great: the correction when the depth is one-tenth the span being 1\ per cent. 191. Effects of Insufficient Resistance to Shearing. If the central part of a beam be cut away as shown at Z in Fig. 144, the strength of the beam will be diminished and its deflection increased. This will be true even if there be only a narrow longitudinal slot at the neutral surface, but the weakening is greater the more material is cut away, the con- dition of the beam in an extreme case becoming that of an N girder (Art. 25) without diagonal bracing. Imperfect union of the central parts will have the same effect in a less degree : thus if two beams be laid one upon another and bolted together the strength of the com- pound beam will be less than that of a solid beam of the same depth. Wooden ships not unfrequently exhibit weakness due to this cause, and to counteract it diagonal riders of iron are introduced to take part of the shearing force. 372 STIFFNESS AND STEENGTH. [PART iv. Theoretical considerations would lead us to conclude that in timber beams the deflection due to shearing is relatively much increased by the flexibility of the transverse sections, the modulus of rigidity being relatively small in most kinds of wood. This conclusion, however, does not as yet appear to have been experimentally verified. The ordinary formula for resistance to bending cannot be applied in such cases without risk of serious error, and the same remark applies with still greater force to the formula of Art. 189, which gives the distribution of shearing stress which will be determined mainly by the relative resistance to shearing of the parts of the section. 192. Economy of Material in Girders. It has been shown already in Art. 159 that a certain ratio of depth to span must be best as regards economy of material, and a calculation will now be given which will illustrate this point. Let us suppose that in order to give sufficient stiffness and stability under the action of lateral forces the mean sectional area C of the web of a flanged girder should be proportional to the shearing force on the section multiplied by the ?'th power of the depth A, and let A be the area of each flange, then the total area S is 2 A + C and the moment of resistance to bending approximately, where c is a co-efficient. Writing this equation it will be seen that for a given value of M, S is least when M-I r f w r 3S 'rTT* / - = 2(fTT)- In a girder with openwork web S= C(r+l), but the value of M is the same. Assume now F=f . C, where F is the shear on the section and /'is a co-efficient much less than the resistance to shearing, on account of various additional straining actions (Art. 188) which have to be con- sidered then by substitution, M^r-.Fh. On replacing M by pFL, where L is the span and ja a co-efficient connecting the shear and the bend, the best ratio (N) of span to depth will be determined. If the load be uniformly distributed N ~sT It is probable that in most cases r = 2 nearly, but that the value of CH. xv. ART. 193.] SHEARING AND TORSION. 373 ///' will vary according to the type of girder from 3 to 4 for a con- tinuous web. For an openwork web the formula is slightly modified. The limiting span of a girder of uniform section is readily found, proceeding as in Ex. 13, page 319, to be 4r A. ~r+l* N' The weight of a smaller girder of the same type is found as in Ch. IV. 193. Joints and Fastenings. Among the most important cases of shearing are those which occur in joints and fastenings of all kinds. Such questions are generally very complex, considered as purely theoretical problems, and the direct results of experience are always required at every step to interpret and confirm theoretical conclusions. When two pieces butt against each other the pressure is transmitted by contact only, and fastenings are therefore required not for trans- mission of stress but merely to retain the pieces in their relative positions. With tension it is otherwise ; it is still necessary to have surfaces which press against one another, and these can only be obtained by the introduction of fastenings which transmit stress laterally, and are therefore subject to shearing and bending. The parts of a joint should be so proportioned as to be of equal strength. One of the simplest examples is that of a pin joint connecting two bars in tension as in a suspension chain with bar links. Fig. 1 (Plate VIII.) shows a pair of bars of rectangular section connected together by links C and D united as shown by pins passing through eyes at their extremities. In suspension chains there are generally four or five bars placed side by side, but the principle is the same in any case. The pull on the chain is balanced by the resistance to shearing of the pins, which have besides to resist bending. Let d be the diameter of the pins, b the breadth, t the thickness of one of the bars, t' the thickness, b' the breadth of the links which for equality of strength, that is to say, of sectional area, will be connected by the equation Let / be the co-efficient of strength for tension, then |/ (Art. 230) will be the co-efficient for shearing, whence remembering that the maximum shearing stress exceeds the mean in the ratio 4 : 3 as shown above According to this estimate the area for shearing should be five-thirds the area for tension, but the true ratio is probably not so great : the calculation supposes that the sides of the pin are subject to normal stress alone, whereas the tangential stress due to friction must be considerable. 374 STIFFNESS AND STRENGTH. [PART iv. Besides the strength of iron such as is used for pins is greater than that of plates. As the calculation applies only to stress within the elastic limit, it is impossible to test it by experiment. In practice the areas are made nearly equal when nothing else is considered except resistance to shearing. When, however, such a joint is actually pulled asunder it frequently gives way in quite a different manner before shearing com- mences. Imagine a cylinder pressed down into a semicircular hollow which it very exactly fits, and let the material be elastic and soft compared with the cylinder, then, reasoning as in Art. 115, page 241, it appears that the stress between the surfaces will be given by the equation p= Po .coB6 t and if P be the pressing force, / the length, 4:P = P or p Q = ^ff If the pin fits the eye exactly the pressure will follow this law so long as the tension is small. As the tension increases, however, the pressure becomes more uniformly distributed over the semi-cylinder, because the eye-hole tends to contract laterally as the links of a chain of rings would do under tension. The other extreme supposition would be to suppose it uniformly distributed, then p p .dl = P or Po^-ji- The actual pressure will be intermediate between these two values. If PQ be too great the metal crushes under the pressure. The theoretical limit to p Q will be considered hereafter; for the present it will be sufficient to say that the experiments of Sir C. Fox * have shown that the curved area should be at least equal to the sectional area under tension, that is to say we ought to have To satisfy these conditions we must have for the ordinary case where the thickness of the eye is the same as that of the rest of the bar d = f 6 : t = |6 approximately. The first of these gives the diameter of pin recommended by Sir C. Fox and other authorities ; the second gives the greatest thickness of link for which this diameter gives sufficient resistance to shearing, but the thickness in actual examples of suspension links is generally considerably less. The pin has also to resist bending, but of small amount in the present example. The sides and end of the eye are subject to tension, but it is not uniformly distributed, the question being similar to that of a thick hollow cylinder under internal fluid pressure. The mode in * Proceedings of the Royal Society, vol. xiv. , p. 139. CH. xv. ART. 193.] . SHEARING AND TORSION. 375 which the eye crushes and then fractures transversely by tension, is shown in Plate VIII. , and further described in Chapter XVIII. In rivetted joints the question is further complicated by the friction between the plates united by the rivets. On the subject of joints and fastenings the reader is referred to Prof. W. C. Unwin's Machine Design. EXAMPLES. 1. Find the diameter of a shaft for a twisting moment of 1,000 inch-tons ; stress allowed being 3| tons per square inch. Ans. Diameter =11 '34". 2. From the result of the previous question deduce the diameter of a shaft to transmit 5,000 H.P. at 70 revolutions per minute. Maximum twisting moment = f the mean. .4ns. 16-37". 3. The angle of torsion of a shaft is not to exceed 1 for each 10 feet of length. What must be the diameter for a twisting moment of 100 inch-tons modulus of transverse elasticity, 10,500,000? Compare the result with the diameter determined from consideration of strength, taking a co-efficient of 3| tons. Ans. Diameter determined from consideration of stiff- ness = 6 '2". Diameter from consideration of strength = 5 "2". 4. Show that the resilience of a twisted shaft is proportional to its weight. . / 2 Volume Ans. Resihence=Ti=- x ----- - -- 5. Compare the strengths of a solid wrought-iron shaft and hollow-steel shaft of the same external diameter assuming the internal diameter of the hollow shaft half the external, and the co-efficient for steel 1^ times that for iron. Ans. 32/45. 6. The external diameter of a hollow shaft is double the internal. Compare its resistance to twisting with that of a solid shaft of the same weight and material. Ans. Strength is greater in the ratio = 1'443. 7. A pillar, whose sectional area is 1^ square feet, is loaded with two tons. Find in Ibs. per square inch the intensity of the tangential stress on a plane inclined at 15 to the axis of the pillar. Ans. tangential stress =5 '18 Ibs. 8. In a single rivetted lap joint, the pitch of the rivets being three diameters or six times the thickness of the plates, find, 1st, the mean stress on the reduced area; 2nd, the shearing stress on the rivets ; and, 3rd, the mean direct stress between rivet and plate : the tension of the joint being 4 tons per square inch of the original area, and the friction between the two surfaces of the plate in contact neglected. Ans. Mean tension on reduced area - =6 tons. Shearing stress on rivet - - 7 '6 tons. 4 x pitch x thickness . _ , Mean direct stress -JT-* ... . - =12 tons per sq. in. diameter x thickness 9. In a beam of I section with flanges and web which may be considered as rectangles, the thickness of each flange is one-sixth the outside depth of the beam, and the breadth twice the thickness. The thickness of the web is half that of the flanges : find the ratio of maximum to mean shearing stress on the section. Ans. -^-. 10. In the last question find the fraction of the whole shearing force which is taken by the web. Ans. 80 per cent. 11. Find the moment of resistance and angle of torsion of an iron bar 1 inch square, 5 feet long, assuming /= 3|, (7=5,000 in. tons. Ans. T='G77 inch tons. = 3. 12. Find a formula for the resilience, under torsion, per cubic inch of a bar of rect- angular section. 376 STIFFNESS AND STRENGTH. [PARTIV.] 13. Show that the weight in Ibs. of a shaft to transmit a given horse power at a given number of revolutions is the value of X being given as in Ch. XVIII., the proper co-efficient of resistance to shearing being used. The rest of the notation is explained on page 356. The distance to which power can be transmitted by shafting with a given loss by friction is given by Ex. 18, p. 262, when the angle of torsion is immaterial, but in practice is generally limited by the necessity of having sufficient stiffness. The bend- ing and twisting of shafts is considered in Chapters XVIL, XVIII. 14. If a bar of square section be sheared diagonally show that the mean shearing stress on the neutral surface is equal to the mean shearing stress on the section. Also find where the mean shearing stress on a longitudinal section parallel to the neutral surface is a maximum and the ratio of maximum to mean. Ans. At a distance from the neutral surface equal to one-eighth the depth. Ratio =1 '125. Note. If the shear on the transverse section in a direction perpendicular to the neutral axis be assumed uniform at points lying on a line parallel to the neutral axis, the maximum shear will be l'125\/2, or about 1*6 times the mean at points lying on the edges of the section. 15. A crank arm of rectangular section 6 in. x 12 in. is acted on by a twisting moment of 300 inch-tons, find the stress produced at (1) the middle of the long side and (2) the middle of the short side in tons per square inch. Ans. g 1 =2'78 ; q z = 2'25. 16. In the last question, suppose the section further to be acted on by bending moments of 100 inch-tons in the plane of the crank, and 150 inch-tons about the shorter axis, find the normal stress produced at the points mentioned. CHAPTER XVI. IMPACT AND VIBEATION. 194. Preliminary Remarks. General Equation of Impact. Hitherto the forces applied to the body or structure under consideration have been imagined to have been originally very small, and to have increased gradually to their actual amount. This is seldom exactly the case in practice, while it frequently happens that the load is applied all at once, or that it has a certain velocity at the instant it first comes in contact with the body. Such cases may all be included under the head of IMPACT, and will form the subject of the present chapter. When a body in motion comes into contact with a second body against which it strikes, a mutual action takes place between them which consists of a pair of equal and opposite forces, one acting on the striking body, the motion of which it changes, the other on the body struck, which it in general moves against some given resistance. Certain changes of figure and dimensions, or, in other words, strains are likewise produced in both bodies, in consequence of the stress applied to them. The simplest case is where the impact is direct and the resistance to motion has some definite value, as, for example, where a pile is driven by the action of a falling weight. Here let R be the resistance which the pile offers to be driven, that is to say, the load which, resting steadily on the pile, would just cause it to commence to sink ; let W be the falling weight, h the height from which it falls, x the space through which the pile sinks in consequence of the blow ; then the mutual action between the pile and the weight at the instant of impact consists of a pair of equal and opposite forces R. The whole height through which the weight falls is h + x, and the space through which the resistance is overcome is x ; hence, equating energy exerted and work done, we have This equation shows that any resistance, however great, can be over- 378 STIFFNESS AND STRENGTH. [PART iv. come by any weight, however small ; and also, that the force of the blow, as measured by the space the pile is driven, is proportional to its energy. We have however assumed that the whole energy of the blow is employed in driving the pile, whereas some of it will be expended in producing vibrations and in damaging the head of the pile and the bottom of the weight. As the pile is driven deeper, the resistance to being driven increases, and at length becomes equal to the crushing stress of the material : the pile then sinks no farther, the whole of the energy of the blow being wasted in crushing. This last is also the case of impact of a flying shot against a soft plastic substance, which exerts during deformation a definite force uniform or variable which brings the weight to rest in a certain space. Suppose V the velocity of the shot, x the space, and R the mean resistance which the substance offers, then the kinetic energy of the shot is WV^ftg, while the work done is Ex^ equating which V' 2 W.-=K*. Here the whole energy of the blow is spent in producing changes of figure in the body struck ; but if the striking body had been soft, and the body which it struck hard and immovable, the energy of the blow would have been employed in producing change in the shape of the striking body. Thus we may write down as the general equation of impact Energy of blow = Work done in overcomiug the resistance to movement of the body struck + Work done in the internal changes in the striking body + Work done in internal changes in the body struck. Which of these three terms is the most important will depend on the relative magnitude of the resistance to movement, and the crushing stress of the materials of the two bodies. If either body have a sensible motion after impact, the corresponding kinetic energy must be taken account of in writing down the equation, as will be seen farther on. 195. Augmentation of Stress by Impact in Perfectly Elastic Material. We now proceed to apply the equation to the case which most immedi- ately concerns us, namely, that of impact on perfectly elastic material, including in this the effect of a load which is applied all at once. We will suppose a structure or piece of material of any kind resting on immovable supports, and struck by a body harder than itself, so that we may neglect all changes produced in the striking CH. xvi. ART. 195.] IMPACT AND VIBRATION. 379 body. Generally in both bodies there will also be produced vibra- tions, of the nature of those constituting sound, which absorb a certain amount of energy, but this we shall neglect. The whole energy of the blow then is supposed expended in straining the structure, or piece of material, struck by the blow. Now the effect of impact is to produce a mutual action S, which represents a force applied to the structure at some definite point. In consequence of this the structure suffers deformation, and the point of application moves through a space x. The resistance to deforma- tion is proportional to x, because the limit of elasticity is not exceeded ; it therefore commences by being zero, and increases gradually till the velocity of the striking body is wholly destroyed. The mean value of the resistance is therefore one-half its maximum value. During the first part of the period occupied by the impact the mutual action S is greater than the resistance, and during the second part less, as will be explained fully presently ; but, when the maximum strain has been produced, the mean value during the whole period must be exactly equal to the mean resistance, the weight and the structure being at rest. The state of rest is only momentary, for the strained structure will immediately, in virtue of its elasticity, commence to return to its original form ; but for the moment, a strain has been produced, which is a measure of the effect of the blow, and which must not exceed the powers of endurance of the material. Let now R be the maximum resistance, and let the blow consist in the falling of a weight W^ through a height h above the point where it first comes in contact with the structure; then h + x is the whole height fallen through, and it follows from what has been said that The resistance R may also be described as the "equivalent steady load," being the load which, if gradually applied at the point of impact, would produce the same stress and strain which the struc- ture actually experiences. We most conveniently compare it with W by supposing that we know the deflection 8 which the structure would experience if the striking weight W were applied as a steady load at the point of impact ; we then have, since the limits of elasticity are not exceeded, x_R L 1*MT Substituting the value of x we get & 2R "2h 380 STIFFNESS AND STRENGTH. [PART iv. Let the height h be n times the deflection 8, then solving the quadratic, the positive root of which alone concerns us, Fig.i48 an equation which shows how the effect of a load is multiplied by impact. 196. Sudden Application of a Load. A particular case is when 7^ = 0, then E = ^W. So that if a load W is suddenly applied to a perfectly elastic body, from rest, not as a blow, it will produce a pressure just twice the weight. This case is so important that we will consider a special example in detail. Let a long elastic string be secured at A. If a gradually increasing weight be applied the string will stretch, and the weight descend. Let the load required to produce any given extension be represented by the ordinates of the sloping line B NN 2 (Fig. 148). Next, instead of applying a gradually increasing load, let a weight W represented by J5 if be applied all at once to the unstretched string. The string will of course stretch, and the weight descend. When it has 2 reached B (Fig. 146) the tension of the string pulling upwards, being represented by BN, will be less than W acting downwards. Moreover, in the descent B Q B an amount of energy has been exerted by the weight represented by the area of the rectangle B M MB. At the same time the work which has been done in stretching the string is represented by the area of the triangle B NB. The excess of energy exerted over work done has been employed in giving velocity to the descending weight, and is stored as kinetic energy in the weight. On reaching B^ the tension of the string is just equal to the weight, but the stretching does not cease here. The weight has now its greatest velocity, which corresponds to an amount of kinetic energy represented by the triangle B^M^M^ Although any further extension of the string causes the upward pull of the string to be greater than the weight W, yet the weight will go on descending until the energy that it has exerted is equal to the work done in stretching the string; then the kinetic energy will be exhausted and the weight will be brought to rest. This will occur when the area of the triangle B Q N 2 B 2 equals the area of the rectangle that is when B<>N<, = 2JB 9 M 9 , or CH. xvi. ART. 197.] IMPACT AND VIBRATION. 381 We thus see that the tension of the string produced by the sudden application of the load is twice that due to the same load steadily applied. The string will not remain extended so much as B^B^ for now the upward pull of the string, exceeding the weight, will cause it to rise. On reaching JB 1 it will have the same velocity upwards that it had on first reaching B 1 downwards. This will carry it to B , from which it will again fall, and so on. Practically the internal friction due to imperfect elasticity, and the resistance of the air, will soon absorb the energy and bring the weight to rest at B r 197. Action of a Gust of Wind on a Vessel. Another interesting example of the way in which the sudden application of a load aug- ments its effect is furnished by the case of a vessel floating upright in the water and acted on by a sudden gust of wind, a question which, though not strictly belonging to this part of the subject, involves exactly the same principle. First, suppose no wind pressure, but that a gradually increasing couple is applied to heel the vessel. If along a horizontal line (Fig. 149) angles of heel be marked off, such as ON, and for those points ordinates such as NL, are set up to represent on some convenient scale the magnitude of the couple required to produce that angle of heel, a curve OL will be obtained, which we have already (p. 184) called the curve of Statical Stability of the ship. Now suppose a steady wind pressure to be gradually applied. It will produce on the masts and sails a definite moment, on account of which the ship will incline to a certain angle, such that the ordinate of the curve of stability corresponding to that angle will represent the moment of the wind pressure. So long as the wind is constant, she will remain inclined at that angle. Next suppose the same wind pressure to be suddenly applied all at once, as by a gust to the ship floating upright at rest. The ship will heel over, and until she is inclined to some extent the wind moment will be greater than the righting moment, and the excess will cause the ship to acquire an angular velocity. Accordingly, when she arrives at the angle of heel corresponding to the moment of wind pressure on the stability curve, she does not come to rest, but inclines farther, until 382 STIFFNESS AND STRENGTH. [PART iv. the energy exerted by the wind pressure is all taken up in over- coming the righting moment through the angle of inclination. The work thus done is represented by the area of the curve of stability standing above the angle of heel reached. Let OW^ represent the magnitude of the wind moment. The ship will incline until the area OL 2 N 2 = area OW^KN V or area O^F 1 L 1 = area L^L^K that is, if the moment of wind pressure remains undiminished as the ship heels, which will hardly be true in practice. Suppose the moment of wind pressure OW^ to be such that the area 0/F OJ L = the area L L 2 L' Q . In this case the sudden gust of wind will carry the ship to such an angle ON' Q that she will not again return ; and the smallest additional pressure of wind will capsize the ship, although that same wind pressure applied gradually would incline the ship to the angle ON, only. 198. Impact at High Velocities. Effect of Inertia. Returning to the general case of impact against a perfectly elastic structure (Art. 195), let us now take the other extreme case in which the height through which the weight falls is great compared with the deflection 8 due to the same weight gradually applied ; then, since n is great, our equation becomes which may be written in either of the forms (2). The first form shows that the stress produced by the impact is pro- portional to the square root of the energy of the blow, and the second, that the deflection occasioned by the fall of a given weight is propor- tional to the square root of the fall, or, what is the same thing, to the velocity of impact. These results are exact when the impact is horizontal, and the last has been verified by experiment. It is to be remembered that the limits of elasticity are supposed not to be exceeded ; when a rail or carriage axle is tested by a falling weight, as is very commonly done, the energy of the blow is generally much in excess, and the piece of material suffers a great permanent set, the resistance is then approximately constant instead of increasing in proportion to the deflection. The effect of the blow is then more nearly directed proportional to its energy. It will be seen presently how small a blow matter is capable of sustaining without injury to its elasticity. The effect of a blow, on a structure or piece of material as a whole, CH. xvi. ART. 198.] t IMPACT AND VIBRATION. 383 is diminished, on account of its inertia, by an amount which is greater the greater the velocity of impact, but which varies according to the relative mass and stiffness of its parts. In the act of yielding the parts of the body are set in motion, and the force required to do this is frequently greater than the crushing strength of the materials, so that a part of the energy of the blow is spent in local damage near the point of impact. Figure 150 shows a narrow deep bar AB, the ends of which rest in recesses in the supports, which prevent them from moving horizontally, but do not otherwise fix them. The bar carries a weight Q in the centre, against which a second weight W moving horizontally strikes with velo- city V. The bar being very flexible horizontally, the weight Q at the first instant of impact moves as it would do if free ; that is, the two weights move onwards together with a common velocity v fixed by the consideration that the sum of the momenta of the two weights is the same before and after impact, so that Fig.150 The energy of the two weights after impact is showing that the energy of the blow has been diminished in the pro- portion W-.W+Q. The loss is due to the expenditure of energy in damage to the weights. If now, instead of a weight Q attached to the centre of a flexible bar, we suppose the bar less flexible and of weight Q, the effect of the blow is diminished by the same general cause, but not to the same extent : the diminution may be estimated by replacing Q in the preceding formula by IcQ, where k . is a fraction to be found approximately by calculation (Ex. 8, p. 399), or determined by ex- periment. In a series of elaborate experiments made by Hodgkinson on bars struck horizontally by a pendulum weight, it was found that k was ^. We are thus led to separate the energy of a blow into two parts : w* v* _ k.wq v^ 1 fT + kQ' 2g *~ IF + kQ' 1g The first of these strains the structure or piece of material as a whole, and the second does local damage at the point of impact. Hence the great difference which exists between the effect of two blows of the 384 STIFFNESS AND STRENGTH. [PART iv. same energy, one of which is delivered at a low, and the other at a high velocity. At high velocities most of the energy is expended in local damage; at low velocities most is expended in straining the structure as a whole. If the body which is struck be in motion, instead of resting on immovable supports, as in Fig. 150, the energy of the blow will be diminished. This case has been considered in Ch. XL, p. 270, where it is shown that the energy of the collision is F where V is the relative velocity of the bodies. Of this a part represented, as before, by replacing Q by kQ is spent in local damage and the rest in straining the structure as a whole. The exceptional case where, as in the collision of billiard balls, the limit of elasticity is not exceeded at the point of impact, need not be here considered. The energy of local damage is, then, not wholly dissipated in internal changes : a part is recovered during the restitu- tion of form which occurs in the second part of the process of impact, and increases the action on the structure as a whole. In ideal cases the whole may be thus recovered, but, in practice, a portion is always employed in producing local vibrations, and finally dissipated by internal friction. 199. Impact when the Limits of Elasticity are not Exceeded. Resilience. The effect of impact on perfectly elastic material may also be dealt with by considering the amount of energy stored up in the body in consequence of the deformation which each of its elementary parts have suffered. We have already seen that when a piece of material is subjected to a simple uniform longitudinal stress of intensity p, the amount of work U done by the stress is p 2 U = v x Volume. '2ij Let w be the weight of a unit of volume of the material, and W the weight of the body considered, then we may write our question U=W .H where H is a certain height given by and the whole elastic energy of the body may be measured by this height, which is the distance through which the body must fall to do an equivalent amount of work. CH. xvi. ART. 199.] IMPACT AND VIBRATION. 385 If for p we write / the elastic strength of the material, then we obtain what we have already called the Resilience of the body, and H becomes what we may call the "height due to the resilience," which, for each material, has a certain definite value, given in feet in Table II., Ch. XVIII., for various common materials. Now in cases of impact where the limit of elasticity is not exceeded, the whole energy of the blow is spent in straining the material or structure, and hence that energy must not, in any case, exceed the resilience. Thus, on reference to the table, it will be seen that in ordinary wrought iron the height is given as 2 ft. 9 in., from whence it follows that in the most favourable case a piece of iron will not stand a blow of energy greater than that of its own weight falling through about 3 feet, without being strained beyond the elastic limit. If the parts of the body are subject to torsion, about 50 per cent, may be added to these numbers, but, on the other hand, they are subject to large deductions on account of the inequality of distribution of stress within the body. Only a portion of the body is subjected to the maximum stress, the rest is strained to a less degree, and consequently has absorbed a less amount of the energy of the blow. Thus, for example, a beam of circular section, even though it be of "uniform strength" (Art. 161), has only one-fourth the resilience of a stretched bar of the same weight, because it is only the particles on the upper and lower surfaces which are exposed to maximum stress, the central parts having their strength only partially developed. We now draw two very general and important conclusions. (1) When a body or structure is exposed to a blow exceeding that represented by its own weight falling through a very moderate height, a part, or the whole, is strained beyond the elastic limit. (2) When a body or structure is not of uniform strength through- out the excess of material is a cause of weakness. On reference to Table II., Ch. XVIIL, it will be seen that an exception occurs to the first principle in the case of the hardest and strongest steel; but, as a rule, the property of ductility or plasticity is essential to resistance to impact. Bodies which do not possess it are generally brittle. In good ductile iron and soft steel the non-elastic part of the resistance to impact will be seen here- after to be at least 1,000 times the elastic part, assuming both equally developed through all parts of the material. These remarks apply to a single blow ; the effect of repetition will be considered hereafter. As an example of the application of the second principle we may mention the bolts for armour plates invented by the late Sir W. Palliser. In these bolts the shank is turned down to the diameter C.M. 2 B 386 STIFFNESS AND STEENGTH. [PART iv. of the base of the thread so as to be of equal strength throughout. (See Ex. 4, p. 305.) 200. Free Vibrations of an Elastic Structure. If a structure be loaded within the limit of elasticity and the load be suddenly re- moved, the elastic forces being unbalanced set the structure in motion and vibration ensues. The vibrations are described as "free" being uninfluenced by any external cause and take place in times which depend only on the inertia of the structure and the intensity of the elastic forces, while their extent is arbitrary being fixed by the magni- tude of the original deformation. In the absence of friction the total energy of the structure must remain constant, a principle expressed by the equation Kinetic Energy + Elastic Energy = Constant. The effect of friction is gradually to dissipate the energy so that the vibrations speedily die out unless kept up by external forces. This action, however, is for the present neglected. The simplest kind of vibration is that in which the deformation is of such a character that the elastic energy can be expressed in terms of a single varying quantity which may be either linear as in the deflection of a beam (page 332), or angular as in torsion (page 361). In all such cases, as will be seen on reference to the pages cited, the elastic energy is cz 2 where z is the varying quantity and c a constant co-efficient. Also the different points of the structure have velocities which are in a fixed proportion to each other, and also to dz/dt the rate of change of the varying quantity in question. This rate of change may be described as the velocity of vibration and denoted by V. The kinetic energy will therefore be bV^ where b is a second constant co-efficient, and the equation of energy becomes Wi + cs i = Constant. This kind of motion has already been studied in Art. 103, Chapter VIII., and on reference to page 204 it will be seen that the period of vibration (T Q ) is given by a rule which includes both the simple examples there considered and applies to all cases. Whatever kind of vibration is dealt with the process of determining TQ is very similar, and the first example to be considered is that of the vibration of a loaded bar. (1) In Fig. 151 a long flexible elastic bar is shown, to the middle CH. xvi. ART. 200.] IMPACT AND VIBRATION. 387 and ends of which weights are attached ; the fraction 1 - /? of the whole weight W being placed in the middle, and the fraction J/3 at each end. The bar is slightly bent into an elastic curve in a horizontal plane and then left to itself, being supposed resting on a smooth table or suspended by vertical strings from two points NN called " nodes " which remain at rest during the motion, lying as they do in a line passing through the centre of gravity of the weights. The weight of the bar itself is supposed small enough to be neglected. The bar at any instant will be bent into a curve which is the same as the deflection curve of a beam supported at the ends and loaded in the middle; hence if z be the versed sine of that curve Elastic Energy - 24 . z^ (p. 332). The weights /3W, (\ - ft)W are at distances (I - P)s and fa re- spectively from NN, and their velocities are therefore (1 - p) V and /3 V respectively, where For dz/dt is the relative velocity. Hence Kinetic P^nergy = - F" 2 , = WPQ-P)?*. (See also p. 270.) The equation of energy is therefore jri j- ' 2 = Constant, whence applying the general rule given above, The period is obviously unaffected by placing additional weight at the nodes for these points are at rest. Suppose then that of the total load W the fraction (\-a}W is placed at the nodes and the remainder aW distributed as before, we shall then have T -" It is often more convenient to consider the "frequency," that is, the number (N ) of vibrations per second. The vibrations considered 388 STIFFNESS AND STRENGTH. [PART iv. -are complete including both a forward and a backward movement; thus in a reciprocating piece driven by a crank the frequency is the number of revolutions per second of the crank. On this under- standing 1 where K is a numerical coefficient depending on the distribution of the load. The smallest value K can have is 12-5, which occurs when a=l, /3 = J, but if half the load be concentrated at the nodes the result is increased more than 40 per cent., becoming 17-7. If the moment of inertia I x vary with x the distance from one end of the section considered, then / must be understood to refer to the middle section and K must be divided by a numerical factor JT, the square of which is given by the equation {a = ^l}, derived from the formula for the elastic energy given on page 331, in terms of the bending moment M which in this case varies as x. If I x increases on going from the ends to the middle, this divisor is greater than unity and the value of K is diminished ; if, for example, I x oc x the divisor is -Jl-6 or 1-21. In Fig. 152 the bar is loaded at the ends and two intermediate points : the two halves are then bent in opposite directions and the bar left to itself. There are now three nodes NNN instead of two W Fig. 152 W as in the preceding case. The time of vibration may be investigated as before, when it will be found that the same formula applies but with a greater value of K. Similarly there may be bending vibrations with four or more nodes, the vibration being quicker the greater the number. If the weight be distributed continuously instead of being concen- trated in given points, the formula for the frequency is still of the same form, but the calculation of K is more difficult. The case of a uniform bar has been thoroughly studied by writers on Acoustics, and in Lord Rayleigh's Treatise on Sound full details will be found. With no other load than its own weight the value of K is about CH. xvi. ART. 200.] IMPACT AND VIBRATION. 20'2 for two nodes, instead of 17 '1 as found above in the case of a concentrated load, and for a greater number of nodes i about 2-225 {2t-l}2. If the transverse section of the beam be of sensible magnitude a correction for "rotatory insertia" is necessary since these sections have a motion of rotation as the beam bends and unbends. The value of K is evidently diminished by this cause and in the case of a vessel or other larger girder-like structure the correction would be considerable. The case of a vessel has of late attracted considerable attention and the value of the constant C in the formula, N= has been determined experimentally by Herr Otto Schlick* who gives for Torpedo Boat Destroyers, (7=157,000. Large Mail Steamers, (7=143,500. Merchant Steamers, (7=128,000. These values are for complete vibrations per minute with two nodes, hence assuming E= 10,000 tons (see Art. 224) the corresponding values of K are about 26, 24, and 21 respectively. They increase with the fineness of lines of the vessel and, as might be expected, differ from the value (20'2) for a uniform rod, partly from the causes already pointed out, and partly from the influence of the water in which the vessel floats which virtually increases her inertia. Some further remarks on these points will be found in the Appendix. (2) Let us next consider a weight W resting on an elastic plat- form or support of any kind, and let 8 be the deflection which may be calculated by methods explained in previous chapters of this work, or if convenient may be found by observation. Let the weight and inertia of the structure itself be neglected for the present, then the equation of vibration will be as before + cz 2 = Constant. Now cz 2 is the elastic energy of the structure, and therefore putting Thus c is determined in terms of 8, and by the general rule already employed (p. 386), * Transactions of the Institution of Naval Architects, vol, xxxv., 1894. 390 STIFFNESS AND STRENGTH. [PART iv. showing that the period is the same as that of a simple pendulum (p. 204) of length 8 reckoned in feet : a rule of very general application being true for example for a beam with ends either fixed or free, loaded with a weight placed at any point ; or for a weight suspended by a spiral spring. The formula differs only in form from that previously obtained on page 387 for the case of a loaded bar, and may be deduced from it by putting a=l and replacing /3(l - f$)W by W, to represent the case where the ends are fixed by concentrating a very heavy load there. (3) In similar structures the deflection due to a given load similarly placed varies inversely as the linear dimensions, and therefore the period of free vibration varies inversely and the frequency directly as the square root of the dimensions. An analogous rule applies to vessels, for in similar vessels El/ W varies as the length, and therefore Elf Wl z varies inversely as the square of the length. Hence in similar vessels the frequency of free vibration varies inversely as the length. 201. Forced Vibrations. If a structure be subject to a load of in- variable amount the only vibrations which can occur are of the kind described in the last article as "free," and the periods are perfectly definite. But if the magnitude of the load be subject to a periodic change the deformations of the structure will also change, corresponding vibrations being set up which may be described as "forced." The period of such forced vibrations is that of the load, while their extent depends on the relation which that period bears to the period of free vibration. When the varying load has acted upon the structure for any considerable time these forced vibrations alone exist and in any case admit of being separately studied. Fig. 153 shows a long flexible bar loosely fixed at the ends carrying a weight W concentrated in the middle. The weight is vibrating under the action of a force which goes through a periodic change being always proportional to the distance z of the vibrating weight from the central line. The maximum value of z being supposed z lt and that of the force Q the actual force in any position will be Qz/z lf We proceed to calculate z l supposing that we know the CH. xvi. ART. 201.] IMPACT AND VIBRATION. 391 period of free vibration (T ) and the period of the varying load (T). It will be seen presently that if TT . Taking the first case /A Q ~K* T /") "\ 0>/ -- h k v ' 1 J of which the first term is the elastic energy of the bent bar and the second represents energy derived from the force Q. If now V be the velocity of the vibrating weight the equation of energy will be TX7J71 1 /A'QJPT -}z* = Constant. If Q were zero we should obtain the equation of free vibration from which it only differs in the co-efficient of z 2 . The general rule {p. 386) already stated therefore gives ifi 7*- 48AV T ^' the quantity Z Q in the right hand equation being the deflection due to a steady load Q. Hence the extent of the vibration is determined by z-, = If T>T Q z l becomes negative, the interpretation of which is that Q must then be taken as acting outwards instead of inwards. It then operates as a resistance to vibration, and thus by diminishing the intensity of the forces restoring equilibrium increases the period. If T=T the periods are said to be "synchronous." The effect of synchronism is that a force Q, however small, produces vibrations of indefinite extent : energy accumulating at each repetition of the force. This result is limited in practice by the effect of friction which absorbs the energy as fast as it is supplied. On reference to Art. 103 it will be seen that the value of z is z l cos 6 where 6 is the angle made with the central line by a uni- formly rotating radius. Or if t be the time reckoned for convenience from the instant when = 90, z = z^. sin 27T^ ; S = Q . sin 2?r ~, where S is the force needful to keep up the vibration. The effect of friction is to diminish the extent of vibration and to cause it to lag behind the variation of the force. 392 STIFFNESS AND STRENGTH. [PART iv. The results of this article are applicable directly to any case in which the inertia of the structure can be regarded as concentrated in a point to which the varying force is applied. When the inertia is distributed the value of will be reduced, but the general char- acter of the results remains the same ; there is always a certain critical speed or speeds at which by synchronism with some particular mode of vibration of the structure, excessive vibration is produced by a load which, if steady, would have no sensible effect. Any approach to these speeds must of course so far as possible be avoided. If the load be originally resting quietly on the structure and then begin to fluctuate, the resulting vibration will in the first instance be a combination of the forced vibration of period T with a free vibration of period T Q ; which will be represented by the equation f t T n t } Z=2A sin 27r^ - -^ . sin 27r^- V, the extent of the free vibration being determined by the consideration that dz/dt is zero when t is zero. When the periods are commensurable this represents vibrations of varying amplitude recurring in regular phases ; but as before stated the free vibrations will generally be speedily extinguished by the effect of friction. 202. Examples of Fluctuating Loads. Let us now consider some examples. (1) The reciprocating parts of machines, especially steam engines, give rise to periodic forces the magnitude of which has already been investigated in Art. 144, p. 283, the period being a revolution. If N be the revolutions per second, it appears from the formulte there given that the maximum value Q of the force arising from a recipro- cating piece of weight W and stroke 2a is To fix our ideas imagine the engine to stand upon a horizontal platform, the time of free vibration of which is T corresponding to a frequency JV . Then when the engine is working the extent of the vibration will be - in which the quantity z will be the deflection produced by Q when resting quietly upon the platform if the weight of platform and engine can be regarded as concentrated below the cylinder. The vibration becomes excessive when N approaches N Q . CH. xvi. ART. 202.] IMPACT AND VIBRATION. 393: In the case of a vessel the revolutions (N) of the engines in similar vessels at corresponding speeds vary inversely as the square roots of the lengths, while the frequency of free vibration as already pointed out varies inversely as the length, hence the ratio N/N varies as the square root of the length. In small vessels the revolutions are not generally sufficient to produce vibration of sensible amount, but in large vessels vibration with two, three, and sometimes with four nodes occurs. The same is the case in torpedo boats on account of their excessive speed. According to Herr Schlick the revolutions must not approach the frequency of free vibration within 10 or 12 per cent. Vibration due to this cause may in great measure be avoided by a proper system of balancing as has been explained in the article already cited. The example there considered is that of a loco- motive in which the necessity for balancing arose at a very early stage, and its principles, therefore, have been long understood. For an approximately perfect balance, as there pointed out, the alternating couples must be considered as well as the alternating forces. In vessels the extent to which vibration is due to the reciprocating parts of the engines has only recently been recognized, and the subject of balancing has acquired great importance. (2) When a vessel rolls in still water her period of unresisted rolling (page 207) is mg where m is the metacentric height, and r the radius of gyration.. Suppose now a weight Q, say of a number of men, be moved from the centre line to one side of a vessel at rest : the corresponding angle of heel (< ) will be given by the equation T) Q = Wm tan = Wm < , nearly, 2i where B is the beam and W the displacement. Let now the men move backwards and forwards from port to starboard and back again, the period of the double movement being T. The result will be forced rolling of period T and extent < a (suppose) which in the first instance will be accompanied by free rolling in period T Q . The free rolling, however, may be supposed to have been extinguished by hydraulic resistance. Assuming this, at any angle of heel there will be a couple due to the men given by "earl. 394 STIFFNESS AND STRENGTH. [PART iv. If T>T the men will always be moving outward when that side of the vessel is below the horizontal, and the moment due to them diminish the righting couple so as to lengthen the natural period. If T 9 ) sin . cos 6. The resultant stress may be found in direction and magnitude by Fig. 154 compounding these results, but it is better to proceed by a graphical construction. On the perpendicular set off OQ to represent p v and Oq to represent p. 2 ; also draw the ordinates QM parallel to p 2 , and qP parallel to p l to meet in P. Then PM=Oq.sine=p 2 7^7 'AC' Whence it follows that the intensity of the stress on AC due to p l is represented by OM, and that due to p 2 by PM. If then we join OP we shall obtain the resultant stress on AC in direction and magnitude. It is easily seen that P lies on an ellipse of which p v p 2 are the semi-axes. This ellipse is called the Ellipse of Stress. If the pair of stresses p v p 2 have opposite signs, then Oq'=p 2 must be set off on the opposite side of 0, and OP' the radius vector of the ellipse lies on the other side of OM, but in other respects the construction is unaltered. When p v p 2 , are equal the ellipse C.M. 2c 402 STIFFNESS AND STEENGTH. [PART iv; becomes a circle ; if they have the same sign the stress is the same in all directions in magnitude and direction like fluid pressure ; if they have opposite signs, as in the chapter on^Torsion, the intensity is the same, but the angle of inclination P'OQ, called the "obliquity" of stress, is variable, being always twice QOM. 205. Principal Stresses. Axes of Stress. We now propose to show that any state of stress in two dimensions (Art. 208) may always be reduced to a pair of simple stresses such as we have just considered. For, drawing the same figure as in the last article, let us inquire the effect of replacing p lt p 2 by other stresses of any magnitude and in any directions. Whatever they be, they evidently must have given tangential and normal components, of which, reasoning as in a former chapter, we know that the tangential must be equal and of opposite tendency. Let the equal tangential components be p t and the normal com- ponents p n and p' n . Consider the Fig.155 equilibrium of the triangular portion ABC (Fig. 155), and let us determine under what conditions it is possible that the stress on AC should be a normal stress only, without any tan- gential component. Resolve parallel to BC ; then, if p be that normal stress,. p . AC . cos =p t . BC+p n . AB or p-p n =Pf tan # Similarly, resolving parallel to AB, whence, subtracting one equation from the other, p n - p' n = p t . (cot 6 ~ tan 6) = 2p t . cot 26 ; or tan 20= . Pt , . This equation always gives two values of 6 at right angles, showing that two planes at right angles can always be found on which the stress is wholly normal. The magnitude of the stress on these planes is found by multiplying the equations together, when we get the " the roots of which, p v p 2 , are the stresses required. Having deter- mined p v p 2 , the ellipse of stress can now be constructed by the method of the last article. CH. xvii. ART. 206.] STRESS, STRAIN, AND ELASTICITY. 403 Every state of stress in two dimensions then can always be repre- sented by an ellipse, the semi-axes of which are called Principal Stresses, and their directions the Axes of Stress. The particular case in which p' n is zero is one of constant occurrence in practical applications. If q be the shearing stress, the equations may then be written Pn t a u20='2q (1); p(p-p n } = f (2). Of the roots of the quadratic the greater has the same sign as that of p n , and the other the opposite. Also, we find by dividing the two equations for p by one another, from which it appears that of the two values of furnished by (1) the one less than 45 must correspond to the greater value of p. Hence the major principal stress is of the same kind as p M and in- clined to it at an angle less than 45. 206. Varying Stress. Lines of Stress. Bending and Twisting of a Shaft. In proving the two very important propositions just given we have assumed (1) that the stress was uniform, throughout the region including the portion of matter we have been considering ; (2) that gravity or any other force acting not on the bounding surface, but on each particle of the interior, may be neglected. It is however to be observed that by taking the portion of matter small enough, both these suppositions may be made, in general, as nearly true as we please : the first, because any change of stress must be continuous, and therefore becomes smaller the less the distance between the points we consider; the second, because any internal force is proportional to the volume, while any force on the boundary of a piece of material is proportional to the surface of the piece. Now the volume of a body varies as the cube, and the surface as the square of its linear dimensions, and it follows that the internal force vanishes in comparison with the stress on the boundary when the dimensions diminish indefinitely. Hence these propositions are still true as respects the state of stress at any given point of a body, even though the stress be variable, and notwithstanding the action of gravity. When, however, we consider the variation of stress from point to point, gravity must be considered. Thus, for example, in the case of a fluid the action of gravity does not prevent the pressure from being the same in all directions, but it does cause the pressure to vary from point to point. When the stress varies from point to point, both the intensity and 4/04 STIFFNESS AND STRENGTH. [PART iv. the direction may vary ; thus, for example, in a twisted shaft the intensity of the stress at any point varies as the distance from the axis, and the direction of the stress varies according to the position of the point, the principal stresses making an angle of 45 with the axis of the cylinder. The axes of stress in this case always touch certain lines which give, at each point they pass through, the direc- tion of the stress at that point. These lines are called Lines of Stress ; in a simple distorting stress, or, in other cases where the principal stresses are of opposite signs, one is a Line of Thrust, the other a Line of Tension. In a twisted shaft of elastic material the lines of stress are spirals traced on a cylinder passing through the point considered, the spirals being inclined at 45 to the axis. If the shaft be bent as well as twisted, the maximum normal stress at any point of the transverse section is given by the equation p nSS - (Art. 155), where M is the bending moment and r the radius. The shearing stress at the external surface due to a twisting moment T is given by q = ^ s (Art. 184). Combining these two together we get, by solving the quadratic for the principal stresses, /- M >/j which gives the principal stresses at that point of the shaft where the stress is greatest. The maximum stress is the same as would be given by a simple twisting moment equal to M + (?-*)-!*; whence p = 5-77, or -2-77, a result which shows that the web is much more severely strained than the flanges. The lines of stress are found from the equation for 0. The direct effect of any load resting on the upper flange must be provided for separately by vertical stiffening pieces. 208. Remarks on Stress in General. We have hitherto been con- sidering only the stress on planes at right angles to a certain primary plane, to which we have supposed the stress on every plane to be parallel. In most practical questions relating to strength of materials this is sufficient, since, though stress frequently exists on the primary plane, it is usually normal and of relatively small intensity. Thus, for example, in a steam boiler there is stress on the internal and external surface of the boiler due to the pressure of the steam and the atmosphere ; but it is of small amount compared to the stress on planes perpendicular to the surface. We therefore content ourselves with a statement without demonstration of corresponding propositions in three dimensions. (1) Any state of stress at a point within a solid may always be reduced to three simple stresses on planes at right angles. (2) The resultant stress on any plane due to the action of three simple stresses at right angles to each other is always represented in direction and magnitude by the radius vector of an ellipsoid. The first of these propositions may be regarded as the last step in a process of analysis, by which we reduce all external forces acting on a structure of any kind : first, into a set of forces acting on each 406 STIFFNESS AND STRENGTH. [PART iv. piece of the structure; and second, into forces acting on each of the small elements of which we may imagine that piece composed ; and lastly, into three forces at right angles acting upon the element, of which one in practical cases is usually small. All questions in Strength of Materials, then, ultimately resolve themselves into a consideration of the effects of forces so applied. One method of conceiving the effect of three such forces is to imagine each separated into two parts, one of which is the same for all, being the mean value of the three ; while the other is compressive for one and tensile for the two others, or vice versa. In isotropic matter (Art. 210) the first set produces change of volume only, and may be called the "volume-stress," or, as no other stress can exist in fluid bodies at rest, a " fluid " stress. The second is a distorting stress, consisting of three simple distorting stresses tending to pro- duce distortion in the three principal planes. EXAMPLES. 1. A tube, 12 inches mean diameter and \ inch thick, is acted on by a thrust of 20 tons and a twisting moment of 25 foot-tons. Find the principal stresses and lines of stress. Taking a small rectangular piece with one side in the transverse section, we find one face acted on by a normal stress of 1'06 tons per square inch due to the thrust, and a tangential stress of 2 '66 tons due to the twisting. Substituting these values forp, t , pt, and observing that the stress on the other face is wholly tangential, we find from the quadratic Major principal stress = 3 '24 (thrust) ; Minor principal stress =2 '18 (tension). Lines of stress are spirals, the lines of tension inclined at 50^ to the axis, and the lines of thrust at 39|. 2. A rivet is under the action of a shearing stress of 4 tons per square inch, and a tensile stress, due to the contraction of the rivet in its hole, of 3 tons per square inch. Find the principal stresses. Ans. Major principal stress=5'8 tons (tension) ; Minor principal stress=2'77 tons (thrust). 3. The thrust of a screw is 20 tons ; the shaft is subject to a twisting moment of 100 foot-tons, and, in addition, to a bending moment of 25 foot-tons, due to the weight of the shaft and its inertia when the vessel pitches. Find the maximum stress and compare it with what it would have been if the twisting moment had acted alone. Shaft 14 inches diameter. Ans. Major principal stress=2'9, Ratio=l*32; Minor principal stress =1*6. 4. A half-inch bolt, of dimensions given in Ex. 6, p. 261, is screwed up to a tension of 1 ton per square inch of the gross sectional area. Assuming a co-efficient of friction of '16, find the true maximum stress on the bolt while being screwed up. Ans. Principal stresses = 2 and *35 tons. 5. It has been proposed to construct cylindrical boilers with searns placed diagonally instead of longitudinally and transversely. What is the object of this arrangement, and what is the theoretical gain of strength ? Ans. Increase of strength = 26^ per cent. 6. A thick hollow cylinder is under the action of tangential stress applied uniformly all over its internal surface in directions perpendicular to its axis, the cylinder being en. xvii. ART. 209.] STEESS, STRAIN, AND ELASTICITY. 407 prevented from turning by a similar stress, applied at the external surface. Find the principal stresses and lines of stress. Ans. The principal stresses are equal and opposite, forming a simple distorting stress, of intensity varying inversely as the square of the distance from the centre. Lines of stress equiangular spirals of angle 45. 7. In Ex. 9, page 375, suppose the beam so loaded that the maximum stress due to bending is 3 tons per square inch, and the total shearing force divided by the sectional area of the web 4 tons per square inch : find the principal stresses at points immediately below the flanges. Ans. Principal stresses 4 and 1*9 tons per square inch. 8. In any state of stress at a point in a body show that the sum of the normal stresses -on three planes at right angles is the same however the planes be drawn. SECTION II. STRAIN. 209. Simple Longitudinal Strain. Two Strains at Eight Angles. We now go on to consider the changes of form and size which are produced by the action of stress. Such changes, it has already been said, are called Strains. In uniform strain every set of particles lying in a straight line must still lie in a straight line, and two lines originally parallel must still be Fig.156 parallel. The lengths of all parallel lines are altered in a given ratio l+e : 1, where e is a quantity, in practical cases very small, which measures the strain in the direction of the line considered. Two sets of parallel lines, however, will not in general remain at the same inclination to each other, nor will their length alter in the same ratio. Thus the sides of a cube remain plane, and opposite sides are parallel, but the parallelepiped is not generally rectangular, and its sides are not equal. The simplest kind of strain is a simple longitudinal strain in which all lines parallel to a fixed plane in the body are unaltered in length, while all lines perpendicular to that plane remain so: that is to say a simple change of length, the breadth and thickness remaining unaltered. Fig. 156 shows an extensible band OBCD, in which OB is fixed, while CD moves to C'D', the breadth being in the first instance un- altered, and the length altered so that CC' = e l .BC. 408 STIFFNESS AND STRENGTH. [PART iv. If any line AEF be traced in the band parallel to BC, the points EF will shift to E'F' positions in the same line, such that for since the strain is uniform the change of length of all parts of the band is the same. If, however, we draw a line QL inclined at an angle to BC, that line will shift to Q'L', a position such that QL has not increased in so great a ratio, and is not inclined to BC at the same angle as before. We are about to determine the actual change of length and angular position of QL by finding that of a parallel AP drawn through A. It has been already remarked that parallel lines in uniform strain must suffer the same strain. Now AP shifts to AP' such that If now the angle PAP' ( = i) be so small that i z may be neglected compared with i, and i compared with unity, and therefore AP' - AP = PP'. cos 6 = e 1 .AP. Thus the strain (e) in the direction of AP is e = e^. cos 2 0. Also, it is clear that P7 PP' i = t^L = ^ . gin 9 = ^ . sin . cos B. A By these formulae the changes of length and angular positions of all line& in the band are determined. Next draw a line AQ perpendicular and equal to AP, and let AQ' be the position into which it moves in consequence of the strain ; we find for are very small, the effect of e lt e 2 taken together must be the sum of those due to each taken separately ; then we find for the change of length and position of any line AP, e = e l . cos' 2 + 2 . sin 2 ; i = (e l - g.,)sin . cos 6, results which may be applied as before to show the changes of dimension and the distortion of a square traced anywhere in the band. We have here regarded the angle i as a measure of the distortion a square suffers in consequence of the strain. If, however, we drop Q'M perpendicular to AF, we have . Now AM is the space through which the line L'Q' has shifted parallel to itself in consequence of the strain, and we see therefore that the angle i also gives a measure of the magnitude of this shifting. By some writers this is called "sliding." It is also called "shearing strain." If we compare the equations we have just obtained for strain with those previously obtained in Art. 204 for stress, we find them identi- cal ; and hence it appears that, so long at least as the strains are very small, all propositions respecting stress must also be true, mutatis mutandis, with respect to strain. Thus, for example, a simple distor tion must be equivalent to a longitudinal extension accompanied by an equal longitudinal contraction; and, again, every state of strain can be reduced to three simple longitudinal strains at right angles to each other, and represented by an ellipsoid of strain. The simple strains are called Principal Strains, and their directions Axes of Strain. Strain, like stress, generally varies from point to point of the body : but the relations here proved still hold good at each point, and we have Lines of Strain just as we previously had Lines of Stress. SECTION IH.-*-CONNECTION BETWEEN STRESS AND STRAIN. 210. Equations connecting Stress and Strain in Isotropic Matter. So far we have merely been stating certain conditions which stress must satisfy in order that each element of a body may be in equilibrium, and certain other conditions which strain must satisfy if the body is continuous. We now connect the two by considering the way in which stress produces strain, which differs according to the nature of the material. We first consider perfectly elastic material (see Art. 147), and suppose that material to have the same elastic properties in all 410 STIFFNESS AND STRENGTH. [PART iv. directions, in which case it is said to be isotropic. Metallic bodies are often not isotropic. as will be seen hereafter (Ch. XVIII. ). Suppose a rectangular bar under the action of a simple longitudinal .stress p v then there results (Art. 148) a longitudinal strain ^ given where E is the corresponding modulus of elasticity. Accompanying the longitudinal extension we find a contraction of breadth that is a lateral strain of opposite sign, of magnitude l/m tb the longitudinal strain where m is a co-efficient. The contraction in thickness will be qual, because the material is supposed isotropic. Hence the effect of the simple longitudinal stress p l is to produce three simple longitudinal strains at right angles, _ . _. _ * l ~E' mE' 3 ~ mE' Next remove p lt and in its place suppose a simple stress p 2 applied in the direction of the breadth of the bar ; we have by similar reasoning the three strains, Ps Pz P* e, = - -% ; e 9 =* ; e*= - -~. mE ' E ' mE And similarly removing p 2 and replacing it by p 3 acting in the direction of the thickness, iJSL. e - -A.. e -P* &l ~ mE' mE' e *~ E' These three sets of equations give the strains due to p lt p 2 , p 3 , each acting alone ; and we now conclude that if all three act together we must necessarily have with two other symmetrical equations. Hence it appears that the effect of three principal stresses, and consequently of any state of stress whatever on isotropic matter, is to produce a strain, the axes of which coincide with the axes of stress, and in which the principal strains are connected with the principal stresses by the equations just written down.* The product Ee l is the simple tensile stress which would produce the strain e v a quantity to which special importance is attached when e^ is the greatest of the three principal strains and in consequence the maximum elongation in any direction at the point considered. * The form in which these equations are given is that employed by Grashof. For practical application it is more convenient than any other. CH. xvn. ART. 211.] STRESS, STRAIN, AND ELASTICITY. 411 The value of Ee l in this case is frequently described as the " equivalent simple tensile stress." 211. Elasticity of Form and Volume. The value of the constant m may be found directly by experiment, though with some difficulty, on account of the smallness of the lateral contraction which it measures ; but it may also be found indirectly, by connecting it with the co-efficient employed in a former chapter to measure the elasticity of torsion. For if we subtract the second of the three equations just obtained from the first, we get m -= Now referring to Art. 204, we find Pt= (ft - J 2 >pm , cos 0, 2i = 2(e l - e 2 ) sin . cos 0, where p t is the tangential stress on a pair of planes inclined at angle 6 to the axes, and 2i is the distortion of a square inclined at that angle to the axes of strain. Since now the axes of strain coincide with the axes of stress, we must have Pl = ^iZA_ = I ^ .E 2t 2( 1 - 2 ) 2m + l an equation which, compared with Art. 183, shows that the co-efficient of rigidity C must be Experiment shows that in metallic bodies C is generally about f #, whence it follows that m lies between 3 and 4. In the ordinary materials of construction the comparison cannot, however, be made with exactness, because such bodies are rarely exactly isotropic and homogeneous. The Talue of m for iron is supposed to be about 3 J. Again, if we add together the three fundamental equations, we find Now the volume of a cube, the side of which is unity, becomes when strained (1 + 6-^(1 + e 2 )(l +e z ), and therefore the volume strain is i + e 2 + e z wnen t Qe strains are very small. Hence, if we separate the stress into a fluid stress N and a distorting stress (Art. 209), we have N= =-T n .-Ex Volume Strain, o(m 2) 412 STIFFNESS AND STRENGTH. [PART iv. and the co-efficient m measures the elasticity of volume. The two constants C and D, which measure elasticity of distinctly different kinds, may be regarded as the fundamental elastic constants of an isotropic body. The ordinary Young's modulus E involves both kinds of elasticity. 212. Modulus of Elasticity under various circumstances. Elasticity of Flexion. When the sides of a bar are free the ratio of the longitudinal stress on the longitudinal strain is the ordinary modulus of elasticity E;. but the equations above given show that, when the sides of the bar are subject to stress, the modulus will have a different value. For example, let the bar be forcibly prevented from contracting, either in breadth or thickness, by the application of a suitable lateral tension, p 2 ( =p s ), then e, e are both zero, and m m whence we obtain for the magnitude of the necessary lateral stress and for the corresponding extension of the bar m 2 - m - 2 Ee, = -- Pi- rn 2 -m ^ 1 Hence the modulus of elasticity is now ^m(m-l) ' This constant A is what Rankine called the direct elasticity 7 of the- substance : it is of course always greater than E. For m = 4, A = \E : form = 3, A=\E. If the bar be free to contract in thickness, but not in breadth, we have p 3 and e 2 zero, and the equations become 7-r m' 2 - 1 whence we find &e l =p l . - ^ > so that the value of the modulus of elasticity is m2 F W- 1 K In a similar way if p 2 , p 3 have any given values the modulus can be found. >R=mR CH. xvii. ART. 212.] STRESS, STRAIN, AND ELASTICITY. 413 It will now be convenient to examine an important point already referred to in the theory of simple bending, that is to say the assumption (Art. 153) that the modulus of elasticity E was the same as in the case of simple tension, > notwithstanding the lateral connection of the ele- mentary bars, into which we imagined the whole beam split up. If these elementary bars were pre- vented from contracting freely, as they would do if separated from each other, the modulus could not be the same. In fact, however, there is nothing in their lateral connection which prevents them from doing so. Figure 157 shows, on a very exaggerated scale, the form assumed by a transverse section AGED originally rectangular, cutting a series of longitudinal sections orignally parallel to the plane of bending in the straight lines shown. Assuming the upper side stretched as in Fig. 122, page 306^ these lines all radiate from a centre 0' above the s beam, which bends transversely, while the originally straight horizontal layers are cut in arcs of circle struck from the same centre. The upper side of Fig.is? the beam contracts and the lower side expands, and reasoning exactly in the same way as in Art. 153 when we derived the formula for the longitudinal curvature, we find a corresponding formula for the trans- verse curvature, p = m, whence it follows immediately that R = mR. In order that this transverse curvature of the originally horizontal layers shall not be inconsistent with the reasoning by which the formula for bending is obtained, all that is necessary is that the deviation from a straight line shall be small as compared with the distance of the layer from the neutral axis. Let u be that deviation, then (see Art. 163) if b be the breadth and h the depth, _ ~ ~ Now the stress being within the elastic limit p/E is very small, for example take the case of wrought iron, for which p/E is not more than T ^Vo th > an d suppose wi = 4, 30,400. ft ~~ 19,200V 414 STIFFNESS AND STRENGTH. [PART iv. It is obvious that u must be always very small compared with y,. except very near the neutral axis, unless b be very large compared with h, and we conclude therefore that when a beam is bent within the limit of elasticity, the lateral connection of the parts cannot have any sensible influence on its resistance to bending, unless its breadth be great. Experience shows, however, that a broad thin plate remains straight in the direction of the breadth when bent longitudinally, and cannot therefore be supposed free to expand or contract laterally except near the edges. In this case, then, there must be a normal stress (p') at every point of a longitudinal section parallel to the plane of bending and this stress must be proportional to the corresponding stress (p)' on the transverse section being given by the equation, P p = m Change of breadth being thus prevented the elasticity of flexion (p. 412) becomes 2 m being from 7 to 1 2 per cent, greater than Young's modulus. 213. Remarks on Shearing and Bending. When a beam is subject< to bending without shearing the only assumption made in the usual theory -given on page 307 is that of complete freedom to expam and contract laterally ; but in general there is also a shear on each section and in consequence tangential as well as normal stress at each point. Hence if two plane sections be taken before the beam is bent, those sections not only rotate about their neutral axes as in Fig. 122 on the page cited, but are also distorted and the con- sequences of this distortion will now be briefly considered. _k K (1) Fig. 158 shows a longitudinal sectioi of a bent beam, the plane of bending bein| as before a plane of symmetry, and NN th( geometrical axis as in Fig. 122. The dott straight lines KPNK as before show th( positions of two transverse sections when bend- ing alone exists, and simply rotate about axe through NN to meet in a centre of curvature not shown in the present figure. The ful ', I curves kpNk show the intersections of the 1 longitudinal section with the actual sections after distortion by the action of the shearing stress at each point Let us now suppose the shearing force to be constant, that is, the same Pig. 158 CH. xvii. ART. 213.] STRESS, STRAIN, AND ELASTICITY. 415 on all sections, as when a beam is fixed horizontally at one end and loaded at the other with a given weight, then as in other analogous cases (pp. 299, 306) we may suppose the shearing stress and con- sequently the distortion the same at corresponding points of the two sections; that is to say, the two curves will be exactly alike and the deviation Pp from the straight line will be the same for both. Hence pp the actual length of a longitudinal layer of the beam is the same as PP the length which it would have had if the shearing stress had been absent. The actual form of the distorted section is very complex, no line in it remaining straight but in general becoming a curve of double curvature ; it is clear, however, that the same reasoning applies to every pair of corresponding points and not merely to points lying in the central plane. Hence the changes of length of all the elementary bars into which the beam is analysed are the same as if there were no shearing, and reasoning as on page 307, we arrive at the same general equations, p__M_E y~ I~ K for the normal stress and the curvature as in simple bending. We conclude, therefore, that these equations must be true notwith- standing the distortion produced by shearing, provided only the shearing force be constant. The truth of the simple reasoning here given is borne out by a complete investigation of the bending and shearing of a beam which like the corresponding investigation for torsion we owe to the late M. St. Venant.* This investigation, based on the supposition of complete freedom to expand and contract laterally (see last article), shows that the usual equations are exact when, and only when, the shearing force is constant. In any case of continuous loading the shearing force is zero at sections of maximum moment, and there is consequently no distortion there, so that at such sections the equations still apply. Where the load is concentrated at one or more points, there will always be shearing and often of great magnitude, but in these cases if not absolutely constant it varies slowly in consequence of a relatively small continuous load between the sections at which the load is concentrated. Hence the equations may be regarded as substantially exact in most practical cases where it is necessary to determine the resistance of a beam or girder to bending. Some qualifications of this statement have already been given in preceding articles of this- book (Art. 189), and it may be further added that when a section * History of Elasticity, vol. ii., Part I., p. 58. 41 G STIFFNESS AND STRENGTH. [PART iv. of maximum moment occurs in the neighbourhood of the ends of the beam or of a concentrated load additional strength may in some cases be required. The local stress due to the direct action of a -concentrated load is frequently very considerable,* though its effect in weakening a solid beam is probably not in any proportion to its magnitude. At the ends of a beam additional strength is generally required for constructive reasons. (2) The transverse curvature of the originally horizontal layers of a beam of rectangular section subject to bending and shearing has the effect of altering the distribution of the shearing stress, which is not uniform along lines parallel to the neutral axis but is less at the centre than at the outer surface. The mean along the neutral axis is 1| times the mean over the whole section, but the actual value is less than this at the centre and greater at the outer surface. This inequality of distribution laterally is in the first instance due to the elevation of the sides of the beam (Fig. 157) above the centre which is caused by the transverse curvature. So long as there is no shearing the curvature and therefore the elevation remains the same for all sections, but when the curvature changes the elevation also varies and produces a corresponding excess distortion at the sides. The whole action is very complex and cannot be reduced completely to calculation in any simple way, but some further remarks will be found in the Appendix. When the depth is not less than 2J times the breadth this effect may be disregarded, but in a square section the difference is 6 per cent., and as the breadth increases becomes much greater.! In other types of section as already stated there is often a large discrepancy between the mean and the true maximum, apart from the effect of transverse curvature. Complete results have been obtained for a circle and some other forms. These calculations of St. Venant, however, only apply to sections of a beam the outer surface of which is free from stress. The direct action of the pressure on the sides of a pin which is being sheared most probably tends to equalize the shear on the section, and the provisional method, already described, when properly checked, is perhaps the best approximation attainable. 214. Thick Hollow Cylinder under Internal Pressure. The equations connecting stress and strain in combination with suitable equations * The Influence of Surface Loading on the Flexure of Beams. By Prof. C. *A. Carns-Wilson. Proceedings of the Physical Society of London, December, 1891. t History of Elasticity, vol. ii., Part I., p. 68. CH. xvn. ART. 214.] STRESS, STRAIN, AND ELASTICITY. 417 expressing the continuity of the body and the equilibrium of each of its elements are theoretically sufficient to determine the distribution of stress within an elastic body exposed to given forces, and in particular to determine the parts of the body exposed to the greatest stress, and the magnitude of such stress. The most important cases hitherto worked out, in addition to those considered in preceding chapters, are the torsion of non-circular prisms and the action of internal fluid pressure on thick hollow cylinders and spheres. For M. St. Venant's investigations on torsion we must refer to Art. 185, page 360, and the authorities there cited. We shall only consider the comparatively simple case of a homogeneous cylinder. Fig. 159a shows a longitudinal section of a hollow cylinder open at the ends, which are flat : the cylinder contains fluid which is acted on by two plungers forced in by external pressure so as to produce an internal fluid pressure p r Fig. 159& shows the same cylinder in . Fig.l59a transverse section : imagine a cylindrical layer of thickness t, this thin -cylinder will be acted on within and without by stress which symmetry shows must be normal ; let these stresses be p and p' and the internal and external ;radii of the thin cylinder be ? and r'. Now, if p' the external pressure had existed alone, a compressive stress q would have C.M. 2D 418 STIFFNESS AND STRENGTH. [PART iv. been produced on the material of the cylinder given by the equation (see Art. 150) p'r' = qt; and if the internal pressure had existed alone, we should have had a tensile stress given by pr = qt; hence when both exist together, we must have p'r'-pr = qt, where q is the stress on the material of the cylinder on a radial plane in the direction perpendicular to the radius reckoned positive when compressive. Clearly t = r' - r, and therefore proceeding to the limit we may write the equation which is one relation between the principal stresses p, q at any point of the cylinder. We now require a second equation, to get which it is necessary to consider the way in which the cylinder yields under the application of the forces to which it is exposed. The simplest way to do this is to assume that the cylinder remains still a cylinder after the pressure has been applied : if so, it at once follows that points in a transverse section originally, remain so, or, in other words, that the longitudinal strain is the same at all points. It is not to be supposed that there is anything arbitrary about this assumption : no other, apparently, can be made if the ends of the cylinder are free, the pressure on the internal surface exactly uniform, and the cylinder be homogeneous and free from initial strain. For when this is the case,. there is no reason why the cylinder should be in a different condition in one part of its length than in another. If the ends are not free, or if the pressure is great in the centre, the middle of the cylinder will bulge, but not otherwise. It is also clear that the total pressure on a transverse section must be zero because the ends are free, and hence it is natural to suppose that it is also zero at every point of the transverse section, an assumption which we shall presently verify. For greater generality we in the first instance suppose it a constant quantity p . The equations connecting stress and strain therefore become m m m CH. xvn. ART. 214.] STRESS, STRAIN, AND ELASTICITY. 419 where e v 2 , e 3 are the compressions in the direction of the radius, the direction perpendicular to the radius in the transverse section, and the direction of the length, respectively. Of these the last is constant, as just stated, and therefore p + q = constant = 2^ is the second equation connecting p, q. Substituting for q, we find or r-r + 2p = 2ci. dr f Multiply by r and integrate, then p = J + Cj, and consequently q = c x |, where c t> is a constant of integration. The two constants, c v c 2 , are now determined by consideration of the given pressure within and without the cylinder. If n be the ratio of the external radius to the internal radius E, we have at the internal surface and at the external surface r = nR) from which two equations we get Substituting these values in the equation for q, the negative sign in this formula indicates that the stress is tensile, as we might have anticipated. The formula shows that the stress decreases 71^+1 2l> from 2 _ .pi at the internal surface to Yz 1 ! at *^ e ext e rna l surface. The mean stress is obtained from the equation (Art. 150) hence the maximum stress is greater than the mean in the ratio 7i 2 +l :n+I, and it is clear that it can never be less than p Y The minor principal stress at the internal surface is p l and (omitting p Q ) the so-called simple equivalent tensile stress can be found. 420 STIFFNESS AND STRENGTH. [PART iv. Verification of Preceding Solution. The radial strain (ej and the hoop strain (e 2 ) are given by the above equations in terms .of the stress. Now these changes of dimension are not independent, but are connected by a certain geometrical relation which it is necessary to examine in order to see whether it is satisfied by the values we have found. Returning to the diagram, suppose the internal radius of the elementary ring repre- sented there to increase from r to s, and the external radius from r' to s' ; then or, since the thickness of the ring changes from t d, . . *=*,.<<*) This relation must always hold good, in order that the rings after strain may fit one another, and should therefore be satisfied by our results. On trial it will be found that it is satisfied, and we conclude that the solution we have obtained satisfies all the condi- tions of the problem, and is therefore the true and only solution, subject to the conditions already explained. For further remarks on this question, see Appendix. 214A. Strengthening of Cylinder by Rings. Effect of great Pressures. The stress within a thick hollow cylinder under internal fluid pressure may be equalized, and the cylinder thus strengthened by constructing it in rings, each shrunk on the next preceding in order of diameter. For a cylinder so constructed will be in tension at the outer surface and compression at the inner surface before the pressure is applied, and therefore after the pressure has been applied will be subjected to less tension at the inner and more tension at the outer surface than if it had been originally free from strain. It is theoretically possible to determine the diameters of the successive rings so that the pressure shall be uniform throughout. The principle is important, and fre- quently employed in the construction of heavy guns. When the limit of elasticity is overpassed the formula fails, and the distribution of stress becomes different. If the pressure be imagined gradually to increase until the innermost layer of the cylinder begins to stretch beyond the limit, more of the pressure is transmitted into the interior of the cylinder, so that the stress becomes partially equalized. If the pressure increases still further, the tension of the innermost layer is little altered, and in soft materials longitudinal flow of the metal commences under the direct action of the fluid pressure. The internal diameter of the cylinder then increases per- ceptibly and permanently. This is well known to happen in the cylinders employed in the manufacture of lead piping, which are exposed to the severe pressure necessary to produce flow in the lead. The cylinder is not weakened but strengthened, having adapted itself CH. xvn. ART. 215.] STRESS, STRAIN, AND ELASTICITY. 421 to sustain the pressure. Cast-iron hydraulic press cylinders are often worked at the great pressure of 3 tons per sq. inch, a fact which may perhaps be explained by a similar equalization, 215. Elastic Energy of a Solid. If a cube of side unity be under the action of normal stresses p v p 2 , p z on its faces the elastic energy will evidently be e v 2 , e 3 being as before the strains given by the general equations connecting stress and strain (p. 410). In most cases one of the stresses Pz will be small enough to be neglected, then substituting for e v # 2 , m j Suppose now these principal stresses^, p 2 are due to the action of normal stresses p n , p n ' on oblique planes combined with a tangential stress q as on page 403, then on solving the quadratic given on the page cited and substituting for p lt p 2 , m m which, using the value of C the co-efficient of rigidity given on page 411 becomes TJ-Pn +K 2 PnPn , f_ 2E mE + 2C* Thus the elastic energy per unit of volume at any point of a solid is, as might be expected, the sum of that due to the normal stress and the tangential stress taken separately. This important principle holds good for each particle, and therefore for the whole, of a beam subject to bending and shearing, a shaft subject to bending and twisting, as well as many other cases. As an example of the use of the formula take the case of the deflection of a beam due to shearing considered in Art. 190. The beam being supposed sup- ported at the ends and loaded with a weight W, its deflection will be SCT/JFand the part due to shearing will therefore be where dF is an element of volume. Taking for simplicity the case of a uniform transverse section the formula may be written q*AlCf dA_ [f dA '~~ where dA is an elementary portion of the transverse section, / the span, and q Q the mean stress W[A, as on page 368. The integral taken over the whole transverse section is a numerical factor bv which 422 STIFFNESS AND STRENGTH* [PART iv. <$ must be multiplied to get the actual deflection due to shearing. Take for example a tube the shear at each point P of the annular section of which was found on page 368 (see Fig. 146). q = 2q oc A/p t - that is, at corresponding points q is proportional to the radius vector as already stated. Replacing as before (p. 359) qp at the outer surface by q^ or 2T/A we get a general formula for the angle of torsion of a nest of similar tubes CH. xvn. ART. 216.] STRESS, STRAIN, AND ELASTICITY. 423 In the case of a rectangular tube the integral is easily evaluated, for evidently if the sides be b and c, (ds (b c\ &*+(* / = 4( -4-T ) =4. - A ^4:0 -7H, }p \c b) A A 2 where / is the polar moment of inertia showing that the general formula given on page 361 is applicable if 4n- 2 is replaced by the somewhat larger number 48. Hence, as might be expected, the tubes are somewhat less rigid than the solid shaft. For an ellipse of semi-axes b and c it is easily shown that | = 7rf- + ^=3T 2 . "t- - = 4T 2 .-7o ]p \c bj A A 2 thus verifying the formula. EXAMPLES. 1. When the sides of the bar are forcibly prevented from contracting, show that the necessary lateral stress is given by p,= Be, where B= -^- -. This constant B is what Rankine called the " lateral " elasticity m 2 - m - 2 of the substance. 2. With the notation of the preceding question prove that 3. In a certain quality of steel =30,000,000; (7=11,500,000; find the elasticity of volume and the values of A and B, assuming the material to be isotropic. Ans. m=3^; D= 25,555, 000. 4. The cylinder of an hydraulic accumulator is 9 inches diameter. What thickness of metal would be required for a pressure of 700 Ibs. per square inch, the maximum tensile stress being limited to 2,100 Ibs. per square inch? Also, find the tensile stress on the metal of the cylinder at the outer surface. Ans. Thick ness =1*84"; Stress = 1,000 Ibs. per square inch. 5. If the cylinder in the last question were of wrought iron, proof resistance to simple tension 21,000 Ibs. per square inch, at what pressure would the limit of elasticity be overpassed? m = 3'5. (See Art. 229.) A us. 6,400. 6. Find the law of variation of the stress within a thick hollow sphere under internal fluid pressure. By a process exactty like that for the case of the cylinder (page 418) it is found that the equation of equilibrium is *. The equation of continuity is the same as that for a cylinder (Art. 214), and the equations connecting stress and strain are now 424 STIFFNESS AND STEENGTH.. [PART iv. We can now by elimination of q, reduction, and integration obtain the constants being found as in the cylinder. 7. The cylinder of an hydraulic press is 8 inches internal and 16 inches external diameter. If the pressure be 3 tons per sq. inch, find the principal stress at the internal and external circumference. ^TAl inner circumference { JJj | == f <. 8. In the last'question find the] "equivalent simple tensile stress" (p. 411), assuming m=3'5. Ans. 5'86 and 2 tons. 9. In examples 15, 16, page 376, find the "equivalent simple tensile stress" at the points indicated, assuming as before 7/2=3*5. CHAPTER XVIII. Fig.lGO MATERIALS STRAINED BEYOND THE ELASTIC LIMIT. STRENGTH OF MATERIALS. 217. Plastic Bodies. If the stress and strain to which a piece of material is exposed exceed certain limits its elasticity becomes imper- fect, and ultimately separation into parts takes place. We proceed to consider what these limits are in different materials under different circumstances : it is to this part of the subject alone that the title " Strength of Materials " is, strictly speaking, appropriate. Reference has already been made (Art. 147) to a certain condition in which matter may exist, called the Plastic state, which may be regarded as the opposite of the Elastic state, which has been the subject of pre- ceding chapters. In this condition the changes of size of a body are very small, as before ; but if the stress be not the same in all directions the differ- ence, if sufficiently great, produces continuous change of shape of almost any extent. Some materials are not plastic at all under any known forces, but many of the most important materials of construction are so, more or less, under great A' inequality of pressure. Fig. 160 shows a block of material which is being compressed by the action of a load P applied perfectly uniformly over the area AB. Let the intensity of the stress be p, then so long as p is small the compression is small and proportional to the stress ; but when it reaches a certain limit the block becomes visibly shorter and thicker. This limit depends on the hardness of the material, and the value of p may be called the " co-efficient of hardness." In an actual experiment the friction of the surfaces between which the block is compressed holds the ends together, so that it bulges in the middle, as in Fig. 166, p. 435, which represents an experiment on a short cylinder of soft steel. In c c D' 426 STIFFNESS AND STRENGTH. [PART iv. the ideal case the cross section remains uniform, changing throughout inversely as the height, as expressed by the equation where A is the area and y the height of the block. In a truly plastic body p the intensity of the stress remains constant, and therefore the crushing load P varies as A y that is inversely as y. This is the same law as that of the compression of an elastic fluid when the compression curve is an hyperbola, and we therefore conclude {Art. 90) that the work done in crushing is U = Py .log e r=pAy\og e r = pP r \o^ r, where r is the ratio of compression and Fthe volume. Certain qualities of iron and soft steel will endure a compression of one-fourth or even of one-half the original height, and amounts of energy are thus absorbed which are enormous compared with the resilience of the metal. To illustrate this, suppose that plasticity begins as soon as the limit of elasticity / is overpassed, then for^> we must write/, and by Art. 149 the resilience for a volume V is /2 Resilience = ^-r, . V. The ratio which the work just found bears to the resilience is therefore o rr Ratio = r- . log"?'. In wrought iron for a compression of one-fourth the height (r= 1*333) this is about 800. The actual ratio must be much greater, because, as we shall see presently, the hardness of the material increases under stress. If lateral pressure of sufficient magnitude be applied to the sides of the block, the longitudinal force being removed, the effect is elongation instead of compression, contraction of area instead of expansion. The magnitude of the lateral pressure is found by imagining a tension Applied both longitudinally and laterally of equal intensity. Such a tension has no tendency to alter the form of the block, being analogous to fluid pressure, but it reduces the lateral pressure to zero, while it introduces a longitudinal tension of the same amount, which has the same value as the longitudinal compression of the preceding case. We see then that in every case a certain definite difference of pressure is required to produce change of shape in a plastic body, the direction of the change depending on the direction of the difference. The work done is found by the same formula as before, r meaning now the ratio of elongation. In the process of drawing wire the lateral pressure is applied by -CH. xvin. ART. 219.] STRENGTH. 427 the sides of the conical hole in the draw-plate, which are lubricated to reduce friction, and the force producing elongation in the wire is the sum of the tensile stress applied to draw the wire through the hole and the compressive stress on the sides. The work done is given by the -same formula as before, p being now the sum in question. 218. Flow of Solids. When a plastic body changes its form the process is exactly analogous to the flow of an incompressible fluid, which indeed may be regarded as a particular case. In the solid the distorting stress at each point at which the distortion is going on has a certain definite value which in the fluid is zero. The experimental proof of this is furnished by the Fig. 161. experiments of M. Tresca, of which Fig. 161 shows an ex- ample. Twelve circular plates of lead are placed one upon another in a cylinder, which has a flat bottom with a small orifice at its centre. The pile of plates being forcibly compressed, the lead issues at the orifice in a jet, and the originally flat plates assume the forms shown in the figure. The lines of separation, indicating the position of particles of the metal originally in a transverse section, are quite analogous to the corre- sponding lines in the case of water issuing from a vessel through an orifice in the bottom. Tresca's experiments were very extensive, and showed that all non-rigid material flowed in the same way. Lead approaches the truly plastic condition ; the difference of pressure necessary to make it flow being always about the same. Tresca ascribes to it the value of 400 kilogrammes per square centimetre, or about 5,700 Ibs. per square inch;* but it is probably subject to considerable variations. The manufacture of lead pipes, the drawing of wire, and all the processes of forging, rolling, etc., by which metals are manipulated in the arts, are examples of the Flow of Solids. 219. Preliminary Remarks on Materials. Stretching of Wrought Iron -and Steel. Materials employed in construction may roughly be divided into three classes. The first are capable of great changes of form without rupture, and, when possessing sufficient strength to resist the * The co-efficient employed by Tresca, and called by him the "co-efficient of fluidity," is half that used in the text. It is the magnitude of the distorting stress necessary to produce flow. See also note in Appendix. 428 STIFFNESS AND STRENGTH. [PART iv. necessary tension, may be drawn into wire. The last property is called ductility, and this word may be used to describe the class which we shall therefore call Ductile Materials. The second, being incapable of enduring any considerable change of this kind, may be described as Rigid Materials. The third are in many cases not homogeneous, but may be regarded as consisting of bundles of fibres laid side by side ; they may therefore be described as Fibrous Materials ; they are generally of organic origin. We shall commence with the consideration of ductile materials, and more especially of WROUGHT IRON AND STEEL. Accurate experiments on the stretching of metal are difficult to make, the extensions being very small and the force required great. If levers are used to multiply the effect of a load or to magnify the extensions, errors are easily introduced. If the levers are dispensed with, a great length of rod is necessary and a heavy load, the manipulation of which involves difficulties. The experiment we select first for description was made by Hodgkinson on a rod of wrought iron '517 inch diameter, 49 feet 2 inches long, loaded by weights placed in a scale pan* suspended from one end. The load applied was increased by equal increments of 5 cwts. or 2 667 '5 Ibs. per square inch of the original sectional area of the bar ; each application of the load being made gradually, and the whole load removed between each. At each application and removal the elongation was measured so as to test the increment of elongation, both temporary and permanent, occasioned by each load. If the rod were perfectly elastic the temporary increments should be equal and the permanent elongations (usually called " sets ") zero. The annexed table shows part of the results of this experiment, the first column giving the load, the second the total elongation, the third the successive increments of the elongation, the fourth the total permanent set. On examining the table we see that, after some slight irregularities at the commencement due to the material not being perfectly homo- geneous, the increments of elongation are nearly constant till we reach the eighth load of 21,340 Ibs. per square inch, after which the increments show an increase at first moderate and subsequently very rapid. Further, the permanent set, which at the commencement is * Being one of the best of its kind of old date this experiment has often been quoted. For the original description, see the Report of the Commissioners appointed to enquire into the Application of Iron to Railway Structures. H. XVIII. ART. 219.] STRENGTH. 429 very minute and increases very slowly, at the same point shows a corresponding increase indicating that the observed increase is almost wholly due to a permanent elongation of the bar, the temporary STRETCHING OF A WROUGHT- IRON ROD, 49 FEET 2 INCHES LONG. LOAD. ELONGATION IN . INCHES. . INCREMENT OF ELONGATION. PERMANENT SET. 2667 -5 x 1 2667-5 0485 0485 x 2 5335 1095 -061 x 3 8003 1675 058 0015 x 4 10,670 224 0565 -002 x 5 13,338. 2805 0565 0027 y. 6 16,005 337 0565 003 x 7 18,673- 393 056 004 x 8 21,340 452 . -059 0075 x 9 24,008 5155 0635 0195 x 10 26,675 598 - '0825 049 xll 29,343 760 162 1545 x 12 32,010 1-310 550 667 increase following approximately the same law as before. Notwith- standing this the bar is not torn asunder till a much greater load is applied. The table shows the results up to a load of 32,000 Ibs. per square inch, but rupture did not occur till a load of 53,000 Ibs. was applied. The extension at the same time increased to nearly 21 inches, being more than forty times its amount at the elastic limit. We conveniently represent the results graphically by setting off the elongations as absciss* along a base line with corresponding ordinates to represent the stress, thus obtaining a curve of "stress and strain" (Fig. 162). The curve will be seen to be nearly straight 452 up to a stress of 22,000 Ibs. and then to bend sharply, becoming nearly straight in a different direction. A curve of permanent set 430 STIFFNESS AND STRENGTH. [PART iv. may also be constructed which is seen to follow the same general law. Accompanying the increase of length of the bar we find a contrac- tion of area within the elastic limit, however, this is so small as to escape observation. Outside the limit it be- comes visible, consisting in the first instance of a more or less uniform contraction at all or nearly all points, followed by a much greater contraction at one or sometimes two points where there happens to be some local weak- ness.* Within the elastic limit the density of the bar diminishes, but by an amount so small that the fact is rather known by reason- ing than determined by experiment. Outside the limit there is a permanent diminution which is perceptible, though still very small. Thus beyond the elastic limit the bar draws out, changing its form like a plastic body with- out sensible change of volume. The bar finally tears asunder at the most contracted section, as shown by the annexed figure (Fig. 163} representing an experiment by Mr. Kirkaldy on a bar of iron 1 inch diameter, in which the contraction of area was 61 per cent., and the elongation 30 per cent., ultimate strength 58,000 Ibs. per square inch of original area, 146,000 Ibs. per square inch of fractured area. The contraction of section in good iron and soft steel is 50 or 60 per cent. 220. Breaking-down Point. The foregoing experiment may, as far as it goes, be taken as a type of a multitude of such experiments which have been made on wrought iron and steel, which show that a tolerably well-defined limit exists, within which the extension is proportional to the pull and the sets are very small, but beyond this limit the process of stretching can only be completely studied by aid of a machine. A full description of various types of testing * On . this point see Preliminary Experiments on Steel by a Committee of Civil Engineers, London, 1863. On account of the uncertainty of the amount of contraction at various points, the ultimate extension may sometimes be an imperfect measure of the ductility of the iron, even when the pieces are of the same length and sectional area. CH. XVIII. ART. 220.] STRENGTH. 431 164. machines will be found in Professor Unwin's treatise on Testing, from which we take such particulars as are necessary for our present purpose. Fig. 164 is a diagram showing the essential parts of one of these machines : BAK is a lever balanced on knife edges at A and carrying a weight W, which can be moved along it by a screw ; E is an hydraulic cylinder, the plunger of which is connected with the lower end of the test piece FD. The short end B of the lever is connected with the upper end of FD, and the weight W is thus balanced by hydraulic pressure. When making the experi- ment water is pumped into the cylinder and a gradually increasing pull is thus applied to the test piece. This pull is measured by continuously moving the weight W by a screw, so as to keep the lever horizontal^ stops C being provided to prevent it from moving far in either direc- tion. The extensions are measured with great accuracy by a suitable apparatus, which not unfrequently automatically traces a curve of stress- and strain. Fig. 165 shows roughly the form of curve obtained, the straight line AB representing the elastic part of the process already described. After passing the point B the curve ~^X bends away from the straight line, F but the deviation is not large till a sharply defined point C is reached at which the curve is nearly hori- zontal, showing that a considerable stretch has occurred while the load ' remains nearly the same. The Fig - 165 - suddenness of this drawing out which is so characteristic of wrought iron and soft steel, is not dis- tinctly perceived in the original way of making the experiment, because the load is not applied continuously. The point at which it occurs is described as the "yield-point," or "breaking-down point." The term "limit of stability," though in some respects preferable, is not so often used. When a bar is stretched in the workshop without recourse to delicate measurements this point may often be recognized by the falling-off' scale and the obvious extension accompanied by lateral 432 STIFFNESS AND STRENGTH. [PART iv. contraction which then occurs. It marks the end of the elastic stage and the commencement of the plastic stage in the process of stretching, and in the roughest class of experiments is the apparent " limit of elasticity," a term which may conveniently be applied when a closer specification is not necessary. After passing C the stress goes on continuously increasing till a second point E is reached, where the curve is once more horizontal, and now if the stretching is carried still further it is found that to prevent the lever from resting on the lower stop, W must be moved gradually back again, showing a continuously diminishing stress till the bar tears asunder as already described. This part of the process for obvious reasons is not perceived when the experiment is made in the absence of a machine. The interpretation appears to be that as far as the point E the contraction of area is taking place throughout the whole length of the test piece, while the part EF of the curve represents the stretching which takes place after local contraction has begun. 221. Real and Apparent Tensile Strength. The ordinates of a curve of stress and strain as usually plotted represent the total pull divided by the original sectional area of the test piece, and the greatest ordinate at E gives the ultimate strength as usually reckoned. It is clear that this is not the real tenacity of the material, for the sectional area has diminished considerably during stretching, and to meet this difficulty the area at fracture was formerly often employed as a divisor instead of the original area, the result obtained being called the "real tensile strength." This, however, gives much too large a value, for the real stress must, at least approximately, be the actual pull at any point divided by the actual area at that point. For this reason the contraction of area is now employed by most authorities exclusively -as a measure of the ductility of the material without reference to its tenacity. Nevertheless the actual stress on the contracted area is much greater than the apparent, and hence it follows that if the form of the piece be such as partly or wholly to prevent contraction the apparent strength will be increased. For example, if two pieces of the same bar be taken and one turned down to a certain diameter, while in the other narrow grooves are cut so as to reduce the diameter to the same amount at the bottom of the grooves, the strength of the grooved piece will be found to be much greater than that of the piece the diameter of which has been reduced throughout, and this can only be explained by observing that the length of the reduced part of the grooved bar is insufficient to permit contraction to any considerable extent. This is a point to be CH. xvin. ART. 222.] STRENGTH. 433 noticed in considering experimental results. * The form of the specimen tested may have much influence. Further, since the limit of elasticity is the point at which flow commences, and since the flow is due to difference of stress, it follows that the same causes must raise the limit of elasticity, and thus we are led to the conclusion that there are two elements constituting strength in a material, first, tenacity, and, secondly, rigidity. In some materials, such as these we are now considering, the tenacity is much greater than the rigidity, and in them the limit of elasticity will depend on the rigidity and will have different positions according to the way the stress is applied. It will lie much higher, and the apparent strength will be much greater when lateral stress is applied to prevent contraction. 222. Increase of Hardness by Stress beyond the Elastic Limit. In clay ^,nd other completely plastic bodies a certain definite difference of pressure is sufficient to produce flow : in iron, copper, and probably other metals, however, as we have just seen, this is not the case, the metal acquiring increased rigidity in the fact of yielding to the pressure. Thus the effect of stress exceeding the elastic limit is always to raise the limit, whether the stress be a simple stensile stress or whether it be accompanied by lateral pressure. All processes of hammering, cold rolling, wire drawing, and simple stretching have this effect. If a bar be stretched by a load exceeding the elastic limit and then removed, on re-application of a gradually increasing load we do not find a fresh drawing out to commence at the original elastic limit, but at or near the load originally applied.! If the load be further increased drawing out recommences. Hence, whenever iron is mechanically " treated " in any way which exposes it to stress beyond the elastic limit, contraction is prevented and the apparent strength is increased ; for example, iron wire is stronger than the rod from which it is drawn ; when an iron rod is stretched to breaking, the pieces are stronger than the original rod. It is riot certain that the real strength of materials is always increased by such treatment ; perhaps in some cases the con- trary, for we know that the modulus of elasticity and specific gravity are somewhat diminished. On the other hand there are cases in which the increase of strength is greater than can be accounted for in * See Experiments on Wrought Iron and Steel, by Mr. Kirkaldy, p. 74. 1st edition. Ulasgow, 1862. t Styffe, On Iron and Steel, p. 68. t The raising of the limit of elasticity by mechanical treatment of various kinds has long been known : in the case of simple stretching the effect appears to have been first noticed by Thalen in a paper, a translation of which will be found in the Philosophical Magazine for September, 1865. C.M. 2 E 434 STIFFNESS AND STRENGTH. [PART iv.. this way. On annealing the iron it is found to have resumed its original properties, a circumstance which indicates that the increased rigidity is due to a condition of constraint which is removed by heating the metal till it has assumed a completely plastic condition. This process of hardening and annealing may be repeated a number of times without altering the yield-point, and it has recently been suggested that hardening by application of stress is analogous to the hardening of steel by heating and sudden cooling, and may be due to a similar change of molecular arrangement.* In considering the effect of impact, the diminution of ductility occasioned by the application of stress beyond the elastic limit is a most important fact to be taken into account (see Art. 232). Working iron or steel hot has generally the effect of increasing both its strength, and its ductility. 223. Compression of Ductile Material. In a perfectly elastic material compression is simply the reverse of tension, the same changes of dimension being produced by the same stress, but in the reverse direction. Also in a plastic body a given difference of stress produces flow, whether the stress be tensile or compressive; hence in ductile metals we should expect to find the modulus of elasticity and the limit of elasticity nearly the same in compression as in tension. These con- clusions are borne out by experiment. In the case of wrought iron and. EXPERIMENT BY SIR W. FAIRBAIRN ON A BLOCK '72 INCH DIAMETER OF SOFT BESSEMER STEEL. TOTAL LOAD. = P. HEIGHT OF BLOCK. = ?/. CRUSHING STRESS. , _ p y 997 16-7 92 37'8 201 865 42-9 23-3 797 45-9 26-3 731 47'4 29'5 672 48-9 32-6 613 49-4 35'8 574 50-6 39-3 535 51-9 41-0 505 50-8 REMARKS. The apparent ultimate tensile strength of this steel was 36 tons, its limit of elasticity 22 tons per square inch. Modulus of elasticity 30,300,000 Ibs. Ratio of contraction -41. * Effect of Repeated Straining and Heating. Society, Vol. Ivii. "W. C. UNWIN, Proceedings of the Royal CH. xvm. ART. 223.] STRENGTH. 435 steel, experiments on the direct compression of a bar are more difficult to carry out than experiments in tension, the bars are necessarily of limited length, and must be enclosed in a trough to prevent lateral bending ; minute accuracy is therefore hardly attainable. A consider- able number have, however, been made, from which it appears that the modulus of elasticity anft the limit of elasticity are nearly the same in the two cases.* The metal yields beyond the limit by a process of flow of the same character as in tension, but expanding laterally instead of contracting. This is especially seen in experiments made by the late Sir W. Fairbairn in 1867, and somewhat earlier by Mr. Berkeley, on the compression of short blocks of steel. In both, the blocks were pieces of round bars, of height somewhat greater than the diameter, and the results were very similar. The annexed table gives the results of one of Sir W. Fairbairn's experiments. Column 1 gives the actual load laid on ; column 2 the corresponding height of the block, both given directly by the experi- ments ; column 3 is calculated by dividing the product of load and height by the original sectional area and height, and represents the crushing stress per square inch of the mean sec- Fig . 166 tional area. If the block did not bulge in the centre (Fig. 166) this would be the actual crushing stress, which, however, must in fact be less. The table shows that after a compression of about one-third, the crushing stress remains nearly constant at about 50 tons per square inch. The experiment terminated at a compression of one-half. This kind of steel then is perfectly elastic up to 22 tons per square inch, is partially plastic between 22 and 50, and behaves as a plastic body under a difference of stress of 50 tons per square inch. The point at which the material becomes perfectly plastic may be described as the " limit of plasticity," it probably corresponds to the point where the load is a maximum and local contraction begins (p. 423) in a stretched bar. The compression of iron blocks has been less thoroughly studied than that of steel, but it is known that the results are similar although the strength and the ultimate ratio of compression are much less. Set becomes sensible at about 10 tons per square inch, and the ultimate strength is from 40,000 to 50,000 Ibs. per square inch if lateral flexure be prevented. * Perhaps the best set of experiments are those made by the "Committee of Civil Engineers." See their Preliminary Report already cited, pp. 7-13. 436 STIFFNESS AND STRENGTH. [PART iv. The bulging which occurs when a short block of ductile material is compressed is due to the instability of a cylindrical flow of the metal, and would probably occur even if there were no friction between the block and the compressing surfaces. Fracture occurs by lateral tearing asunder along longitudinal cracks when the height is small. When of greater height the block crushes by lateral bending. In wrought iron the ratio of length to diameter at which lateral bending commences is about 3 and the corresponding crushing stress is 36,000 Ibs. per sq. inch, remaining independent of the length until the ratio reaches about one-third of the values given on page 342, after which the length begins to influence the crushing load as described in the chapter cited. In tubular struts this limit is about 15. 224. Bending within and beyond the Elastic, Limit. Since wrought iron and steel are nearly perfectly elastic when the stress applied is not too great, it follows that the formulae already obtained for the moment of resistance to bending and deflection of a bar must be true for these materials so long as the stress does not exceed the elastic limit determined by tension experiments of the kind just described. (1) Very careful experiments were made by M. Styffe* on the deflection of bars of small size, 4 feet long, which fully confirm this conclusion ; the value of the modulus of elasticity deduced from the observed deflection by the formula given on page 324 of this work closely agreeing with the value found by stretching the same bar. When smaller values are obtained by experiments on bending it is now recognized that this is due to the effect of shearing discussed on page 370, which, when neglected, may reduce the apparent value of the modulus by 20 per cent, or more. Some recent experiments by Messrs. Read and Stanbury, described in a paper which will be further referred to presently, give a modulus (apparent) of about 10,000 tons for beams of channel and Z section. In the case of a broad thin plate the modulus in bending should be greater than that in tension or compression (p. 414), but this theoretical conclusion appears as yet not to have been verified. In built-up beams the modulus, as might be expected, is still further reduced : thus Rankine in his Civil Engineering states that the value for large girders is on the average 17,500,000 Ibs. or about 8000 tons. In the paper just referred to the authors make a very interesting com- parison between the observed deflection of a vessel and the result of a careful determination by graphical integration of the differential equation of the deflection curve. Two different vessels tested in this * See Styffe, On Iron and Steel, already cited. CH. xviii. ART. 224.] STEENGTH. 437 way gave nearly the same value for the modulus which was found to- be about 10,000 tons.* (2) Again, it has been repeatedly explained in the earlier part of this book that the lateral connection of the several layers into which we imagine a beam divided has no influence on the stress produced by bending so long as the limit of elasticity is not exceeded. But when the limit is passed, the connection between those layers which are most stretched and compressed with those layers which have not yet lost their elasticity prevents their contraction jand expansion, and so raises the limit of elasticity in accordance with the general principle explained in Art. 221. Thus, the limit of elasticity lies higher, and the apparent elastic strength is greater in bending than in tension. In Fairbairn's experiment quoted above the same steel was tested in tension, com- pression, and bending. The elastic limit in bending was 30 tons, in tension 22 tons. The magnitude of the difference will depend on the form of transverse section, and on the ductility of the material. Ac- cording [to Mr. Barlow it may reach 50 per cent, in a rectangular section.! The case of cast iron will be referred to further on. (3) As soon as the elastic limit is passed, the stress, at points near the surface, no longer varies as the distance from the neutral axis. It does not increase so fast because the extension or compression is not accompanied by a proportionate increase of stress. Hence a partial equalization of stress is produced, and the maximum stress for a given moment of resistance is reduced. To illustrate this it may be interesting to make a calculation of the effect of equalization bjr supposing^ that under a bending moment very slowly and steadily applied beyond the elastic limit, the metal behaves like a truly plastic material throughout the transverse section, so that the stress is uniform. Referring to the formula on page 309, we have in which we must now, instead of assuming that p varies as y, suppose p a constant. Then M=2p.Ay where A is the area of the part of the section which lies on either side of the neutral axis and y the distance of its centre of gravity from that axis. For the same value of the modulus this gives a moment of re- sistance in a rectangular section 50 per cent, greater than if the material had been elastic. How far any apparent increase of strength due to equalization or lateral connection may be regarded in practice is * On the Relation leticeen Stress and Strain in Vessels, by T. C. Read and G. Stanbury. Transactions of the Institute of Naval Architects for 1894, Vol. xxxv., p. 372. \Phil Trans., 1855-57. 438 STIFFNESS AND STEENGTH. [PART iv. uncertain. A failure of elasticity must have taken place at certain points in order that there may be any increase at all, arid in cases where the load is frequently reversed the bar must be weakened. (See Art. 230.) CAST IRON AND OTHER RIGID MATERIALS. 225. Stretching of Cast Iron. The phenomena attending rupture by tension of cast iron are essentially different from those described above for the case of ductile metals. This will be sufficiently shown by an experiment, also made by Hodgkinson, on a bar of this material 50 feet long, T159 inch diameter. The experiment was made in the same way as that already described on the wrought-iron rod,* and the results are shown in the annexed table. The first four loads were applied as before, by increments of 5 cwt., here equivalent to 531 Ibs. per square inch; the whole load, after measurement of the elonga- "^255 tion, being completely removed, and the permanent set measured. After the fourth load the increment was 10 cwt., and this was carried on till the bar broke at a stress of Fig. 167, STRETCHING OF A CAST-IRON BAR, 50 FEET LONG, 1-159 INCH DIAMETER. LOAD IN LBS. PER SQUARE INCH. ELONGATION IN INCHES. INCREMENT OF ELONGATION. PERMANENT SET. 1. 531 024 024 x2='048 Perceptible. 2. 1,062 0495 0255x2= -051 0015 3. 1,592 0735 024 x2=-048 002 4. 2,123 09828 0247x2= -05 14 0045 5. 3,185 1485 0503 0105 6. 4,246 200 0515 0155 7. 5,308 255 055 022 8. 6,370 313 058 028 9. 7,431 374 061 037 10. 8,493 435 061 046 11. 9,554 504 069 056 12. 10,616 572 068 067 13. 11,678 648 076 0795 14. 12,739 728 080 095 15. 13,801 816 088 1115 16. 14,863 912 096 132 17. 15,924 1-000 088 ~ * Report of Commissioners on the Application to Railway Structures, p. 51. t-H. xvin. ART. 226.] STKENGTH. 16,000 Ibs. per square inch. The third column as before shows the increments of elongation, which, after a stress of 5,308 Ibs. per square inch, or the breaking load, has been reached, show a gradual increase till actual rupture occurs. The results of the experiments are graphically exhibited in the annexed diagram (Fig. 167) of stress, strain, and permanent set. The form of the curve is different from that of wrought iron, showing no point of maximum curvature, because in this material the bar does not draw out. Hodgkinson experimented on a large variety of different kinds of iron, and expressed his results by a formula, which may be written p = Ee(l-ke), where, as before (Art. 148), p is the stress, e the extension per unit of length, E the ordinary modulus of elasticity, and k a constant. The term ke here expresses the defect of elasticity of the bar. From .the results of his experiments we find the average values =14,000,000; &=209. Oast iron, however, is a material of variable quality, and the value of these constants may have a considerable range. Up to one-third the breaking load it may be regarded as approximately perfectly clastic, but the limit is by some authorities placed much higher. 226. Crushing of Rigid Materials. In the ductile metals the effects of compression are nearly the reverse of those of extension, as has been sufficiently shown in previous articles, but in cast iron this is by no means the case. Hodgkinson experimented in this question with great care and accuracy, testing pieces of iron of exactly the same quality under compression and tension to enable a comparison to be made. The bars were enclosed in a frame and tested by direct compression. Hodgkinson expressed his results by a formula, which may be written p = Ee(l-ke), the symbols having the same meanings as before, and the values may be taken as =13,000,000; & = 40. The smaller value of k indicates that the elasticity under compression is much less imperfect under the same stress. Short cylinders of the metal were also crushed, and the crushing load found to be five times the tensile strength or more. It thus appears that in compression cast iron is six times stronger than in tension, and this is true not merely of the ultimate resistance but in great measure also of the elastic resistance, for the elasticity of 440 STIFFNESS AND STRENGTH. [PART iv.. the metal is not sensibly impaired until one third the crushing load is reached. The manner in which crushing occurs is shown in the accompanying figure ; instead of bulging out like a ductile metal, oblique fracture takes place on a plane inclined at 45 or rather less to the axis, being (approximately) the plane on which the shearing stress is a maximum (Fig. 168). Great resistance to compression, as compared with tension, and sudden fracture by shearing obliquely or by splitting longitudinally are char- acteristics of all non-ductile materials, of which cast iron may be taken as a type. They are, in, fact, materials the tenacity of which is much less than the rigidity. In rigid materials crushing takes place not only by oblique shearing but also by longitudinal cracks. UN WIN (Testing of Materials, p. 419) finds that the mode of crushing and the resistance to crushing are much influenced by the material on which the specimen rests. When bedded on a soft material, the lateral flow of this material supplies by friction a transverse force on the base of the specimen, in consequence of which it crushes by longitudinal cracks at a smaller load than if the bed were hard, in which case oblique shearing occurs. This is a highly interesting observation, but it would be premature to say that all cases of crushing by longitudinal cracks can be explained in this way. 227. Breaking of Cast-Iron Beams. When a cast-iron bar is bent till the tensile stress at the stretched surface exceeds one-third the tensile strength of the material, the defective elasticity of the metal causes a partial equalization of stress on the transverse section as in the case of wrought iron. Besides this, the elasticity being much more perfect under compression than under tension, the equalization is greater on the stretched side than on the compressed side, and the neutral axis moves towards the compressed side of the beam. For both these reasons the moment of resistance to bending is greater for a given maximum tensile stress than it would be if the material were perfectly elastic. Thus it follows that if the co-efficient in the ordinary formula for bending be assumed equal to the tensile strength of the material, the calculated moment of resistance will be less than the actual moment of rupture of the beam by an amount which is greater for a rectangular section than for an I section. The value of the co-efficient in the formula which corresponds to the actual breaking weight is known as the "modulus. CH. xviii. ART. 228.] STRENGTH. 441 : of rupture " or the " bending strength " of the material, a quantity^ greater than the simple tensile strength in a ratio which varies accord- ing to the type of section. A very complete set of experiments on the breaking of cast-iron bars was made in 1888-9 by Professor Bach, who shows that the ratio ranges from 1*45 in an I section and 1'75 in a rectangular section with side vertical to 2'1 in a circular or H section, and 2'35 in a square section with diagonal vertical ; these numbers naturally being slightly different in different qualities of iron. The experimental result is always greater the more material is concen- trated in the neighbourhood of the neutral axis, and this circumstance renders it almost certain that the increased apparent strength of cast iron in bending is simply due to the causes above mentioned and not, as has often been supposed, to any influence of curvature on the strength of the metal. Two examples (10, 11, page 462) which are given at the end of this chapter will serve to show how great an effect is produced by equalization combined with a moderate shift of the neutral axis. SHEARING AND TORSION. COMPOUND STRENGTH. 228. Shearing and Torsion. We now pass on to cases where the- ultimate particles of the material are subject not to a simple longi- tudinal stress, but to stress of a more complex character. The simplest case is that of a simple distorting stress where the stress consists of a pair of shearing stresses (Fig. 140) on planes at right angles, or what is the same thing (Art. 183) of a pair of equal and opposite longitudinal stresses (Fig. 141) on planes at right angles. Examples of this kind of stress occur in shearing, punching, and twisting. Experiments on shearing are subject to many difficulties and are often not conducted in such a way as to satisfy the conditions necessary for uniformity of distribution of stress on the section. Moreover they necessarily give the ultimate resistance only without reference to the limit of elasticity. The whole process of shearing and punching is very complex, being at the commencement of the operation usually accompanied by a flow of the metal similar to that already referred to. Thus, when a hole is punched in a thick plate the punch sinks deep into the plate before the actual punching takes place, the metal being displaced by lateral flow, and the piece ultimately punched out being of less height than the thickness of the plate.* Separation takes place in the first instance by the formation of fine- cracks inclined at 45 to the plane of shearing. In soft materials the * On this subject see M. Tresca's paper cited above, and two articles in the Journal of the Franklin Institute. 442 STIFFNESS AND STRENGTH. [PART iv. surfaces slide past each other and separate, but in harder materials there is a strong tendency to the formation of an oblique fracture. In wrought iron and steel the ultimate resistance to shearing, though varying considerably, may be taken as about three-fourths the ulti- mate resistance to tension of the same material. The question of a -theoretical connection between the elastic strengths in the two cases is considered further on. Experiments on torsion are not numerous, and many of those which exist are not experiments on simple twisting, but on a combination of bending and twisting. Such experiments would be of great value if accompanied by corresponding experiments on simple twisting and bending made on similar pieces of material. It is known, however, that in the ductile metals the elastic resistance to torsion is less than the resistance to tension. A series of experiments on torsion made by Prof. Thurston give some interesting results. * Curves are drawn, the absciss* of which represent angles and the ordinates twisting moments, and the form of these curves shows that in some cases defective homo- geneity causes a great deficiency in the elasticity at small angles of torsion. In general, however, the curves closely resemble the ordinary curve of stress and strain, already given for a stretched bar, being nearly straight up to a certain point and then curving towards the axis. The formula for the angle of torsion of shafts given on page 361 has been tested by Bauschinger, in the case of square and circular sections by comparison with experiments made by him in 1878 on 13 pair of test pieces of iron and steel of various degrees of hardness, the mean result of the whole agreeing well with the formula. Some pieces of cast iron of rectangular and elliptic section showed, as might be expected, a less perfect agreement. In twisting, as in bending, after passing the elastic limit, the stress at each point of the section, instead of varying as the distance from the centre, as it must do in perfectly elastic material, varies much more slowly so as to become partially equalized. Hence the twisting moment corresponding to a given maximum stress is greater than it would be if the elasticity were perfect. In the case where the equalization is perfect it is easy to show that the twisting moment is increased in the proportion 4 : 3, a result first given in 1849 by Prof. J. Thomson. The curves given by Thurston show that in many cases an approximately constant twisting moment was reached indi- cating that nearly complete equalization must have existed. * See Paper on Materials of Machine Construction, read before the American Society of Civil Engineers, 1874. No diameters are given, except for the woods, so that the stress corresponding to the limit of elasticity cannot be found. Stretching Inch-Tons per Cubic MATERIAL. Ultimate Strength. Yield- Point Elonga- tion Inch. per Cent. To Yield- To T. C. S. Point, Fracture. Iron Bars, - 25 22 19 15 20 8-65x10-3 4-33 Iron Plates, 22 19 17 14 10 7-54 1-8 Soft Steel ( 15 to '3 per cent. \ of Carbon), - - / 30 23 18 25 12-5 6-5 Medium Steel ("3 to '5per\ cent, of Carbon), - / 35 22 15 18-6 ,, 4-6 Hard Steel ('5 to 75 per\ cent, of Carbon), - / 45 30 8 34-6 3-2 Cast Iron, - n 45 6 7x1 0' 3 Lead, - 11 _ Sheet Copper, 18J . . Cast Copper, 8A Fir, 5i 27 Oak, - -.- 2 1 The yield-point in the ductile metals is given in the next column ; it is from -5 to '7 of the ultimate strength, being generally about '6. Ductility is commonly measured by the total elongation up to fracture reckoned as a percentage of the original length of the test-piece. To render the measure definite, the length of the test-piece shall be some given multiple of the diameter, generally 8 or 10. The ultimate strength and ductility of steel vary according to the amount of carbon it contains in such a way that the sum of the two remains nearly constant, being about 53 in the example given in the table. In steel compressed in a fluid state by the Whitworth process the constant sum would be about one-third greater. The total amount of work done in stretching till fracture occurs may be separated into two parts, one before and the other beyond the yield- point. The first is calculated approximately by supposing the elasticity perfect up to the yield-point, and furnishes a rough maximum estimate of the resilience. The second may be obtained graphically from CH/XVIII. ART. 238.] STRENGTH. 459 a diagram of stress and strain, or calculated by a formula given by Professor Kennedy, based on the supposition that the curve is a para- bola. If x is the fractional elongation, p the yield-stress, / the ultimate strength, n + 'lf Work done = ^. y -V^. o The last two columns of the table give the two parts of the work, the first part being a small fraction the numbers are multiplied by 10~ 3 to avoid fractions. The whole is only a small part of the total resist- ance to impact of a cubic inch of the material, because only a part of the test-piece is fully stretched. If we consider the crushing of a small block, the result is many times greater ; in the example given in the table on page 434, the total work done is about 32 inch-tons per cubic inch. The value given for cast iron is obtained by integration from Hodgkinson's equation of the curve of stress and strain. In many kinds of cast iron the work done in fracture would be 2 or 3 times greater, but in any case is a small fraction. ADDENDA. 237. Principle of Similitude. When geometrically similar test-pieces of similar material are stretched till fracture occurs, the percentage of elongation is the same, the pull is proportional to the sectional area arid the work done to the volume of the piece. This law, which has been proved by the experiments of M. Barba, is merely a particular case of a general principle which applies to all similar and similarly loaded pieces, whether or not the limit of elasticity has been over-passed. To produce similar deformations, whether in stretching, bending, crushing, or in any other way, the load must be proportional to the sectional area and the work done to the volume of the pieces. It has been verified by experiments on the crushing of stone by M. Bauschinger, and a number of other examples will be found in a small treatise by Professor Kick.* Any deviations from this law should be due to differences of material and mode of manufacture between small pieces and large ones. In framing semi-empirical formulae for cases in which exact formulae are unattainable this law should be borne in mind. Thus in the case of pillars the formulae proposed by Hodgkinson do not satisfy the law, and should be rejected in favour of some formula, such as Gordon's, which does satisfy it. 238. Expansion and Contraction. We conclude this division of our work by giving some explanation of the effect of changes of tempera- ture, a subject too important to pass by unnoticed. * Das Gesetz der Proportionalen Wider&tande, Leipzig, 1885. 460 STIFFNESS AND STRENGTH. [PART iv. When a homogeneous body, free from initial strain, is uniformly heated throughout its whole mass, it undergoes a change of linear dimen- sions which is the same in every direction, being given by the equation t e ~K* where / is the rise of temperature and K a quantity which in a given material is roughly approximately constant for a moderate rise of temperature, being for degrees Fahrenheit given by the annexed table. LINEAR EXPANSION OF METALS. K 100 K METAL. K H a- ^W Wrought Iron, - Cast Iron, - 147,500 162,000 ll-3 20 7 6-5 1-26 77 Copper, - Brass, - - - - 104,500 95,400 '16 25 35 18 If the change of dimension is forcibly resisted the stress produced is the same as, reversed, would be required to produce that change in the absence of a change of temperature. Thus if a heated bar be prevented from contracting as it cools through f , the tensile stress will be produced. The change of temperature corresponding to 1 ton per sq. inch is here K/E, a quantity given in degrees Fahrenheit in the third column of the table. If the body be unequally heated internal stress will generally be produced, even if there be no external resistance to deformation. In exceptional cases, however, there may be no internal stress, and of these one of the most simple will now be considered. Let us suppose that heat is flowing steadily through a flat plate of thickness y inches in consequence of a difference of temperature t of its faces. The quantity of heat which flows per sq. ft. per minute is where o- is a co-efficient of conductivity given in the fourth column of the table. The outer surface of the plate being hotter than the inner expands more, and if there be no external resistance, the plate bends without internal stress into a spherical form. If R be the radius, reasoning as in Art. 153, p. 307, 1 e t F Curvature = -^ = - = -JT- = n- R y Ky traction diminishes. Fig. 173 shows a pipe of some size through an orifice in the flat end AB of which water is being forced, issuing into the atmosphere. The co-efficient k is found to depend on the proportion which the area of the original orifice A bears to that of -the pipe S, because the smaller S is, the less is the angle of convergence. This has been expressed by an empirical formula due to Rankine which may be written which will be found to give k = '618 when S is infinite, as is nearly the -case for a simple orifice as explained above, while for smaller values k ^Proceedings of the Institution of Civil Engineers, vol. Ixxxiv. h lt h 2 , then it appears from Art. 237 that V_/ ; ,Po-Pi. V_/ 3 ,Po-Pz 2g~ l + w > 2g~' 2 + ~w where p is the pressure on the surface CC. Take now some convenient line DD at a depth Z below the water surface CC, and z lt # , be the elevation of the section above this- datum level so that then the above equations may be written o 9 2g w ~ ] w 2g w " 2 ' This result shows that if u, p, z be the velocity, pressure, and elevation for any section of the pipe, + - + z = Constant. 2g w Each of the terms of this equation represents a particular kind of energy : the first is energy of motion, the third energy of position, the second is energy due to pressure, the origin of which will be further explained in the next chapter. The equation therefore shows that the total energy of the water remains constant as it traverses the pipe, and is accordingly the algebraical expression of the Principle of the Conservation of Energy. It supposes that no energy is lost by frictional resistances, and that any change in the internal motions- of the particles amongst themselves may be disregarded. The word "head," the origin of which we have already seen, is frequently employed for the energy per unit of weight. (See Appendix.) An important consequence of this principle is that where the sectional area of the pipe is least, and consequently the velocity greatest, there the pressure is least. Hence it follows that the velocity cannot exceed a certain limiting value u, found by putting p = 0. At an elevation z above datum level At a greater velocity a negative pressure would be required to pre- serve the continuity of the fluid mass, and under these circumstances the water breaks up with consequences to be hereafter considered. It further appears that water can flow through a closed passage against a difference of pressure, provided the area of the passage vary so as to permit a corresponding reduction of velocity. An. CH. xix. ART. 244.] ELEMENTARY PRINCIPLES. . 477 example of this occurs in the case of the discharge through a trumpet- shaped mouthpiece. In Fig. 175 water enters from a vessel at KK, an orifice provided with a mouthpiece, which first Fig-. 175. contracts to DD, and then expands to EE where ^ the jet enters the atmosphere. The pressure at EE is that of the atmosphere, and therefore at DD is less than that of the atmosphere, that is, less than it would be if the trumpet were cut off at the neck. Hence the discharge is increased by the addition of the expanded portion. If the water issued into a vacuum the jet would not expand to fill the wide mouth of the trumpet, which would not in that case have any influence on the discharge. The increased discharge and partial vacuum at DD have been verified by experiment.* SECTION II. MOTION OF AN UNDISTURBED STREAM. 244. Distribution of Energy in an Undisturbed Stream. Vortex Motion. If the reservoir in the last article be imagined to supply a stream running in a channel of any size either closed or open, that stream, if undisturbed by any of the causes mentioned hereafter, may be supposed made up of an indefinite number of elementary streams, each of which moves as it would do in a closed pipe, as just described, without in any way intermingling with the rest. The forms of these ideal pipes depend solely on the form of the channel in which the stream is confined. The equation '++**+& 2g w w applies to the motion in every pipe, and from it we may draw two important conclusions. In the first place, it may be written in the form p-Po 7 u " . = Zl Z -pr- ') w 2g' and therefore the pressure at any point is less than if the water were at rest by the height due to the velocity at that point. Again, the equation interpreted as in the last article shows that the energy of all parts of the fluid is the same, or, as we may otherwise express it, the energy of the fluid is uniformly distributed. From either way of stating the result it appears that the pressure is greatest where the velocity is least, and conversely. Now, if the water move in curved lines in a horizontal plane, each particle of water is at the instant moving in a circle, and to balance its centrifugal force (Art. 131) the pressure on its outer surface must be greater than * Readers to whom the subject is new are recommended to pass on at once to Art. 248, p. 483. 478 . HYDEAUL1CS. [PARTY. that on its inner. It follows therefore that, if a channel is curved so as to alter the direction of the stream, the pressure increases as we go from the inner side of the channel to the outer ; while, on the other hand, the velocity is greatest at the inner side and least at the outer. The change is the greater the sharper the bend, for the centrifugal force is greater. In open channels the change at the surface where the pressure is constant is in elevation instead of in pressure. The magnitude of the change can be calculated in certain cases (see Appendix), of which we can only here consider one which is of special importance. If the particles of water describe circles about a common vertical axis, the elementary streams will form uniform rings, the centrifugal force of which can be calculated as in Art. 145, page "288. The resultant force on the half ring is employing the notation of the article cited given by F2 P = w.2A. . 9 This is balanced by an excess pressure on the outer surface of the half ring, and if that excess be Ap the -corresponding resultant force is Ap . 2r, as shown on page 302. Equating this to P 4to.!?.4'.>i. 9 r The ring is supposed of breadth unity, and for A we may write the thickness of the ring, which may be called Ar. Dividing by this, and proceeding to the limit dp_w F 2 dr g r an equation from which the pressure can be found if the law of velocity be given. If the fluid rotated about the axis like a solid mass, V would vary as r ; but the case now to be examined is that in which V varies inversely as r, as expressed by the equation Vr = Constant = k. Substitute and integrate, then replacing k by Vr, it will be found that t jL&;sj w 2g w 2g where the suffix refers to a given point where the pressure is p^ and the velocity V . This result shows that the energy is uniformly distributed, and we infer that if the direction of a moving current is changed so that the particles of water describe concentric circles, the velocity varies inversely as the distance from the centre. CH. xix. ART. 245.] ELEMENTARY PRINCIPLES. A mass of rotating fluid is called a " vortex," and in the case just considered the vortex is described as "free," because the motion is that which is naturally produced (comp. Art. 273, p. 534). A free vortex is necessarily hollow, for to hold the water together a negative pressure would be required near the axis of rotation, but the hollow may be filled up by water moving according to a different law. 245. Viscosity. When the motion of a mass of water is free from sudden changes of direction, loss of energy takes place only through the direct action of viscosity, a property of fluids which it will now be necessary briefly to consider. In Fig. 160, page 425, a block of plastic material is represented, and it was explained that to produce change of form a certain difference of pressure was necessary, depend- ing on the hardness of the material. In a fluid a similar difference of pressure is necessary to produce a change of form at a given rate, and the magnitude of the difference is proportionate to the rate. If u be the rate at which the height of the block is diminishing and the breadth increasing, each reckoned per unit of dimension, the thickness remaining constant, p = 2cu, where c is a co-efficient called the "co-efficient of viscosity." Or to express the same thing differently, if w be the rate at which a small rectangular portion of the fluid is distorting, as in Fig. 140, p. 352, q the corresponding distorting stress, q = c . w. Hence, when a fluid moves, any change of form requires an amount of work to be done which is proportionate to the speed at which the change takes place. In a free vortex the rate of distortion is twice the angular velocity of the particles round the axis, and varies inversely as the square of the distance ; the changes of shape are therefore very rapid near the centre, and energy is consequently dissipated much more rapidly than in the stream from which the vortex is produced. In the case of water the viscosity is so small that such changes of form as occur in an undisturbed stream are not rapid enough to absorb any large amount of energy. For example, in the discharge from orifices in a thin plate the loss of head is only 5 or 6 per cent. It is only when the water is disturbed by the neighbourhood of a rough surface over which it moves or in other ways described further on, that large quantities of energy are dissipated and fractional resistances of great magnitude produced. 480 HYDRAULICS. [PART v. 246. Discharge from Large Orifices in a Vertical Plane. When the orifices through which water is being discharged from a reservoir are not small compared with the head and the dimensions of the reservoir, the question becomes more complicated. If the plane of the orifice be vertical the velocities of the several parts of the stream are not the same as in the case, so far as can be judged by the eye, when the orifice is small. On the contrary the velocity of that part of the stream which issues from the lower part of the orifice is visibly greater than that proceeding from the upper part. Hence it follows that the centre of gravity of the fluid issuing in a given time, to which the head is measured, is not on the same level as the centre of the contracted section, but lies below it. The corresponding point on the section may be described as the Centre of Energy. Also the velocity of the centre of gravity of the fluid is not the same as the velocity of mean flow Q/A, and the internal motions of the stream, even when undisturbed, are of sensible mag- nitude and cannot be neglected. To find the discharge therefore we must consider separately each of the elementary streams of which the whole stream may be imagined to be made up, and obtain the result by integration. To illustrate these points let us consider the comparatively simple <;ase of a rectangular orifice ABCD (Fig. 176) from which water is being discharged from a reservoir, the level from which the head is measured being LL. The stream contracts on efflux, and the contracted V ' V .-'...I.. L-.B section may be supposed rectangular. The position and dimensions of this section it will be necessary to suppose known by experiment ; let its breadth be b, and let its upper and lower sides be at depths Y l , Y 2 below LL. Divide the area into horizontal strips, and consider any one at depth y, then the velocity will be given by the formula &~2gy. The quantity discharged per second will be given by }^2 r, which by integration gives This determines the discharge, which is the same as with the mean velocity of flow, 7/ _ Q_ ,- 2 Y% - Y* - CH. xix. ART. 246.] ELEMENTARY PRINCIPLES. 481 The height due to this velocity is 4. /F^_F^\2 ; '=-(T;^)' which corresponds to a point called by Rankine the Centre of Flow, which lies somewhat above the centre of the section. When the head is measured to this point the discharge in the absence of hydraulic resistances is determined as if the orifice were small. Again, the kinetic energy of the water discharged per second will be , U= w\ bv.fT dy = w . b \/2g I Jr> *9 which by integration gives By dividing U by wQ we get the depth of the centre of gravity of the fluid discharged per second below LL, that is to say, the true head h is 5 The velocity of the centre of gravity which is the true velocity of delivery is y= \^y__ , 3 Y* - Y^ '''*'l' arid the energy of translation on delivery is a quantity less than the whole energy wQh by the energy due to internal motions. In attempting to estimate the effect of the internal motions due to hydraulic resistance this method of analysis appears the most exact in principle. Practically, however, it is always necessary to obtain the discharge as above in terms of the dimensions of the orifice itself, and then allow for contraction and hydraulic resistance by a suitable co-efficient of discharge. Some additional examples will be found at the end of the present chapter. Again, if the dimensions of the orifice be not small compared with the surface of the water in the vessel from which the discharge takes place, this surface will sink with a velocity V which is of sensible magnitude. If the area of the surface be S and that of the contracted section A Q , the discharge will be Q = A v = SF, an equation which determines V. The water will now have a velocity V before descending through the height h, and the equation of energy is therefore v* -7' 2 = 2gh. C.M. 2 H 482 HYDRAULICS. [PART v. This may be written if we please ,,2 J72 T A+ 2? showing that in addition to the actual head h we must consider the virtual head V' 2 l^g due to the initial velocity of the water. In many hydraulic questions it is inconvenient or impossible to measure the head from still water. It is then measured from some point where the water is approaching the orifice with a velocity determined by observation. The actual head h must then be increased by the height due to the velocity of approach! 247. Similar Motions,- When an incompressible fluid flows steadily through a pipe of small transverse section it was shown in Art. 243 that the total head is given by the equation while the discharge is Q = Au = A u , where the suffix refers to some given point. Imagine now a precisely similar pipe constructed on an enlarged scale through which a fluid of different density is flowing, and let it be similarly placed relatively to the datum level ; then, if large letters be used to denote the corresponding quantities in the large pipe, To each point in the small pipe will correspond a point in the large one ; then at corresponding points if n be the ratio of enlargement, Z=nz, the sectional areas are in the ratio n 2 : 1, and the velocities must be in some fixed ratio depending on the relative discharge. Let us suppose the velocity-ratio to be Jn : 1, the velocities are then said to correspond. Since in this case U 2 = nu 2 we find from the above equations . o _.. W 2g w Thus at corresponding velocities the difference of pressure-head at any two points of the large pipe is n times the difference at corresponding points of the small pipe. And if the pressure-ratio be n : 1 at any one pair of corresponding points it will be the same at any other pair. In the motion of an undisturbed stream as already explained the complete stream may be analyzed into distinct elementary streams in each of which the flow goes on as it would in an isolated pipe : the CH. xix. ART. 248.] ELEMENTARY PRINCIPLES. 483 forms of these elementary streams depending on the form of the surfaces within which the fluid is enclosed and by the uniformity of pressure on such parts of the surface as are free. Let us now suppose we have two streams, the boundary surfaces and elementary streams of which are similar and similarly placed, the ratio of enlargement being as before n:l. Further let the velocities be in the ratio Jn;}, and the pressure-heads at some one pair of corresponding points in the ratio n : 1 ; then the pressure-heads at every other pair of corresponding points will be in the same ratio, or in other words, the distribution of pressure will be the same. If then we have any actual motion on a small scale it must also be possible on a large scale when the velocities correspond. Such motions are said to be similar. In similar motions at corresponding speeds the distribution of pressure will be the same and conversely. Let us take as an example the discharge of water from an orifice considered in Art. 246. Imagine two tanks, one large the other small, with similar orifices similarly placed with corresponding depths of water so that the heads are in the proportion n : I ; then the principle of similar motions enables us to say that in the absence of hydraulic resistance, the co-efficients of contraction and discharge must be the same in the two cases, the velocities must be in the ratio Jn : 1, and the discharge in the proportion n-^Jn : 1. Strictly speaking however we must suppose the atmospheric pressures in the ratio n: 1, a restriction which is probably not actually necessary (p. 471). Further if we consider any small area in the boundary surface of the small motion and a geometrically similar and similarly situated small area in the large motion, either the total pressures on those areas or their resolved parts in any given direction will when divided by the weight of a cubic foot of fluid be in the proportion n 3 : 1 ; and therefore will be in the proportion of the total weights of fluid in the two cases. The principal application of this very important principle is in the theory of the resistance .of ships : it being equivalent to saying that in similar vessels at corresponding speeds the resistances (if any) when not influenced by causes of the nature of friction, must be in the proportion of their displacements. The needful qualifications of this principle and the mode of making use of it will be briefly noticed in the Appendix. SECTION III. HYDRAULIC RESISTANCES. 248. Surface Friction in General. We now proceed to study experi- mentally some of the more important causes of hydraulic resistance. 484 HYDRAULICS. [PART v. Fig. 177 shows a thin flat plate AB with sharp edges completely Fig.i77. immersed in the water. The plate is moving edge-ways through the water with velocity V, then a certain resistance R is experienced which must be overcome by an external force. This resistance consists in a tangential action between the plate and the water, and so far is analogous to the friction between solid surfaces, but it follows quite different laws, which may be stated as follows : (1) The friction is independent of the pressure on the plate. (2) It varies as the area of the surface in contact with the water. (3) It varies as the square of the velocity. These laws are expressed by the formula where / is a co-efficient which, as in the friction of solid surfaces, is described as the " co-efficient of friction." The value of this co-efficient depends on the degree of smoothness of the plate. Thus, for example, in some experiments, to be described presently, on thin boards moving through water it was found that the co-efficient was '004 for a clean varnished surface, and -009 for a surface resembling medium sand- paper, the units being pounds, feet, and seconds. The first of these laws, so far as is known at present, is always strictly fulfilled, but to the second and third there are certain limi- tations, as in the ordinary laws governing the friction of solid surfaces. In the first place, if the velocity be below a certain limit the water adheres to the surface, arid its velocity relatively to the surface is some continuous function of the distance from the surface so that the stream does not break up. This will be further referred to hereafter; for the present it is sufficient to say that the resistance then follows an entirely different law, varying nearly as the velocity instead of the (velocity) 2 . The limiting velocity, however, at which this is sensibly the case is so low that in most practical applications the effect may be disregarded. In the second place, it is supposed that the water glides over all parts of the surface with the same velocity ; but if the surface be any considerable length the friction of the front portion of the surface on the water furnishes a force which drags the water forward along with the surface and so diminishes the velocity with which it moves over the rear portion. The friction is thus diminished, and in large surfaces very considerably diminished. Thus Mr. Froude, experimenting on a surface 4 feet long, moving at 10 feet per second, found the value of / given above, but when the length was 20 feet and upwards, those values were diminished to '0025 and *005 respec- CH. xix. ART. 248.] ELEMENTARY PRINCIPLES. 485 tively. Increasing the length beyond a certain amount produces very little change, and within a certain limiting length the effect is insen- sible. These limits must depend on the speed, but no exact observa- tions have been made on this point. The power of the speed to which the friction is proportional has, however, been found to be diminished on long smooth surfaces, as shown below. The skin friction of vessels, on which the resistance chiefly depends at low speeds, is much diminished by the effect of length. Experiments on surface friction were made by Colonel Beaufoy. They formed part of an elaborate series of experiments on the resist- ance of bodies moving through water, carried out during many years in the Greenland Dock, Deptford. Beaufoy employed the formula to represent his results, and for the index n obtained the values l - 66, 1*71, 1'9 in three series of experiments. The standard experi- ments on the subject are however due to the late Mr. Froude : they were made on boards f\ inch thick, 19 inches deep, towed edgeways through the water. The boards were coated with various substances so as to form the surface to be experimented on. The following table gives a general statement of Froude's results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resist- ance at other speeds is easily calculated. Length of Surface, or Distance from Cutwater, in Feet. Nature of Surface. 2 Feet. 8 Feet. 20 Feet. 50 Feet. A B c A B C A B C A B C Varnish, 2-00 41 390 1-85 325 264 1-85 278 240 1-83 250 226 Paraffin, - 1-95 38 370 | 1-94 314 260 1-93 271 237 Tinfoil, - 216 30 295 1-99 278 263 1-90 262 244 T83 246 232 Calico, - 1-93 87 725 1-92 626 504 1-89 531 447 1-87 474 423 Fine Sand, 2-00 81 690 2-00 583 450 2-00 480 384 2-06 405 337 Medium Sand, 2-00 90 730 i 2-00 625 488 2-00 534 465 2-00 488 456 Coarse Sand, - 2-00 110 880 ! 2-00 714 520 2-00 588 490 Columns A give the power of the speed to which the resistance is approximately proportional. Columns B give the mean resistance per square foot of the whole surface of a board of the lengths stated in the table. Columns C give the resistance in pounds of a square foot of surface at the distance stern ward from the cutwater stated in the heading. 486 HYDRAULICS. [PART v. To the three laws already mentioned may be added : (4) In different fluids the friction varies as the density of the fluid. The grounds for this statement will be seen further on. It amounts to saying that surface friction is a kind of eddy resistance (p. 504). If we assume this, the laws of friction between a fluid and a surface are expressed by the equation The co-efficient of friction / is now distinguished from the friction per square foot given in the table above. We have already seen that it is not constant, and it is now known that in addition to the circumstances already mentioned, it varies according to the tem- perature of the fluid, diminishing in water apparently as much as 1 per cent, for each 5 F. rise of temperature. 249. Surface Friction of Pipes. When water moves through a pipe the friction of the internal surface causes a great resistance to the flow. Fig. 178 shows a pipe of uniform transverse section (not necessarily circular) provided with two pistons, AB, A'B', at a distance x enclos. Fig.178. ing between them a mass of _ A _ V _ water. The pistons and in- p 1 eluded water move forward _____________ together with velocity v under the action of a force E, required on account of the friction of the pistons and of the water on the pipe. Omitting piston friction the force R will be given by T> r c< r R =fw8- =f. wsx, 2g ' '2g where S is the wetted surface and .s- the perimeter. If we imagine the pipe full of water moving through it with velocity v, the force R is supplied by the difference of the pressures p, p' on the pistons, and therefore, if A be the sectional area The quantity A/s may be replaced by m and is described as the " hydraulic mean depth " of the pipe, a term derived from the case of an open channel to be considered hereafter. In the ordinary case of a cylindrical pipe m = ^d. Further, we may reduce the pressures to feet of water by dividing by w, and thus obtain for the difference of pressure h' -i, f X V 2 fi = / . , m '2g CH. xix. ART. 250.] ELEMENTARY PRINCIPLES 487 the value of the co-efficient / being determined by special experiment on pipes. This formula for the head necessary to overcome surface friction is continually in use. The formula gives directly the head necessary for a length x of the pipe, when the water, by being enclosed between pistons, is constrained to move over the surface with a given velocity : when the pistons are removed and the water flows freely it represents the facts very imperfectly. The central parts of the stream move quicker than the parts in immediate contact with the pipe, and besides, though the circumstances are different, we cannot be sure that the velocity over the internal surface is not affected in the same way as in the case of a moving surface. The value of / has therefore to be obtained by special experiment, and the results of such experiments show that it varies very greatly according to the condition of the internal surface, and partly also on the diameter and velocity, the value being greater in small pipes than large ones, and at low velocities than high ones a point considered further on. (See page 495.) For the present we assume -0075 as roughly repre- senting the facts when there is no special cause for increased resistance. For a pipe of circular section, length I, we have therefore where for 4/ we commonly assume the value '03. 250. Discharge of Pipes. The velocity v is the actual velocity with which the water moves, so that i^/Zg is the energy of motion of each pound of the water. The loss of energy by friction is the same as that of raising the water through a height h\ and is therefore equal to the energy of motion when H.= 33 nearly, that is, a length of pipe equal to 33 diameters absorbs an amount of energy equivalent to the whole energy of motion of the water. In pipes of any length, therefore, the effect of friction is very great, so much so that the size of a pipe is principally fixed by the loss of head which can be permitted. It is easily seen that to deliver water with a given velocity the loss varies inversely as the diameter, and that to deliver a given quantity it varies inversely as the fifth power of the diameter; thus, the smallest permissible diameter is fixed almost entirely by the value of h', which may be supposed already known. 488 HYDRAULICS. [PART v. The quantity discharged per second is given us by the formula Q = Av and on substitution this becomes All dimensions are here in feet, and Q is in cubic feet per second. If we require gallons per minute for a diameter of d inches, the formula will be &=G m ^ f where C is a constant connected with 4/ by the equation 4-736 For 4/=-03, this gives (7=27'3, but for clean iron pipes not less than 9 inches in diameter the value 30 may be employed. 251. Open Channels. Keturning to Fig. 178, suppose the pipe, instead of being horizontal, is laid at an angle 6 (see Fig. 179, next page), so that the difference of level of the two ends in y = I . sin Q y then the difference of pressure-head is p -p - I v 2 r - 1 = f _ __ nj w J 'm'2g yt and therefore may be made zero if the slope of the pipe be . 1 tf h' sm0=/. - x- = T . in 2g I But if the pressure be constant we may remove the upper surface of the pipe and thus obtain the case of an open channel. The quantity m is now the sectional area of the channel divided by the wetted perimeter, and is therefore the actual depth in a very board shallow channel, but in other cases less in a ratio dependent on the form of section. As before stated it is described as the "hydraulic mean depth " of the channel. We can now find the velocity and discharge of a stream of given dimensions and fall, provided that we know the value of /, or conversely the size of channel for a given discharge and fall. The value of/, however, varies for the same reasons as in pipes which indeed apply with still greater force, so that the limits of variation are wider. The average value does not differ very widely from -0075, already adopted for pipes; but to obtain results of even moderate accuracy a special study of the experiments on the subject is necessary, which will not be attempted in this treatise. CH. xix. ART. 253.] ELEMENTARY PRINCIPLES. 489 252. Virtual Slope of a Pipe. If the pipe be laid at any other angle the pressure will not be constant, and the mode in which it varies is best seen by a graphical construction. Suppose small vertical pipes Aa, Bb to be placed at points A, B of the pipe we are considering (Fig. 179), then (if they enter the water square, without being bent pi J7g towards the direction of motion) the water will rise in them to a level representing the pressure in feet of water at these points. If there were no fric- tion the level would be the same in both and the difference (bk in the figure) therefore represents the loss by friction. Now draw a horizontal line through b, and take c on it, so that ac = AI> = l, then the angle caN is given by the equation h sin ^ = y, and is therefore the slope of a channel of the same length and hydraulic mean depth which would give the same discharge. This angle is therefore called the VIRTUAL SLOPE of the pipe. At any point P in the pipe, the water would rise to the level of the corre- sponding point p in the virtual channel, found by taking ap = AP. The construction would of course fail if h' were equal to, or greater than I, but this case does not occur in practice ; on the contrary, in pipes as in channels the angle i is nearly always small. The virtual slope is frequently one of the data of the question. The line ac is- variously described as the "pressure line," "line of virtual slope," or "hydraulic gradient." The pipe need not be straight; it may be curved or be laid in sections at different slopes, there will still be a continuous hydraulic gradient, provided the diameter be the same throughout ; but if the sections be of different diameters each section will have its own slope. In practice care must be taken that the pipe does not rise above its hydraulic gradient, for otherwise there will be a partial vacuum : the pipe then acts as a syphon which is liable to fail on account of leakage and the presence of air in the water. 253. Loss of Energy by Eddies and by Broken Water. We now proceed to consider other causes of frictional resistance. In Fig. 180 two streams of water, moving with different velocities, 490 HYDEAULICS. [PART v. converge towards each other and unite into one. Each stream, so far as can be judged by the eye, moves originally without disturbance in the manner described in Art. 244. On union, however, near the junction indicated by the dotted line SS in the figure, small depres- sions are observed, which move for some distance along with the stream, and then disappear. On examination these depressions are found to consist of small portions of the fluid in a state of rotation, the speed of rotation being greatest at the centre and gradually dying away towards the circumference. A motion of this kind was called a " vortex" in Art. 244, and in the present case is also described as an " eddy " ; it is independent of the general motion of the stream, and its energy is therefore of the internal kind. The disappearance of the eddies thus formed is due to viscosity, the effect of which is much greater in the eddy than in the stream as already explained. After the eddies have disappeared the two streams are found to have become a single one, moving with a velocity intermediate between those of the streams which form it, but possessing less energy. Theoretically there is nothing to prevent two streams of a perfect fluid from moving side by side with different velocities, but such a motion is always unstable, and will not long continue without the formation of eddies by a sudden change of direction (Art. 244) in small portions of the fluid which separate from the rest. The instability is greater the more nearly perfect the fluid is. Whenever the water in motion inter- mingles with water at rest, or moving with a different velocity, internal motions of a complex kind are produced, representing a considerable amount of energy of the internal kind which is virtually lost even before its final dissipation by fluid friction. Again, in order that a mass of water may form a continuous whole, sufficient pressure must exist on the bounding surface to prevent the pressure at any point within the mass from becoming zero, as explained in Art. 243. If this condition is not satisfied the water breaks up more or less completely, and the result is a confused mass with complex internal motions rapidly disappearing as before by fluid friction. When waves break on a beach, or when paddles strike the water and drive it upwards in a mass of foam, the process takes place on a large scale before our eyes ; but the same thing occurs in most cases where the velocity of a mass of water is suddenly changed, and of this we will now consider some examples. CH. xix. ART. 253.] ELEMENTARY PRINCIPLES. 491 Fig.lSla. Fig. I8la shows a jet of water filling a tank. Here the water pouring in possesses the kinetic energy Wv~j'2g due to the original velocity of the water, and the height from which it falls into the tank. If it be of some size as compared with the tank the water will be com- pletely broken up ; if it be small it will penetrate the water in the tank without much apparent disturbance at the surface : in either case the result is a mass of water at rest as a whole, so that its energy is all of the internal kind. If the jet be shut off" the water rapidly settles down to rest, the whole energy is then dissipated by fluid friction. Fig. 1816 shows a bucket moving horizontally, bottom foremost, with velocity V, while a horizontal jet moving with greater velocity strikes Fig.isit. it centrally : the bucket is then filled with broken water which .pours out under the action of gravity. In water-wheels a series of buckets are filled in succession, and the broken water carried on with the wheel. Here if the bucket were at rest the loss of energy would be, as before, Wv 2 /2g : but as it is moving with velocity V, the striking velocity on which the breaking depends will be v - V, and the loss of energy is where W is the weight of water acted on in the time considered. Both these cases may be treated as examples of the collision of two bodies considered on page 270, one of the bodies being indefinitely great. The energy of collision is employed in breaking up the water. It is represented in the first instance by internal motions, and sub- sequently dissipated by fluid friction. Fig. 182 represents a pipe which is suddenly enlarged from the diameter ed to the diameter ab. The water is moving through the 492 HYDEAULICS. [PART v. small part of the pipe with velocity v, and, on passing through ed spreads out so as to fill the larger part. At some distance from the enlargement it moves in a continuous mass with velocity V, but in its immediate neighbourhood we have broken water, as in the case of the bucket, from which it only differs in the enclosure of the water in a casing. The loss or energy per unit of weight may be expected to be the same as before, and is therefore a formula which gives us the "loss of head." If the sectional areas- of the two parts of a pipe be A, a the discharge is so that if m be the ratio of areas, m The coefficient of resistance is therefore (m-lf or (1-1/m) 2 , according as the velocity to which it is referred is that in the large pipe or that in the small one. Instead of the water moving from a small pipe into a large one, we may have the converse case of a suddenly contracted pipe as in Fig. 183. The loss here is due to precisely the same cause, namely a sudden enlargement, which is produced as follows. In the figure the stream of water moving with velocity u contracts on passing through Fig. 183. cd nearly as it would if the small part of the pipe were removed, as in Fig. 173, p. 474, until it reaches a contracted section KK, and is then moving with a velocity v which is greater than u in the ratio of the area of the large pipe to the contracted area KK. The loss of head in this part of the process is not large. After passing KK, however, an expansion takes place to the area of the small pipe, and this is accompanied by breaking up, the space between the contracted jet and the pipe being filled up with broken water. In Fig. 1 84 we have the extreme case, in which the large pipe CH. xix. ART. 253.] ELEMENTARY PRINCIPLES. 493 is a vessel of any size. We thus obtain the case of a pipe with square-edged entrance which has already been referred to .^^W\\llll/l . Fig.i84. in Art. 242. Another modi- fication is that of a diaphragm in a pipe, as in Fig. 1 85. The small pipe is here larger than the orifice through which the water enters, and in the figure we have simply a single pipe divided into parts by a diaphragm with an orifice in the centre. The stream of water, after passing the contracted section KK, expands to fill the pipe. In cocks when partially closed, a loss of head of the same kind occurs, which may be increased to any extent by closing the cock further. In all these cases the loss of head may be calculated approximately by means of the formula for a sudden enlargement, but the ratio of enlargement is not known exactly, on account of the uncertainty of the value of the co-efficient of contraction to be as- sumed. Losses of head v of this kind are indeed always subject to varia- tion within certain limits from accidental -causes; in general and on the average the quantity of water broken up will bear a certain proportion to the whole quantity passing, and in consequence we have the general law of hydraulic resistance stated on page 472, but the ratio may vary from time to time, and cannot be stated with precise accuracy. The causes of this uncertainty will be clearly understood on considering somewhat more closely the manner in which the loss takes place. In Figs. 183, 185 two plane surfaces at right angles meet at a, forming an internal angle through which water is flowing. The particles of water there describe curves which are all convex towards ffi, and in conformity with the general principle explained in Art. 245, the pressure must increase and the velocity diminish on going towards a. The water then moves slowly and quietly round the angle without disturbance. But when compelled by the general movement of the stream to move round an external angle such as kea in Fig. 182, the case is very different ; the particles then describe curves which are 494 HYDRAULICS. [PART v. concave round e ; and consequently the pressure diminishes in going towards e, while the velocity increases. To hold the particles of water in contact with the surface, an infinite pressure would be required in the other parts of the fluid. The particles of water therefore leave the surface at e, and describe a path ea', regaining the surface farther on ; ea' is then described as a " surface of separation," as it separates the moving mass of water from a portion enclosed within it which is in a state of violent disturbance. Such are the surfaces shown in Figs. 182-186. It is not, however, to be supposed that these surfaces are sharply defined, and that they permanently separate different masses of water. On the contrary, no such equilibrium is possible ; the surfaces are continually fluctuating, and a constant interchange takes place between the so-called " dead " water and the stream. In this intermingling eddies are produced nearly as in the comparatively simple case of two streams given on page 490. The process is always essentially the same, and consists in sudden changes of direction being communicated to parts of the stream which become detached from the rest. 254. Bends in a Pipe. Surface Friction. In some other cases the process of breaking up by which energy is lost is less obvious, and the ratio is subject to greater variations. When a pipe has a bend in it, if the internal surface of the pipe were perfectly smooth and free from discontinuity of curvature, there would be no disturbance of the current of water, which would flow as described in Art. 248. These conditions, however, are not satisfied by actual bends in pipes, and there is always a loss of head due to them in addition to the loss by surface friction. This loss can only be determined by experiment, but it is easy to conjecture that the loss will be proportional to the angle through which the pipe is bent, and that it will be greater the quicker the bend, that is, the smaller the radius of the bend is as compared with the diameter of the pipe. The extreme case of a bend is a knee, but the loss is not in this case proportional to the angle of the knee, but follows a complex law. For details respecting bends and knees the reader is referred to the treatises cited at the end of this chapter, but some common examples are given in the table on page 496. In the case of surface friction the loss of energy is represented in the first instance by eddies formed at the surface and thrown off. In almost all practical cases of the motion of water in pipes and channels, even when to all outward appearance quite undisturbed, the fluid is in fact in a state of eddy motion throughout, and dissipation of energy at CH. xix. ART. 254.] ELEMENTARY PRINCIPLES. 495 every point is going on much more rapidly than would be the case if the motion were of the simple kind described in Art. 248. The quantity of water broken up, however, is not generally in a fixed proportion to the quantity passing, for reasons already partly indicated in Art. 249. In the first place, as in the case of a board moving edgewise through water, the friction per sq. ft. is proportional to the n il1 power of the velocity, where n is an index which, in smooth surfaces, is somewhat less than 2. Secondly, the disturbance caused by the friction at a given velocity is less at some distance from the surface than in its immediate neighbourhood, and hence the central portion of the water in the pipe is less disturbed than the boundaries, and that the less the greater the size of the pipe. The loss of head therefore at a given velocity is less in large pipes than small ones. The various experiments on the discharge of pipes have been very thoroughly examined by Professor Unwin,* who has shown that they are represented very closely by a formula originally given in a slightly different form by Hagen, , I v*~ y where x and y are small fractions measuring the deviation from the . simple formula already used, and /x is a co-efficient. The values of fji, x, y stated below are selected from a number of cases given by Professor Unwin in the paper already cited : the values of ^ being for diameters in feet. KIND OF PIPES. f* x y "Wrought Iron (Gas), 0226 21 25 New Cast Iron, 0215 168 05 Cleaned Cast Iron , - 0243 168 o Incrusted Cast Iron, - 044 16 o The result for cleaned cast iron is equivalent to taking in the simple formula -n^ 4/=-p, (d in inches) Ju> a formula which may be used for any case of a clean surface not of the smoothest description. In the smoothest surfaces the value of /A is slightly smaller, and the velocity must also be considered; but as in the case of open channels a special study of the experiments on the subject is necessary to obtain fairly accurate results. In very rough surfaces the value of 4/ may be doubled. The value '03 employed in preceding articles, allows a certain margin for incrustation, except in pipes less than 2 or 3 inches in diameter. For Darcy's formula see Appendix. * Formulae for the Flow of Water in Pipes. Reprinted from Industries, 1886. 496 HYDRAULICS. [PART v. 255. Summation of Losses of Head. The total loss of energy due to a number of hydraulic resistances of various kinds is found by adding together the losses of head due to each cause taken separately. The velocity of the water past each obstacle will not generally be the same for all, and it is then necessary to select some one velocity from which all the rest can be found by multiplication by a suitable factor for each obstacle. If n be this multiplier the loss of head will be where V is the velocity selected for reference. The value of Fis then found for motion under a given head H by the formula The various values of F already given are collected with some additions in the annexed table : CO-EFFICIENTS OF HYDRAULIC RESISTANCE. NATURE OF OBSTACLE. VALUE OF F. REMARKS. Orifice in a Thin Plate. 06 Square-edged Entrance of a Pipe. 5 Sudden Enlargement of a Pipe in the ratio m : 1. (m-1) 2 Referred to Velocity through large part of Pipe. Bend at right angles in a Pipe. 14 Radius of Bend = 3 x Diameter of Pipe. Quick Bend at right angles. 3 Radius of Bend Diameter of Pipe. Common Cock partially closed. 75, 5-5, 31 Handle turned through 15, 30, 45 from position when fully open. Surface Friction of a Pipe the length of which is n times the diameter. 4/. n For a clean Cast-iron Pipe d inches diameter. . '036 4/=-6/T- \'d Knee in a Pipe at right angles. Unity. In Bends the coefficient is pro- portional to the angle of the Bend, but in Knees the law is much more complex. H. xix. ART. 256.] ELEMENTARY PRINCIPLES. 497 SECTION IV. PRINCIPLE OF MOMENTUM. 256. Direct Impulse and Reaction. The generalized form of the second and third laws of motion, described as the Principle of Momentum in Chapter XL of this work, may be employed with great advantage when the motion of water in large masses is under consideration, because the total momentum of a fluid mass depends solely on the motion of the centre of gravity (p. 267), and not on the very intricate motions of the parts of the fluid amongst themselves. Further, the energy dissipated by fractional resistances is accounted for by these internal motions, or by the mutual actions of the fluid particles, and the total momentum is therefore independent of these resistances. Hence it follows that results may be obtained which are true notwithstanding any frictional resistances, and in some cases the loss of energy by them may be determined a priori. Also the pressures on fixed surfaces may be found which do no work, and to which therefore the principle of work does not directly apply. Fig. 186 shows a jet of water striking perpendicularly a fixed plane of infinite extent, and exerting on it a pressure P. The magnitude of this pressure is found by considering that the plane exerts an equal and opposite pressure on the water, which changes its velocity. The water originally moving with velocity v, spreads out laterally, and any motion which it possesses is parallel to the plane. In time t the impulse is 7 J /, and the change of momentum is Mvt, where M is the mass of water delivered per second. Equating these we have where W is the weight of water delivered per second. If the plane be smooth, and gravity be neglected, the motion of the water will be continuous ; but if it be rough to any extent, so that breaking-up occurs, the result will still be correct, provided only the roughness be symmetrical about the axis of the jet. And the action of gravity parallel to the plane does not affect the question. In Fig. 187 we have the converse case of water issuing from a vessel with a lateral orifice. Here the water, which originally was C.M. 2 1 498 HYDRAULICS. [PART v. Fig.187. at rest, issues with velocity v, and the momentum generated in time t is Mvt. To produce this momentum a corresponding impulse is required, which is derived from the resultant horizontal pressure P of the sides of the vessel upon the water. We have as before Wv A pressure equal and opposite to P is exerted by the water on the vessel this is described as the " reaction " of the water ; and if the vessel is to remain at rest, must be balanced by an external force supplied by the supports on which it rests. A remarkable connection exists between the change of pressure on the sides of the vessel consequent on the motion and the co-efficients of contraction and resistance. First, suppose the water at rest, the orifice being closed, then the value of P is zero, and the pressure on the area of the orifice is w . A . h, the notation being as in Art. 240. When the orifice is opened the pressure on that side is diminished, first, by the quantity w . A . h ; secondly, by an unknown diminution S due to the motion of the water (p. 477} over the surface near the orifice. Now . A . h= the notation still being as in the article cited. Replacing A by kA we obtain Since S is always positive the least value of k is If there be no frictional resistance k= - 5, and this is the smallest value k can have under any circumstances. For a small pipe projecting inwards as in Fig. 172, p. 473, these conditions are approximately realized, the water being at rest over the whole internal surface of the vessel. Fig.188 257. Oblique Action. Curved Surfaces. When a jet impinges obliquely on an indefinite place (Fig. 188), the water spreads out laterally as before, but the quantity varies according to the direction. In the absence of friction the velocity of individual particles is the same as that of the jet in whatever direction the water passes. At the same time the velocity of the whole mass of water parallel to the plane cannot CH. xix. ART. 257.] ELEMENTARY PRINCIPLES. 499 be altered by the action of the plane, and is therefore v . cos 6, where is the angle the jet makes with the plane. It immediately follows that any small portion of water diverging from the centre of the jet at an angle < with the jet must be balanced by another portion diverging in the direction immediately opposite, and the quantities so diverging must be in the ratio 1 - cos : 1 + cos , being inversely as the changes of velocity parallel to the plane. But if the circum- stances be such that breaking-up takes place, the motion of the water parallel to the plane will be undetermined, and in general there will be a tangential action on the plane of the nature of friction. The normal pressure on the plane is in all cases the same, being given by the formula W P = Mv .sin 6 = .v. sin 6. g If the surface on which the water impinges be curved it is necessary to know the average direction and magnitude of the velocity with which the water leaves the surface. In the absence of friction, as already noticed, the velocity of the individual Pig-189 particles is unaltered unless the water be enclosed B r in a pipe so that the pressure can be varied a case for subsequent consideration ; the direction, however, will depend on the way in which the water is guided. In cases which occur in practice it will generally be found either that the whole of the water is guided in some one direction, or that it leaves the surface in all directions symmetrically. Taking the first case, suppose the original velocity (v) of the water to be represented by OA (Fig. 189), and the final velocity to be diminished to V by friction, and altered in direction so as to be represented by OB. Then the change of velocity in the most general sense of the word (p. 265) is represented by AB. If this be denoted by v the change of momentum per second is g The resultant pressure on the surface is parallel to AB and numerically equal to P. In applications to machines the curved surface is frequently a vane which is not fixed, but moves with a given velocity ; the pressure can then be found by a simple addition to the diagram. Through draw 00', representing the velocity (u) of the moving surface in direction and magnitude, then 0' A represents the velocity with which 500 HYDKAULICS. [PART v. the water strikes the surface. Considering the vane as fixed, the velocity is now estimated with which the water would leave it, and O'B' drawn to represent it : the change is now AB' instead of AB. If the absolute velocity is required with which the water leaves the surface, it may be found simply by joining OB', which will completely represent it ; the change of velocity being AB', whether the velocities are absolute or relative to the moving surface. The cup vane AC A (Fig. 190), against which a small jet of water impinges centrally, may be taken as an example where the water spreads in all directions symmetrically. If OA be tangent to the vane at A , making an angle 6 with the centre line of the jet, the water leaves the vane in the direction OA with unaltered velocity (neglecting friction). The resultant pressure P is in the direction of the jet, and the velocity in that direction is altered from v to v cos in the opposite direction, so that the change of velocity is v(l + cos 0). Thus we have 258. Impulse and Reaction of Water in a Closed Passage. When the water is moving in a closed passage the resultant pressure to be considered in applying the principle is not merely that on the sides of the passage, but also that on the ideal surfaces which separate the mass of water we are considering from the complete current. In the previous cases the pressure of the atmosphere on the free surface bounding the fluid was the same throughout, and was balanced by an equal pressure of the surface against which it impinges, which is not included in the preceding results. This is now no longer the case. An important example is that of the sudden enlargement in a pipe already referred to in Art. 253. In Fig. 182, page 491, take ideal sections, KK, kk of the large and small portions of the pipe, and consider the whole mass of water between them. This mass is acted on (1) by the pressure (p) on the transverse section kk, (2) by the pressure (P) on the transverse section KK, and (3) by the pressure of the sides of the pipe. If we resolve in the direction of the length CH. xix. ART. 259.] ELEMENTARY PRINCIPLES. 501 of the pipe, the only part of (3) which we need consider is the pressure (p) on the annular surface, ae, bd, the area of which is A - a, and the whole resultant pressure is therefore PA -pa -p' (A a) in the direction opposite to the motion of the water. Now let W be the weight of water delivered in one second, then in that space of time W passes from the small pipe, where its velocity is v, to the large pipe, where it has a velocity V, so that if we equate the resultant pressure to diminution of momentum, W i PA-pa-p>(A-a) = ^(v-V=t a formula which may be written p v V(v-V\ '- w w g u m being as in Art. 253 the ratio of enlargement. Let now H be the total head in the large pipe and h in the small one, then subtracting (v 2 - F 2 )/2g from both sides and re-arranging the terms Zg W \ 171; Comparing this result with that obtained in the article cited, it appears that the value of the loss of head there given is a necessary consequence of supposing p =p', but cannot otherwise be correct. That the pressure in the broken water at ae, bd is nearly equal to the pressure in the small pipe may be considered probable a priori, independently of the experimental verification which the formula has received. SECTION V. RESISTANCE OF DEEPLY IMMERSED BODIES. 259. Eddy Resistance. The subject of the resistance of ships is outside the limits of this treatise, for the ship moves on the surface of water, exposed to the atmosphere, on which waves are produced ; whereas in the branch of mechanics now under consideration, the water is supposed to move within fixed boundaries. A certain part of the subject, however, may properly be considered as belonging to Hydraulics. If a body be deeply immersed in a fluid, that part of the fluid alone which is in its immediate neighbourhood will be affected by its motion, and the question is not essentially different from the cases already considered of the movement of water in pipes and channels. Fig. 191 shows a parallelepiped abed moving through water in the direction of its length, the face cd being foremost. To an observer 502 HYDRAULICS. [PART v. Fig. 191. whose eye travels along with the body the water will appear to move past the solid in a stream of indefinite extent. At some distance away the action of the solid is insensible, but it becomes in- creasingly great as the solid is approached, and is greatest for that part of the water which moves in immediate contact with it. At c and d eddies are formed in passing round the corners exactly as in the case at the same points in Figs. 183, 184 the stream in fact is suddenly con- tracted in the same way as in passing from a large pipe to a small one, the diminution of area in this case being the transverse section of the solid. After this the water moves in actual contact with the solid until it reaches the corners ab, when it describes the curves aS, bS, meeting in S (see p. 494), after which it forms a continuous stream as before. The two curves enclose between them a mass of eddying water exactly similar to the eddies at a and b in Fig. 182 the stream, in fact, suddenly expands, just as in passing from a small pipe to a large one, the increase of area being in this case the sectional area of the solid. The eddies thus formed during the passage of the solid through the water absorb energy, which must be supplied by means of an external force, which drags the body through the water. This kind of resistance to the movement of a body through water is called Eddy Resistance, and may be almost entirely avoided by employing " fair " forms, that is by avoiding all discontinuity of curvature in the solid itself, and in the junction of its surface with the direction of motion. The way in which it is created by the action of the eddies will be discussed further on. A general formula for eddy resistance is derived thus. As already stated the water suffers no sensible disturbance at a certain distance from the solid. If then we imagine a certain plane area A attached transversely to the solid, and moving with it, all the water affected by the solid will pass through this plane, and its quantity will be QAF, where V is the velocity. In similar solids this area must be pro- portioned to the sectional area S of the solid, so that we write A = cS, where c is a constant depending on the form. Of this water a certain fraction will be disturbed by eddies, and the velocity of each particle H. xix. ART. 259.] ELEMENTARY PRINCIPLES. 503 of water will be some fraction of the velocity of the solid. Hence it follows that the energy U generated per second in the production of eddies must be where c' is a co-efficient. Now this amount of energy is generated by means of a force which drags the solid through the water, at the rate of V feet per second, notwithstanding an equal and opposite resistance E. We have then RV=cc'wS.~, or dividing by V, and replacing cc' by a single constant k, The co-efficient k is to be determined by experiment for each form of solid. In the case of the parallelopiped shown in the figure, the value of k depends little on the length, unless it be so short that the eddies at the corners cd coalesce with those in the rear of the solid, and it then becomes the same as that of a plate moved flatwise. Further it is nearly the same, if the transverse section be circular instead of square, and does not greatly differ from unity. For the flat plate it is greater and may be taken as 1 *25. It must be remarked, however, that resistance of this kind is very irregular, and may vary considerably even in the course of the same experiment. To reach a permanent regime it is necessary that the velocity should be perfectly uniform through a run of considerable length, a condition most nearly attained in the experiments made by Beaufoy (p. 485), and recently by Mr. R. E. Frpude at the Admiralty works. Their results are 1'13 and I'l respectively, but by some authorities much larger values are given. The same remarks apply to the case of a sphere for which the value may be taken as about '4. For a cylinder moving per- pendicular to its axis it is probably about '5. In all cases the value of k is independent of the units employed. It is also to a great extent independent of the kind of fluid, being roughly approximately the same for example in air as in water ; but this would not hold good for fluids of very different viscosity ; nor is it even approximately true for high speeds in air, because the compressibility of the air affects the question. The same remarks apply to the co-efficient (F) of hydraulic resistance employed above. It has been found that co-efficients of surface friction are greater in salt water than in fresh in the ratio of the densities of these 504 HYDRAULICS. [PART v. fluids, as we might anticipate, since surface friction is a kind of eddy resistance. Let us now consider more particularly the way in which the resistance is produced. When a solid rests in any given position in a fluid the resultant horizontal pressure over the whole surface is zero, or in other words, if the solid be divided by any vertical plane the resultant pressure on the rear half is equal and opposite to that on the front. When the solid is set in motion in a given direction, the current of fluid passing it is separated by it into parts, which may be regarded as distinct streams having a single point or a line of points on the front of the solid at which the division takes place. At these dividing points the fluid is reduced to rest relatively to the solid, and (p. 477) the pressure there exceeds the hydrostatic pressure which would exist were the solid at rest by the quantity wV^fog. As each stream gliding over the surface moves away from the points of division its velocity in- creases, and consequently the excess pressure diminishes, till at length at a certain distance it vanishes. Over a certain area, then, in front of the solid, the resultant horizontal pressure is in excess of that which would exist were the solid at rest. Now, in the absence of eddies, the streams on uniting again behind the solid would be brought to rest at one or more points of union lying in corresponding positions on the hinder surface, and in consequence there would be a corresponding excess pressure behind which would be found exactly to balance the excess in front, so that there would be no resistance to movement. Take, for example, a solid, the front and rear of which are exactly alike ; if there were no dissipation of energy of any kind, the motion of the fluid in front and rear would necessarily be the same, for no alteration is conceivable merely by reversing the direction of movement. The difference between front and rear consists in the instability of the motion in the rear, in con- sequence of which the streams do not fully unite on the surface of the solid, but leave a space between filled with eddies which lower the pressure there, reducing it in general below the hydrostatic pressure which would exist were the solid at rest. Any eddies which are pro- duced at sharp corners like c, d (Fig. 191) lower the pressure in the streams, and the reduction is ultimately transmitted to the rear of the solid, and takes effect in the same way. There is a strictly analogous difference between the motion in a pipe through a sudden contraction (Fig. 183) and a sudden enlargement (Fig. 182). The co-efficient k is frequently regarded as the sum of two parts m and 11, of which the first represents the plus pressure in front, and the CH. xix. ART. 260.] ELEMENTARY PRINCIPLES. 505 second the minus pressure or suction in the rear ; the terms plus and minus being used with reference to the hydrostatic pressure which would exist were the solid at rest. The eddies have little influence on the co-efficient m, which, when the motion is perfectly steady and uniform, is necessarily less than unity, and can in many cases be approximately calculated ; they chiefly affect the co-efficient n, which (on the same supposition) would otherwise be equal and opposite to m, but actually has a certain value only capable of being determined by experiment, or inferred from its value in some similar case (Ex. 8, p. 510). It has a maximum possible value depending on the depth of immersion, for the minus pressure evidently can never be greater than the hydrostatic pressure due to the depth. 260. Oblique Moving Plate. The case of a flat plate moving obliquely through a fluid may now be briefly mentioned, being of great technical importance. The plate, in the first instance, is sup- posed rectangular, of indefinite breadth, and immersed in an infinite fluid, through which it moves in a line perpendicular to its longer side. Turning to Fig. 188, p. 498, suppose the jet represented to be of indefinite breadth, perpendicular to the plane of the paper, then the difference between this and the present case consists in the isolation of the jet and the infinite extent of the plane. These circumstances, however, make no difference in the character of the motion in front of the plane ; the current of fluid passing is still divided into two, as indicated in the figure, the points of division lying on a line per- pendicular to the plane of the paper, which is parallel to the longer axis of the rectangle. The streams are of different magnitudes, that which makes an acute angle (0) with the current being the smaller, for reasons given in the article cited, which apply also to the present case. Hence the line of division moves away from the centre when is diminished, and when becomes very small approaches nearly to the edge of the rectangle. The line of division, however, always exists, and along it the excess pressure is wF 2 /2g as already described. The total excess pressure upon the front of the plane is, as before, P = .F.sine, 9 only in the present case we do not know directly the quantity of water which is acted on. If we write W=w.SV S will be the unknown area of an ideal isolated jet, which would pro- duce the same effect and , yi P = w . S . . sin 0, g 506 HYDRAULICS. [PART v. a formula which may be written where ^ is the area of the plate and /z a co-efficient depending on the quantity of water acted upon. The value of the plus portion of the co-efficient of resistance is now /* . sin 6. Behind the plane, eddies are formed, the effect of which is represented by the minus portion n of the co-efficient. The total co- efficient k is now p . sin + n. To determine k two methods may be adopted : (1) By direct experiment on planes set at various angles in a stream, various formulae have been obtained, of which, perhaps, the best is that devised by Duchemin, and adopted by Poncelet in the second edition (1839) of the Mfcanique Industrielle, namely, 2.sin6> , where & is the value of k when the plate is at right angles to the stream. (2) By methods of calculation which cannot be explained here, Lord Kayleigh has shown that the plus portion of the co-efficient is 2?r . sin ~4-f7r.sin0' it being pre-supposed that behind the plane there is an indefinite mass of fluid at rest relatively to the plane, and separated from the moving current by fixed surfaces of separation. The actual value of m may probably be nearly the same as in the actual case where eddies are formed, but the minus part of the total co-efficient, which does not exist in the ideal case, must still be found by experiment. If = 90, m becomes '88, and adopting l - 25 as the value of k, n is found to be '37. When is very small it will be seen that ^ becomes con- stant, being equal to 7r/2 or T57, a conclusion which might have been foreseen, for at small angles there appears no reason why the effective breadth of the current of water acted on by the plate should vary. The suction at the back of the plate has the same general effect as the excess pressure in front, namely, of deflecting a current of water, the breadth of which is approximately constant for small values of 0. Thus, when is small (not exceeding 10 or 15), the value of k is a sin where a is constant. The value of a was taken by Froude as l - 7 for thin flat plates, but there can be little doubt that it is much greater when the back of the plate is convex, so that the eddies extend over the whole area, instead of being localized at the back of CH. xix. ART. 261.] ELEMENTARY PRINCIPLES. 507 the leading edge. According to Ducherain's formula, it will be seen that a = 2& , or about 2 '5. It must be remembered that the resistance considered in the present article is the force normal to the plate. The resistance in the direction of motion is obtained by multiplication by sin 0, and to it must be added the component in the direction of motion of the tangential force on the plate. If the plate is very thin and perfectly flat on both sides, the tangential force is due to surface friction only being at small angles nearly the same as if it moved edgewise ; but otherwise it will be much greater, and must be ascertained by experiment. The ratio which it bears to the normal force is much less variable, and may be taken as '00- r ). The value given by Froude is '0047. From what has been said it is clear that the line of action of the normal pressure on the plate does not pass through the centre ; if therefore it be mounted on an axis parallel to the longer side the plate cannot be in equilibrium if the axis passes through the centre, but will always tend to place itself perpendicular to the direction of motion. This is also true for a square or circular plate, and so far as is known the value of k in this case is not very different. 261. Pressure of a Current against an Obstacle. When an obstacle is placed below the surface of a stream a pressure is experienced by the obstacle which is due to the same causes as when a solid moves through still water, and, since the relative motion is the same in the two cases, should be given by the same formula V' 2 P = kwS~-. -9 In fact, however, the cases are often very different, because a uniform steady current is seldom to be met with in nature. The motion of the water is often unsteady and almost always disturbed by eddies due to the neighbourhood of the boundaries or other solid bodies. Experience shows that the value of k is generally considerably greater than in the case of motion through still water. For a flat plate fixed at right angles to a stream Dubuat found k to be 1 '86, and this estimate being confirmed by other experimentalists, has been very generally accepted. The irregularity and uncertainty characteristic of experiments on fluid resistance, when the solids exposed to its action are of unfair form, is especially marked in the case of wind pressure for sufficiently obvious reasons. This question, together with that of the resistance of the atmosphere to moving bodies, is outside the range of this work, but a short statement of results will be found in the Appendix, in which a brief account is also given of the theory of the resistance and propulsion of ships. 508 HYDRAULICS. [PART v. EXAMPLES. FIRST SERIES (SECTIONS I. AND III.). 1. The injection orifices of the jet condenser of a marine engine are 5 feet below the surface of the sea, and the vacuum is 27 inches of mercury : with what velocity will the water enter the condenser, supposing three-fourths the head lost by f rictional resistances ? Also find the co-efficients of velocity and resistance and the effective area of the orifices ta deliver 100, 000 gallons per hour. Ans. Velocity =23*6' per second; Area=27 sq. inches. 2. Water is discharged under a head of 25' through a short pipe 1" diameter with square-edged entrance ; find the discharge in gallons per minute. Ans. 66J. 3. Water issues from an orifice the area of which is '01 sq. feet in a horizontal direc- tion and strikes a point distant 4' horizontally and 3' vertically from the orifices. The head is 2' and the discharge 25 gallons per min. ; find the co-efficients of velocity, re- sistance, contraction, and discharge. Ans. c= '816, F='5, k='72, C='59. 4. j The wetted surface of a vessel is 7,500 sq. feet, find her skin resistance at 8 knots and the H.P. required to propel her, taking the resistance to vary as F 2 with a co-efficient of '004. Ans. Resistance = 5, 500 Ibs., H.P.=135. 5. The diameter of a screw propeller is 18', the pitch 18', and the revolutions 91 per min. Neglecting slip find the H.P. lost by friction per square foot of blade at the tips, taking a co-efficient '008 to include both faces of the blade. Ans. Friction = 65 Ibs. per square foot. H. P. = 10 '6. 6. Two pipes of the same length are 3" and 4" diameter respectively : compare the losses of head by skin friction (1) when they deliver the same quantity of water, (2) when the velocity is the same. Ans. Ratio =4 '21 and 1 '33. 7. Water is to be raised to a height of 20' by a pipe 30' long 6" diameter : what is the greatest admissible velocity of the water if not more than 10 per cent, additional power is to be required in consequence of the friction of the pipe ? Ans. 8|' per sec. 38. Two reservoirs are connected by a pipe 6" diameter and three-fourths of a mile long. For the first quarter mile the pipe slopes at 1 in 50, for the second at 1 in 100, while in the third it is level. The head of water over the inlet is 20 feet and that over the outlet 9 feet. Neglecting all loss except that due to surface friction, find the dis- charge in 'gallons r per min., assuming /='0087. Ans. v. =3'43 f.s. Discharge = 253 gallons per min. 9. A river is 1000' wide at the surface of the water, the sides slope at 45, and the depth is 20' ; find the discharge in cubic feet per sec. with a fall of 2' to the mile,, assuming /='0075. Ans. 154,000. 10. A tank of 250 gallons capacity is 50' above the street. It is connected with the street main, the head in which is 52' by a service pipe 100' long : find the diameter of the pipe that the tank may be filled in 20 min. What must the head in the main be to- fill the tank in five min. with this service pipe? Ans. d=l'f>". Head in main = 82'. 11. Water is discharged from a vessel by a long pipe : show that the discharge is the same for all pipes of the same length and diameter with the discharging extremity in the same horizontal line. Draw the hydraulic gradient and examine the case of a syphon. 12. In question 2 suppose the pipe instead of being short to be 25" long, find the discharge, assuming for surface friction /= '01. Ans. 52. 13. A horizontal pipe is'reduced in diameter from 3" to J" in the middle, the reduction being very gradual. The pressure head in the pipe is 40', what would be the greatest velocity with which the water could flow through it, all losses of head being neglected ?' Ans. 1'4' per sec. CH. xix.] ELEMENTARY PRINCIPLES. 509 H. A pipe 2" diameter is suddenly enlarged to 3". If it discharge 100 gallons per min., the water flowing from the small pipe into the large one, find the loss of total head and the gain of pressure head at the sudden enlargement. State the two values of the co-efficient of resistance. Ans. Loss of head =". F=l'56 or '31. Gain of pressure =1' 2". 15. In the last question suppose the water to move in the reverse direction. Find the loss of head and the change of pressure consequent on the sudden contraction, assuming the co-efficient of contraction to be '66. Ans. Loss of head =7%". Diminution of pressure =2' 5f". 16. A horizontal pipe 30' long is suddenly enlarged from 2" to 3" and then suddenly returns to its original diameter. Length of each section =10'. Draw the hydraulic gradient when the pipe is discharging 100 gallons per min. into the atmosphere, assuming as co-efficient of surface friction 4 /='03. Find the total loss of head. Ans. Total loss of head =10' 2^". 17. A pipe contains a diaphragm with an orifice in it the area of which is one-fifth the sectional area of the pipe. Find the co-efficient of resistance of the diaphragm, assuming the contraction on passing through the orifice the same as that on efflux from a vessel through a small orifice in a thin plate. Ans. F=4Q. 18. Find the loss of head in inches due to a bend through 45 of radius 6" in a pipe 2" diameter, the velocity of the water being 12' per sec. Ans. 2". 19. In question 1 suppose the ship moving at 10 knots and the orifice of entry so arranged as to cause no additional resistance : find the velocity of delivery. Ans. Addi- tional head =4 '42' : velocity =25' per sec. 20. Water is supplied by a scoop to a locomotive -tender at a height of T above the trough. Assuming half the head lost by frictional resistance, what will be the velocity of delivery when the train is running at 40 miles per hour, and what will be the lowest speed of train at which the operation is possible ? Ans. 36' per sec. ; 14^ miles per hour. 21. If m be the hydraulic mean depth of a channel of rectangular section, sides in the ratio n : 1 ; show that the h.m.d. of a circular section of the same area is 1 \TT \n 22. A^pipe is suddenly enlarged to double its diameter (1) all at once, (2) by two stages ; compare the losses of head, the stages in (2) being arranged so that the loss may be the least possible. Ans. Ratio =4. 510 HYDRAULICS. [PART v, EXAMPLES. SECOND SERIES (SECTIONS II., IV., AND V.). 1. A stream of water delivering 500 gallons per min. at a velocity of 15 feet per sec. strikes an indefinite plane (1) direct, (2) at an angle of 30 : find the pressure on the plane. Ans. (1) 39 Ibs. ; (2) 19^ Ibs. 2. Employ the principle of momentum to prove the formula on page 478 for the resultant centrifugal force of one-half a rotating ring of fluid. 3. A plane area moves perpendicularly through water in which it is deeply immersed : find the resistance per sq. foot at a speed of 10 miles per hour. Deduce the pressure of a wind of 20 miles per hour using the same co-efficient. Ans. Resistance = 269 Ibs. Wind pressure = 1'312 Ibs. 4. Compare the resistance of an area moving flatwise through the water with it& resistance moving edgewise so far as due to surface friction, the co-efficient for which is 004. Ans. Ratio = 312. 5. Water is being discharged from a tank with vertical sides, by a sharp-edged rectangular notch 8 inches wide, the lower edge of which is 4 inches below the level of still water. Co-efficient of discharge, '6. Find the discharge in gallons per minute. Ans. 154. NOTE. A notch is treated as an orifice the upper edge of which is at the still water level. Hence in the formula of page 481, 6 is to be taken as 8 inches, Yj zero, and Y z 4 inches. Contraction and hydraulic resistance are then allowed for by multiplication by the co-efficient which varies to some extent according to the proportions which the head and the breadth of the tank bear to the width of the notch. 6. Obtain a formula for the discharge from a triangular notch with sides inclined at an angle 6 to the vertical, the apex being downwards and at a depth h below still water. Ans. 0s2gc.*/ty,'tU?,A*. NOTE. The co-efficient of discharge c varies somewhat with the angle being about '6 when the angle is 45 : but by the principle of similar motions (p. 482) will be nearly independent of the head in a notch of moderate size, a considerable practical advantage. 7. When a sphere moves in a straight line through a fluid the velocity with which the fluid glides over the surface, at a point the angular distance of which from the central line is 0, is f . sin 6. Assuming this, find the plus portion of the co-efficient of resistance. Ans. f. 8. In the last question assuming the motion in front the same as before notwith- standing the formation of eddies at the rear : and further, assuming the suction to extend over the same area as the excess pressure with a co-efficient the same as for a flat plate, find k. Ans. k= '386. REFERENCES. For further information on subjects connected with the present chapter, the reader is referred to a treatise on Hydraulics by Professor W. C. Unwin, M.I.C.E., forming part of the article Hydro-Mechanics in the " Encyclopaedia Britannica." CHAPTEE XX. HYDEAULIC MACHINES. 262. Preliminary Remarks. Hitherto the energy exerted by means of a head of water has been supposed to be wholly employed in over- coming frictional resistances, and in generating the velocity with which the water is delivered at some given point. We now proceed to consider the cases in which only a fraction of the head is required for these purposes ; the remainder then becomes a source of energy at the point of delivery by means of which useful work may be done. A machine for utilizing such a source is called an Hydraulic Motor. Hydraulic energy may exist in three forms, according as it is due to motion, elevation, or pressure. In the first two cases it is inherent in the water itself, being a consequence of its motion or its position as in the case of any other heavy body. In the third it is due to the action of gravity or some other reversible force, sometimes on the water itself, but oftener on other bodies, as, for example, the load on an accumulator ram. The water is then only a transmitter of energy and not directly the source of it. As, however, the energy transmitted is proportional to the weight of water delivered, just as in the two other cases, the water is, as before, described as possessing energy. The energy per unit of weight is called "head," as sufficiently explained in the preceding chapter, and the "total head " is the sum of the "velocity head," the "actual head," and the "pressure head." Hydraulic motors are classed according to the mode in which the water operates upon them, which may be either by weight, or by pressure, or by impulse, including in the last term also "reaction." Most hydraulic motors are capable of being reversed, and then become machines for raising water, commonly described as Pumps. SECTION I. WEIGHT AND PRESSURE MACHINES. 263. Weight Machines. To utilize a head of water, consisting of an actual elevation (h) above a datum level at which the water can be delivered and disposed of, a machine may be employed in which 512 HYDKAULICS. [PART v. the direct action of the weight of the water, while falling through the height h, is the principal motive force. The common overshot water-wheel (Fig. 2, Plate III., p. 141) may be taken as a type. Here the driving pair is a simple turning pair, and the driving link is the force of gravity upon the falling water, which acts directly on buckets open to the atmosphere. If G be the delivery in gallons per minute, the energy exerted in foot-pounds per minute is E=lOGh. The head h is here measured from the level of still water in a reservoir which supplies the wheel. If v be the velocity of delivery to the wheel, the portion v 2 /2g is converted into energy of motion before reaching the buckets and operates by impulse. In a wheel of this class, therefore, the water does not operate wholly by weight. The speed of the wheel is limited to about 5 feet per second by the centrifugal force on the water, which, if too great, causes it to spill from the buckets. It will be seen hereafter that the velocity of the water should be about double this, so that v is about 10 feet per second, and the part of the fall operating by impulse is therefore about 1'5 feet. The remainder operates by gravitation, but a certain fraction is wasted by spilling from the buckets, and emptying them before reaching the bottom of the fall. More than one half the head operating by impulse is always wasted (Art. 270), and this class of wheels is therefore only suitable for falls exceeding 10 feet. The great diameter of wheel required for very high falls is incon- venient, but examples may be found of wheels 60 feet diameter and more. The efficiency of these wheels under favourable circumstances is -75, arid is generally about -65. In "breast wheels" the buckets are replaced by vanes which move in a channel of masonry partially surrounding the wheel. The water is admitted by a moveable sluice through a grating of fixed blades in the upper part of the channel. The channel is thus filled with water, the weight of which rests on the vanes and furnishes the motive force on the wheel. There is a certain amount of leakage between the vanes and the sides of the channel, but this loss is not so great as that by spilling from the buckets of the overshot wheel. The efficiency is found by experience to be as much as *75. As the diameter of the wheel is greater than the fall a breast wheel can only be employed for moderate falls. In both these machines the water virtually forms part of the piece on which it acts. This link of the kinematic chain forms one element of the driving pair, while that attached to the earth forms the other. CH. xx. ART. 264.] MACHINES. 513 In the overshot wheel the water is contained in open buckets, in the breast wheel it is contained in a closed chamber or channel. A third class of weight machines is referred to farther on under the head of pumps. 264. Hydraulic Pressure Machines in Steady Motion. A water wheel of great diameter is a slow-moving cumbrous machine, and for heads of 100 feet and upwards it is therefore necessary to employ a pressure or an impulse machine. Such machines are also often more convenient for low falls. In pressure machines the driving link is compressed water, which is forced between the elements of the driving pair by some source ot the energy which supplies the necessary head. The head is sometimes an actual elevation either natural or artificial : in the docks at Great Grimsby the hydraulic machinery is operated from a tank placed on a tower 200 feet high. It is however difficult to get a considerable pressure in this way, and an apparatus called an Hydraulic Accumulator is therefore generally resorted to. Two forms occur, of which one is shown in Plate IX. In the first a plunger or ram is forced into -a cylinder by heavy weights placed in a plate-iron cage suspended from it and stayed by iron rods. The accumulator is supplied by pumps generally worked by steam, which is the ultimate source of the energy, the accumulator merely serving the purpose of a store of energy which can be drawn on at pleasure. For ordinary hydraulic machinery the pressure is limited to 750 Ibs. per square inch from the difficulty of obtaining pipes of sufficient strength and of working slide valves under heavy pressures. In machines for riveting and other special purposes, however, pressures of 1,500 Ibs. per square inch and upwards are employed. The accumulator then consists of a cylinder B (Fig. 1, Plate IX., p. 525), loaded with ring weights EE, sliding on a fixed spindle F, divided into two lengths of which the upper portion is of smaller diameter than the lower. In either form the accumulator provides a store of compressed water which can be supplied by suitable pipes to any number of machines, placed often at considerable dis- tances. A head of 1,700 feet is thus readily obtained, and for special purposes much more : differ- ences of level may therefore be disregarded as of small importance, and the water j considered as operating wholly by pressure. C.M. 2K 192< 514 HYDRAULICS. [PART v. The driving pair of the machine forms a chamber of variable size which is alternately enlarged by the pressure of the water, and con- tracted to expel it. In most cases it is a simple cylinder C and piston B (Fig. 192) : the water is admitted by a port from a pipe L, transmitting it from the accumulator at pressure p. Let the piston move through a space x, let A be its area, then Energy exerted =pAx =p . X, where X is the volume swept through by the piston. If w be as usual the weight of a cubic foot, w . X is the weight of water which enters the cylinder as the piston moves through the distance x, and therefore Energy exerted per Ib. of water = - = pressure-head in cylinder. This might have been anticipated from what was said in the last chapter as to the meaning of the term " head," and in fact it is equally true if the driving pair be not a simple piston and cylinder, but of any other kind. The head in the cylinder is less than that in the accumulator, on account of the friction in the supply pipe and other frictional resist- ances, and it is on the action of these resistances that the working of the machine depends. Let V be the velocity of the piston in its cylinder, p Q pressure in accumulator, F the co-efficient of hydraulic resistance referred to the velocity of the piston (Art. 255), then, neglecting differences of level, also the heights due to velocities of working and accumulator pistons, If the machine be moving steadily the pressure p will be equal to the useful resistance which the piston is overcoming, increased by the friction of the piston in its cylinder. Thus p and p will be known quantities, a certain definite velocit}'- V Q will then be determined, which may be described as the "speed of steady motion " : it is givea by the equation Since the hydraulic resistances may be increased to any extent at pleasure by the turning of a cock, it follows that the speed of an hydraulic pressure machine can be regulated at pleasure. Further, if the resistance to the movement of the piston be diminished, the speed will increase only by a limited amount, and can, under no- circumstances, be greater than is given by CH. xx. ART. 265.] MACHINES. 515 which can be regulated as before. The surplus energy is here absorbed by the frictional resistances, and an hydraulic pressure machine there- fore possesses the very important, and for many purposes, valuable characteristic that it contains icithin it brakes which work automatically. 265. Hydraulic Pressure Machines in Unsteady Motion. Although the speed of a pressure engine cannot exceed a certain limit, which is easily found, yet it does not follow that the limit will ever be reached. When the engine starts, the piston and the water in the pipes have to be set in motion, the force required to do this is so much subtracted from that available to overcome resistances. A con- siderable time therefore elapses before a condition approaching steady motion can be obtained. In Figs. 178, p. 486, water is supposed flowing through a pipe with a velocity u. Two pistons at a distance x enclose water between them, as in Art. 249, then the difference of pressure p 1 -p 2 in the case of steady motion is simply balanced by the surface friction, but in un- steady motion is partially employed in accelerating the flow of the water. Neglecting friction the acceleration g will be given by the formula (ft - where A is the sectional area of the pipe and W is the weight of the water between the pistons. Replacing W by Ax . w, as in the preceding article Pi~P* = x t w " g' which gives a simple formula for the change of pressure-head due to inertia. Now if nA be the area of the working piston, the velocity of the water in the pipe is n times the velocity of the piston, and the accelerations are necessarily in the same ratio ; and hence it follows that the difference of pressure-head between cylinder and accumulator due to an acceleration g' of the piston is for a length of pipe I w g In addition to this the piston itself requires a certain pressure to accelerate it. Let q Q be the "pressure equivalent to that weight," being the actual weight divided by the area, as in Art. 109, p. 224, then the pressure due to inertia is ?=. ty But if S, S' be the corresponding values of S, V2 77'2 y 2a and if the ordinates be taken near together the area in question will be- nearly KN . NN'. We have therefore, by division, 1? ' *7 A I 1 \9 KZ A ,. T , wA(n-\Y KN= Nh --~W ' That is, if a number of equidistant ordinates be drawn near together the ratio of consecutive ordinates is constant. The curve may be roughly traced from this property; it is identical with the curve already drawn in Art. 123, p. 252, except that it is a linear instead of a polar curve. The mean resistance to recoil is given by the equation (S + P }1 = Energy of Recoil, where / is the distance traversed. It would, of course, be advantageous to have a uniform resistance to recoil, because the maximum pressure in the compressor would be diminished and less strain thrown on CH. xx. ART. 268.] MACHINES. 521 the gear. This is the object of the various modified forms of the compressor, in which the orifices are not of constant area, but become smaller as the recoil proceeds. In order that the resistance may be constant we must have so that (n - l)/^is constant. Further, since the retardation is uniform W ' where x is the distance from the end of the recoil. It appears therefore that the orifices should vary in such a way that (n-l)' 2 x should be constant. Descriptions of two forms of compressor, with varying orifices, will be found in the Gunnery Manual. Instead of a sliding pair we may employ a turning pair. This is the common "fan" or "fly" brake used to control the speed and absorb the surplus energy of the striking movement of a clock, or in other similar cases. A friction dynamometer (p. 279) was designed by the late Mr. Froude for the purpose of measuring the power of large marine engines, in which the ordinary block or strap surrounding a shaft or drum is replaced by a casing in which a wheel works. Vanes attached to the wheel and the fixed casing thoroughly break up a stream of water passing through the casing. Any amount of energy may thus be absorbed without occasioning any considerable rise of temperature. Siemens' combined brake and regulator has been mentioned already on page 278. 268. Transmission of Energy by Hydraulic Pressure. Energy may be distributed from a central source, and transmitted to considerable distances with economy by hydraulic pressure. The delivery in gallons per minute of a pipe d" diameter is . (Art. 250). Assume now that the pipe supplies an hydraulic machine at a distance of I feet from an accumulator in which h is the head. Further, suppose that n per cent, is lost by friction of the pipe, then the power trans- mitted in foot-lbs. per minute is and the distance to which N horse-power can be transmitted with a loss- of n per cent, is in feet 1,800,000** 522 HYDEAULICS. [PART v. With the usual pressure in accumulators of 750 Ibs. per square inch, or 1700 feet of water, this gives the simple approximate formula 1-3300$. 'Thus, for example, 100 horse-power may be transmitted by a 5" pipe to .a distance of 4 miles, or 10 horse-power by a 1" pipe to a distance of 220 yards, with a loss by friction not exceeding 20 per cent. The diameter of pipe is limited by considerations of strength and cost. The power of a motor supplied by a given pipe does not increase indefinitely as its speed increases, but is greatest when one-third of the head is lost by friction.* The maximum possible power is therefore given by the formula H.P. = 220 - (approximately). This is of course two- thirds the value of N in the preceding formula. 269. Pumps. If the direction of motion of an hydraulic motor be reversed by the action of sufficient external force applied to drive it, while, at the same time, the direction of the issuing water is reversed so as to supply the machine at the point from which it originally proceeded, we obtain a machine which raises water instead of utilizing a head of water. Every hydraulic machine therefore may be employed to raise water as well as to do work, and most of them actually occur in this form; they are then called PUMPS, though in some cases this name would not be used in practice. Much of what has been said about motors applies equally well to pumps : the principal difference lies in the fact that the useful resistance which the pump overcomes is always reversible, whereas in the motor this is not necessarily the case. The principles of action and the classification of hydraulic machines are, in the main, the same in both cases. Some points omitted while considering motors as being of most importance in pumps, and certain differences of action between the two will now be briefly noticed. Certain machines occurring principally as pumps will be mentioned. (1) If the direction of motion of an overshot wheel be reversed a machine is obtained which is known as a "Chinese Wheel." It picks up water in its buckets and raises it to a height somewhat less than the diameter of the wheel. This machine is little used, but a reversed breast wheel is frequently employed in drainage operations, under the name of a "scoop," or "flash" wheel. The working pair is here * This result was pointed out to the writer by Mr. (now Prof. ) Hearson. It appears to be little known. OH. xx. ART. 269.] MACHINES. 523 a turning pair, but in the chain pump we find an example in which one of its elements is a chain passing over pulleys. The chain is endless and is provided with flat plates fitting into a vertical pipe, the lower end of which is below the surface of the water, and through which the water is raised. In the common dredging machine the closed channel (p. 512) is replaced by buckets. In a third class of weight machines the water occupies the moveable chamber and forms with it a kinematic pair with only one solid element, while it forms with the link attached to the earth, a working pair which has also but one solid element. The Archimedian screw, and certain varieties of " scoop " wheel, in which the water enters the scoop at the circum- ference of the wheel and is delivered at the centre, are examples of this kind. (2) The most common forms of pumps are the "lift" or "force" pumps, which consist of a chamber which expands to admit the water to be lifted and contracts in the act of lifting; they are therefore pressure machines like those considered in Arts. 264-5, but reversed. The name " pump " originally applied to these machines alone. Fig. 196 shows a common lift pump. A is a cylinder at a certain height Aj above the water to be raised, C is a piston working in the cylinder by the action of which the water is lifted. The piston has orifices in it which permit the water to pass through. p^ 19e The orifices are closed by a valve, as is also the opening at the bottom of the cylinder. These valves are simple "flaps" which open on hinges to permit the water to pass upwards, but close the passage to motion in the -opposite direction, thus acting as a ratchet (p. 158). Assuming the piston at the bottom of its stroke, at rest olose to the bottom of the cylinder, let it be supposed to rise; the valve b will rise and allow air to pass if any. After several strokes the air will be nearly exhausted, and if h^ be not too great the empty space will be filled with water raised from the tank by atmospheric pressure. Thus the water will pass into the cylinder closely following the piston. At the top of the stroke the piston commences to descend, b closes arid a opens, allowing the water to pass above the piston. This water is now raised by the piston to any required height. In force pumps the process is the *! % same, but the water passes out through an orifice in the bottom of the cylinder instead of through the piston ; the raising of the water above the level of the cylinder is done in the down stroke instead of the up. 524 HYDEAULICS. [PARTY, The difference between this action and that of a pressure motor lies mainly in the valves, which here open and close automatically by the action of the water instead of by external agency. Further, the pump wholly or partly works by suction, a method by no means peculiar to pumps, for it also occurs in motors, but nob so frequently. The height of the water barometer is 34 feet, but the height to which a pump will work by suction is not so great. When the piston is at the bottom of its stroke there must, for safety, always be a certain clearance space below. This space always contains air, the pressure of which diminishes as the piston rises, but cannot be reduced to zero. Further, a certain pressure is required to overcome the weight and friction of the valve before it opens. At least 3 feet of the lift is absorbed in this way, and generally considerably more. To obtain a high vacuum for scientific purposes, air pumps are specially designed to meet these difficulties. Also, leakage must be allowed for and the diminution on account of friction and inertia, which will be considerable if the speed be too great or the pipes too small, as will be understood on reference to Arts. 264-5, all of which applies to pumps as much as to motors. It is hardly necessary to observe that power is neither gained nor lost by the use of suction ; it simply enables the working cylinder to be placed above the water to be lifted, an arrangement which is in most cases convenient. The limit in practice is about 25 feet. Pumps are commonly, but not always, single-acting ; they are worked by the direct action of a reciprocating piece, or by means of a rotating crank. In the first case, when independent, a piston acted on by steam or water pressure is attached to a prolongation of the pump plunger : a crank and fly-wheel is often added, as in Fig. 4, Plate II., p. Ill, to control the motion and define the stroke. When driven by the crank three working cylinders, placed side by side with a three-throw crank, are commonly used, in order to equalize the de- livery, and so to avoid the shocks due to changes of velocity. An air-chamber, forming a species of accumulator, may also be used with the same object. An arrangement of pumps, as applied by Messrs. Donkin & Co. to raise water from a well 200 feet deep and force it to a height of 143 feet above the engine-house, may be mentioned as an example. A set of lift pumps at the bottom of the well worked by "spear" rods from the surface, are combined with a set of force pumps in the engine-house itself. The speed of these pumps is about 80 feet per minute, and they deliver about 600 gallons per minute. Pumps almost always have a certain "slip," that is they deliver less water than corresponds to the piston displacement and number of To face page 525. < To face page 525. OH. xx. ART. 269.] MACHINES. 525 strokes : in this example the slip was 1 2 per cent. The efficiency of the pumps and mechanism of the engine was found to be 66 per cent, by careful experiments.* In raising water from great depths in mines, force pumps at the bottom of the mine are used, worked by heavy " spear " rods from a beam engine at the surface. The weight of the rod supplies the motive force during the downward stroke of the pump ; while the engine, which is single-acting, raises the rods again during the down- ward stroke of the steam piston. DESCRIPTION OF PLATES IX. AND X. In order further to illustrate the action of water-pressure machines Plates IX. and X. have been drawn. Fig. 1, Plate IX., shows the differential accumulator described on page 513. In Fig. 2 is represented an hydraulic crane, designer! by Sir W. Armstrong, for lifting weights of 2 to 3 tons. In it the hydraulic power is applied to rotate the crane as well as to lift the weight. In order to effect the lift the high-pressure water from the accumulator is admitted to the cylinder A, and forces out the plunger B. There are two pulleys at a and two at 6. One end of the chain is secured to the cylinder A, it is led round 6, then round a, again round 6, then under the second pulley at a up through the hollow crane post on to the weight as shown. The effect of this arrangement is that any movement of the plunger B is at the hook multiplied four times. If B is simply a plunger working in a stuffing box, then the expenditure of energy is always the same whatever weight is being lifted, and the amount must be equal to that which corresponds to lifting the maximum possible weight. This is an objection which is common to all such machines. The surplus energy is expended in overcoming frictional resistances (p. 514). To mitigate this evil, in cranes of high power the plunger has a piston end, which fits a bored cylinder, and is provided with a cup leather, as shown in Fig. 3. The sectional area of the plunger is about one-half that of the cylinder. If a light weight is to be lifted, water is admitted to both sides of the piston, and the difference of the pressures, equal to what would be exerted on a simple plunger, is available for effecting the lift. When it is required to lift a heavy weight water is admitted to the side C only of the piston, the annular space D being put in communication with the atmosphere. Thus the full pressure due to the area of the piston is exerted with the corresponding expenditure of water. For the purpose of rotating the crane a pair of cylinders, E, are provided, of which one only is shown in the figure. The thrusting out of the plunger F of one of them foy the pressure of the water causes the other to be drawn in by means of a chain which passes around a recessed pulley secured to the crane post. In Plate X., Figs. 1 and 2 show the construction of Downton's Pump, so much used -on board ship. In the barrel work three buckets with flap valves, as shown in Fig. 2. The rods to which the upper and second buckets are attached are necessarily out of centre The rods to the lower buckets pass through desp stuffing boxes in the buckets .above, and thus the buckets are maintained from canting seriously. The movement -of the bujkets is effjcbel by a thrae-throar cnnk, the crank pins, which are not round, being set at 120 apart. These pins fit and work in a curved slot in the bucket rod heads. Assuming the admission of no air but water only from below, the discharge * Minutes of Proceedings of the Institution of Civil Engineers, vol. Ixvi. 526 HYDEAULICS. [PART v. of the pump will at each instant equal the displacement of the fastest upward moving bucket. Accordingly the rate of discharge may be represented by a curve, as in Fig. 3. If the slot in the rod head were straight and the pin round, then, the crank moving uniformly, in direction shown, the velocity of discharge would be represented by the radii from O to the dotted curve BABABA, which is made up of parts of three circles, the position of the radius being that of either of the three cranks. The effect of the curved slot is to diminish the maximum and increase the minimum discharge, as shown by the full curve B'A'BA'B'A'. Figs. 4 and 5 of this Plate are sections of the hydraulic engine referred to on page 518, employed to rotate a capstan. It need only be further added that a single rotating valve V suffices for admission and exhaust of all three cylinders. The high-pressure water is supplied by the pipe P to the passage S surrounding the valve and exhausted from the cylinders through the central passage. EXAMPLES. 1. In estimating the power of a fall of water it is sometimes assumed that 12 cubic feet per second will give 1 H.P. for each foot of fall: what efficiency does this suppose in the motor ? Ans. '72. 2. An accumulator ram is 9 inches diameter, and 21 feet stroke ; find the store of energy in foot-lbs. when the ram is at the top of its stroke, and is loaded till the pressure is 750 Ibs. per square inch. Ans. 1,000,000 foot-lbs. 3. In a differential accumulator the diameters of the spindle are 7 inches and 5 inches ; the stroke is 10 feet : find the store of energy when full, and loaded to 2,000 Ibs. per square inch. Ans. 377,000 foot-lbs. 4. A direct-acting lift has a ram 9 inches diameter, and works under a constant head of 73 feet, of which 13 per cent, is required by ram friction and friction of mechanism. The supply pipe is 100 feet long and 4 inches diameter. Find the speed of steady motion when raising a load of 1,350 Ibs., and also the load it would raise at double that speed. Ans. Speed=2 feet per second. Load =150 Ibs. 5. In the last question, if a valve in the supply pipe is partially closed so as to increase the co-efficient of resistance by 5, what would the speed be ? Ans. 1*6 f.s. 6. Eight cwt. of ore is to be raised from a mine at the rate of 900 feet per minute by a water-pressure engine, which has four single-acting cylinders, 6 inches diameter, 18 inches stroke, making 60 revolutions per minute. Find the diameter of a supply pipe 230 feet long, for a head of 230 feet, not including friction of mechanism. Ans. Diameter =4 inches. 7. Water is flowing through a pipe 20 feet long with a velocity of 10 feet per second. If the flow be stopped in one-tenth of a second, find the intensity of the pressure produced, assuming the retardation during stoppage uniform. Ans. 62 feet of water. 8. If X be the length equivalent to the inertia of a water-pressure engine, F the co-efficient of hydraulic resistance, both reduced to the ram, r the speed of steady motion ; find the velocity of ram, after moving from rest through a space x against a constant useful resistance. Also find the time occupied. 9. An hydraulic motor is driven from an accumulator, the pressure in which is 750 Ibs. per square inch, by means of a supply pipe 900 feet long, 4 inches diameter ; what would be the maximum power theoretically attainable, and what would be the velocity in the pipe at that power ? Find approximately the efficiency of transmission at half power. Ans. H.P. =240 ; v=22 ; efficiency = '96 nearly. CH. XX. ART. 270.] MACHINES. 527 10. A gun recoils with a maximum velocity of 10 feet per second. The area of the orifices in the compressor, after allowing for contraction, may be taken as one-twentieth the area of the piston : find the maximum pressure in the compressor in feet of liquid. Ans. 560 to 594. 11. In the last question assume weight of gun 12 tons ; friction of slide 3 tons ;. diameter of compressor 6 inches ; fluid in compressor water ; find the recoil. Ans. 4 feet 2^ inches. 12. In the last question find the mean resistance to recoil. Compare the maximum and mean resistances each exclusive of friction of slide. Ans. Total mean resistance 4 '4 tons. Ratio=2'2. SECTION II. IMPULSE AND EEACTION MACHINES. 270. Impulse and Reaction Machines in General. The source of energy may be a current of water or the head may be too small to obtain any considerable pressure, and it is then necessary to have some means of utilizing the energy of water in its kinetic form. A machine for this purpose operates by changing the motion of the water and utilizing the force to which the change gives rise. If the water strikes a moving piece and is reduced to rest relatively to it r the machine works by "impulse," and if it be discharged from a moving piece, by "reaction." There is no difference in principle between these modes of working, and both may occur in the same machine. In either case, the motive force arises from the mutual action between the water and the piece which changes their relative motion. Machines of this class are also employed for high falls when the low speed of pressure machines renders their use inconvenient or impossible. The water is then allowed to attain a velocity equivalent to a considerable portion of the head immediately before entering the machine, so that its energy is, in the first instance, wholly or partially converted into the kinetic form. The simplest machine of this kind is the common undershot wheel r consisting of a wheel (Fig. 197) pro- Fig. 197. vided with vanes against which the water impinges directly. Let the velocity of periphery of the wheel be F, then the water after striking the vanes is carried along with them at this velocity. If, then, the original velocity of the water be v, the diminu- tion of velocity due to the action of the vanes will be v - V. Let W be the weight of water acted on per second, then the impulse on the wheel must be usi v _y\ P = ) .528 HYDRAULICS. [PART v. but if A be the sectional area of the stream, this being the weight of water per second which comes in contact with -all the vanes taken together, .-. P = W Av(v-V). 9 The power of the wheel is PV foot-lbs. per second, and the energy of the stream is Wv 2 /2g r therefore Efficiency = This is greatest when V=fyo and its value is then -5, showing that the wheel works to best advantage when the speed of periphery is one-half that of the stream, but that the efficiency is low, never exceeding '5. Such wheels may be seen working a mill floating in a large river, or in other similar circumstances, but they are cumbrous and, allowing for various losses not included in the preceding investigation, their efficiency is not more than 30 per cent. In the early days of hydraulic machines, they were often used for the sake of simplicity or, as in the example shown in the figure, from a want of comprehension of their principle.* In mountain countries, where unlimited power is available, they are still found. The water is then conducted by an artificial channel to the wheel, which sometimes revolves in a horizontal plane. When of small diameter their efficiency is still further diminished. In overshot wheels and other machines operating chiefly by weight the head corresponding to the velocity of delivery is partly utilized by impulse, and the speed of the wheel is determined by this considera- tion. In all cases of direct impulse, if h is that part of the head operating by impulse, the speed of maximum efficiency is or in practice somewhat less, and at that speed at least half that head is wasted. The great waste of energy in this process is due partly to the velocity V with which the water moves onward with the wheel, and partly to breaking-up during impulse. It is in fact easy to see that one-fourth the head is wasted by each of these causes. To .avoid it, the water must be received by the moving piece against which it impinges without any sudden change of direction, and must be discharged at the lowest possible velocity, effects which may be produced by a suitably-shaped vane curved so as to deflect the water * See Fairbairn's Millwork and Machinery, from which this figure is taken, vol. i. , p. 149. c'H. xx. ART. 270.] MACHINES. 529 gradually and guide it in a proper direction. The principle on which such a vane is designed may be explained by the annexed diagram. In Fig. 1 98 A B is a vane moving with velocity V in a given direction, against which a jet strikes. Drawing a diagram of velocities, let Oa represent v, the velocity of the jet, and let 00' represent V. Then as before (p. 499) O'a represents the velocity of the jet relatively to the vane, and, in order that the water may impinge without shock, the tangent to the vane at A must be parallel to O'a. The vane is now curved so as gradually to deflect the water, in doing which there is a mutual action between the jet and the vane furnishing the motive force which drives the wheel. If the water leave the vane at B) its velocity relatively to the vane is represented by O'b drawn parallel to the vane at B, and somewhat less than O'a in magnitude, to allow for friction, unless the water be enclosed in a passage, when it will bear some given proportion to O'a. The absolute velocity with which the water moves at B is now represented by Ob, and this may be arranged to deliver the water in a convenient direction with a velocity just sufficient to clear the wheel and no more. Two examples of the use of such vanes may now be mentioned. (1) In the Pelton wheel recently introduced in America, the buckets of an ordinary vertical water wheel, receiving a jet of water tangen- tially under a considerable head, are divided in the middle, and each half curved so as to form a cup or pocket facing the jet. The inner edges of the two halves are now united so as to form a dividing edge, upon which the jet impinges centrally and by which it is separated into two parts, each diverging laterally and then turning through an angle of nearly 180. The double pocket with its dividing edge is not essential, a simple cup vane (Fig. 1 90, p. 500) would suffice ; but it probably renders the jet less liable to breaking up from unsteadiness or in consequence of the angular motion of the bucket. A wheel of this kind at the Comstock mines, Nevada, U.S., works under a head of 2,100 feet with a velocity of periphery of 180 f.s.* Their efficiency is very considerable, in many cases exceeding 80 per cent. (2) Of much older date are the vanes applied to \ vertical water wheels by Poncelet in order to utilize as far as possible a head of moderate amount. The water in this case impinging below the wheel at A, ascends to B, and then while the vane is moving onwards * The Practical Engineer, June 17, 1892. C.M. 2 L 530 HYDKAULICS. [PARTY, descends again to A under the action of gravity. The velocity of the water relatively to the wheel is thus reversed : O'b being approxi- mately equal and opposite to O'a. In all impulse and reaction machines there is a speed of maximum efficiency which, as in the simple case first considered, is given by the formula r=lcj*jh, where k is a fraction depending on the type of machine. 271. Angular Impulse and Momentum. The most important of these machines are those in which the change of motion produced in the water is a motion of rotation, and it is need- ^. _ >v ful to consider that form of the principle of momentum which is applicable to such cases. In Fig. 199, W is a weight describing a circle round with velocity V -, then the product of its momentum by the radius r is called the "moment / of momentum " of the weight about 0. If \ s y represent an axis to which W is attached rigidly, we may imagine it turning under the action of a force P at a radius R. The moment of P multiplied by the time during which it acts is called the " moment of impulse." During the action of P the weight will move quicker and quicker and the motion is governed by the principle expressed by the equation Moment of Impulse = Change of Moment of Momentum. If L be the moment of P, then taking the time as one second, L Change of Moment of Momentum per second. This equation is true, not only for a single weight and a single force, but also for any number of weights and any number of forces. As in other forms of the principle of momentum it is also true, not- withstanding any mutual actions or any relative movements of the weights or particles considered. Further, any radial motions which the particles possess may be left out of account, for they do not influence the moment of momentum. A particular case is when L = O y then the moment of momentum remains constant, a principle known as the Conservation of Moment of Momentum. The terms " moment of momentum" and "moment of impulse" are often replaced by " angular momentum," "angular impulse." A weight rotating about an axis is capable of exerting energy in two ways. First, it may move away from the axis of rotation, overcoming by its centrifugal force a radial resistance which it just overbalances. CH. xx. ART. 272.] . MACHINES. 531 Secondly, it may overcome a resistance to rotation in the shaft to which it is attached. In either case the work done will be represented by a diminution in the kinetic energy of the weight. If the shaft be free, the diminution of kinetic energy must be equal to the work done by the centrifugal force, and it may be proved in this way, that if V be the velocity of rotation of the weight, r the radius, Vr = Constant, an equation equivalent to the conservation of the moment of momentum. Conversely, energy may be applied to a rotating weight either by moving it inwards against its centrifugal force, or by a couple applied to the axis of rotation. In turbines both modes of action occur together as we shall see presently ; and the employment of the principle of momentum, though not necessary, is on the whole the most convenient way of dealing with the question. 272. Reaction Wheels. Fig. 200 shows a reaction wheel in its simplest form. GAG is a horizontal tube communicating with a vertical tubular axis to which it is fixed, and with which it rotates. Water descends through the vertical tube, and issues through orifices at the extremities of the horizontal tube so placed that the direction of motion of the water* is tangential to the circle described by the orifices. The efflux is in opposite directions Fi #- 200. from the two orifices, and a reaction is pro- duced in each arm which furnishes a motive force. There are two methods of investigating the action of this machine which are both instructive. Frictional resistances are, in the first instance, neglected. (1) Let the orifices be closed, and let the machine revolve so that the speed of the orifices in their circular path of radius r is V. Centrifugal action produces a pressure in excess of the head h existing when the arms are at rest, the magnitude of the excess in feet of water being This is so much addition to the head, which now becomes This quantity H may also be considered as the head "relative to the moving orifices " estimated as on p. 475. When the orifices are opened, the water issues with velocity v given by 532 HYDRAULICS, [PART v. thus the water issues with a velocity greater than V, and after leaving the machine has the velocity vV relatively to the earth. The energy exerted per Ib. of water is A, and this is partly employed in generating the kinetic energy corresponding to this velocity. The remainder does useful work by turning the wheel against some useful resistance, so that we have per Ib. of water " and, dividing by h, -, ffi . V(v-V) 2F Efficiency = * _ / = - _ . gh v+F (2) A second method is to employ the principle of the equality of angular impulse and angular momentum already given in Art. 271. Originally the water descends the vertical tube without possessing any rotatory motion, but after leaving the machine it has the velocity v -V its angular momentum is therefore for each Ib. of water, Angular Momentum = - -- - . r. Now according to the principle the angular momentum generated per second is also the angular reaction on the wheel which, when multiplied by F/r, the angular velocity of the wheel, gives us the useful work done per second. Performing this operation, and dividing by the weight of water used per second, we get per Ib. of water Useful Work = V ( V ~ V \ 9 This is the result already obtained, and the solution may now be completed by adding the kinetic energy on exit. From the result it appears that the proportion which the waste work bears to the useful work is v - V : 2 F, which diminishes indefinitely as v approaches F; but in this case the velocities become very great, since v 2 - F 2 is always equal to 2gh. The frictional resistances then become very great, so that in the actual machine there is always a speed of maximum efficiency which may be investigated as follows: Let F be the coefficient of hydraulic resistances referred to the orifices, then 2 > (l+F) g =B=h+%- g . The useful work remains as before, and therefore . .hmciency J v z_ a fraction which can readily be shown to be a maximum when CH. xx. ART. 273.] MACHINES. 533 which value of v, when substituted in the preceding equation, will give the value of V in terms of h for maximum efficiency. The existence of a speed of maximum efficiency is well known by experience with these machines. In general it is found to be about that due to the head, so that 7* = 2gh, a value which corresponds to ^='125, and gives an efficiency of '67. This is about the actual efficiency of these machines under favourable circumstances ; of the whole waste of energy two-thirds, that is two- ninths of the whole head, is spent in overcoming frictional resistances, and the remaining one-third, or one-ninth the whole head, in the kinetic energy of delivery. The reaction wheel in its crudest form is a very old machine known as " Barker's Mill." It has been employed to some extent in practice as an hydraulic motor, the water being admitted below and the arms curved in the form of a spiral. These modifications do not in any way affect the principle of the machine, but the frictional resistances may probably be diminished. 273. Turbine Motors. A reaction wheel is defective in principle, because the water after delivery has a rotatory velocity in consequence of which we have seen a large part of the head is wasted. To avoid this, it is necessary to employ a machine in which some rotatory velocity is given to the water before entrance in order that it may be possible to discharge it with no velocity except that which is absolutely required to pass it through the machine. Such a machine is called in general a TURBINE, and it is described as "outward flow," "inward flow," or " parallel flow," according as the water during its passage through the machine diverges from, converges to, or moves parallel to, the axis of rotation. Fig. 20 la shows in plan and section part of an annular casing forming a wheel revolving about an axis XX through which water is flowing, entering at the, centre and spreading outwards. The water leaves the wheel at the outer circumference. Fig. 20 Ib is similar, but the flow is inward instead of outward. If we consider a section aa made by a concentric cylinder of length y and radius ?, the flow will be Q = u. 27m/, where u is the radial velocity or, as we may call it, the " velocity of flow." The area of the section (27my) may conveniently be called the "area of flow." The 'value of Q is everywhere the same, and therefore ury must be constant. It is generally desirable to make u constant or 534 HYDEAULICS. [PART v. nearly so, and then the form of the casing is such that ry is constant. Whether this be so or not, the value of u can always be calculated at any radius for a given wheel with a given delivery. The water which at any given instant is at a given distance r from the axis may be considered as forming a ring RR, which rotates while at the same time it expands or contracts according as the flow is outward or inward. The velocity of the periphery of this ring may be described as the "velocity of whirl," and if it be called V, the moment of momentum of a ring, the weight of which is W, is W M= -. vr. 9 If the wheel has no action on the water, this quantity cannot be altered, and we must then have vr = Constant. The water then forms what we have already called a "free vortex" (Art. 244), with the addition of a certain radial velocity u, in con- sequence of which the rings change their diameter. The paths of the particles of water are then spirals, the inclination of which depends on the proportion between u and v. The case now to be considered is that in which the moment of momentum of the rotating rings is gradually reduced during their passage through the wheel by the action of suitable vanes attached to it. An impulse is thus exerted on the wheel which furnishes the motive force. The moment of this impulse is given by the equation, where wQ is the weight of all the rings passing through the machine H. xx. ART. 273.] MACHINES. 535 in a second, and the suffixes 1, 2 refer to entrance and exit respec- tively as indicated in the figures for the two cases of outward flow and inward flow. In this article the turbine is supposed to work to best advantage when the water is discharged without any whirl, that is when # 2 = 0, and putting aside friction the only loss then is that due to the velocity of flow u, which may be made small by making the wheel of sufficient breadth at the circumference where the water is discharged. In practice there are of course always frictional resistances, but, for given velocities, the impulse on the wheel is not altered by them, so that the moment of impulse is always given by the above equation. Suppose, now, h the effective head found from the actual head by deducting (1) the height due to the velocity of delivery, (2) the friction of the supply pipe and passages in the wheel, (3) the loss (if any) by shock on entering the wheel ; then Work done per second = wQh. But, if Fj be the speed of periphery of the wheel at the radius i\ where the water enters, V-Jr-^ is the angular velocity of the wheel, and L. V^i\ is the work done per second. We have then for the oase where there is no whirl at exit V\ v i ~ ffh> The effective head h in this formula includes (1) a part equivalent to the useful work, and (2) a part equivalent to the frictional re- sistances to the rotation of the wheel, such as friction of bearings and friction of the water surrounding the wheel (if any) on its external surface. This last item is often described as " disc friction." If H be the actual fcead, the efficiency, apart from external friction, is T?C ' ^ ^1 V 1 Efficiency = 3 = 7 U. The whirl before entrance is communicated by fixed blades SB, curved, as shown in the figures, so as to guide the water in a proper direction on entrance to the wheel. It is the use of these guide blades which characterizes the turbine as distinguished from the reaction wheel. The whirl at different points, either in the wheel or outside it, depends on the angle of inclination of the vanes or guide blades to the periphery. These blades are so numerous that the water moves between them nearly as it would do in a pipe of the same form. If 6 be the angle such a pipe (Fig. 202) makes with the periphery at any point at which the water is flowing through it with velocity U, the radial and tangential components of that velocity will be U . sin 6 and 536 HYDRAULICS. [PART v U . cos#. The first of these is always the velocity of flow u, whether the pipe be fixed or whether it be attached to the revolving wheeL Fig. 202. In the fixed pipe the second is the velocity ? M /* c of whirl which we may call v\ and for motion along a fixed guide blade before entering the wheel, r - co * i j. n V =U. COt 0. In the moving pipe, however, it is the velocity of whirl relatively to the revolving wheel, and this is V-v, therefore Case I. Suppose the vanes of the wheel are radial at the circum- ference where the water enters. In order that the water may have no velocity of whirl relatively to the wheel on entrance, and that the water may enter without shock, we must then have v'=T lt that is, the value of for the fixed guide blade at entrance should be given by Further, the water should be discharged without whirl, that is, v should be zero at the circumference where the water leaves the wheel, hence tan # 2 = ^. '2 The inclination of the fixed blades at entrance, and of the vanes at entrance and exit is thus determined. At intermediate points it would be desirable that it should so vary that vr should diminish uniformly from entrance to exit in order that the action of all parts of the vane upon the water may be the same. This condition would completely determine the form of the vane, but, in practice, any "fair" form would be a sufficient approximation. Supposing the vanes thus designed v I =F l , and the speed of periphery of the wheel, at the circumference where the water enters, is then given by the simple formula a value which applies to the outer periphery of an inward-flow and the inner periphery of an outward-flow turbine. The flow through the wheel instead of being radial may be parallel to the axis, and in this case the formula is still applicable if V^ be taken as the mean of the speeds of the outer and inner peripheries. Case II. In drawing Figs. 201a, 2015, it has been supposed that the vanes are radial at entrance, but this restriction is not necessary ; they CH. xx. ART. 274] MACHINES. 537 may be supposed inclined at a given angle to the periphery. The speed of periphery of the wheel may then be reduced by a proper choice of the angle (Ex. 11, p. 549). Many forms of outward-flow turbines exist, of which the best known was invented by Fourneyron, and is commonly known by his name. The inward-flow or vortex turbine was invented by Prof. James Thomson. Parallel flow, often described as Jonval turbines, from the name of the original inventor, are also a common type (com- pare p. 544). For descriptions and illustrations of these machines the reader is referred to the treatises cited at the end of this chapter. The efficiency of turbines when working under the best conditions is as much as 80 per cent. Their action will be further investigated in the Appendix. 274. Turbine Pumps. Impulse and reaction machines are always reversible, and every motor may therefore be converted into a pump by reversing the direction of motion of the machine and of the water passing through it. If, for example, in the reaction wheel of Fig. 200 we imagine the wheel to turn in the opposite direction with velocity V, while by suitable means the water is caused to move in the opposite direction with velocity v-V, so as to enter the orifices with velocity v, it will flow through the arms to the centre and be delivered up the central pipe. The only difference will be that the lift of the pump will not be so great as the fall in the motor on account of frictional resistances. So, any turbine motor is at once converted into a turbine pump by reversing the direction of its motion and supplying it with water moving with a proper velocity. An inward-flow motor is thus converted into an outward-flow pump, and conversely. No inward-flow pump appears as yet to have been constructed, though it has occasionally been proposed. The " centrifugal " pump so common in practice is, of course, always an outward-flow machine. The earliest idea for a centrifugal pump was to employ an inverted Barker's Mill, consisting of a central pipe dipping into water connected with rotating arms placed at the level at which water is to be delivered. This machine, which must be carefully distinguished from the true reversed Barker's Mill mentioned above, operates by suction. Its efficiency, which may be investigated as in Art. 272, is very considerable (Ex. 4, p. 548), but there are obvious practical incon- veniences which prevent its use in ordinary cases. The actual centrifugal pump is a reversed inward-flow turbine. 538 HYDRAULICS. [PART v. All that was said about motors in the last article applies equally well to pumps, and the same formula applies, V being the speed of rotation of the wheel, now usually called the "fan " and v that of the water, both reckoned at the outer periphery where the water issues. The quantity h is now the gross lift found by adding to the actual lift, the head corresponding to the velocity of delivery, the friction of the ascending main, the friction of the suction pipe and passages through the wheel into the main, and the losses by shock at entrance and exit. A pump, however, works under different conditions from a motor, and corresponding differences are necessary in its design. The energy of a fall can, by proper arrangements, be readily converted, wholly or partially, into the kinetic form without any serious loss by frictional resistances, and the water can, therefore, be delivered to the wheel with a great velocity of whirl to be afterwards reduced by the action of the wheel to zero. When such a motor is reversed, the water enters without any velocity of whirl, and leaves with a velocity, the moment of momentum corresponding to which represents the couple by which the wheel is driven. To carry out the reversal exactly, this velocity ought to be reduced to as small an amount as possible in the act of lifting. Now the reduction of a velocity without loss of head is by no means easy to accomplish, and (see Appendix) always requires some special arrangement. In Thomson's inward-flow turbine, when reversed, the water is discharged with a velocity of whirl which is equal to the speed of periphery F", and given by the formula The corresponding kinetic energy represents at least half the power required to drive the pump, and if it be wasted, as was the case in some of the earlier centrifugal pumps constructed with radial vanes, the efficiency is necessarily less than '5, and in practice will be at most 3. One method of avoiding this loss is to cause the wheel to revolve in a large "vortex chamber," at least double the diameter of the wheel from the outer circumference of which the ascending main proceeds. The 'water before entering the main forms a free vortex, and its velocity is reduced one-half as it spreads radially from the wheel ; three-fourths the kinetic energy is thus converted into the pressure form. The speed of periphery in pumps of this class is that due to half the gross lift. Assuming their efficiency as -65, the gross lift is found by an addition of 50 per cent, to the actual lift. CH. xx. ART. 274.] MACHINES. 539 Many examples of vortex-chamber pumps exist, but they are com- paratively rare, probably because the machine is more cumbrous ; in practice a different method of reducing the velocity of discharge is generally employed. Instead of the vanes being radial at the outer periphery, they are curved back so as to cut it at an angle #, given by the formula (p. 536) F-v = u.cot 6, the velocity of whirl is thus reduced from V to kV, where k is a fraction, and the speed is then If the efficiency be supposed '65, and the velocity be reduced in this way to one-half its original value, this gives about 10^/H for the speed where H is the actual lift. The greater speed is a cause of increased friction as compared with the vortex-chamber arrangement, but on the other hand the friction of the vortex is by no means inconsiderable, and this is so much subtracted from the useful work done. The centrifugal pump in this form was introduced by Mr. Appold in 1851, and is commonly known by his name. Another important point in which the pump differs from the motor is in the guidance of the water outside the wheel. In the motor there are four or more fixed blades which guide the water to the wheel ; but in the pump the outer surface of the chamber surrounding the wheel forms a single spiral guide blade. The whole of the water discharged from the wheel rotates in the same direction, and in order that the discharge may be uniform at all points of the circumference the sectional area of this chamber should increase uniformly from zero at one side of the ascending main to a maximum value at the other side. In some of the earlier designs of centrifugal pumps it was supposed that some of the water would rotate one way, and some the other, but in fact all the discharged water rotates with the wheel, and the passage should be so designed as to permit this, the area corresponding to the proposed velocity of whirl. There are, however, examples in which the water is discharged in all directions into an annular casing, and guided by spiral blades parallel to the axis of rotation. (See a paper by Mr. Thomson, Min. Proc. Inst. C.E., vol. 32.) Centrifugal pumps work to best advantage only at the particular lift for which they are designed. When employed for variable lifts, -as is constantly the case in practice, their efficiency is much reduced and does not exceed -5. It is often much less. 540 HYDRAULICS. [PART v. 275. Approximate investigation of the Efficiency of a Centrifugal Pump. (1) Few centrifugal pumps utilize more than a small fraction of the energy of motion possessed by the water at exit from the wheel, and an investigation of their efficiency on the supposition that this energy is wholly wasted is therefore of considerable interest. Let h be the actual lift, and let all frictional losses except that specified be neglected ; then, if u be the velocity of flow, and v the velocity of whirl at exit, the loss of head is (u 2 + v 2 )/2g, and the gross lift is Substituting this value of h in the formula for V, and replacing u by its value (V - v)tan 6, we obtain Adding J F 2 to each side, and re-arranging the terms, a formula from which we find . h, F 2 - Efficiency == This result shows that the efficiency is greatest when v= V . sin 6 and on substitution we find Maximum efficiency = sec 2 0(1 - sin 6) = ^ - r ^. The speed of maximum efficiency is found from the equation i^o 2 = which gives F 2 = (l The proper velocity of flow is tt F . tan 0(1+ sin 0), and the area of flow through the periphery of the wheel should be made to give this velocity with the intended delivery. At any other speed V the velocity of flow will be given by and the efficiency may be found by the preceding formula. (2) In the preceding investigation .it is supposed that the whole of the energy of motion on exit from the wheel is wasted, and it follows as a necessary consequence that the efficiency is much greater when the CH. xx. ART. 275.] MACHINES. 541 vanes are curved backward than when they are radial. This con- clusion has been verified experimentally, and till recently has been very generally accepted, yet there can be no doubt that so great a waste is not a necessity in a pump with radial vanes but is in great measure a consequence of improper design of the chamber surrounding the wheel and its connection with the delivery pipe. Let A be the sectional area of the chamber at a point the angular distance of which from the point of junction with the delivery pipe is < : then if there be no whirlpool chamber . A- A * AQ '^ the chamber consisting of a simple spiral passage the section of which increases uniformly from zero to its maximum value A Q . The fan will now discharge uniformly at all points of its periphery with a radial velocity u connected with v the velocity of entrance to the delivery pipe by the equation A v = Suj where S is the area of flow. The junction with the delivery pipe must be knife-edged next the wheel and form a continuation of the spiral passage gradually expanding till the full size of the pipe is reached. The area A will generally be such that v is less than the speed of periphery V at any ordinary speed of working, and the water issuing from the radial vanes with velocity of flow u and of whirl V will intermingle with water which has simply the smaller velocity v with which the water moves through the spiral passage. The consequent loss of head may be taken as {u 2 + (F- v) 2 }/2g. The other resistances for the purposes of this calculation are taken as due, (1) to surface friction of pipes and passages, (2) to losses at entrance to the wheel, and (3) to the gradual enlargement after entering the delivery pipe. By suitably curving the vanes at the inner periphery (2) may be reduced and made to depend only on the velocity of flow u, which is proportional to v. We have therefore where /3 is a co-efficient, whence we find This is greatest when v = p g, ^= ^ , iff * and the corresponding Maximum efficiency = 1 - . ^ Q , 2i L + p 542 HYDEAULICS. [PART v. By increasing the size of the suction and delivery pipes and the area of outflow S, the resistance of these pipes and of the passage through the wheel can be reduced to a small amount, while the part of the co-efficient /3 which measures the friction of the spiral passage outside the wheel can hardly exceed -35, or at most -4. Little is known as to the loss in a gradual enlargement, but in many cases, as for example in a trumpet-shaped orifice, it is small : in any case it can only be a fraction of that due to a sudden enlargement. If then the pre- cautions mentioned above are taken in designing the chamber, the value of j3 will not exceed 2 and may probably be capable of being reduced to unity, giving an efficiency ranging from '66 to '75. From one-third to one-half the energy of motion on exit from the wheel is now utilized. Of the waste- work from two- thirds to one-half is due to the sudden change of velocity on entrance to the spiral passage and the rest to surface friction. Curving back the vanes has the effect of reducing the velocity of whirl only when the area S is small enough to increase the velocity of flow u to an amount which causes a considerable loss of head on passing through the wheel. The speed of periphery -is also increased, and for these reasons it is probable that, especially at high lifts, a properly designed pump with radial vanes is more efficient. If a whirlpool chamber be added the spiral passage now forms part of the chamber, and care must still be taken that at the junction with the delivery pipe no obstruction is offered to the rotation of the water. The influence of the form of the vanes is further discussed in the Appendix. When a centrifugal pump is started the fan is filled with water which, in the first instance, rotates as a solid mass with the fan. If the radius of the inner periphery be m times that of the outer where rn is a fraction, it will not commence to deliver water till the speed reaches the value But when once started, the speed may be reduced below this value without stopping the delivery, provided that some of the energy of motion on exit from the wheel is utilized. This has been observed to occur in practice, and it will serve as a test of efficiency. 276. Limitation of Diameter of Wheel. For a given fall in a motor or lift in a pump the diameter of wheel in a turbine is in many cases limited, because some of the frictional resistances increase rapidly * with the diameter. Let u as usual be the velocity of flow, d the diameter, b the inside CH. xx. ART. 276.] MACHINES. 543 effective breadth of the wheel at exit after allowing for the thickness of the vanes ; then the delivery in cubic feet per second is Q = Su = ubrrd. Now, if the breadth b be too small as compared Vith the diameter, the surface friction of the passages through the wheel will be too great, as in the case of a pipe the diameter of which is too small for the intended delivery. Thus b is proportional to d : also, we have seen that u in most of these machines is proportional to V lt that is to *Jh, and it follows therefore, by substitution for b and u that 6=CaVE, where C is a co-efficient. If the wheel be wholly immersed in the water the surface friction (Ex. 8, p. 548) is relatively increased by increasing the diameter. On investigating how great the diameter may be without too great a loss we arrive at the same formula. Where it is of importance to have as large a diameter as possible to reduce the number of revolutions per minute, the diameter of wheel in a pump or a turbine is therefore found by the 'formula If G be the delivery in gallons per minute, h the actual fall in feet, d the external diameter also in feet, the value of c for an outward- flow turbine is about 200. This formula is frequently used in the case of a centrifugal pump with a value of the constant not differing greatly from that just given : but it must be understood that it is only suitable for a pump in which the velocity of whirl at exit from the wheel is reduced by curving back the vanes and increasing the velocity of flow as already described. When the vanes are radial the velocity of flow may be reduced at pleasure. If now D be the diameter of the suction pipe determined for a given delivery Q in the usual way (pp. 488, 495), UQ the velocity in this pipe, Q = u . Trbd = U Q . -D 2 . If u be proportional to U Q) and, as before, b proportional to d, this shows that d should be proportional to D. Assuming U = UQ and b = d the ratio d/D is 2 : but a somewhat larger value is probably desirable, at least for high lifts. It should be observed that in this case the diameter of fan does not depend on the lift, but only on the delivery. 544 HYDRAULICS. [PART v. Centrifugal pumps cannot generally be employed for very high lifts, partly because it becomes increasingly difficult to utilize the energy of motion on exit from the wheel, and partly on account of disc friction. The fan rotates much faster than the wheel of a turbine, and the disc friction is consequently much greater. 277. Impulse Wheels. The formula V* r gk> which gives the speed of a turbine wheel in terms of the effective head, when the vanes are radial at entry, also gives the velocity of whirl at entrance, and therefore shows that, of the whole head employed in driving the wheel and producing the velocity of flow, one-half operates by impulse. When the vanes are not radial (p. 537), a [certain fraction, depending on the inclination and sometimes less than one-half, operates by impulse. The remainder operates by pressure, and turbines of this class are consequently not simple impulse, but impulse-pressure machines. It is necessary therefore that the wheel should revolve in a casing, and that the passages should be always completely filled with water. The diameter of wheel is then limited as explained in the last article, and for a small supply of water and a high fall the number of revolutions per minute becomes abnormally great. This consideration and the necessity o adaptation to a variable supply of water render it often advisable to resort to a machine in which the passages are actually or virtually open to the atmosphere. The whole of the energy of the fall is then converted into the kinetic form before reaching the wheel, and consequently operates wholly by impulse. A wheel of this kind approaches closely in principle to the Poncelet water wheel mentioned in Art. 270, but is often still described as a "turbine," because the water is guided by fixed blades before reaching the wheel. A common example is a Girard turbine with axial flow. The flow of the water is here parallel to the axis of the wheel, spiral guide blades being ranged round the circumference of a cylinder like the threads of a screw in order to give the necessary whirl to the water before entrance. The wheel is provided with a similar set of spiral vanes curved in the opposite direction, which reduce it to rest as it passes through. In the French roue h, poire the wheel is conical, the water enters at the circumference, and, guided by spiral vanes, descends to the apex where it is discharged. Impulse wheels, which are sometimes described as " Girard " turbines even when the flow is radial, appear not to be so efficient as a pressure- turbine working at its best speed. The Pelton wheel (p. 529) may CH. xx. ART. 279.J MACHINES. 545 be taken as an exception. On the other hand their efficiency is very little diminished by a considerable falling off in the supply of water, ^-nd this advantage is so great that they are much employed in cases where the supply of water is subject to variation. The propulsion of ships is effected by machines, which are virtually impulse wheels reversed. The subject is outside the limits of this work, but some information respecting it will be found in the Appendix. 278. Stress due to Rotation. Machines of the class considered in this section often work under a very considerable head, and the speed of periphery determined by the formula becomes very great. Thus in the example of the Pelton wheel given on p. 529, the head is 2100 feet, and the speed of periphery 180 f.s. If A be the stress on the rim of the wheel due to centrifugal action, reckoned in feet of material, as on p. 81 and elsewhere in this treatise, the formula for the centrifugal stress produced given on p. 288, may be put in the form V* = g\ and thus A is simply proportional to the head. In the example just mentioned, this gives a value of A of about 1000 feet. In fans and other machines working with elastic fluids, the head is often much greater. The greatest permissible value of A is given, subject to the qualifications there stated, in the Table on p. 455. SECTION I II. MACHINES N GENERAL. 279. Equation of Steady Flow in a Rotating Casing. -^-When water moves in a pipe or passage of any kind rigidly attached to a wheel or drum rotating about a fixed axis : a general equation can be found for steady flow, as in the case of a fixed pipe considered in the last chapter. Referring to Fig. 202, p. 536, let the pipe there represented be fixed in any position to a rotating wheel, and consider a point in the pipe, the velocity of which is where n is the number of revolutions per second, and r is the distance from the axis. Let the velocity of flow through the pipe at this point be 7, and resolve this velocity into U.cosO along the periphery of the circle described by the point, and U . sin 6 perpendicular to this periphery. In the question considered on the page cited this second component was radial, the pipe lying in a plane perpendicular to the C.M. 2 M 546 HYDBAULICS. [PART v. axis of rotation. We now take the general case in which the pipe is inclined to this plane, and U . sin 9 is consequently the resultant of a radial velocity and an axial velocity, each independent of the velocity of rotation. As before, the component U . cos is the velocity of whirl relatively to the rotating pipe and v=F-U.cose will be the absolute velocity of whirl. Let Q be the flow per second, and consider two points in the pipe specified by the suffixes 2 and 1. Then if L be the couple applied to the part of the pipe between these points = y (V2-Vl)> and since V\r is the same at all points, being the angular velocity 27m, Energy exerted per second = (#2^2 ~ y : ^i)- This is the amount of energy exerted per second on the water as it passes from the point 2 to the point 1, arid is employed in increasing the head. Now the absolute velocity ( K) is given by the equation Hence if as usual p/w be the pressure-head and z the elevation, change of head will be Multiplying this by wQ and equating it to the energy exerted the terms containing vV disappear, and omitting the suffixes 7~To T/'y --+z+ ^ ---- = Constant, w 2g which is the general equation of steady flow. The equation may also be written p U 2 n F 2 + z + -7T- = Constant + , w 2g 2g' showing that the total head in the pipe is increased in consequence of the rotation by the quantity F 2 /2<7, which is the so-called "head due to centrifugal force." If H^ be the head outside the casing before the water enters, then from the value of the absolute velocity given above it appears that PI. !S ~ - CH. xx. ART. 279A.] MACHINES. 54T where the suffix 1 refers to the point of entrance. Hence by sub- stitution - l -. w 2g g Similarly if H 2 be the head after leaving the casing, p U*-F>_ ^V, w + 2g 7~ ;f ' where the suffix 2 refers to the point of exit. When there is no loss of head by hydraulic resistances within the casing HI -H.^= V ^, & which, as before, gives the head employed in driving the wheel in a motor; or, when negative, the increased head created by the external forces driving the wheel in a pump. The losses of head by hydraulic resistance are determined directly from the velocity U just as if the casing were at rest. In questions relating to turbines and centrifugal pumps the general equation here given is often very useful. 279A. Similar Hydraulic Machines. If two machines, whether motors or pumps, are imagined differing only in scale, the heads of water or lifts as the case may be being in the same proportion, the velocities for a given efficiency in the absence of friction by the general principle of similar motions (p. 482) will be as the square roots of their linear dimensions. The same will be true approximately when hydraulic resistances are taken into account. Taking for example the formula *'-*4,* d y' which gives the loss of head in a pipe, we see at once that the losses of head by pipe friction in the two cases compared will be the same fraction of the actual head or lift, and therefore the efficiencies will be the same ; and the same argument applies to all the hydraulic resistances. The efficiency on the small scale, however, will be relatively diminished because the value of 4/ is greater in the small scale motion just as the skin friction of a model is relatively greater than that of a vessel. The delivery in similar machines at corresponding speeds varies as h$ and the power as h?, where h is the linear dimension or head. In comparing ventilating or blowing fans with centrifugal pumps this principle must be borne in mind. Unless the fan be of great size ts action is only comparable with that of a centrifugal pump of great lift and small delivery. 548 HYDRAULICS. [PART v. EXAMPLES. 1. In a reaction wheel the speed of maximum efficiency is that due to the head. In what ratio must the resistance be diminished to work at four-thirds this speed, and what will then be the efficiency ? Obtain similar results when the speed is diminished to three-fourths its original amount. Ans. Efficiency ='63 or '64. Ratio ='84 or 114. 2. Water is delivered to an outward-flow turbine, at a radius of 2 feet, with a velocity of whirl of 20 feet per second, and issues from it in the reverse direction at a radius of 4 feet, with a velocity of 10 feet per second. The speed of periphery at entrance is 20 feet per second, find the head equivalent to the work done in driving the wheel. Ans. 24 '22 feet. 3. In a Fourneyron turbine the internal diameter of the wheel is 9^ inches, and the outside diameter 14 inches. The effective head (p. 535) is estimated at 270 feet; find the number of revolutions per minute. Ans. 2,200. NOTE. These data are about the same as those of a turbine erected at St. Blasien in the Black Forest. 4. An inverted Barker's Mill (p. 537) is used as a centrifugal pump. If the co- efficient of hydraulic resistances referred to the orifices be '125, show that the speed of maximum efficiency is that due to twice the lift, and find the maximum efficiency. Ans. Maximum efficiency '75. 5. A centrifugal pump delivers 1,500 gallons per minute. Fan 16 inches diameter. Lift 25 feet. Inclination of vanes at outer periphery to the tangent 30 U . Find the breadth at the outer periphery that the velocity of whirl may be reduced one-half, and also the revolutions per minute, assuming the gross lift 1 times the actual lift. Ans. Breadth = inch. Revolutions =700. 6. In the last question find the proper sectional area of the chamber surrounding the fan (p. 541) for the proposed delivery and lift. Also examine the working of the pump at a lift of 15 feet. Ans. 24 sq. inches. 7. A jet of water moving with a given velocity, strikes a plane perpendicularly. Find how much of the energy of the jet is utilized in diiving the plane with given speed. Determine the speed of the plane for maximum efficiency, and the value of the maximum efficiency. Ans. peed of maximum efficiency = one-third that of jet. Maximum efficiency = ^ 8 T . 8. Assuming the ordinary laws of friction between a fluid and a surface, and .supposing that any motion of the fluid due to friction does not affect the question : find the moment of friction (L}> and the loss of work per second (7), when a disc of radius a rotates with speed of periphery V. Ans. L=f. ~ .a 3 V 2 ; U=f. 2 . a 2 . V 3 . o o 9. If the rotating disc in question 8 be surrounded by a free vortex of double its diameter, show that the loss by friction of the vortex on the flat sides of the vortex chamber is 2| times the loss by friction of the disc. 10. Show that the loss of head by surface friction in the spiral passage (p. 541) of a centrifugal pump is the same as in a passage of uniform transverse section of the same area of length about 2^ times the diameter of the fan. 11. The vanes of a turbine wheel are inclined at an angle i to the radius at entrance, the angle being measured in the direction of motion of the wheel. Find the speed of CH. xx.] MACHINES. 549 periphery for no shock at entrance and no whirl at exit. Also find the necessary con- nection between the angles of the vanes at entrance and exit and the angle of the guide blades. Ans. With the notation of Case I., p. 536, and further supposing ? > 2 /r 1 =?>?, cot 2 = m (cot 6 1 - tan i). REFERENCES. The subject of hydraulic machines is very extensive, and it is impossible within the- limits of a single chapter to do more than give a general idea of their working. For- descriptive details and illustrations the reader is referred, amongst other works, to GLYNN. Power of Water. Weales' Series. FAIRBAIRN. Millwork and Machinery. Longman. COLTER. Water-Pressure Machinery. Spon. BARROW. Hydraulic Manual. Printed by authority of the Lords Commis- sioners of the Admiralty. As a text-book on turbines and pressure engines may be mentioned BODMER. Hydraulic Motors. Whittaker. 1889. CHAPTER XXL ELASTIC FLUIDS. 280. Preliminary Remarks. An elastic fluid under pressure is a source of energy which, like a head of water in hydraulics (p. 470), may be employed in doing work of various kinds by a machine, or simply in transferring the fluid from one place to another. In hydraulics we commence with the case of simple transfer, but the density of gases is so low that, unless the differences of pressure considered are very small, the inertia and frictional resistances of the fluid employed in a pneumatic machine have little influence : it is the elastic force which is the principal thing to be considered. In studying pneumatics, therefore, we commence with machines working under considerable -differences of pressure and then pass on to con- sider the flow of gases through pipes and orifices together with those machines in which the inertia and frictional resistances of the fluid cannot be neglected. SECTION I. MACHINES IN GENERAL. 281. Expansive Energy. The special characteristic of an elastic fluid is its power of indefinite expansion as the external pressure is diminished. While expanding, it exerts energy of which the fluid itself is, in the first instance, the source, whereas the energy exerted by an incompressible fluid is transmitted from some other source. Expansive energy is utilized by enclosing the fluid in a chamber which alternately expands and contracts ; the common case being that of a cylinder and piston. Fig. 203 represents in skeleton a cylinder and piston enclosing a mass of expanding fluid. Taking a base line aa to represent the stroke, set up ordinates to represent the total pressure S on the piston in each position ; a curve 1 Q2 drawn through the extremities of these ordinates is the Expansion Curve. Reasoning as in Art. 90, p. 182, [CH. xxi. ART. 281.] PNEUMATIC MACHINES. 551 the area of this curve represents the energy exerted as the piston moves from the position 1, where the expansion commences, to the position 2, where it terminates. One common case was considered in the article cited, namely, that in which the expansion curve is a common hyperbola. This is included in the more general supposition, where y is the distance of the piston from the end a of its stroke, and n is an index which, for the particular case of the hyperbola, is unity. Most cases common in practice may be dealt with by ascribing a proper value to n; for air it ranges between 1 and 1*4, and for steam it is roughly approximately unity. The suffixes indicate the points at which the expansion commences and terminates. If, now, E be the energy exerted during expansion, This formula may be written in the simpler form l-n 71-1 in applying which, the terminal pressure $ 2 is supposed to have been previously found from the equation It is, for brevity, convenient to write y2 = ry\> S 2 = fi.S lt where r is a number known as the " ratio of expansion," and p. is a 552 ELASTIC FLUIDS. [PART v_ fraction which may be described as the "pressure fraction" connected with r by the formula * The formula for E then takes the simpler form * n-i n-i The product pr employed for simplicity in this and other formulae- which follow is given by the equation If 7i = 1 the formula fails and is replaced by The value of E is here, in the first instance, expressed in terms of the total pressure on the piston, but, reasoning as in Art. 264, p. 513, we may, if P be the pressure, replace S by PA, and Ay by V, so that S^ is replaced by P 1 V r In "rotatory" engines and pumps the expanding chamber is not a simple cylinder and piston, but is formed from a turning pair. Or, more generally, the chamber pair may be formed from any two links of a kinematic chain which it may be convenient to select for the purpose. In its last form the formula is applicable in every case. If the expansion curve be not given in the form supposed, the value of E is determined graphically by measuring the area of the curve, in doing which, when the chamber is not a simple cylinder, the base of the diagram must represent the volume swept out by the chamber pair, and the ordinates the pressures, per unit of area. 282. Transmitted Energy, -The energy exerted by an elastic fluid consists not merely of that derived from the expansive power of the fluid pressing against the piston, but also of that which is transmitted in the same way as would be the case if it were incompressible. The fluid is supplied from a reservoir, which may either be an accumulator in which it is stored by the action of pumps, or a vessel in which, by the action of heat, it is generated or its elasticity increased. In any case, so long as the cylinder remains in communication with the reservoir the fluid enters at nearly constant pressure, and energy is exerted on the piston just as in the water-pressure engine. During this period of admission the energy exerted is * The symbol x was used for the pressure-fraction in former editions ; the change to u. has been made. to avoid confusion, x being used for the " dryness-fraction " of steam later on in this chapter. en. xxi. ART. 283.] PNEUMATIC MACHINES. the notation being as in the last article. It is usually convenient to- express volumes in cubic feet and pressures (p) in Ibs. per square inch. We must then replace P by 144p. The whole energy exerted on the piston is now . n - ILT U=L+E=L.- , n-l which for the case of the hyperbola becomes U=L(l+log e r). The mean pressure on the piston is conveniently denoted by p m , and is represented in the figure by the ordinate of the line mm so drawn that the area of the rectangle ma is equal to the area of the diagram. Its value is given by the formula? p-. n - ur 1 4- log, r *-=rT-i ; *=* -- ^ A reservoir filled with an elastic fluid at high pressure is an accumu- lator, the absolute amount of energy stored in which is the expansive energy or the total energy according as the pressure is not, or is, maintained by the addition of fresh fluid in place of that discharged, the expansion being supposed indefinite in either case. With the law of expansion already supposed, when n is greater than unity,. fir vanishes when the expansion curve is prolonged indefinitely. The total absolute energy is then where V^ is the whole volume of fluid considered. When n is not greater than unity, U^ is infinite. 283. Available Energy. Of the whole amount of energy thus cal- culated only a part is available for useful purposes, because in practice there is always a " back " pressure P on the working piston, or, more generally, on the sides of the chamber in which the fluid is enclosed. In overcoming this, the work PtfV^ is done, and nothing is gained by prolonging the expansion beyond the point at which the terminal pressure P 2 has fallen to P . The corresponding ratio r is given by the formula The available energy is found by writing r = r , /^ = /^ i n the value of U, and subtracting P ?' V^ P l J^/VQ. This result is . being the difference of the values of U when the expansion commences. ELASTIC FLUIDS. [PART v. -at 1 and at 0. It is always finite, and is graphically represented by the area LI Ob, shaded in the annexed figure. In the transmission and storage of energy by elastic fluids this quantity plays the same part as the "pressure-head" in hydraulics, o which indeed it reduces if n be supposed very great, r unity, and / Fig. 204. V-^ the volume of a Ib. of water. It is the energy of a* given quantity of fluid due to a given difference of pressure, for which, as before, the term "head" may be used when the quantity considered is 1 Ib. Two cases may now be mentioned which are of special importance. (1) Let the reservoir contain air at pressure P reckoned in atmospheres of 14*7 Ibs. per sq. inch, or 2116 Ibs. per sq. ft., and let n= 1'4, then U, - Z7 = 3-5 x 2116(P 1 F 1 -P o r o ), from which we find, writing P 1 = P, P = 1 , and substituting for F" Available energy - 7400 (P - P*) V, where V is the volume of the weight of air considered. (2) Let n= 1 instead of 1'4, then which gives Available Energy = 2 1 1 6 V . P log e P. In either case by putting V= 1 we get the available energy per cubic foot of compressed air, which, it should be observed, depends solely on the pressure. The available energy is here calculated on the supposition that the reservoir is kept constantly full. When the reservoir is not kept full the only available energy is the expansive energy, less the work done in overcoming P through the volume V Q - V Y This is graphically represented by the curvilinear triangle N()l in Figs. 203 or 204, and is most conveniently given by the formula Available Energy = U l - U Q - (P l - P ) V v 284. Cycle of Mechanical Operations in a Pneumatic Motor Mechanical Efficiency. Motors operating by the pressure of an elastic fluid may IOQ described generally as Pneumatic Motors. They are either supplied from an accumulator, as in hydraulic motors of the same class, or they CH. xxi. ART. 284.] PNEUMATIC MACHINES. 555 may be heat-engines serving as the means by which heat energy is utilized. In either case the mechanism of the motor is the same, and consists of a chamber which expands to admit the fluid and contracts to discharge it, with a proper kinematic chain for utilizing the motion of the chamber pair. In water-pressure engines the contraction to expel the water from the chamber is not considered, because all pressures are reckoned above the atmosphere, and the pressure in the accumulator is so great that small differences of pressure may be disregarded. With elastic fluids it is commonly different : the " exhaust " of the chamber must be taken into account. Returning to Fig. 203, suppose that the piston has reached the end of its stroke, the cylinder is then filled with fluid of a certain pressure p 2 which may be supposed known. Let now a valve be opened allow- ing the cylinder to communicate with the atmosphere, or with a reservoir containing fluid at a lower pressure p Q . The fluid in the cylinder then rushes out into the reservoir, and the pressure in the cylinder speedily subsides to ^> ; the fluid expands in this process, but its expansive energy is wasted in producing useless motions in the air which afterwards subside by friction. After subsidence let the piston be moved back by an external force applied to it which supplies the energy necessary to overcome the " back " pressure p Q . The fluid is discharged from the chamber, and so long as the communication with the exhaust reservoir is open the pressure remains constantly p . We represent this on the diagram by drawing a horizontal line 66, the ordinate of which is p . The work done in overcoming back pressure is 144/> F 2 and is represented on the diagram by the rectangle ba ; this is so much subtracted from the energy exerted by the motor. Thus the volume of the chamber goes through a cycle of changes alternately expanding and contracting. During expansion energy is exerted, the corresponding mean pressure p m is the "mean forward pressure." During contraction work is done, and the corresponding mean pressure is the "mean back pressure." The difference between the two is the "mean effective pressure" which measures the useful work done, as shown by the equation Useful work = (p m -j9 )144F 2 , and is graphically represented by the area of the closed figure Ll2bb. In most cases the moveable element of the chamber pair divides the chamber into two parts, one of which expands while the other contracts, and conversely : the motor is then described as " double- acting." The force acting on the moving piece is then the difference between the forward pressure in one chamber and the back pressure 556 ELASTIC FLUIDS. [PART v_ in the other, and when the stress on the parts of the machine is to- be considered this is the effective pressure upon which the stress depends (p. 230). For all other purposes, however, the back pressure is to be taken as just explained. If the pressure p v p Q in the supply and exhaust reservoirs be given, and also the form of the expansion curve, the only waste of energy in this process arises from incomplete expansion. Imagine the ex- pansion curve prolonged to the point o where it meets the back pressure line, arid suppose the stroke lengthened so as to reach this point, then additional work would be done by the fluid which would be represented graphically by the area of the curvilinear triangle '2ob. This area represents energy lost by unbalanced expansion, and to- avoid it the expansion must be "complete," that is, the fluid must be allowed to expand till its pressure has fallen to ^> , the pressure, in the exhaust reservoir, a condition seldom fulfilled in practice, because the loss by friction and other causes becomes disproportionately great. Leaving this out of account, a pneumatic motor is capable of exerting only a certain maximum amount of energy, quite irrespec- tively of the nature of its mechanism, but dependent only on the pressures between which it works and the nature and treatment of the fluid. A motor which reaches this maximum power may be described as mechanically perfect, and the ratio of the actual useful work done to the theoretical maximum may be described as the MECHANICAL EFFICIENCY of the motor. In practice the back pressure is greater than p Q the pressure in the exhaust reservoir itself, the excess being due to the resistance of the passages connecting it with the cylinder. It depends on the speed of piston, the density and nature of the fluid together with the dimensions and type of the passages. No satisfactory formula has been found for it, but its value must be supposed known in each individual case. In Fig. 205 the ordinate of the horizontal line 33 Fig. 20a represents the actual back pressure p B while the other lines are the same as in Fig. 203 : then the shaded area 03320 represents OH. xxi. ART. 285.] MACHINES IN GENERAL. 557 W^ the waste work at exhaust due to incomplete expansion and excess back pressure for a given terminal pressure p y It is given by the formula (P* - P 9 ) V, - 1 n-l which becomes if n= 1, The waste at exhaust may also frequently be conveniently expressed by an equivalent pressure j? upon the piston. Dividing by V 2 we find For a given value of the expansion index n this is independent of the initial pressure and of the nature of the fluid. Hence for given values of the pressure-fractions pjp^ p Q /p 3 , the fractional loss at exhaust is smaller the higher the initial pressure, a very important principle which will frequently be referred to further on. The waste at exhaust here considered is, at least in condensing steam engines, the principal mechanical loss in pneumatic motors, apart from leakage, but there are also some minor losses by a portion of the fluid being retained in the "clearance" space of the chamber after the exhaust is completed, and by the "wire drawing" due to the resistance of the passages connecting it with the supply reservoir. These, however, are details which cannot be considered here. The theoretical maximum is clearly the same as the store of energy in the fluid used (already found in previous articles), which for brevity will be denoted by A. The consumption of fluid (neglecting clearance and leakage) is oije cylinder full, at the terminal pressure, in each stroke. 285. Pneumatic Pumps. A pneumatic like an hydraulic motor may be reversed by applying power to drive it in the reverse direction, and the machine thus obtained is a Pump which takes fluid at a low pressure and compresses it into a reservoir at high pressure. The cycle in the pump is the same as the cycle in a motor, but the operations take place in reverse order. As the chamber expands fluid is drawn in from the low-pressure reservoir and energy is exerted on the piston by the original " back " pressure : as the chamber contracts the fluid is compressed till it reaches the pressure p lt when a valve 558 ELASTIC FLUIDS. [PARTY. opens and admits it to the high-pressure reservoir. There is, however, this important difference, namely, that the process of unbalanced expansion in the motor cannot be reversed ; and therefore, if the pump is to operate on the same weight of fluid, the volume of the working cylinder must be enlarged so that the expansion curve may start from o. If this be supposed, the compression curve will, for the same fluid treated in the same way, be identical with the expan- sion curve of the motor. If there were no unbalanced expansion the motor would be exactly reversible, and the condition of a motor being mechanically perfect may therefore be described by saying that it must be mechanically reversible. The difference of working of the valves in pumps and motors has already been referred to in Art. 269. The work done in pumping air into a reservoir is the same apart from resistances as the available energy U U Q found in a previous article (p. 554), but, for reasons to be explained hereafter, it is generally advisable to express it in terms of the volume of air used at atmospheric pressure, the formula for compression to P atmospheres absolute then becomes Work done = 7400 F (P f -l) ( = 1'4), where V Q is the volume of atmospheric air consumed, and, as before, the work done per cubic foot depends solely on the pressure. Air pumps are still more frequently employed for the purpose of exhausting a chamber, in which case the atmosphere is the high-pressure reservoir into which the low-pressure air in the chamber is forced. The formula for U- U is, with change of sign, directly applicable to this case, V being the volume of low-pressure air and P the pressure expressed as a fraction of an atmosphere. In either case if the pressure in the chamber or reservoir is not maintained constant the formula must be modified as before explained. Examples are given at the end of the chapter. In all pneumatic motors a pump is required to replace the fluid in the supply reservoir. Unless the motor be a heat engine this pump must be driven by external agency, and the whole process is one of storage, transmission, and distribution of energy, a subject briefly considered further on. 286. Indicate Diagrams. The pressure existing in the chamber of a pneumatic machine may be graphically exhibited by means of an instrument called an Indicator. In steam engines especially its use is indispensable to enable the engineer to study the action of the steam. PLATE XI. Fig. I To face page 559. CH. xxi. ART. 286.] THERMODYNAMIC MACHINES. Figs. 1 and 2, Plate XL, show an indicator in elevation and section. S is a drum revolving on a vertical axis, A is a cylinder communicating with the steam cylinder, the pressure in which is to be measured. P is. a pencil connected by linkwork with a small piston H so as to move with it up or down in a vertical line. The piston is pressed down by a spring which measures the pressure, while the drum, by means- of a cord passing over pulleys and connected with the steam piston,, revolves through arcs exactly proportional to the spaces traversed by it. A card is folded round the drum, and as the engine moves a curve is traced by the pencil upon it which shows the pressure at each point of the stroke. In practice many precautions are necessary to secure accuracy in the diagram ; the more so the higher the speed, because the friction and inertia of the parts of the indicator, together with unequal stretching of the cord and inaccuracy in the reducing motion connecting the drum with the steam piston, may give rise to serious errors. To diminish the effect of inertia the stroke of the indicator piston is made short and multiplied by linkwork. In the example shown (Crosby's patent) the spring applied to the drum to keep the cord tight has a tension which increases as the drum rotates from rest. This increase compensates for the inertia of the drum, and is said to give a more nearly uniform tension of the cord. Fig. 206 shows an indicator diagram taken in this way from the high pressure cylinder of a compound engine. BB is the atmospheric line drawn on the card by the indicator Pig. 206. j _B pencil when the cylinder communicates with the atmosphere. AA is. the vacuum line laid down on the diagram at a distance below BB, which represents the pressure of the atmosphere, as found by the barometer, reckoned on the scale of pressures. Then on the same scale any pressure shown by the indicator is the absolute pressure when measured from A A. The figure drawn is a closed curve bearing a general resemblance to the diagram (Fig. 203), which was drawn to represent the cycle of operations of a motor. The principal difference is that the corners of the theoretical diagram are rounded off in the actual diagram, an ,560 ELASTIC FLUIDS. . [PARTY. -effect principally due to the valves closing gradually instead of instantaneously. Also at the end of the return stroke a certain amount of steam is retained in the cylinder and compressed behind the piston, causing the very considerable rounding off observable at the left-hand corner. In every case the mean effective pressure may be determined graphically by measuring the area of the diagram and dividing by the length of the stroke. This, with the number of revolutions per 1' determines the horse-power for an engine of given dimensions, and the consumption of steam in cubic feet per 1' for each horse-power thus "indicated" can be found. The weight of steam used, however, cannot be found without measurement of the feed water used, because the steam always contains an unknown amount of water mixed with it. 287. Brake Efficiency. The indicated horse-power (I.H.P.) deter- mined as in the preceding article, is subject to a deduction consequent on the friction of the mechanism of the engine, and the power actually -delivered is the Brake Horse-Power (B.H.P.), which can, at least theoretically, be measured by a suitable dynamometer, and which in -small motors actually is frequently measured by a "friction brake" {p. 279). The ratio of the two is the frictional or brake efficiency. The term "mechanical efficiency" is commonly employed with re- ference to the external (frictional) mechanical waste alone, but the internal mechanical waste considered in Art. 284 may also properly be included in the meaning of the word. The external waste by friction of mechanism, as will be seen on reference to page 259, may be represented by a pressure / on the piston given by the formula /=*+/< where / is the load on the engine reduced to unit of area of the piston -and e, / are constants. To the remarks made on the page cited, it may be added that recent researches * show that these constants, though nearly independent of the load, increase with the initial pressure of the steam, that is, they are greater for high ratios of expansion than for low. They also increase with the speed, but no definite law of increase with initial pressure and speed has been discovered. In many types of non-condensing engine the friction is independent of the load, that is, the co-efficient e is the zero and the friction pressure / may reach 3 or 4 Ibs. per sq. inch. * Manual of the Steam Engine, by R. H. Thurston. Part I., second edition, p. 560. "Wiley & Sons, New York, 1892. CH. xxi. ART. 288.] THERMODYNAMIC MACHINES. 561 In any case the pressure / is equivalent to an increase in the back pressure, and the lowest value which the terminal pressure pt, can have consistently with economy is Hence also the external and the internal mechanical wastes are subject to nearly the same laws. The most important part of the external waste is approximately constant, and may be included, if we please, with the corresponding part ( W^ of the internal waste calculated in Art. 284. SECTION II. THERMODYNAMIC MACHINES. 288. Cycle of Thermal Operations in a Heat Engine. So far all that lias been said applies equally well to all pneumatic motors, though its most important application may be to the case where the fluid serves as the means whereby mechanical energy is obtained through the agency of Heat. We now go on to consider very briefly the principles which apply especially to heat engines. In heat engines the pump necessary to replace the fluid in the supply reservoir, or discharge it from the exhaust reservoir, is worked by energy derived from the working cylinder, so that the engine is self-acting. Now, if the condition of the fluid were the same in the pump as it is in the working cylinder, as much energy would be required to drive the pump as is supplied by the motor, or in practice, more ; a necessary condition therefore that any useful work should be done is that, by the agency of heat, the condition of the fluid should be changed so that its mean density, while being forced into the supply reservoir, shall be greater than when doing its work in the working cylinder. Hence, the fluid must be heated in the supply reservoir, and cooled in the exhaust reservoir, and therefore in every heat engine, in addition to the cycle of mechanical operations, there is a cycle of thermal operations consisting of an alternate addition and subtraction of heat ; the heat in question being supplied by a body of high temperature and abstracted by a body of low temperature. In non-condensing steam engines the pump is the feed pump which supplies the boiler with the fluid in the state of water; in the boiler heat is supplied which converts it into steam of density many hundred times less than that of water. The pump is in this case very minute, and requires a trifling amount of energy to work it. In condensing engines we have, in addition, the air pump. In air engines the compressing pump is generally a conspicuous part of the apparatus and requires a large fraction of the power of C.M. 2 N 562 ELASTIC FLUIDS. [PART v. the motor to drive it; because the changes of density due to the alternate heating and cooling are comparatively small. 289. Mechanical Equivalent of Heat. Heat and mechanical energy are mutually convertible ; a unit of heat corresponding to a certain definite amount of mechanical energy which is called the "MECHANICAL EQUIVALENT " of heat. The statement here made is the First Law of the Science of Thermodynamics, and it shows that quantities of heat may be expressed in units of work, and, conversely, quantities of work in units of heat. In dealing with questions relating to heat and work, a common unit of measurement must be selected. In most cases the thermal unit is adopted, and quantities of work reduced to such units by division by the mechanical equivalent of heat. Until recently the numerical value of the equivalent was taken as 772 in British units, but it is now recognized that this is somewhat too- small. In this work the value 780 will be employed, which is just one per cent, greater, and quantities of work in foot pounds are therefore reduced to thermal units by division by 780. Thus the horse-power of 33,000 ft. Ibs. per minute becomes 42-3 thermal units per minute or 2538 per hour. In heat engines the cycle of thermal operations consists of an alternate addition of heat (Q) and subtraction of heat (B), so that, if W be the useful work, W=Q-R, that is, the work is done at the expense of an equivalent amount of heat which disappears during the action of the engine. In steam engines this has been tested experimentally by measuring the heat supplied in the boiler and the heat discharged from the condenser. The difference should be, and in fact is found to be, the thermal equivalent of the work done by the engine. The ratio WjQ is usually called the " absolute," or sometimes, for reasons we shall see presently, the "apparent" efficiency of the engine, but would be much better described as the Co-efficient of Performance. It is always a small fraction : in the best steam engines, for example, it seldom exceeds '18 losses connected with the furnace and boiler not being included. Supposing as on page 577, A the theoretical maximum for a pneumatic motor working between the given limits of pressure, the ratio WjA, which we will call e t is the " mechanical " efficiency. 290. Mechanical Value of Heat. In stating the first law of thermodynamics nothing is said about the temperature at which the CH. xxi. ART. 290.] THERMODYNAMIC MACHINES. 563 heat is used. In other words, the mechanical equivalent of heat i& just the same whether the temperature be low or high. Yet common experience tells us that the value of heat for mechanical purposes depends very much on this circumstance. The heat discharged from the condenser of a condensing steam engine, or with the exhaust steam of a non-condensing engine, is of little value for the purposes of the engine. So obvious is this fact that the first attempts at connecting the work done by a heat engine with the heat supplied to it may be partly described as attempts to show that temperature, not quantity, was equivalent to energy, heat being supposed as indestructible as matter. It is now known, however, that difference of temperature is not in itself energy, but merely an indispensable condition that heat may be capable of being converted into work. The power of a heat engine depends on difference of temperature, being greater, the greater that difference is ; but in all cases only a fraction of the heat supplied is converted into mechanical energy. In the converse operation of converting mechanical energy into- heat it is possible, by employing it in overcoming frictional resist- ances, to obtain an amount of heat equal to the energy employed, but such processes are always irreversible. The only way of converting heat into work is by means of a heat engine in which the rejection of heat at low temperature is as essential as the supply of heat at high temperature. Difference of temperature is wasted if heat be allowed to pass from a hot body to a cold one without the agency of steam, air, or some other body, the density of which is changed by its action. When once wasted it cannot be recovered, a fact of common experience which is expressed in other words by a second thermodynamic principle. SECOND LAW. Heat cannot pass from a cold body to a hot one by a purely self-acting process. By a "self-acting" process in this statement is meant any process which is of the nature of a perpetual motion being independent of any external agency. By the employment of mechanical energy drawn from external bodies, heat may be made to pass from a cold body to a hot one, the amount of energy required being greater the greater the difference of temperature. And the method sometimes employed of raising steam, without the use of a furnace, by means of heat derived from the exhaust steam condensed in a solution of caustic soda, shows that energy derived from chemical affinity may serve the purpose. But, if no energy is employed, no heat will pass. 564 ELASTIC FLUIDS. [PARTY. Difference of temperature must therefore be carefully utilized, and since the smallest difference of temperature is sufficient to cause heat to pass from a source into the air or steam which exerts energy, it at once follows that the process of conversion of heat into work will be most efficient if all the heat be supplied while the fluid has the temperature of the source of heat, and all the heat rejected while it has the temperature of the body which subtracts heat. These are the conditions of maximum efficiency, and if they are satisfied it is possible to show that a mechanically perfect motor (p. 576) supplied with heat Q will exert the energy Jj, T 2 , being the temperatures of addition and subtraction of heat, reckoned from the "absolute" zero, a point 460 below the ordinary zero of Fahrenheit's scale. This is true whatever be the nature of the heat engine employed for the purpose, and no more heat can be converted into work under any circumstances. An engine which satisfies these conditions may be described as "thermally perfect." If two bodies be at the same temperature heat may be made to flow in either direction from one to another, the actual direction being determined by a difference which may be made as small as we please : that is, the process is reversible. Hence the conditions of maximum thermal efficiency may also be described by saying that the cycle of thermal operations must be " thermally reversible." And the condition that an engine may be both mechanically and thermally perfect may be completely described by stating that the engine is reversible. Whichever way we adopt of stating the result it follows at once that a unit of heat has a certain definite MECHANICAL VALUE given by the equation where T lt T Q are the temperatures between which it can be used. When reckoned in thermal units M is also often called the AVAILABLE HEAT. If, instead of the whole amount of heat Q being supplied at the same temperature T 19 the fractions q } , q. 2 , q s , ... are supplied at the several temperatures r J\, T 2 , T 3 , ..., the temperature of abstraction of heat remaining the same, the mechanical value of the whole is the sum of the mechanical values of each of the several parts taken separately. On expressing this principle algebraically it will be found that the mechanical value of the whole is now in thermal units T - T CH. xxi. ART. 291.] THERMODYNAMIC MACHINES. 565 where T m is the average temperature of supply given by the equation J_-i + 2. f , rn rp '7' ' ' ' ' 1 m * 1 * 2 In many cases the whole or a part of the heat is supplied at a uniform rate as the temperature rises or falls, as, for example, when a mass of hot air is employed as a source of heat by cooling it at constant pressure. The exact value of the mean temperature of heat so supplied may be found by integration, but unless the change of temperature is excessive, the mean in question is very approximately the arithmetic mean of the highest and lowest temperatures. If then a quantity of heat Q be supplied at a uniform rate as the temperature rises from 1\ to .T lt the part of that heat mechanically available will be a useful formula which we shall have occasion to use presently. 291. Available Heat of Steam. When steam is formed from water supplied to a boiler the temperature of the boiler is connected with the pressure by a perfectly definite law, so that when the pressure is known the temperature can be found, and conversely. The results are well known, and given by a table which, being generally accessible, need not be reproduced here. The process may be separated into two parts (1) the raising of the feed water from T , the temperature at which it enters, to T } , the temperature of the boiler ; and (2) the formation of steam at the constant temperature T l . The quantity of heat supplied during the first stage is approximately 1\ - T Q , and the rate at which it is supplied is approximately uniform. Its mechanical value is therefore, putting T 2 = T G in the general formula given above, During the second stage, if the steam formed be saturated and free from moisture, the quantity of heat supplied is commonly called the "latent heat of evaporation," and is given for each pound of steam by the well known formula 1 -966--71(* 1 = 2lS*), where ^ is the temperature Fahrenheit. Since the whole of this is supplied at the temperature 1\( = ^ + 460), the corresponding mechanical value is The mechanical value of the whole heat supplied is now 566 ELASTIC FLUIDS. [PARTY. In using this formula the lower temperature T must correspond to PQ the pressure in the condenser as shown by the vacuum gauge, or assumed for the purposes of the calculation. In a non-condensing engine T Q must correspond to the pressure of the atmosphere, which in this case is the exhaust reservoir, that is, it must be supposed 212 + 460 or 673. In perfect engines the mechanical value of the heat supplied is also the available energy of the fluid used, which is thus obtained from the temperatures of supply and rejection of heat without the necessity of knowing the form of the expansion curve, which always must be such that its area, as in preceding articles, represents the energy in question. The available energy is therefore given by the formula A close approximation, however, to the available energy may be ob- tained by considering the form of the expansion curve (see Appendix). This leads to the very simple formula Po where P v is the product of the pressure and the volume of dry saturated steam at the lower limit of pressure p Ql a quantity found in thermal units by a formula given further on or from a table of the properties of saturated steam. If the steam be superheated 6, the additional heat supplied will be \Q thermal units nearly, and the rate of supply will be approxi- mately uniform. The corresponding mechanical value will therefore be, putting r 2 = Tj + in the general formula, The whole available energy is now M Q + M 1 + M Z , but the increase is relatively small, the actual economy due to superheating being not due to this cause but to a reduction in cylinder condensation, as will be further explained presently. 292. Thermal Efficiency. If an engine be mechanically perfect the work done per unit of heat will be simply the mechanical value, if the conditions of maximum efficiency are satisfied. In general, however, some of the heat will be supplied at a lower temperature than the source of heat, and some will be abstracted at a higher temperature than that of the refrigerator. When this is the case difference of temperature is wasted, and there is a corresponding loss of thermal > , M and, therefore, would have been increased in the proportion T l + T Q :T 1 . As, however, the supply of heat at rising temperatures cannot practically be avoided, the available energy is considered to be that which remains after deduction of the corresponding loss. A part of the loss may be regained by use of a properly constructed feed-water heater, but the resulting gain is most conveniently estimated independently. The standard of comparison in heat engines therefore is not always an ideally perfect engine, but is fixed with reference to the result which could be attained in an engine of that type if all its working arrange- ments were perfect. In practice the engine will not be either mechanically or thermally perfect; its efficiency (W/M) will then be the product (ek) of the mechanical efficiency (WjA) and the thermal efficiency (A/M). The efficiency thus calculated is estimated relatively to an engine which is mechanically and thermally perfect, and may be described as the 568 ELASTIC FLUIDS. [PARTY, " relative " or " true " efficiency, as distinguished from the " absolute " or "apparent" efficiency defined in a former article. To estimate the efficiency of a heat engine without any reference to the temperatures between which the heat can be used is very misleading. The true efficiency of the best condensing steam engines is about 65 per cent., instead of 18 per cent, as it appears to be merely from the quantity of heat used. The standard of comparison is, however, for reasons which have just been pointed out, generally to some extent conventional, and consequently varying estimates of the efficiency may be made. 293. Compound Engines. The working fluid may be discharged from one contracting chamber into a second which simultaneously expands. In many cases an intermediate reservoir is employed, which receives the fluid from the first chamber and supplies it to the second ; the two chambers are then virtually separate, and form two distinct motors, the power of which can be separately calculated. The sum of the two is the power of the compound motor; it is necessarily the same as if the fluid had been used with the same expansion curve between the same extreme pressures in a single chamber ; except that the frictional resistance of the passages between the chambers and the intermediate reservoir represents a certain loss of energy in the compound motor which does not occur in the simple one. When there is no intermediate reservoir there is no distinct period of admission or expansion in the low-pressure chamber, but the power may still be determined graphically for each chamber, and the results added. The process of compounding may be carried further by the employment of triple and quadruple expansion. In every case the energy of the fluid is the same, and cannot be affected by the mechanism employed to utilize it, unless its density or elasticity be altered by contact with the sides of the chamber in which it is enclosed. In steam engines, however, the action of the sides of the cylinders has great influence by condensing steam as it enters the cylinder. The liquefied steam is re-evaporated towards the end of the stroke as the temperature of the steam falls, but the process is nevertheless a very wasteful one. The action is greater the greater the degree of expansion employed, because the range of temperature is greater, and the gain by expansion is thus in great measure neutralized or even converted into a loss. By employing two cylinders instead of one the expansion is divided into two parts each of moderate amount, and liquefaction may be diminished. Moreover for constructive reasons the excessive expansion necessary to obtain CH. xxi. ART. 294.] THERMODYNAMIC MACHINES. 569 the full advantage of high-pressure steam cannot be carried out in a single cylinder. Compound engines are therefore being used more and more wherever economy of fuel is a consideration, and in marine practice have almost superseded the simple engine. The principal losses in steam engines are (1) a mechanical loss due to incomplete expansion, and (2) a thermal loss due to liquefaction. One of these cannot be diminished without increasing the other ; but considerable economy may be effected by the use of a " steam jacket," by the employment of superheated steam, and by compounding. 294. Useful Work of Steam. The relation between the pressure (P) and the volume (v) of dry saturated steam is expressed by the equation pj* = Constant, from which is readily derived the formula log (Pv) = 1-7882 + '0607 logp, which gives in thermal units the value of Pv for 1 Ib. of dry saturated steam of pressure p Ibs. per sq. inch. The logarithms are here common, not hyperbolic. In the formula for W given in Art. 285 (p. 557), the value to be used for P Z V 2 can ^ e obtained by calculating P 2 v 2 for the terminal pressure p 2 , and then multiplying by x 9 the dry ness fraction of the steam at release. The index n of the expansion curve may for this purpose be taken as 10/9, and we thus obtain for the waste work at exhaust the formula The remaining losses may conveniently be expressed as a fraction 1 - k of the available heat (M) of the steam for the given boiler and vacuum pressures^, p Q . The useful work of 1 Ib. of steam is then W=kM-W Q . The value of the co-efficient k depends mainly on thermal losses, of which the principal is cylinder condensation, but it also includes leakage and the minor mechanical losses already referred to. Thus, in com- pound engines, k is diminished by the losses by clearance and leakage in all the cylinders and by wire drawing between the cylinders. The quantity (\-k)M may conveniently be described as the "missing work," representing, as it does, mainly, losses which cannot be detected by the indicator alone, but only by measurement of the feed-water. In a given engine W, W^ and M can be derived from data furnished by experiment, and hence k can be found. Examples of this calculation for engines of various types will be found in the author's work on the 570 ELASTIC FLUIDS. [PARTY. steam engine,'*' from which it appears that unless there is some special cause of waste, the " missing work " is from. 20 to 30 per cent, of the available energy ; that is, the value of k is from '7 to '8. By assuming values of k, and x. 2 the dryness fraction of the steam at release, the consumption of steam in Ibs. per I.H P. per hour can be determined for given pressures ; for since one horse-power is 2538 thermal units per hour, 2538 Lbs. per I.H.P. per hour = The fraction x 2 is less variable than k, and may generally be assumed at '8, unless there be some special cause of waste; and thus in the most economical engines commonly occurring in practice, the consump- tion of steam will be found approximately by writing k = *8, 2 = '8 in the preceding formulse. But where cylinder condensation is excessive, as, for example, is the case in small engines running at a low speed, the consumption may be double the amount thus given. On the other hand it may be somewhat less when superheated steam is used, mainly because cylinder condensation may in this way be greatly reduced. 295. Efficiency and Performance of Steam Engines. We have already described the ratio WfM (p. 567) as the efficiency of the steam ; it is given by the formula py -^ M = *--M- If expansion be carried to the greatest extent which can in any case be advantageous (Art. 287), p 2 will be a given quantity, and JF may be taken as constant. The efficiency for a given vacuum is then greater the higher the boiler pressure ; that is, an increase in the boiler pressure has not only the effect of increasing the energy of the steam theoretically available, but it renders it possible to utilize a greater fraction of it. On the other hand, an improvement in the vacuum increases the available energy, but as it also increases W$ in a much greater proportion, the efficiency is lowered. This is a necessary con- sequence of the low pressure of the steam requiring large cylinders and great friction for a given power. The energy theoretically available from heat employed at temperatures much below 212 can only be made use of without great waste by means of some fluid which is more volatile than water. In non-condensing engines the fraction W Q fA may be and generally is small ; their efficiency therefore may be as much as '75, or, in special cases, more. * The Steam Engine considered as a Thermodynamic Machine, third edition, p. 322, Spon, 1895. CH. xxi. ART. 296.] THERMODYNAMIC MACHINES. 571 The efficiency of an engine is not generally greatest when expansion is carried to the extreme limit fixed by the back pressure and by friction, because the value of 1 - k is greater than it would be if a smaller expansion had been employed; this is specially the case in single- cylinder engines working at a moderate speed. The expansion which can be usefully employed in practice is further limited by considera- tions of cost; interest on capital, as Professor Thurston has pointed out, being a " waste " which ought to be taken into account. The general question of steam engine economy is far too large and important to consider in detail in the present work, but the foregoing observations may be of service in drawing attention to the principal points to be studied. 296. Reversed Heat Engines. A heat engine like an hydraulic motor may be reversed, and then becomes a machine for drawing heat out of cold bodies and supplying it at a higher temperature just as a pump takes water from a low and discharges it at a high level. Most heat engines occur in their reversed form, being employed as "refrigerating" or, to use the phrase employed in Germany, "cold" machines in the artificial production of ice, or the maintenance of a low temperature in a chamber for the preservation of articles of food. If the heat engine be perfect the reversal will be exact, the same thermodynamic machine, or as for brevity we might perhaps describe it, the same THEKMO being a heat motor or a heat pump according to the direction in which it is driven. As in hydraulic machines, however, the reversal in practice will not be perfect, and certain con- structive differences between the motor and the pump will generally be rendered necessary by the different conditions under which they work. The refrigerating machines most in use are the air machine, which operates by the compression and subsequent expansion of atmospheric air, and the ammonia-compression machine. The first of these, which is a reversed air engine, we shall have occasion to refer to presently. The second, which is much employed in the manufacture of ice, will now be discussed in illustration of the fore- going remarks. In making ice by this method, the water to be frozen, originally, of course, at the atmospheric temperature (1\), is contained in chambers forming divisions of a refrigerating tank filled with brine, at a tempera- ture below the freezing point, by which it is first cooled to 32 and finally frozen. The heat thus received by the brine, together with that which leaks into the tank from surrounding bodies, is then abstracted by the evaporation of liquid anhydrous ammonia contained in a coil of 572 ELASTIC FLUIDS. [PART v- piping immersed in the tank. The liquid in question is highly volatile, its vapour having a pressure of over four atmospheres at the tempera- ture 32 F. As fast as it is formed the resulting ammonia gas is drawn into a double-acting compressing pump, by means of which its pressure is raised : the temperature at the same time rising above that of the atmosphere. When the pressure has reached a certain limit, ranging from 8 to 12 atmospheres, a valve opens, and the gas passes into a second coil of piping surrounded by circulating water of atmospheric temperature, by which it is condensed once more into liquid. To carry out the process perfectly it would now be necessary to admit the liquid into an expansion cylinder, where its pressure would gradually fall while driving a piston. A portion of the liquid would then evaporate, and the temperature would be reduced till it had fallen to T , the temperature of the evaporating coil in the brine tank. This part of the process not being practicable, the high-pressure liquid is actually allowed to rush through a small connecting pipe into the coil, thus completing a continuous cycle. The difference this makes will be considered further on ; for the present we suppose the expansion cylinder to exist, and to be connected with the crank shaft by which the compressing pump is driven. If now R be the heat abstracted from the refrigerating tank at temperature T Q and U the energy exerted in driving the crank shaft, the heat transferred to the circulating water by the condensing coil will be the final result of the process being that a quantity of heat R passes from the temperature T Q to the higher temperature T x by the agency of a certain amount of mechanical energy U, which is converted into heat in the process. Further, assuming the temperatures T l and 1\ of the coils to differ very slightly from the temperatures of the atmosphere and the tank, every step of the process will be exactly reversible, and when reversed the machine becomes a heat motor, of which U must be the mechanical value of the heat Q and E the heat rejected. Hence the relations between U, Q, R must be the same in the two cases, that is if U be the mechanical energy necessary to abstract the heat R*. The quantity of heat may be described as the REFRIGERATING VALUE of the energy U. CH. xxi. ART. 297.] THERMODYNAMIC MACHINES. 573 In the actual machine the expansion cylinder is omitted, and the energy required to drive the crank shaft is correspondingly increased ; while a portion of the liquid ammonia is none the less evaporated as it rushes into the evaporating coil without drawing heat from the brine, so that the heat abstracted is not increased. If then E' be the heat actually abstracted from the freezing water for a given amount of energy U, R will be less than R from this cause, as well as from leakage of heat and other losses. The ratio R'jR is the efficiency of the machine, which in this case includes the friction of the mechanism, and in good machines of this class appears to be about 40 per cent. But, as in motors, the standard of comparison is to some extent conventional, because it is possible to make various practical estimates of the " refrigerating value " of the energy employed. The foregoing sketch, necessarily very brief, of the action of thermo- dynamic machines is all that can be attempted in the present work. We now pass on to consider more particularly the transmission of energy by elastic fluids and the flow of gases through pipes and orifices. SECTION III. TRANSMISSION OF ENERGY, FLOW OF GASES. 297. Internal Energy. Internal Work. The distinction between internal work and external work was pointed out in Art. 92, p. 186, and the corresponding distinction between internal and external energy of motion in Art. 133, p. 268. These distinctions are principally im- portant in fluids, because the extreme mobility of their parts renders internal motions, of great magnitude, of common occurrence. We have already seen in Chapter XIX. how energy is dissipated by the internal action of liquids ; in gases the same dissipation occurs, and is even more important. In liquids the absorption of energy is almost completely irreversible, but in gases it is not so. We may have internal energy as well as internal work : the greater part of the expansive energy of a gas being due to internal actions. The state of an elastic fluid is completely known when its pressure and volume are known, but these quantities are capable of any variation we please within wide limits, provided only that we have unlimited power of adding or subtracting heat. If, however, a third quantity, the temperature, be considered, it will be found that the three are always connected together by a certain equation depending on the nature of the fluid, so that when any two are given the third is known. For example, in the so-called " permanent " gases, such as 574 ELASTIC FLUIDS. [PART v. dry air, the equation is very approximately PF=c.T, where T is the temperature reckoned from the "absolute" zero, as in Art. 290, and c is a constant which for pressures (P) in Ibs. per square foot and volumes (V) in cubic feet per Ib. has, for dry air, the value 53'2. The " state " of the fluid is completely known if any two of these three quantities are known, but not otherwise. To produce a given change of state a certain definite amount of work must be done in overcoming molecular resistances ; this is the internal work, and is the same under all circumstances. But in gaseous fluids, the molecular forces being reversible, may tend to give rise to the change of state, and then we have internal energy instead of internal work. Taking the first case : if the change be at constant volume, this internal work will be the total work done ; but in general the volume changes, and in consequence external work is done, the amount of which depends not merely on the change of state, but also on the way in which that change is carried out. The total amount of work is the sum of the internal and external work : it is done by the agency of heat energy supplied from without, so that we write Heat Expended = Internal Work + External Work, the three quantities being expressed in common units. An important application of this equation is to questions relating to- the formation of steam, but this we must pass over, our present object being to consider the flow of gases through pipes and orifices, for which purpose the equation is written Expansive Energy = Internal Energy + Heat Supply, or, in other words, of the whole expansive energy of the fluid, a part is derived from internal molecular forces, and a part from heat supplied from without. If no heat is supplied from without the expansive energy is equal to the internal energy: this case is called "adiabatic" expansion, obtained by writing 7i=T4 in the formulae of Art. 281. More generally, it is shown in treatises on thermodynamics that the internal work done in changing the temperature of a Ib. of air from T! to T. 2 is 7 2 - /!, where I=K r .T=2-5Pr, K v being the specific heat at constant volume, which is 2'5 c. Hence when the temperature falls from 1\ to T 9 the internal energy supplied by the fluid is K v (T l T ^2) an< ^ tne equation becomes for a heat supply Q- CH. xxi. ART. 298.] TRANSMISSION OF ENERGY. 575 Fig. 207. This is the fundamental equation from which all cases may be derived. If the heat be supplied to a permanent gas at a uniform rate as the temperature falls, it may be shown that the law of expansion is PV n = constant, as supposed in Art. 281, and this is generally per- missible with sufficient approximation. The expansion index n then depends upon the proportion which Q bears to E. If Q = E the expansion is hyperbolic, and the whole of the expansive energy is derived from heat supplied from without. The manner in which the expansive energy (E) depends on the heat supply (Q) is well seen by the annexed diagram (Fig. 207). Let, as before, the ordinates of the point 1 represent the pressure and volume before expansion and those of the point 2 after expansion, 1, 2 being the expansion curve. Set downwards N^Z^ N 2 Z 2 , each equal to 2J the corresponding pressure ordinates, and complete the rectangles OZ^ OZ%. Then complete the rectangle Z^Z^ and draw the diagonal SL to meet the vertical through 0. Finally through the inter- section draw // horizontally ; then the rectangle IN 2 will be found to be the Z; difference of the rectangles OZ lt OZ. 2 , and therefore represents the internal energy exerted during expansion. Thus the area 12/7 (shaded in the figure) represents the heat supply : which will depend not only on the points 1, 2, that is, on the change of state of the air, but also on the form of the expansion curve, that is, on the way in which the change takes place. 298. Transmission of Energy by Compressed Air. A reservoir of compressed air furnishes a supply of energy which may be transmitted by pipes to distant points and distributed at pleasure. The losses which occur in the pipes by leakage and friction will be discussed further on; the present article will be devoted to the consideration of the process of compression and expansion. The volume of 1 Ib. of air at the atmospheric pressure is r=-^a=h nearl y> \ where T Q is the absolute temperature. The work done in compressing 1 cubic foot to a pressure of P atmospheres without gain or loss of 576 ELASTIC FLUIDS. [PARTY. heat, and forcing it into a reservoir, is (Art. 285, p. 558) Work done = 7400 (P f -l), while the temperature will rise to T T P^ 1 \~ *- - r If the temperature could be prevented from rising the work done would be reduced to 2116 log e P; but this can only be effectively done by the injection of water in the form of spray into the com- pressing cylinder. No form of w r ater jacket appears to have any considerable effect in the short space of time occupied by the work- ing stroke. After the compression is complete the air may be cooled on its way to the reservoir by passing it through pipes exposing a large surface to the external application of cold water ; an operation which is conducive to economy, for otherwise the hot air in the reservoir will lose heat by radiation and conduction, and the pressure will be reduced. Let us suppose the air thus cooled at constant pressure to temperature T, the work done in forcing it into the reservoir will not be reduced, the only difference is that a part of the admission work will be done outside the reservoir in compressing the cooling air at constant pressure, the total amount remaining the same. Hence, in the absence of spray-injection, the work done per cubic foot of air drawn from the atmosphere is always nearly the same, being given by the above formula, and this conclusion would be correct if the air were heated instead of cooled before entering the reservoir. The compressed air is now conducted by pipes to a corresponding motor at any distance. The air-motor consists of a working cylinder and piston with valves attached, as in the case of a steam engine. Assuming the expansion adiabatic and complete the energy exerted, per cubic foot of compressed air consumed is, as shown on page 554, 7400(P-P^), whatever the temperature. Now if T be the tempera- ture, the density of the compressed air is greater than that of the atmosphere in the ratio PT /T, and therefore we obtain by division Energy exerted = 7400^ (l - (I) \ *o\ \*V / a general formula giving the available energy of an air motor per cubic foot of air drawn from the atmosphere by the compressing pump. For reasons already stated the whole of this will riot be utilized (p. 555). On the other hand, the expansion has been supposed adiabatic, although there can be little doubt that the cooling of air below the atmospheric temperature is greatly hindered by the condensation of vapour mixed with it, and by drawing heat from external bodies. CH. xxi. ART. 299.] FLOW OF GASES. 577 Subject to these remarks the efficiency of transmission will be T /1\^ T Efficiency - - (1) Let the air after compression be cooled to jP , and supplied without re-heating to the motor, the efficiency is now l/P f , and there- fore diminishes rapidly as the pressure increases. The loss is due to change of temperature, and may be greatly diminished by compound- ing the compressing pump so as to compress the air in two or more stages ; the air being thoroughly cooled between each stage. Com- pression by stages is necessary for constructive reasons when the pressure is very high, and, when properly carried out, is economical. It has of late been introduced for economical reasons. (2) If the air is re-heated after transmission before entering the motor the efficiency will be increased as the formula shows. It is true that heat will be spent in raising the temperature of the air, but the corresponding gain of work in the motor cylinder is propor- tionally very large. If T>T l more energy will be exerted in the motor cylinder than is necessary to drive the compressing pump, the whole arrangement operating as a heat-engine. If T< r l\ the arrange- ment operates as a reversed heat-engine, being, in fact, a well-known form of refrigerating machine. The theory of the process considered in this light is given in the author's work on the Steam Engine already cited, in which the principles of thermodynamics are explained at length. 299. Steady Flow through a Pipe. Conservation of Energy. Referring to Fig. 174, p. 475, suppose that the reservoir is closed, and that it contains an elastic fluid at high pressure which is flowing through the pipe. Unless the change of pressure be very small, difference of level may be disregarded as relatively unimportant (p. 550), and we have only to consider differences of pressure, while, on the other hand, we must now remember that, when the pressure changes, energy is exerted by expansion as well as by transmission. The energy transmitted from the reservoir to any point where the pressure is P and volume V is PQ^Q, where the suffix indicates the state of the fluid in the reser- voir. Of this the amount PV is transmitted through the point, and the difference P^V^-PV together with the expansive energy E is employed in generating the kinetic energy which the gas possesses in consequence of the velocity u with which it is rushing through the pipe at the point considered. Thus, if the motion be steady, C.M. 2 O 578 ELASTIC FLUIDS. [PARTY. where Q is the heat (if any) supplied during the passage from the reservoir to the point. If no heat be supplied, ^- + 3 -5 PV=. Constant, J an equation which may also be written j ^- + K p . T= Constant, where K p is the specific heat at constant pressure. If we have to do with any elastic fluid other than a permanent gas, 3-5PF" must be replaced by I + PV, where / is the internal energy, and if the question be such that the elevation of the point considered has any sensible influence, the term z must be added as in the corresponding case of an incompressible fluid. The equation for a compressible fluid, however, is much more general than that for an incompressible fluid, because the internal energy is taken into account, and consequently any energy exerted in over- coming frictional resistances is replaced by an equivalent amount of heat generated. It follows that the equation is true whether there be frictional resistances or whether there be none, provided that the internal motions have time to subside and be converted into heat by friction, and provided that none of the heat thus generated is trans- mitted to external bodies. It sometimes happens that we have to consider cases where a quantity of heat Q is supplied to a permanent gas during its passage from a point 1 to a point 2, we shall then have the equation p , l , an equation which is true, however great the variations of pressure or temperature are, and whether or not there are frictional resistances. 300. Velocity of Efflux of a Gas from an Orifice. The most im- portant applications of the equation for the steady flow of a gas are to the discharge of air or steam from an orifice and to the flow of air through long pipes. In the first case the frictional resistances are small and are con- sequently neglected. It will be desirable to give a method of treating the question which is independent of the general equation. In Fig. 208a ol2k represents the expansion curve for a small portion of the gas as it rushes out of the reservoir A (Fig. 208b) in which it is confined through a small orifice into the atmosphere. The jet contracts at issue to a contracted section kk, nearly as in the case CH. XXI. ART. 300.] FLOW OF GASES. 579 where the fluid is incompressible, and then, in general expands again in some such way as is shown in the figure. The velocity through the Fig. Pig. 2O8a. contracted section may be denoted by u, and the pressure there by P. The area of the contracted section is connected with the area of the orifice by the equation A ^_ 7* A as on page 473, k being a co-efficient. Each small portion of the fluid expands from the state represented by the point o on the diagram to that represented by K; in some intermediate state it will be represented by a point 1 on the expansion curve, and immediately after by 2, a point near to 1. Let u lt u 2 be the corresponding velocities, then r.SP, where w is the mean density and V the mean specific volume repre- sented graphically by the mean of 01, sf2. Hence V. 8P is represented by the area of the strip cut off by these ordinates. Dividing the whole area into strips, the area of each strip represents the corresponding change in v?/2g, so that the total area represents the final value of this quantity. We have then - = Area N OK M = ^ p = h. The quantity h thus found and graphically represented is the "head" due to difference of pressure, as fully explained in Art 283. Assuming the expansion curve PV n = Constant, as before, 2g Ti-1 n- 580 ELASTIC FLUIDS. [PART v. Now, if the expansion be adiabatic n=l'4, and nc/(n-l) is equal to K p , so that the result might have been written down at once from the general equation of the preceding article. Employing the notation of Art. 281, but replacing the suffix 1 by the suffix 0, the velocity of efflux is given by the formula 301. Disc/iarge from an Orifice. The weight of gas discharged per second from an orifice of contracted area A is now found from the formula AU W= -y> where V is the specific volume of the gas at the instant of passing through the contracted section, and therefore supposing A unity the weight per unit of area is given by W* = 9/ n . ^0^0/1 _ nr\ For Fwe now write 7 n-l F 2 x and finally obtain n i In applying this formula /z must be supposed known and r calculated from it by the equation on p. 552. It will be found on examination that as /A diminishes from unity W increases to a maximum value and then diminishes again to zero. That is, if the pressure in the throat of the jet at the contracted section be diminished the discharge does not increase indefinitely, but reaches a maximum and then decreases. On substitution for r in terms of //, it will be seen that for a given pressure (P ) in the 2 n+1 reservoir W is greatest when /* n /* w is greatest. This will be found to be the case when 2 Xw 3 ! n+ 1 The expansion is adiabatic, and the values of n with the resulting values of //. for maximum discharge are shown in the annexed table. NATURE OF GAS. VALUE OF n. VALUE OF ju.. Dry Air, ------ 1-4 528 Superheated Steam, - - - - 1-3 546 Dry Saturated Steam, 1-135 577 Moist Steam, - 11 582 1 1 CH. xxi. ART. 301.] FLOW OF GASES. 581 The discharge is therefore a maximum when the external pressure is from -5 to '6 the pressure in the reservoir. For dry air it will be found on substitution that the maximum discharge per second per unit of contracted area is M7 3'9P Q _ PQ 1 and for dry steam The pressure P was originally supposed expressed in Ibs. per square foot, but it may now be taken as Ibs. per square inch in the numerator of these fractions, in which case W m will be the discharge per square inch. The diminution of the discharge on diminution of the external pressure below the limit just now given, is an anomaly which had always been considered as requiring explanation, and M. St. Venant had already suggested that it could not actually occur. In 1866 Mr. R. D. Napier showed by experiment that the weight of steam of given pressure discharged from an orifice is really independent of the pressure of the medium into which the efflux takes place ; and in 1872 Mr. Wilson confirmed this result by experiments on the reaction of steam issuing from an orifice. * The explanation lies in the fact that the pressure in the centre of the contracted jet is not the same as that of the surrounding medium. The jet after passing the contracted section suddenly expands, and the sudden change of direction of the fluid particles gives rise to centrifugal forces which cause the pressure to increase on passing from the surface of the jet to the interior on the prin- ciple explained on page 478. This will be better understood by reference to the annexed figure (Fig. 209) Fig. 209. which shows a longitudinal section of the jet at the point where the contrac- tion of transverse section is greatest. * The particles describe curves the radius of curvature of which increases from a small minimum value at the surface k, to an infinite value at the centre. The pressure p increases from that of the medium (TT) at k to a maximum p' at the centre, the increase being very rapid at first and afterwards more gradual. The problem is therefore far more complicated than we have supposed, * Discharge of Fluids, by R. D. Napier. Spoil, 1866. Annual of the Royal School of Naval Architecture for 1874. 582 ELASTIC FLUIDS. [PART v. each small portion of the jet having its own pressure and (conse- quently) its own velocity and density. The results of experiment however suggest that an approximate solution may be obtained by the assumption of a mean pressure in the throat of the jet, with a corresponding mean velocity ; this mean pressure being that which gives maximum discharge in every case in which that quantity is greater than TT. At lower pressure it is to be assumed equal to IT. Adopting this hypothesis we see that whenever steam is discharged into the atmosphere from a boiler the pressure in which is greater than, about, 25 Ibs. per square inch absolute, or 10 Ibs. above the atmosphere, the formula given above for maximum discharge is to be used. If we assume the mean value 252 for \AP ^o fc kis gi yes Pi/70 for the weight discharged from an orifice per square inch of effective area per second, the pressure p l being the absolute pressure in the boiler expressed in Ibs. per square inch. Contraction and friction must be allowed for by use of a co-efficient of discharge, the value of which however is more variable than that of the corresponding co-efficient for an incompressible fluid. Little is certainly known on this point. 302. Flow of Gases through Pipes. Returning to the general equation, we have now to examine the case where air or steam flows through a pipe of considerable length. As in the case of water, the frictional resistances are then so great that most of the head is taken up in over- coming them. The velocities of the fluid are therefore not excessive, and the value of u?/2g varies comparatively little. Now, in the equation - + K P T = Constant, *9 the numerical value of K p is about 184, and therefore a variation of temperature of a single degree will correspond to a great change in . v u 2 /'2g ; it may therefore be assumed that the temperature remains very approximately con- f p stant provided only that the difference of pressure at the two ends of the pipe is not too great compared with its length. In Fig. 210 suppose 1, 2 to be two sections of the pipe at a distance Ax so small that, in estimating the friction, the velocity may be taken at its mean value u ; then the force required to overcome friction is OH. xxi. ART. 302.] FLOW OF GASES. 583 where s is the perimeter and / the friction per square foot, as on page 484. Replacing this by a new co-efficient/', as on page 486, S-f %..*&*.*, z ff in which equation w means the weight of unit-volume of the gas. Now, it was pointed out on page 493 that surface friction was a kind of eddy resistance, and that in the case of water it was proportional to the density. This leads us to suppose that in fluids of varying density, not / but/' is a constant quantity. Replacing w by its equivalent we obtain, suppressing the accent of/, 2 We now apply the principle of momentum which will be expressed by the equation where ^Fis the weight of gas flowing through the pipe per unit of area per second, and the suffixes refer to the two sections in question, the area of which is A. Now, the motion through the pipe being steady, W is the same throughout, so that = W = Constant. By substitution for W and writing H for v?/'2g, an equation is obtained which, when written in the differential form, becomes -.H, m m being the hydraulic mean depth. Next, if T be the temperature, which, as remarked above, is sensibly constant, P = C =tr. ', .'. 8P=-IF. C -^.8u. V u u 2 Substitute again for W and u, we then find On substitution for F.8P, the value of which has just been given, the differential equation becomes integrable by dividing by H, and we obtain on performing the integration in H.I where I is the length of the pipe, and H Q , H the values of u 2 /2g at entrance and exit respectively. In application of this equation the 584 ELASTIC FLUIDS. . [PARTY. term containing the logarithm is small as compared with the rest, and may generally be omitted : also tf-a 2 #0 f a ratio which is known if the pressures are the data of the question. In the case of steam cT is to be replaced by the nearly constant product PV> which is to be taken from a table for its quantity so as to obtain a mean value according to the pressure considered. The value of the co-efficient in small pipes not more than 3 inches diameter is about the same as in the case of water, namely '007, but it appears to diminish much more rapidly as the diameter of the pipe increases. In pipes 12 inches diameter and upwards, with velocities from 10 to 25 f.s., Professor Unwin gives the value -003 as the result of a reduction of a large number of experiments made on the resistance of the Paris air mains.* The equation just found must not be applied to cases in which the difference of pressure is too great compared with the length of the pipe. The friction is then not great enough to prevent the velocity from becoming excessive ; the temperature then sensibly falls, instead of remaining constant as supposed in the calculation. An equation can be found which takes account of the fall of temperature when necessary, but in such cases as commonly occur in practice, the supposition of constant temperature is sufficiently approximate. When the difference of pressure is small, the equation will be found to reduce to the hydraulic formula for flow in a pipe. This case will be considered presently. The head is given by the formula and the power expended in forcing the air through is Wh or PAu log e r ft.-lbs. per square foot of sectional area per 1". The efficiency of the process of compression and expansion has already been considered when compressed air is used for the trans- mission of energy, and it need only be added that the question of leakage is one of great importance. In some cases the method has proved a failure from this cause. It is probably always more difficult to render the joints of a pipe tight under air pressure than under steam pressure ; but experiments by Professor Riedler on the Paris mains showed that the loss may be made very small, not exceeding one per cent, per mile per hour. * Proceedings of the Institution of Civil Engineers, vol. cv., p. 192. CH. XXL ART. 303.] FLOW OF GASES. 585 303. Flow of Gases under Small Differences of Pressure. When the differences of pressure are small and no heat is added or subtracted, a gas flows in the same way as a liquid of the same mean density. In the case of air the mean specific volume is found from the equation cT T V P ~40' the units being feet and pounds, the mean pressure that of the atmosphere, and the temperature measured on Fahrenheit's scale from the absolute zero. At 59 this gives F=13 cubic feet, but the actual volume will vary slightly from variations in the mean pressure. The small differences of pressure with which we have now to do are commonly measured by a syphon gauge in inches of water. One inch of water is equivalent to a pressure of 5-2 Ibs. per square foot. If now AP be the difference of pressure in Ibs. per sq. ft., i the corresponding number of inches of water, the head due to it will be, as in Art. 300, FAP, and therefore The velocity due to this head, or, what is the same thing, the volume discharged per sq. ft. of effective area per second in the absence of frictional resistances, is in cubic feet and the weight-discharge in pounds per second JP-= 166-6^ At 59 e one inch of water gives a head of 67 '5 feet and a discharge of 65-9 cubic feet, or 5O7 Ibs. per second; but at 539 the head is 130 feet and the discharge 91 '3 cubic feet, or 3-67 Ibs., results which show that the effect of a given difference of pressure is entirely different according to the temperature of the flowing air. This is a point which must always be borne in mind in applying hydraulic formulae to the flow of gases. Frictional resistances are taken into account by the employment of a co-efficient as in hydraulics, and, as elsewhere ex- plained, there is reason to believe that the values of these co-efficients are the same, except so far as they may be dependent directly or indirectly on the co-efficient of contraction (p. 473). Co-efficients of contraction are more variable in air than in water, but their average value does not differ widely in the two cases, and may provisionally be assumed the same. 586 ' ELASTIC FLUIDS. [PARTY. In particular, it is well established that the formula for the discharge of a pipe in cubic feet per second (p. 488), applies to air at the low velocities here considered, with the same value of the co-efficient k as in water, that is (d in feet) about 40. The head ti is calculated, as just explained, according to the temperature of the air, for a given difference of pressure. It is sometimes necessary to consider the flow of some gas other than atmospheric air. In approximately permanent gases this is easily done if we know the density of the gas. For example, the density of common coal gas is about '43, air being unity. The value of c in the formula PF= cT is then proportionately increased, but in other respects the formulae are unaltered, the index of the adiabatic curve and the constants 2 '5, 3 '5 which depend on it remaining unaltered. The formula for small differences of pressure may also be employed for the non-permanent gases, such as steam, with a corresponding modi- fication. Pneumatic machines in which the variation of pressure is small are analogous to hydraulic machines, and most of what was said in the last chapter is applicable to them. The common fan, for example, is a cen- trifugal pump, the lift of which is the difference of pressure reckoned in feet of air, that is, at ordinary temperatures, about 67 feet for each inch of water. The speed of periphery is ,Jgli in feet per second, where h is the lift increased, as explained in the case of the pump, on account of frictional resistances and the curving-back of the vanes. Some remarks on the influence of the form of the vanes on the efficiency of a fan will be found in the appendix (note to p. 542). Fans are employed to produce a current of air for the purpose of ventilating a mine, ship, or structure of any kind. In mines they are often 30 feet in diameter or more. The pressure required is here small and the speed moderate. They are also used to produce a forced draught in torpedo boats, or the blast of a smithy fire. The pressure is then 5 inches of water and more, corresponding to a lift of 300 feet and upwards. The speed of periphery is consequently excessive, and for the comparatively great pressures required for a foundry cupola or a blast furnace, it is necessary to resort to some other sort of blowing machine. 304. Varying Temperature. Chimney Draught. If the temperature of the flowing air is varied by the addition or subtraction of heat, its density will be altered during the flow, and it is then necessary H. xxi. ART. 304.] FLOW OF GASES. 587 to know the mean density, in order that we may be able to calculate the " head " due to a given difference of pressure as measured by the water gauge. In Fig. 211 OX, OY are axes of reference parallel to which ordi- nates are drawn as usual to represent volumes and pressures. A given difference of pressure AP is represented by the difference Y ab of a pair of ordinates. The Pi8> - 211< original volume of the air is represented by al. Suppose now that in flowing through a passage of any description the air is heated, as for example in passing through a furnace, the volume increases greatly while the pressure falls slightly; this will be re- presented by the curve 12, terminating at a point 2. The form of the curve will depend on the law of heating, and will be very different according to the state of the fire : if the bars of the grate be blocked by clinker and the surface of the fire be free from special obstruction, most of the frictional resistance and corresponding fall of pressure will occur before the air is heated, and the curve will slope rapidly near 1 and slowly afterwards; while, conversely, if the fire be covered with fresh fuel and the grate bars clear, the reverse may be true. After being heated let theair pass through a, boiler tube, by which heat is abstracted, till it reaches the chimney : the volume then diminishes greatly while the pressure falls slightly, as shown by the curve 23, terminating at a point 3, such that b3 represents the volume of the air in the chimney. The form of the curve 23 will depend on the law according to which the tube abstracts heat. The area of the whole figure al'23b represents the "head" due to the whole difference of pressure AP, and it will now be obvious that this head will vary according to circumstances which cannot be precisely known. Thus the mean density cannot be found except by empirical formulae derived by direct experience, and con- sequently applicable only to the special cases for which they have been determined. It has hitherto been most usually assumed in the case of a furnace and boiler that the mean density was that of the air in the chimney, which amounts to supposing that the forms of the curves 12 23 are such that the area of the rectangle a3 is equal to the area of the whole figure. This is the supposition em- ployed by Rankine, and in many cases it appears to lead to correct results. 588 ELASTIC FLUIDS. [PARTY. In every case of the flow of heated air it must be carefully con- sidered what the mean density will probably be. Its value can often be foreseen without difficulty. It is only in the case of long passages, where the air suffers great frictional resistance while being heated or cooled, that it is uncertain what value to adopt. The draught which draws air through a fire may be produced artificially or by the action of a chimney. In the latter case there is a difference of pressure within and without the chimney at its base due to the difference of weight of a column of air of the height of the chimney at the temperature of the chimney and at that of the atmosphere. Radiation causes the temperature of the air to be less in the upper part of the chimney and so diminishes the draught, the frictional resistances have the same effect. If these be disregarded the draught in inches of water will be where T Q is the temperature of the atmosphere, T that of the chimney, while I is the height of the chimney in feet. The temperatures are here reckoned from the absolute zero. If, for example, the temperature of the chimney be 539 F., and that of the atmosphere 59 F., the height of the chimney required for a draught of 1 inch of water will be about 141 feet, or in practice more on account of friction and radiation. The effect of this draught in drawing air through a furnace or through passages of any kind will vary according to the circumstances which have just been explained. CH. xxi.] EXAMPLES. 589 EXAMPLES. FIRST SERIES (SECTION I.). 1. Find the store of energy in the reservoir of a AVhitehead torpedo. Capacity 5 cubic feet. Pressure 70 atmospheres. Ans. If n=l 2, 420, 000 ft. -Ibs., or 3, 130 thermal units. n = l'4 1,092,000 or 1,414 Ratio of results = '45. 2. In the last question the air is supplied to the torpedo engines by a reducing valve so that the pressure in the supply chamber remains constantly at 13 atmospheres : find the available energy. Ans. If n = 1 1, 900, 000 ft. -Ibs. w=l'4 1,346,000 NOTE. The difference between these results and the preceding is the effect of wire- drawing (resistance of valve). The supply chamber is supposed small. 3. Air is stored in a reservoir the pressure in which is maintained always nearly at 10 atmospheres : find the store of energy per cubic foot of air supplied from the reservoir. Ans. If n=l 48, 700 ft. -Ibs. n=l'4 35,700 Ratio =733. 4. A chamber of 100 cubic feet capacity is exhausted to one-tenth of an atmosphere ; find the work done, assuming n=l. Here if the chamber be imagined to contract, compressing the air still remaining in it, the energy exerted will be due to the pressure of the atmosphere, and the difference between this energy and the work done in compression will be available for other purposes. In exhausting this is reversed. Ans. 142,000 ft. -Ibs. 5. Find the mechanical efficiency of an engine so far as due to incomplete expansion (ratio r) : assuming the expansion hyperbolic. Ans. If R be the ratio of complete expansion, l + log e r-^ Efficiency = : 5^. log e R 6. In the last question obtain numerical results for a condensing engine, taking the back pressure at 2 Ibs. and boiler pressure 60 Ibs. Ans. Ratio of expansion, - 1 2 5 10, Efficiency, - - - '284 '48 '72 '87. 7. Find the comparative mechanical efficiencies in a condensing and a non-condensing engine. Back pressure in condensing engine 2, in non-condensing 16. Boiler pressure 60 and 100. Ratio of expansion 5 in both cases. The engines must here be supposed to have the same lower limit of pressure of 2 Ibs. ; and the result for the non-condensing engine includes the loss by the actual back pressure being 16 Ibs. Ans. '72, '46. 8. Find the loss by wire-drawing between two cylinders from one constant pressure of 60 Ibs. to another constant pressure of 40 Ibs. Expansion hyperbolic. Ans. '405 PV. 9. One vessel contains A Ibs. of fluid at a given pressure PA, and a second B Ibs. of the same fluid at a lower pressure PB . A communication is opened between the vessels And the fluid rushes from A to B ; find the loss of energy. The loss here is the difference between the energy exerted by A Ibs. expanding from VA to V, and the work done in compressing B Ibs. from VB to V: where VA, VB are 590 ELASTIC FLUIDS. [PARTY. the specific volumes of the fluid in A and B, and V that of the fluid after equilibrium has been attained, found from the formula AVA + BVn A + B Hence the loss is very approximately AB (VB-VA}(PA-PB} 10. In a compound engine the receiver is half the volume of the high-pressure cylinder, and at release the pressure in the cylinder is 25 Ibs. per square inch, while that in the receiver is 15 Ibs. per square inch : find the loss of work per Ib. of steam. Obtain the results also when the receiver is double instead of one-half the cylinder. Am. Case I., 1838 ft. -Ibs. Case II., 3873 ,, 11. In a condensing engine find the mean effective pressure and the consumption of steam in cubic feet per I.H.P. per minute at the boiler pressure: being given, back pressure 3, boiler pressure 60 Ibs. per square inch (absolute), ratio of expansion 5. Ans. Mean effective pressure 28*33 Ibs. per square inch. Consumption of steam = 1 '62 cubic feet per minute. 12. If the volume of 1 Ib. of dry steam at the boiler pressure be taken in the preceding question as 7 cubic feet and the liquefaction during admission 20 per cent. ; find the weight of steam consumed in Ibs. per I.H.P. per hour. Ans. 175. 13. Find the H.P. necessary to compress 100 cubic feet of air per minute to a pressure of 7 atmospheres (absolute), the air being drawn from the atmosphere at temperature 60 and forced at constant pressure into a reservoir. Suppose the compression (1) adiabatic, (2) isothermal. Ans. Work per Ib. of air =92 '2 thermal units. = 68-8 H.P. = 16| or 12. 14. In the last question suppose the compression carried out in two stages, the air at each stage being cooled at constant pressure after adiabatic compression : find the work done per Ib. of air. Ans. 79 '2 thermal units. CH. XXL] EXAMPLES. 591 EXAMPLES. SECOND SERIES (SECTION II.). 1. Find the mechanical value of a unit of heat, the limits of temperature being 600 and 60 ; 300 and 100 ; 400 and 212. Ans. 393, 203, 169 ft.-lbs. 2. The limits of temperature in a heat engine are 350 and 60 ; find the thermal efficiency when two-thirds of the whole heat supplied is used between 300 and 100, one-sixth between 200 and 100, and one-sixth between 250 and 100. Ans. '658. 3. In question 6, First Series, on account of a gradual increase in the liquefaction the thermal efficiency at the several ratios of expansion mentioned is assumed as '9, '85, 7, '5 ; find the true efficiency. Ans. "256, '408, '504, '435. 4. In a compound engine the pressure of admission is 100 Ibs. per square inch, the steam is cut off at one-third in the high-pressure cylinder, the ratio of cylinders is 2J ; the back pressure is 3 Ibs. per square inch, the large cylinder 40 inches diameter, and the speed of piston 400 feet per minute. Find the H.P., neglecting wire-drawing and sudden expansion. Ans. 567. 5. In the last question suppose that the engine has a very large intermediate reservoir, and that the cut-off in the low-pressure cylinder is 'o ; find the pressure in the reservoir, neglecting wire-drawing, also the loss per cent, by sudden expansion at exhaust from the high-pressure cylinder, and the percentage of power developed in the two cylinders. Obtain the results also for a cut-off of one-third in the low-pressure cylinder. Ans. Cut off . Cut-off %. Pressure in reservoir, - 26 '7 40 Loss by sudden expansion per cent. , "8 *7 Percentage of power in high-pressure cylinder, - - 46'5 32*4 ,, ,, low-pressure ,, - - 52 '6 57 '6 6. Compare the efficiencies of the simple and compound engine, assuming the liquefaction the same at the best ratio of expansion, which is 5 in the simple engine and 7 in the compound engine, while in the latter 5 per cent, of the work is lost by wire-drawing between the cylinders. Back pressure and boiler pressure in both cases 3 Ibs. and 84 Ibs. respectively. Ans. Gain by compounding 2^ per cent. 7. In question 5, instead of supposing the whole expansion represented by a single hyperbolic curve, assume that at the end of the stroke in the high-pressure cylinder the steam is dry, while at the end of the stroke in the low-pressure cylinder the steam contains 10 per cent, water. Obtain the required result for the cut-off *5 and find the weight of steam used (exclusive of jacket steam) in Ibs. per I.H.P. per hour. Also obtain the results when the steam at the end of the stroke in the high -pressure cylinder contains 30 per cent, water, all other data remaining the same. Ans. Case I. Case II. Pressure in reservoir, - - - - - - 22 '5 14 '9 Percentage of power in high-pressure cylinder, - 55 37 '5 ,, low-pressure ,, 55 62 '5 Lbs. of steam per I.H.P. per hour, 13 16 '5 8. The available heat of a pound of coal is 10,000 thermal units ; find the consumption of coal per I.H.P. per hour in a perfect heat engine working between the limits 600 and 60. 592 ELASTIC FLUIDS. [PART v. 9. The temperature of the atmosphere is 70 and that of a tank in which ice is being made 26. Find the H.P. necessary to drive a perfect ice-making machine, per ton of ice per hour. Latent heat of water =142 ; specific heat of ice= '5. Ans. 14^. 10. Air is heated at constant volume till its temperature is raised from 70 to 300, then expanded to three times its volume at constant temperature. Find the mean temperature of supply. Ans. 247 F. 11. In the last question suppose the air subsequently to expand adiabatically till its temperature has fallen to 70, and then to be compressed at constant temperature till the original pressure is reached. Deduce the co-efficient of performance, and verify your calculation. Ans. Co-efficient= '25. 12. Air at a pressure of 1,000 Ibs. per sq. inch and a temperature of 539 expands to 6 times its volume without gain or loss of heat ; find the pressure and temperature at the end of the expansion. Ans. p=8l, t=27. 13. In the last question suppose the air at the end of the expansion to have a pressure equal to 1^ times that given by the adiabatic law, and heat to be supplied at a uniform rate as the temperature falls ; find the index of the expansion curve and the work done during expansion. Compare the heat supplied with the work done and find the specific heat. (See page 575. ) Ans. =ri74. Specific heat='223. Work done = 82,000 ft. -Ibs. Ratio = '575. 14. Find a formula for the useful work done per Ib. of steam in thermal units with a vacuum of 1'41 inches of mercury absolute, a back pressure of 1J and a terminal pressure of 4 Ibs. per sq. inch ; assuming x. 2 ='S, k='8 (p. 569). Ans. W^M-54. 15. By means of the formula of the preceding question deduce the consumption of steam and the efficiency for the series of pressures stated below. Boiler Pressure. . 350 9'77 '662 180 11-3 -642 84 13-7 -619 60 151 -605 20 22-5 -535 16. Find a formula and deduce numerical results as in the last two questions, assuming a terminal pressure of 8 Ibs. per square inch and k~- '7, all other data remaining the same. (Compare pages 570, 571.) CH. XXL] EXAMPLES. 593 EXAMPLES. THIRD SERIES (SECTION III.). 1. Air is contained in a vessel at a pressure of 25 Ibs. per sq. inch and temperature 70. What will be the velocity with which the air issues into the atmosphere (pressure 15 Ibs. per sq. inch) ? Also find the discharge and the head. Ans. h =13, 420; u= 930 ft. per second. JF=34'26 Ibs. per sq. inch of orifice per minute. 2. In the last question find the initial pressure corresponding to maximum discharge for all external pressures less than that of the atmosphere. Find this discharge. Ans. Pressure = 28*5 Ibs. per sq. inch. Discharge =39^ Ibs. per sq. inch per minute. 3. "What weight of steam will be discharged per minute from an orifice 2 inches diameter, the absolute boiler pressure being 120 Ibs. per square inch? Co-efficient of discharge 7. Ans. 227 Ibs. 4. Air flows through a pipe 6 inches diameter and 4,000 ft. long ; the initial pressure is 20 and the final pressure 15 Ibs. per sq. inch ; temperature 70 ; find the velocities and the discharge. 4/= '03. Ans, Velocity at entrance =39 feet per second. ,, exit =52 feet Discharge = 4 Ibs. per sq. ft. = '78. 5. In the last question find the loss of head and the H.P. required, to keep up the flow. Ans. A'=8,124feet. H.P.=11. 6. Steam at 50 Ibs. rushes through a pipe 3 inches diameter and 100 feet long with a velocity at entrance of 100 feet per second ; find the loss of pressure. 4/= '03. Ans. 1*6. REFERENCES. For descriptive details and illustrations of the mechanism of steam engines the reader is referred amongst other works to THURSTOX. History of the Growth of the Steam Engine. International Scientific Series. Kegan Paul. YEO. Steam and the Marine Steam Engine. Macmillan. RIGG. Practical Treatise on the Steam Engine. Spon. C.M. 2 P APPENDICES. APPENDIX. A. NOTES AND ADDENDA. Notes marked [1890] . . . [1900] have been added on reprinting at the dates mentioned. I. -STATICS OF STRUCTURES. RANKINE'S treatise on Applied Mechanics appeared in 1858. The sixth edition is quoted in the following notes by the letters A.M. PAGE 2. " The word STRESS has been adopted as a general term to comprehend various forces which are exerted between contiguous bodies or parts of bodies, and which are distributed over the surface of contact " (A.M., p. 68). It appears from this that RANKINE'S use of the word is confined to internal forces, but by some writers it is employed for all forces, whether external or internal. Ties and struts are, however, defined as in the text (A.M., p. 132). PAGE 3. The total load on the platform of a timber bridge carrying an ordinary roadway may be assumed as 250 Ibs. per sq. ft., of which 120 represents the weight of a closely packed crowd, and the remainder is the weight of the roadway and platform. The weight of a timber roof (slate {or tile) is from 12 to 24 Ibs. per sq. ft. The travelling load on railway bridges is commonly estimated at 1 ton per foot-run. [1900.] On account of the continued increase in the weight of trains, the fore- going estimate of the travelling load on railway bridges has long ceased to be adequate. The weight of heavy locomotives now ranges from 70 to 100 tons and the equivalent uniform load for a span of 70 to 100 feet is estimated by Mr. Farr (Proc. I.C.E., Vol. CXLL, p. 12) at 1J to 2 tons per foot-run. For short spans it is still greater, the load being concentrated on a limited number of axles. PAGE 14. The diagram of forces for a funicular polygon under a vertical load was (probably) first given by ROBISON in his treatise on Mechanical Philosophy, Vol. I. Dr. Robison died in 1805, and this work is a collection of his papers published in 1822. PAGE 20. In the Saltash bridge the compression member of each girder is a tube of elliptical section 15 feet in breadth, 8 feet in depth. A pair of chains, one on each side, carry the platform. PAGE 20. A. Of the various methods of constructing a parabola the most con- venient is that in which a curve is drawn through the intersections of a set of lines radiating from a point, with a set of equidistant lines drawn parallel to a 598 NOTES AND ADDENDA. fixed line : the radiating lines being drawn so as to cut off equal intercepts on another fixed line. It can easily be proved that this curve is the funicular poly- gon proper to a uniform load without introducing any properties of the parabola. [1892.] The foregoing remark applies to purely graphical methods, but it is much simpler to plot the products AK, BK (p. 39) for equidistant positions of K. PAGE 21. Let P be the vertical tension of the chain at the point P, then, since dyldx = P/H, where H is constant, d 2 y_ 1 dP_io dx~*~H* dx~JT This equation is equally true if iv vary according to any law, and is therefore the general differential equation of a cord or linear arch under any vertical load. Particular cases are : (1) The Common Catenary. Here if m be the weight of a unit of length of the cord, ds an element of arc, the equation then becomes, if H=m . c, &*, Divide by the right-hand member, multiply by dy/dx, and integrate, then 7y y ;] =- / c an equation which, by integration and a proper determination of the constants, gives for the form of the curve (2) The Catenary of Uniform Strength. Here, if T be the tension of the chain atP, dx T=m\ = w\- t "where X is the length equivalent to the stress (p. 80), <*L]L-T_ . fk = i. (fo\* dx*~ H\ dx~\ \dx ) Integrating by the same process as before we find x X as the equation to the curve. In ordinary cases there is very little difference between the catenary and the parabola, and these curves therefore are not of much interest. If the form of an arch be not such as corresponds to the distribution of the load on it, a horizontal force will be necessary for equilibrium, and the investigation of the magnitude of this force is a problem of wider application. Let p be the intensity of this force per unit of length of a vertical ordinate, then H is no longer constant, but is given by dH . dP , dy P -^ = p, also -^ =w and -r=-m dy dx dx H three equations from which p can be found for any distribution of load and form STATICS OF STKUCTUEES. 599 of arch. This is the general problem of the linear arch. For examples see A.M., p. 199. If p = const., w = const., we obtain the ellipse as the proper form of arch to sustain the pressure of a great depth of earth. PAGE 30. On reciprocal diagrams of forces in general the reader is referred to a memoir by CLERK MAXWELL in the Transactions of the Royal Society of Edin- burgh for 1870. The notation used in the text was suggested by HENRICI in the course of a dis- cussion on a paper by CROFTON read before the Mathematical Society in 1871. The figure in the text was drawn at the time by the writer to illustrate the method. The notation was afterwards given by Bow in the treatise referred to. PAGE 33. It is convenient to have a general term for the tendency to separate into parts due to the action of external forces on a structure or part of a structure. The term "straining action" used in the text is taken from Ch. II., Part III., of a treatise on Shipbuilding (London, 1866), edited and in great part written by RANKINE. By some writers this tendency to separate would be called "stress," and for a simple thrust or pull there is no objection to doing so (A.M., p. 132). In more complex cases a separate word is preferable, as the conception is very different. (Comp. p. 295.) PAGE 44. In some of his engines, before the introduction of cast iron, WATT employed a timber beam trussed with iron rods, forming a Warren girder in two divisions with diagonals inclined at about 30 to the horizontal. This is perhaps the earliest example of such a construction. (ROBISON, Vol. II., p. 14.) TAGE 53. See Plate VIII., p. 463. PAGE 56. The method here detailed is given by RANKINE in his work on Civil Engineering (p. 242), who ascribes it to LATHAM. If M be the bending moment, F the shearing force, w the load per foot-run, we have the equations dF which are the symbolical expression of the method. They may be used to find by integration the bending moment and shearing force at any section due to a given load, the constants of integration being found by considering that the bending moment is zero at two points, which must be known if the problem is determinate. (See Art. 38, p. 77. ) PAGE 65. See Ch. II., Part III., of the work on Shipbuilding, cited above. PAGE 66 [1900]. On travelling loads the reader is referred in addition to the works cited at the end of the chapter, to the paper by Mr. Farr already mentioned in the note to p. 3. PAGE 70. The properties of funicular polygons were first thoroughly investi- gated by CULMANN, who based upon them a complete system of graphical calcula- tion. In the semi-graphical methods employed in this treatise the integral calculus, trigonometry, and even, to a great extent, algebra, are replaced by geometrical constructions, but arithmetic is still used, and certain steps of the various processes are conducted by numerical calculations. For example, in Ch. II., the supporting forces of a loaded beam are found by the ordinary process of taking moments. In the modern purely graphical methods every step is taken graphically, whatever the calculation be. For example, the displacement of a 600 NOTES AND ADDENDA. vessel at a given draught, or her stability at a given angle of heel, would be found without the use of arithmetic. The pressure of other matter, and the amount of illustration required, have prevented the writer from making any considerable use of these methods in this treatise. At present they can hardly be considered suitable for an elementary work, though, if graphical calculations were introduced into our schools, the case might be different. A full account of them will be found in the treatises referred to in the text (p. 74). PAGE 72. The property of the funicular polygon expressed by the equation Hy M follows immediately by comparing the equations d?y_u^ d?M _ ~dx~H' ~dy?~ W ' of which one gives the form of the polygon for a given load, and the other the bending moment due to the same load. Another fundamental property is that any two sides of the polygon must meet on the line of action of the load on that part of the polygon which lies between the two sides. When the load is vertical, and represented by a curve, as in Fig. 36a, p. 62, this is equivalent to saying that any two tangents to the curve of moments must intersect on the vertical through the centre of gravity of the area of the curve of loads between the corresponding ordinates. (See p. 328.) The funicular polygon, considered as a line of transmission of stress, will be again referred to in the notes to Ch. XVII. PAGE 79. The theory of linear arches is merely an introduction to the theory of arches in general. Arches are of two kinds (1) the stone or brick arch j (2) the metallic arch. In either case the theorem of the text is of equal import- ance. In a blockwork arch the linear arch corresponding to the load shows the direction and position of the resultant of the mutual action between the blocks, and must therefore (p. 331) fall within the middle third of the arch ring. (AM., p. 258.) PAGE 82. See CLERK MAXWELL'S memoir referred to above (p. 599). PAGE 86. For the effects of changes of temperature, see Ch. XVIII., p. 460. PAGE 88. One of the most remarkable suspension bridges which have been constructed is the East River Bridge at New York, opened in May, 1883. The principal opening of this bridge is 1,600 feet span, the platform 85 feet wide, and 135 feet above the water. Cables, four in number, each of 145 square inches net area, constructed of 19 steel wire ropes, each containing 278 wires. Estimated strength of wire, 170,000 Ibs. per square inch. II. KINEMATICS OF MACHINES. PAGE 93. Referring to Figs. 1, 2, Plate II., p. Ill, it seems clear that the sector pair CD, Fig. 1, differs kinematically much more from the turning pair BA than it does from the sliding pair CD of Fig. 2. The writer, therefore, would have been disposed to classify. the three lower pairs as the " oscillating pair," the "turning pair," and the "screw pair." This, however, would have probably involved more considerable alterations in REULEAUX'S nomenclature than would have been justified in a general elementary treatise. KINEMATICS OF MACHINES. 601 [1892.] Turning pairs are not unfrequently distinguished into lever pairs and crank pairs as in the classification of crank chains (p. 112). [1900.] In a paper read before the Royal Society in 1895, and published in their Transactions (Vol. 187, p. 15), Professor Hearson proposes a classification of mechanisms based on the combination of "turning," " swinging," and "sliding" motions. A new notation is introduced of a very simple and expressive character, which may probably be found of great service in descriptive mechanism. It is pointed out that the three-slide or "wedge" chain (p. Ill) may be bent into a cylinder and then becomes a screw. Other important changes are proposed in the nomenclature and methods employed by REULEAUX, which are worthy of careful attention. Had the author been writing a new book advantage would have been taken of these researches. It may be hoped that Professor Hearson may pursue a subject which has been far too much neglected, and the study of which, in the author's judgment, may be expected to lead to important results. PAGE 95. The three incomplete lower pairs are considered by REULEAUX as higher pairs. The writer here follows GRASHOF (Theoretische Maschinen-Lehre, Band II. ). PAGE 100. Diagrams of velocity are considered generally by CLERK MAXWELL (Matter and Motion, p. 28). The application to mechanism is, so far as the writer is aware, new. The construction of curves of position and velocity of a piston has, for many years past, formed a regular part of the course of instruction at Greenwich, and formerly at South Kensington. PAGE 103 [1895]. In a letter which appeared in Engineering of June 14th, Mr. Archibald Sharp calls attention to a construction for the acceleration of a piston due to Professor Klein of Lehigh University, U.S.A. This con- struction, published in 1891, escaped the author's notice when revising this book in 1892 ; it is much simpler and more useful than that given in the text. (Ex. 11.) Referring to Fig. 48, page 100, imagine a circle described on DP, the con- necting rod, as diameter and a second circle with centre P the crank pin, and radius PT. Let EE be the points of intersection of the two circles, then the chord EE, produced if necessary, will cut the line of centres BDO in a point Z, such that when the crank rotates uniformly OZ _ Acceleration of Piston. OP "Acceleration of Crank Pin. For if the chord EE cut the rod DP in N, and OM be drawn perpendicular to PT to meet PT in M, NM=OZ. cos (p. Now if OP represent the acceleration of the crank pin P, PM will represent the resolved part of that acceleration in the direction of the rod DP; and if / be the acceleration of the cross head D y f . cos will be the resolved part of that acceleration along DP ; and therefore / . cos - PM = Length of rod x (Ang. Vel. ) 2 , in which equation the Ang. Vel. is that of the connecting rod when the angular velocity of the crank is supposed unity. But the angular velocity-ratio of the 602 NOTES AND ADDENDA. rod and crank is, as is well known, PT/PD, and considering the circle on PD as diameter, PN . PD whence it is clear that and /= OZ. [1900.] Referring again to the figure, and employing the usual notation as on page 100, PT _PT PO _cosfl sin0_l cos PD~ PO PZ>~cos0 aia0~n and PM = PO.cos(d + (f>); therefore, supposing the acceleration of the crank pin unity so that / is the acceleration-ratio, 1 sin 2 1 cos 2 . . f= COS 6 - - -- - + - - rr- n cos n 003^0 This is the exact formula for piston acceleration which may also be obtained by differentiation of the exact formula for the velocity. It can be put in several different forms, of which the present appears the simplest. It is arrived at in a different way on p. 229. Since an approximation can be obtained by the binomial theorem in powers of l/n. The first three terms of the series give as a second approximation f 1 . cos 26 -cos 40 /= cos 6 + - cos 20 + . n 4?i 3 Since 4?i 3 would rarely be less than 100 in any practical case it is clear that for all ordinary purposes the first approximation obtained by supposing cos unity is amply sufficient, a point we shall have occasion to mention hereafter. (See note top. 396.) PAGE 105. In Owen's air compressor two such mechanisms (Fig. 50) are placed face to face with the guide A and block D common, a steam piston is connected with d and the air-pump piston with the corresponding point d of the other mechanism. The object is to adapt the pressure of the steam to the varying pressure of the air during compression. PAGE 108. Stannah's pump has been introduced since the publication of REULEAUX'S work. The example there given is a mechanism used in the polish- ing of specula. PAGE 111. The double-slider mechanism, with sliding pairs and turning pairs alternating, is common in collections of mechanisms, but is not often found in practice. It is omitted in REULEAUX'S enumeration. The example given (Rapson's slide) and Stannah's pump were pointed out to the writer by Mr. (now Prof.) Hearson. PAGE 120 [1892]. This article (Art. 56) was numbered 53 in former editions and placed earlier (p. 112). A new article (55, pages 118-120) has been added, partly in order to introduce the conception of an instantaneous centre at once DYNAMICS OF MACHINES. 603 instead of postponing it to Ch. VII., and partly to explain the connection between diagrams of velocity such as are here considered, and graphical methods in common use based directly on the properties of the centre. PAGE 161. The propositions relating to centrodes have long been known, and are, perhaps, stated as clearly by BELANGER in his excellent treatise on kine- matics (Trait^ de Kinematique. Paris, 1864) as by REULEAUX himself. In the author's opinion it is the conception of a kinematic chain which constitutes REULEATJX'S great contribution to the theory of mechanism. It is virtually a complete reconstruction of the whole theory of machines, while the centrodes are only a method of stating results which was already known. Kinematic formula such as are employed by REULEAUX to indicate the component elements of a mechanism, in the same manner as a chemical formula shows the composi- tion of a substance, may be regarded as indispensable, if it be attempted to proceed with the study of descriptive mechanism. (See above Note to p. 93.) [1892.] In the first edition of this work the word centrode was spelt centroid a term now very generally appropriated to the centre of mass of a body. PAGE 170. The author has ventured on the introduction of the terms " driving pair," "working pair." They are simply the natural adaptation of the well- known phrases "driving point " and ** working point " to REIJLEAUX'S theory. PAGE 173. The term "multiple chains" has also been introduced by the author. HI. -DYNAMICS OF MACHINES. The impossibility of a perpetual motion and the practical application of the principle of work were well understood by SMEATON and others of our great engineers of the last century. Smeaton's papers, read before the Royal Society in 1759-82, were long regarded as an engineering text-book by his successors. The language in which their ideas are expressed, however, were not regarded as consistent with NEWTON'S teaching, and this circumstance perhaps concealed the real importance of the ideas themselves. At any rate, although the term "energy" was proposed by YOUNG, no considerable use was made of them by students of mechanical science until the publication by PONCELET, in 1829, of the Introduction a la Me'caniquc Industrielle, a work which has had a great influence on the study of mechanics. The third edition of this work (Paris, 1870), published after PONCELET'S death, will be quoted by the abbreviation Mec. Ind. Poncelet's methods were explained, and considerable additions made to the theory of machines, by MOSELEY in his Mechanical Principles of Engineer- ing (London, 1843). PAGE 182. This method was probably employed for the first time by WATT in his expansion diagram. See ROBISON'S Mechanical Philosophy, Vol. II. It is given by POXCELET (Mec. Ind., p. 66). PAGE 184. The terms "statical" stability, "dynamical" stability, in re- lation to vessels were introduced by MOSELEY (Phil. Trans., 1850). They have been criticized by OSBORNE REYNOLDS, perhaps not without justice, but are too firmly rooted to be displaced. PAGE 185. " Force is an action between two bodies, either causing or tend- ing to cause, change in their relative rest or motion" (A.M., p. 15). The 604 NOTES AND ADDENDA. distinction between internal work and external work is due to PONCELET (Mec. Ind., p. 30). PAGE 187. The language in which writers on mechanics have expressed the distinctive character of frictional resistances has been severely criticized by REULEAUX in his notes to his work on the Kinematics of Machines (Kennedy's translation, p. 595). The author by no means supposes that he can escape this universal censure, for the difficulty of expressing abstract principles in a form to which no objection can be made is almost insuperable. As, however, CLERK MAXWELL remarks in reference to a different question, the language in which a truth may be expressed is less important than the truth itself. Friction always causes energy to disappear, and is never a source of mechanical energy except indirectly through the agency of thermal energy. In mechanics this is a distinction of such fundamental importance that it even justifies, in the author's opinion, the use of such phrases as ' ' loss of energy. " The extension of the term "reversible" from a machine to the resistances which are overcome by the machine has been ventured on, though with some hesitation. The old term "active " can hardly be considered suitable. PAGE 187. "Envisage sous ce point de vue, le principe de la transmission du travail comprend impliciternent toutes les lois de Faction reciproque des forces, sous un e"nonce qui en facilite infmitement les applications a la Mecanique industrielle, qu'on pourrait nommer la Science du travail des forces. Des le premier pas des jeunes Sieves dans 1'etude, cet e'nonce, en effet, se presente a eux comme une sorte d'axiome evident par lui-m^me, et done la demonstration leur semble superflue aussitdt qu'ils ont bien saisi ce qu'on entend par travail mecanique, et qu'il leur est clairement demontre que ce travail, reduit en unite's d'une certaine espece est dans les arts, 1'expression vraie de I'activite' des forces" (Mec. Ind., p. 3). This passage from PONCELET is quoted to show how clearly it was seen, even before the discovery of the conservation of energy in its complete form, that the principle of work ought to be regarded as fundamental, and not merely as a deduction from certain equations. [1895.] In his interesting work on the development of dynamics, Professor Mach traces the ideas of PONCELET to HUYGHENS, and expresses his conviction that the difficulties which the conception of Work encountered were due to unimportant historical circumstances. The development of dynamical science might have proceeded on different lines, and the Principle of Work might have been regarded as fundamental at a much earlier date. There can be on question that it was a great misfortune to engineering science that such was not the case. The absence of due recognition of a principle which is actually forced on all those engaged in mechanical operations was the principal cause of the difference which for so long a period existed, and still does exist to some extent between the mechanics of the engineer and the mechanics of the schools. An American translation of the book here referred to (Die Mechanik in ihrer Entwickdung, Prague, 1883,) appeared at Chicago in 1893 under the title of The Science of Mechanics (London : Watts & Co.). See especially pages 178, 248-251, 272. PAGE 189. The modifications made here in the old statement of the principle of work, as applied to machines, are necessary consequences of REULEAUX'S conception of a kinematic chain. DYNAMICS OF MACHINES. 605 PAGE 193 [1892]. The author has little faith in the utility of "definitions" as applied to such a conception as that of a machine, and the remarks here made which have been somewhat amplified in the present edition must not be understood as an attempt at constructing one. PAGE 194 [1895]. A living agent works to best advantage when exerting a certain effort at a certain speed during a day's work of a certain length. The best effort and speed depend obviously on the strength and training of the individual as well as on the kind of work done. Some examples may here be given for a day's work of 8 hours. EFFORT. SPEED. POWEK. (Lbs.) (Feet per 1'.) (Ft.-Lbs. per 1'.) MAX. Without a Machine, . 33 \ /5280 Working a Crank, . . 22/ \3520 HORSE. Direct Traction, . . 130 250 32,500 Working a Machine, . 100 160 16,000 It will be observed that the power of a horse on this estimate when directly employed is little less than the conventional horse power of 33,000 ft.-lbs. per minute introduced by WATT, it is said, as the result of experiments on the work done by the powerful dray horses employed in London breweries. RANKINE'S estimate (Steam Engine and other Prime Movers, page 89,) of the power of an ordinary horse is much less, being 26,000 ft.-lbs. per 1', and he also gives smaller values for the work of men. When working at best speed the power during a whole day's work is a maximum, but all living agents can work at a much greater speed for a short interval developing from 3 to 5 times as much power as when the work is continuous. Thus strong men working a fire engine at two-minute intervals can develop half-a-horse power or even more. Empirical formulas have been constructed showing the relation between the power exerted at given effort and speed for a given time with that developed under the most favourable circumstances, but they can hardly be considered as satisfactory. PAGE 196. To avoid misapprehension, it may here be stated that in this, as much as in the preceding section, the object is to explain and to verify the principle of work : not in any sense to demonstrate it. PAGE 198. Except in the use of the word "kinetic" instead of "actual," the statement here is in the form given by RANKINE (A.M., p. 500). The author is entirely of (the late) Mr. W. R. Browne's opinion that this is the best form and has always used it himself. The idea of energy being stored in a body in motion perhaps first appears clearly in MOSELEY'S treatise. PAGES 203-207 [1892]. Art. 103 on oscillations has been re-written, with additions. The formula for the length of the simple equivalent pendulum on page 206 was printed incorrectly in the first edition an error corrected in the second. Various other changes and additions have been introduced in the second half of this chapter for the sake of clearness and to make it harmonize better with the rest of the book. The infinite series, by which the time of vibration of a pendulum is given, will be found in most treatises on the kinetics of a particle. See, for example, Price's Infinitesimal Calcidus, Vol. III., p. 549. 606 NOTES AND ADDENDA. PAGE 212. The construction by means of which curves of crank effort are obtained was given by PONCELET, but it does not appear chat any such curves were actually drawn until they were given by ARMENGAUD in his treatise on the Steam Engine. In Fig 97, to save room, the curves are placed half above and half below the base, but otherwise the figure is that of ARMENGAUD ; it is far the most convenient form for applications. PAGE 220. The stress due to centrifugal action on the rim of a wheel is given by a formula (p. 288) which may be written in the simple form V 2 = g\, where X is the length due to the stress (p. 80). A velocity of 80 feet per second gives a length of only 200 feet, or about one-fifth of the stress cast iron would safely bear in tension. The inequality of distribution produced by inextensible arms tying together opposite points on the rim of the wheel doubtless increases the maximum stress ; but the principal reason for the low limit required for safety is the alternate bending backwards and forwards of the arms as energy is alternately stored and restored by the wheel. The speed is occasionally increased to 100 feet per second. The author is indebted to Prof. Unwin for the information that when the wheel is in segments the speed should be limited to 40 feet per second. PAGE 222. The method here given occurred to the author many years back ; but it is believed to have been previously published in Engineering. PAGE 226 [1900]. As regards the effect of the varying speed of the crank on the inertia-pressure of a piston, it will be seen on reference to page 102 that the part of the piston acceleration due to this cause is while the part considered in this article is / 2 = ( cos 6 + - cos 26 \ E \ n ) a Evidently the average relative importance of these parts is measured by the ratio of d Vo/dt, the tangential acceleration of the crank pin to f r 2 /a the radial accelera- tion. If now we write q . F for A V where q is a co-efficient as on page 220, and m . TQ for A where 7 7 is the period and m a fraction, the mean value ofdVJdt for the change considered will be rfF o = l o = 17 dt m ' r o m ' 2ira and therefore Ratio of accelerations = pr^- 2?rm A change of 10 per cent, in the speed taking place in one-tenth of a revolution, or of 25 per cent, in one quarter of a revolution, would make q/m unity, and it is evident that when an engine is running steadily this is an extreme case. In general the value of q/m must be much less. It is true that the maximum value of dV /dt is greater than the mean, and under certain conditions it may be very much greater ; but on the whole it may be said that the average error of the supposition that the crank rotates uniformly is not of much importance. But if the maximum d V [dt occurs near the middle of the stroke, as it often will, where / 2 is small and j\ greatest, the inertia-curve for that part of the stroke will be completely altered. DYNAMICS OF MACHINES. 607 PAGE 228 [1892]. The investigation here given of the effect of the angular motion of a connecting rod has been added to the present (1892) edition. Art. Ill (page 231) on pumping engines has also been added to render the chapter less incomplete. PAGE 229 [1900]. The investigation of the exact value of the inertia-pressure of a connecting rod is given now for the first time. On the representation of a curve of crank effort by a Fourier series, see Note to p. 395. PAGE 240. The Friction Circle was defined and its use explained by RANKINE in his treatise on Milhvork and Machinery, p. 428. PAGE 265 [1895]. The adoption of metric measures in engineering practice was recommended by a committee of engineers 40 years ago, and in 1868 a Bill passed its second reading in the House of Commons by a large majority for rendering it compulsory within three years. In June of the present year a committee of the House has reported in favour of its compulsory introduction in two years. The inconvenience of a change to the present generation of engineers would be very great, but it is probable, even in the absence of compulsion, that the pressure of foreign competition may render it inevitable before very long. From the point of view of abstract science this change, however, is only part of that which is desirable or even necessary : for the metric system just as much as our own is a gravitation system of measurement, that is the unit of force instead of being derived from the unit quantity of matter with due regard to the units of time and space is taken as the force with which the unit quantity of matter is drawn to the ground at a given point on the earth's surface. As explained in the text this renders it necessary to dissociate the unit of inertia from the unit quantity of matter, and to use the words "weight," "pound," "kilogramme," etc., in a double sense since they are applied indis- criminately to forces and to quantities of matter. Most modern writers on mechanics when using gravitation measure seek to avoid ambiguity as regards the term "weight" by confining the use of the word to the force of gravitation and employing the term "mass" to signify the quantity of matter determined by weighing as well as the inertia measured by the quotient W\g. It may be questioned whether the ambiguity thus introduced is not more misleading than the original, and the term "weight" has therefore been used in its old meaning throughout this work. It might be avoided perhaps by calling the quotient Wfg the "Inertia" of the body, but the author has not felt at liberty to introduce a new term in this connection. In the absolute system of measurement the unit of force is dissociated from the unit quantity of matter, and so taken that the units of inertia and quantity of matter become identical. To do this it is only necessary to take as a unit of force the force necessary to generate unit velocity in unit time in the unit quantity of matter. A special name is then given to the unit of force which is now entirely independent of gravitation. In the c.G.S. system this unit is called the Dyne. The system possesses undoubted advantages on the score of clearness and precision, but the practical difficulty of dissociating the unit of force from the unit of quantity of matter would be very great in any case. In the C.G.S. system the difficulty is greatly aggravated by the smallness of the units chosen, the force called 1 kilogramme in gravitation metric measure being no 608 NOTES AND ADDENDA. less than 981,000 dynes. No force commonly occurring in practice therefore can be expressed in dynes without multiplication by some large power of 10 subject to a great liability to error in the index. It is probable, however, that some modified form of the C.G.S. system may ultimately be found which is capable of being practically used. Through the agency of the electrical engineer some of its nomenclature is becoming well known. PAGE 268. The distinction between internal and external kinetic energy is pointed out by RANKINE (A.M., p. 508). PAGE 271. On Governors in general the reader is referred to a paper by CLERK MAXWELL in the Proceedings of the Royal Society, No. 100, 1868. A full account of the principles of construction of centrifugal regulators will be found in Tkeoretische Maschinen-Lehre, Band III., Leipzig, 1879, von. Dr. F. Grashof. PAGE 280 [1900]. The internal balance of forces in machines at rest considered in Art. 142 is only complete in machines which are actually or virtually self-contained (compare p. 193). In motors or machines driven from a motor the balance is rarely perfect because the connection between the driving pair and the working pair is not sufficient to completely close the circuit. Thus in the vertical engine taken as an example, suppose the engine to be employed in driving a screw ; the resistance to the rotation of the screw arises from water outside the vessel and disconnected from it : the moment of crank effort is therefore unbalanced and tends to heel the vessel over : while any variation in that moment in the course of the revolution will furnish a periodic couple tending to produce vibration. PAGE 284. The utility of balance weights, sufficiently heavy to neutralize completely the horizontal forces, is by no means universally admitted. The vertical forces introduced are very great (Ex. 17, p. 291), and, should they synchronize with the period of vertical oscillation of the engine on its springs, most dangerous results might follow. PAGE 285 [1900]. The investigation given in Art. 144A of the effect of the inertia of a connecting rod on the primary and secondary inertia-forces has been added to this edition. PAGE 288 [1900]. As a simple example of a self-balanced engine may be taken the case of a 3-cylinder engine : one cylinder A being between the other two, 5j and B^, but on the opposite side of the crank shaft. The centre lines of the three cylinders are all in the same plane, but crank A is at 180 to cranks B^ B z . For simplicity let the cranks S lt B. 2 be of equal length, but in any proportion to crank A : the ratio of connecting rod to crank being the same in all. Evidently in such an arrangement, the rods will always be parallel and all the reciprocating parts similarly situated. If they be similar and the weights for A and B 1} B 2 jointly, be inversely as the cranks; the centre of gravity of the whole must always lie in the axis of the shaft, and there can be no alternating forces. And if the weights of B, B% are inversely as the distances apart of the cranks there can be no alternating couples. The engine therefore is completely balanced so far as regards the inertia forces, and would commonly be described as "self-balanced," but the balance does not include the forces mentioned in the note to p. 280. STIFFNESS AND STKENGTH OF MATERIALS. 609 IV. -STIFFNESS AND STRENGTH OF MATERIALS. PAGE 317 [1892]. The calculation here given (Art. 161) relating to beams of tmiform strength, presupposes that the transverse section of the beam varies slowly, a condition which of course is very far from being satisfied near the ends of the beam. Any attempt to take into account the variation of section, would, however, only lead to results of much greater complexity without any correspond- ing increase in their utility. The forms obtained are purely ideal, being incapable of being practically used without the addition of material necessary to provide against straining actions other than bending. PAGE 332. If an elastic solid or, more generally, a set of connected pieces of perfectly elastic material, be under the action of any number of forces P ly P%, ..., and any number of couples A/j, M. 2 , ... , in equilibrium, the value of U must be where x lt x z , ... , are the displacements of the points of application of the forces and i lt i. 2 , ... , the angular displacements of the arms of the couples. For if the forces gradually increase from zero, always remaining distributed in the same way, each part of the load (P) will exert the energy ^Px, since the space moved through (x) must clearly be proportional to P. The same argument applies mutatis mutandis to couples. Hence the whole energy exerted must be given by the above formula, and this is always represented by the energy stored up in the system when the parts are perfectly elastic. Now, imagine the solid immoveably fixed at three or more points, and let one of the forces P-^ be increased by a small quantity 5P a , all the other forces retaining their original magnitudes. The effect of this is that the points of application of all the forces move through certain small spaces (dx), and the arms of all the couples through certain small angles (Si). The total additional work done will be But, on differentiating the value of U on the supposition that Pj alone varies, we find and therefore by substitution 8(T=x 1 . 8/V A similar equation is derived by supposing one of the couples to vary, and we obtain the general equations dU_ dU_. dP~ : '' dM~*' that is, the displacements are the partial differential co-efficients of U with respect to the forces. The forces to be considered are partly weights or other loads of known mag- nitude, and partly arise from the stress between the bounding surfaces (real or ideal) of the solid and external bodies. The boundary forces must be consistent with statical equilibrium, but subject to this condition are determined by equations found by differentiating the function U. In particular, when the bounding surface is fixed, the partial differential co-efficients of U with respect to the corresponding forces must be zero. The value of U is then, in most cases (perhaps always), a minimum, as stated in the text. It appears then that whenever the elastic potential can be found and expressed in terms of the external and boundary forces acting on the system, the necessary equations for determining the boundary forces and the deflection produced by the C.M. 2 Q 610 NOTES AND ADDENDA. external forces can all be found by differentiation of U and by the conditions of statical equilibrium. As an example, take the case of a beam loaded in any way and fixed at the ends. Let the beam be AB (Fig. 28), and let the notation be as on pages 40, 41, then (page 331) Substitute for M by the formula on page 41, and integrate between the limits I and o, we find 2EI . U= J (M A * + M A M B + M B 2 )l The integrals are most conveniently expressed in terms of, S the area of the curve of moments (m), z the distance of its centre of gravity from A , and y the height of its centre of gravity above AB. The formula then becomes, dividing by 2, El. U= The potential is thus expressed in terms of the load on the beam and the bending moments at its ends. The latter may have any values we please consistently with statical equilibrium, and the partial differential co-efficients of 7 with respect to M A M B will be the slopes at the ends. In particular, if the ends are fixed horizontally, equations which determine M A M B , and express that the function U is then a minimum. In the particular case of a symmetrical load The value given on page 326 for the particular case of a uniform load will be found to agree with this result. The potential for a continuous beam may be immediately deduced, by addition of the potentials for each span taken separately, in terms of the bending moments at the points of support. The theorem of three moments (page 330) for the case of supports on the same level, then follows at once by differentiating with respect to the moment at the middle point of support. In all cases, differentiation of U with respect to any portion of the external load will give the deflection at the point where that load is applied. In applying this method care must be taken that the supporting forces, in terms of which the potential is expressed, are independent : if they are not, then the equations of statical equilibrium will be conditions subject to which U will be a minimum. To take a simple example, suppose a perfectly rigid four-legged table standing on four similar elastic supports and loaded in any way, then where P I} P 2 ^*3> ^*4 are t ne P ar "ts of the whole load resting 011 each leg, and n is some multiplier. Here the forces P are partly determined by three statical equations for equilibrium of the table, and only one additional equation is found by making U a minimum. STIFFNESS AND STRENGTH OF MATERIALS. 611 This method was explained and applied to a number of examples in some paper by the author, which appeared in the Philosophical Magazine for 1865 ; the demonstrations there given, however, were insufficient. The author at that time supposed it to be new, but it had already been given in a memoir by M. E. F. MEXABREA. Comptes Rendus, vol. xlvi. (1858), page 1056. PAGE 338. The lateral disturbance is here supposed small. With a larger disturbance the pillar would return even if the value of W were equal to 27/ 2 , and with a greater value would bend over into a position of equilibrium given by the formula w=( * e Y. 2 ^ 7 , where 6 is the angle subtended by the circular arc into which the pillar is bent. PAGE 340. When the pillar is absolutely straight and homogeneous and of uniform transverse section, the lateral deflection due to an actual deviation a is given by the formula a , . cos ml and the formula for the effect of deviation becomes ,P V 2 In any actual example, however, this formula would not be exact any more than that given in the text. Each particular example will have its own formula. The result of all such formulas, however, must be nearly the same for a small deviation. Further, a great proportional change in the deviation, always suppos- ing it small, produces little change in the crushing load, and this probably explains why experiment gives tolerably definite values of the crushing load although its precise amount must depend on accidental circumstances. PAGE 339 [1890], The method here adopted of proving EULER'S formulae is to assume the curve in which the pillar bends to be a curve of sines, and then to show that the sectional area is constant, a process which is the converse of that employed in the first edition of this work. Being more simple, the demonstration has been placed in the text instead of being relegated as before to the Appendix. It is worth remarking that the co-efficient of elasticity of flexion employed in them is not, strictly speaking, Young's modulus (E) but E-p where p is the intensity of the stress on the cross section. This, though theoretically interesting, is of no importance in practice, because of the extreme smallness of the ratio p/E in all practical cases. The case where one end of the pillar is fixed and one rounded was first, it is believed, correctly treated by GRASHOF in 1866. In a paper published in the Proceedings of the Cambridge Philosophical Society, Vol. IV., Part II., GREEXHILL has determined the greatest height of a vertical pole which is consistent with stability, that is the greatest length of pole of given diameter which will stand upright without bending over laterally at the summit. Let \ = E/iv be the length due to a stress E for a given material, E being as usual Young's modulus. This method of measuring a stress is explained on page 80. Then for a pole of uniform transverse section of radius a the greatest height is given by the simple formula NOTES AND ADDENDA. in which all the quantities are given in the same units. For a pole of radius a at the base diminishing in section uniformly to zero at the summit so that the longi- tudinal section is triangular the same formula serves, but the co-efficient is v7'63, or 1 - 97 instead of 1 '26. For a pole of pinewood 6' diameter at the base the greatest height is about 90 f$et in the first case and 140 in the second. The stress (wh) on the transverse section at the base must, as appears from what is said in the text, be much less than the crushing stress (/) of the material, a condition which would generally be satisfied. These formulae are of considerable theoretical interest and are applied in the paper cited to questions relating to the growth of trees : it must be remembered, however, that the effect of wind pressure is neglected, a circumstance which limits considerably the practical application of the formulae. PAGE 347 [1890]. A formula, corresponding to EULER'S formula for pillars, has been obtained for the collapse of a flue of unlimited length, by LEVY and HALPHEN. This formula as quoted by GREENHILL in a letter which will be found in the Enqineer for February 1888 is Et where t is the thickness, a the radius, both reckoned in inches, while E as usual is Young's modulus, and p the collapsing pressure. The corresponding form of Gordon's formula deduced as on page 340 will be '-5 p= ~: ct 2 where the " theoretical " value of the constant c is This formula may be expected, with suitable values of the constants, to give the collapsing pressure of a flue, the length of which is so great as to have no sensible influence on its strength. In short lengths the strength is greater as described in the text. Thrust and Torsion. When it is a question of strength only this case is dealt with on the principles explained in Chapter XVII. GREENHILL has, however, pointed out that, if the unsupported length of the shaft be too great, it is necessary to consider its stability. Let P be the end thrust on the shaft, T the twisting moment, then the greatest unsupported length consistent with stability is given by the formula Let P be the greatest load which by EULER'S formula this length of shaft would carry considered as a pillar, and let T be the greatest twisting moment consistent with strength, /being the co-efficient of resistance to twisting, then the formula may be written, supposing d the diameter, or using p , p to represent the stress per square inch of section STIFFNESS AND STRENGTH OF MATERIALS. 613 Asf/E is necessarily a very small fraction and T/T is fractional this shows that there can be very little difference between p and p , so that unless these quantities themselves be small, that is the unsupported length of the shaft very great, the twisting makes no sensible difference in the stability of the shaft. The formula, therefore, in ordinary cases, though theoretically interesting, is not of practical value. It will be found with numerical applications in a paper read by Professor Greenhill before the Institution of Mechanical Engineers in 1883. PAGE 358 [1900]. The formula here given for the torsion of a tube of non- circular section has been inserted in the present edition chiefly in order to give some idea of the reason why the maximum stress in shafts generally occurs at points on the circumference nearest the centre, and as leading directly to a formula for the torsion of shafts which is exact for elliptic sections. It is not probable that it is exact for any other form of section, and in cases like that of a rail where the curvature of the profile is partly concave it can hardly be even approximate. The corresponding formula for the rigidity of a tube under torsion is given on page 422, and also appears now for the first time. PAGE 360. The formulae given in different books for the moment of resistance of a shaft of rectangular section exhibit considerable discrepancies. COULOMB, to whom the formula for a circular section is due, supposed that in every case r=f.-, r \ where I is the polar moment of inertia and r x is the outside radius. In a rectangular section of sides a and b this gives which for a square section of side h becomes T= -2357/. h s . If these results were correct it would appear that a shaft of given sectional area was stronger the more unequal the sides were, a result quite contrary to experi- ence. In a memoir on torsion published in the Memoires de Vlnstitut for 1856, BARRE DE SAINT VENANT investigated the question thoroughly, and obtained the results given in the text. RANKINE (A.M., page 358) gives '281 fh 3 as the result of SAINT VENANT'S calculations without further explanation. This value is greater than that given by COULOMB'S hypothesis, and is certainly too large. [1890, 1892.] That the resistance of a- shaft to torsion was not proportional to the ratio I\r had long been recognized, and prior to the acceptance of ST. VENANT'S results the formula originally given by CAUCHY was much employed. This formula agrees with COULOMB for a square section, the result being about 11 per cent, greater than that given by ST. VENANT. The error diminishes as the inequality of the sides increases and vanishes when the ratio (n) of the sides is very small. The corresponding ratio of strengths of a rectangular and a circular section of the same area is 614 NOTES AND ADDENDA. a result given in former editions of this work after modification, by replacing the constant factor '8353 by '738, so as to make it agree with ST. VKNANT for square sections. The error is then very small for moderate values of n ; but in the present [1892] edition it has been thought advisable to give ST. VENANT'S own formula, the maximum error of which is estimated by him as 4 per cent. [1895.] The error of ST. VENANT'S formula in practical cases may be diminished, as pointed out in the text, by a slight modification of the constants. The formula in common use by the best German technical writers at the present time appears to be which for values of j3 greater than '5 may be considered as a rough approximation, but for small values of j8 gives values of T which are much too small. The investigation given by BACH in his treatise referred to further on is unsatisfactory as the distribution of stress assumed is not shown to be consistent with the corresponding warping of the section. From a letter in Nature (June 1888) by Mr. Dewar it appears that RANKINE'S value was obtained by taking the angle of torsion as a measure of the strain produced. PAGE 367 [1892]. The formula for the distribution of shearing stress on a section has in this [1892] edition been put in a more simple and general form, and its true interpretation pointed out. See also Note to page 416 further on. In an excellent paper, which will be found in the Transactions of the Institution of Naval Architects for 1890, the late Professor P. Jenkins has applied this formula to investigate the effects of longitudinal shearing stress in a vessel. Professor Jenkins was a former distinguished student of the Royal Naval College, and the author is happy to have this opportunity of expressing his regret at his premature decease. PAGE 388 [1895]. The subject of vibration has of late acquired increased im- portance in consequence of experimental investigations which have been made on the vibration of vessels and girder bridges. A brief explanation of the most important points relating to it has therefore been added to the present edition in the articles which follow in the text. On reading Herr Schlick's interesting paper of 1894 the author felt some doubt whether the values of the constants quoted in the text were for complete or single vibrations. In a letter dated April 13th, 1895, Herr Schlick kindly informed him that complete vibrations were meant, and at the same time pointed out that the result would depend very much on the allowances made in calculating the moment of inertia I. In the method actually adopted the sections of angle irons and similar pieces were supposed concentrated in their centre lines, no allowance was made for reduction of area by rivet holes, and no account was taken of bilge stringers, keelsons, etc. It will be observed that the constant is 23 per cent, greater for a torpedo-boat destroyer than for a merchant vessel, showing that the distribution of the weight is a leading element in the question as might be anticipated from what is said in the text. Unequal distribution and concentration of the weight in the neighbour- hood of the nodes increase the frequency and more than compensate for the influence of the causes pointed out which tend to reduce it. To the examples given in the text on the effect of synchronism may be added the case of a locomotive traversing a girder bridge. In order to balance the reciprocating parts heavy counter-balance weights are necessary attached to the STIFFNESS AND STRENGTH OF MATERIALS. 615 driving wheels, and these weights produce periodic vertical forces of great magnitude (Ex. 17, p. 291), the frequency of which is equal to the number of revolutions per 1" of the wheels. When the speed reaches a certain limit experi- ence shows that a bridge over which the locomotive is passing vibrates greatly, an effect due to synchronism between the period of a revolution and the period of free vibration of the structure. The strength of a screw shaft to resist the combination of thrust and torsion to which it is subject is diminished by centrifugal action, a question which has been discussed by Professor Greenhill. PAGE 393 [1892]. In a paper published in the Transactions of the Institution of Naval Architects for 1892, Mr. Yarrow conclusively showed that the vibration of torpedo boats is almost entirely due to the reciprocating parts of the engines and can be got rid of by a proper system of balancing. PAGE 395 [1900]. The statement made in the text (Art. 203) depends on the mathematical proposition that any periodic function of may be fully expressed by a series of sines or cosines of 6, 20, 3d, .... This series is called Fourier's Series, a full and clear account of which will be found in Professor Byerly's Treatise on Fourier's Series (Boston, U.S.A., 1895). In its complete form a Fourier's Series consists (1) of a constant term representing the average value of the function, (2) of a series of sines, (3) of a series of cosines, but it generally can be simplified by considering the nature of the function. In the case considered in this article the function is the acceleration-ratio of a piston that is (p. 602) 1 sin 2 1 cos 2 /=cos0--.- + -. T-, n cos n cos 3 a formula true for all values of 6. This is unchanged when 6 is changed into - 6, so that the series can contain no sines, and only the first term is changed when is changed into TT + 0, so that there can be no cosines of odd multiples of 0. Hence the Fourier Series for / must be f=cos0 + A . cos 20 + JB . cos40+.... The co-efficients A, B ... of a Fourier Series are found by a process of integration when the integrations can be performed, or mechanically by an instrument of the nature of a planimeter, or calculated approximately. In the present case in the note already cited, the series is given to a second approximation, but the first approximation used in the text is always sufficient. If instead of the acceleration-ratio we consider the velocity-ratio, the first approximation to which is ^ = sin + J- sin 20 (p. 102), VQ <&n we find that it is true for all values of and therefore is itself the Fourier Series to the same approximation. For values of between and TT the ratio of the crank effort (R) to the piston pressure (P) is given by the same formula (p. 191), but this is now 110 longer the Fourier Series because it is not true for all values of 0. For values of between TT and 2?r the steam pressure is transferred to the other side of the piston and the formula is --. sin 20. 616 NOTES AND ADDENDA. This discontinuity in the ordinary formulae for the crank effort is one of the- principal reasons why graphical methods are almost exclusively used in such questions. To find the form of the Fourier Series, consider that the crank effort is always in the same direction and its average value is 2P/TT ; the series therefore contains a constant term. Also it is unaltered when 6 is changed into 2ir 6, so that the series can contain no sines and must consist of two series of cosines. The first of these representing sin 6 for values of 6 between and IT and sin 6 between TT and 2?r can contain only even multiples of 6. The second representing sin 26 between the same limits contains only odd multiples of 6. The co- efficients are found by integration. Hence omitting higher terms ^_4fl cos20 cos40 1/COS0 cos 30 \) 7>~w\2 3~ ~I5~ H ~?A~3 5 jj" This formula gives 7? when P is constant. When P varies, the series contains sines as well as cosines ; the odd multiples of 8 as above representing the effect of obliquity. We have therefore R 2 ^- = -+A . cos20 + #. sin 20. +..., L m 7r where P m is the mean pressure and A, B, etc., are constants. The idea of employing a Fourier Series to represent a curve of crank effort is due to Professor Lorenz, and its advantage is that, being true for all values of 0, any number of cranks at any angles may be superposed, and a formula found for the combination. In his paper read before the Institution of Naval Architects, and published in their Transactions for the present year (1900), Professor Lorenz finds the general conditions that the co-efficients of cos 20, sin 20, may vanish so that the series may contain only terms of a higher order. These conditions are taken as being approximately the conditions for greatest uniformity of crank effort, and it is pointed out that they are not inconsistent with the con- ditions necessary for the balance of the inertia forces. Obliquity how r ever is neglected, and, as the formula given above for a constant pressure shows, the effect of obliquity is often very important. The method being new and capable of many applications, it has been thought proper to notice it here. Vibration in a vessel due to inequality in the turning moment on a screw shaft must not be confounded with the bending vibrations due to inequality in the action of the water on the blades of a screw. It is torsional, a kind of vibration which according to Schlick really does occur (Trans. I.N.A., 1895, p. 292). PAGE 396 [1900]. Since 1892 the subject of the vibration of vessels and the balancing of marine engines has been very extensively discussed in papers for the most part appearing in the Transactions of the Institution of Naval Architects. It has been placed on a solid foundation by the experimental researches combined with theoretical investigation of Herr Otto Schlick, some of whose earlier results are given in the text. Also may be mentioned papers by Mr. Mallock, Mr. Macfarlane Gray, Professor Dalby, and others. The author regrets that the limits of this work render it impossible to do more than allude to these researches. Article 203A on the centrifugal whirling of shafts has been added to the present edition, as the subject has of late become one of considerable importance. PAGE 403. A line of stress may be regarded as the geometrical axis of a curved rod which is in tension or compression, as the case may be, under the action of a load perpendicular to itself. The whole solid, therefore, may be conceived as made STIFFNESS AND STRENGTH OF MATERIALS. 617 up of a set of rods, each of which is a rope of linear arch in equilibrium under a transverse load. Each rod transmits stress in the direction of its length. If there be no lateral stress the rods are straight, but otherwise they are curved. In a framework structure loaded at the joints, the bars of the frame may be regarded as lines of stress except at the joints where those lines assume complex forms. The tendency of modern science is to regard all force as being due to the transmission of stress through a medium of some kind, even in such cases as that of gravity, where no medium perceptible to our senses exists. All forces on this conception are represented by a system of lines of stress. PAGE 405 [1892]. Except in beams of / section, the effect of shearing in increasing the maximum stress and strain due to bending is unimportant, even when the length is only two or three times the depth. This remark, however, does not hold good. for materials such as timber, which have a relatively small resistance to longitudinal shearing. Timber beams not unfrequently give way by longitudinal shearing at the neutral surface. PAGE 410. The theory of elastic solids has been much more fully treated with reference to practical application by GRASHOF, SAINT VENANT, and other con- tinental writers than in any English treatise. The author is chiefly indebted to GRASHOF'S work, Die Festiykeits lehre (Berlin, 1866), a new edition of which appeared in 1878. An attempt has been made in the present work to distinguish clearly between those parts of the subject which are necessarily true either exactly or to a degree of approximation which is capable of being exactly calculated, and those parts which depend on hypotheses more or less probable. The first are placed in the present chapter ; the second in the chapter which follows. [1892.] Since the publication (1884) of the present work, a part of the late Dr. Todhunter's History of the Elasticity and Strength of Material* has appeared. The book has been edited and put into its present form by Professor Karl Pearson, Vol. I. containing the early history of the subject, and an exhaustive account of all that has been done up to the year 1850 appeared in 1886. The first chapter of the second volume, entirely written by Professor Pearson, was published in 1889 as a separate work entitled the Elastical Eestarches of Barre de St. Venant. It covers the whole of ST. VENANT'S labours on the subject of Elasticity, extending over a period of 35 years. [1895.] The second volume of the History has since appeared, in which the work just mentioned is incorporated, and to which therefore references are made. The standard German treatise on the technical applications of the theory of elasticity is at present Professor Bach's Elasticitdl und Festiykeit, Berlin, 1890, to which valuable work the author is indebted for information and corrections, especially in Chap. XVIII. PAGE 412. Attempts have been made to prove by theoretical reasoning that, in a perfectly elastic isotropic material, the value of m is necessarily 4, and the demonstration is still considered valid by some authorities, while others consider that such reasoning simply shows that matter is not constituted in the way supposed in the .demonstration. It is difficult to obtain material which is really perfectly isotropic, but all the experimental evidence at present goes to show that m may have various values. PAGE 416. Some other points in the theory of bending may here be noticed: (1) The effect of curvature is that a lateral stress // must exist on, 618 NOTES AND ADDENDA. the longitudinal layers given by the same equation as is used for thick hollow cylinders under internal fluid pressure (page 410), viz., Replacing r by R + y, and p by the value given in the text we find d . , . Ey 5 "*>=!?' and therefore, by integration, W& pr -^ + constant. Since p' is zero at the outer surface where y %h, where p l is the stress due to the bending at the outer surface, and r is replaced by its mean value It. At the neutral surface p' is greatest, but even there has only the very small value This lateral stress is therefore never great enough to have any perceptible influence on the elasticity of the layers. (2) It has been stated on page 299 for the case of tension, page 311 for the case of bending, and page 355 for the case of torsion, that the distribution of stress on any transverse section is the same, however the straining forces are applied to a bar, provided only that their resultant be given in magnitude and position. This may be regarded as a general principle applicable in all cases. Any other distri- bution of stress produced on a transverse section by friction or other external forces applied directly to it will change with great rapidity on passing to transverse sections not directly exposed to such forces. It is, however, generally necessary to provide additional strength at these exceptional sections. PAGE 416 [1895]. Assuming the transverse curvature circular the elevation (u) of the sides of the beam above the centre is given by the formula 6 2 p_ I 2 M _ 36 2 M ''~SmE ' y~8mE ' I ~2mEh* ' A' obtained on page 413, for a rectangular section. Now the value of du/dx is evidently the difference of steepness of a central line traced on the side of the bar and the geometrical axis of the bar. Hence if Aq be the difference of shear at the side and at the centre, g the mean shear over the whole section, du 36 2 For a square section assuming m 4 and remembering that |g is the mean shear along the neutral axis, we find that the difference between the shear at the side where it is a maximum and the shear at the centre where it is a minimum is one-tenth the mean. The maximum then exceeds the mean by about 5 per cent. This rough calculation is given for the purpose of illustrating the remarks in the text, but as the formula for u is not exact when shearing is taken into account, accurate numerical results can only be obtained by ST. VENANT'S calculations. STIFFNESS AND STRENGTH OF MATERIALS. 619 PAGE 417 [1895]. The formulas for plates supported at the edges, and exposed to normal forces, have long been known in two or three simple cases, and are frequently quoted. Unfortunately these formula? have not as yet been sufficiently verified by experiment and have therefore been omitted in the present work notwithstanding the importance of the question. Readers who wish to see the present state of the subject are referred to Professor Bach's small work, Versuche iiber die. Wider-stands fahiykeit ebener Flatten. Berlin, 1891. PAGE 420. The lines of stress for a thick hollow cylinder under internal fluid pressure, and also under the action of tangential stress applied as in Ex. 6, p. 406, will be found to be equiangular spirals, the angle of the spiral depending on the proportion between the fluid stress and the tangential stress. The verification given in the text is necessary because, otherwise, we could not be sure that the assumptions on page 418 were consistent with one another. This is very well shown by supposing the cylinder to rotate and obtaining a solution of the problem when thus modified, assuming the cylinder to remain cylindrical and employing the equation of verification. It will be found that the solution thus obtained can only be true if the stress on the transverse section varies according to a certain law. If the cylinder is long it appears that this must really be the case except very near the ends. The problem of a swiftly rotating circular saw appears not as yet to have been attempted ; it is found by experience that a saw to run at high speed must be hammered so as to be " tight " at the periphery. The same difficulty occurs if the material of the cylinder be not isotropic. PAGE 427. TRESCA'S experiments are described in detail, with a great variety of interesting illustrations in a series of memoirs which have been separately pub- lished (Memoires sur VEcoulement des Corps Solides). The example in the text is taken from the second memoir (Paris, 1869). It is to be remarked that the influence of time was not taken into account. PAGE 435. The modulus of elasticity in compression is found to be less than that in tension in cast iron as well as wrought iron in about the same ratio. This circumstance, together with the equality of the moduli for bending and tension, leads us to conjecture that the effect is due to lateral bending which cannot be wholly prevented by the trough. PAGE 436 [1892. 1895]. When a tube is thin and stiffened so as to prevent flexure as a whole, crushing may take place by local buckling. On the subject of buckling the reader is referred to a paper by Mr. J. A. Yates on the Internal Stresses in Steel Plating due to Water Pressure in the Transactions I.N.A. for 1891 (Vol. XXXII., page 190). Since that time the mathematical theory of buckling has been discussed in some papers by Mr. Bryan, which have appeared in the Proceedings of the Mathematical Society. PAGE 438. The argument of Art. 224 applies equally to any case where stress is not uniformly distributed. In the hydraulic press cylinder the stress is never reversed, and the increase of strength is probably reliable. PAGE 446 [1895]. BACH'S experiments on plates here referred to are noticed on page 614. See also a paper on Bulkheads, by Dr. Elgar, in the Trans. I.N.A., 1893. 620 NOTES AND ADDENDA. PAGE 440. No formula of this kind is anything more than a formula of inter- polation supplying the place of missing experiments. The author is led to make this remark by the elaborate manner in which such formulae are discussed by some writers. The study of WOHLER'S original memoir cannot be too strongly recommended to those interested in the subject. PAGE 449 [1890J. The proposal (which has been partially carried out in practice) to employ a margin instead of a factor of safety in boilers was made by the late Mr. Sennett, formerly engineer-in-chief of the navy, and his views were endorsed on the discussion of his paper (Transactions of the Institution of Naval Architects, Vol. XXIX.) by Mr. A. C. Kirk and Mr. Marshall. To the remarks made in the text it may be added by way of caution that though theoretical reasoning and laboratory experiments may furnish valuable indications of the direction in which to move, yet any steps in the direction of lower factors of safety should be very gradual, and only taken by those who possess the widest knowledge and the greatest experience of nearly similar cases. It is impossible a priori to foresee all the circumstances which may influence the necessary margin of safety. PAGE 456 [1892]. The values given for the resilience of timber in the two earliest editions of this work were much too large. When the yield-point of the ductile metals is regarded as the limit of elasticity, the resilience is given in Table III. on the following page. V. -HYDRAULICS AND HYDRAULIC MACHINP^S. PAGE 473. The standard experiments on the co-efficients of velocity and con- traction in the case of orifices are those made by WEISBACH, and described by him in his treatise Die Experimental Hydraulik (Freiberg, 1S55), to which the reader is referred for details. A short pipe projecting inwards is known as Borda's mouthpiece. The theoretical minimum value of the co-efficient of contraction (5, see p. 498) is closely approached when the pipe is very thin and sharp-edged ; otherwise the value is somewhat larger, say about '55. [1892.] Experiments by BAZIN and others on orifices of large size (7 inches diameter and upwards) give a co-efficient of discharge of '6. MAIR, however, obtained the value '61 in an orifice only 1 inch diameter. PAGE 476. The use of the term " head " for the energy per unit of weight of a fluid is not free from inconvenience the two things not being identical unless the datum level be at the surface of the fluid. PAGE 477. The velocity of the water in any one of the ideal pipes is inversely proportional to the sectional area of the pipe. Now the form of the pipes depends solely on the form of the bounding surfaces, and it follows, therefore, that the velocities of all parts of the stream bear a fixed proportion to each other, depend- ing only on the nature of the bounding surfaces. In the language of the theory of mechanism, the fluid forms a closed kinematic chain. The chain is closed by the pressure of the bounding surfaces, and when the velocity exceeds a certain limit the chain opens. Energy can then no longer be transmitted uniformly to all parts of the fluid, and is no longer uniformly distributed. When energy is unequally distributed, eddies are formed. PAGE 477 [1895]. The steady motion of an undisturbed stream is one of the principal objects of study in treatises on analytical hydrodynamics, and to such HYDRAULICS AND HYDRAULIC MACHINES. 621 works the reader is referred for an account of what is known on the subject. Some important points, however, can be rendered intelligible by methods of more limited scope, and the note which follows, taken from the Philosophical Magazine for February ] 876, is therefore here introduced. The motion of the fluid is sup- posed to be in two dimensions, that is, all particles are supposed to be moving in directions parallel to a given plane taken for convenience as vertical and the motion in all planes parallel to the given plane is the same. In Fig. 212 AB, CD are consecutive lines of motion, commonly described as s u Fig. 212. *' stream lines " lying in the same vertical plane, and representing the section of an elementary stream which flows, as described in the text, without intermingling with the rest of the fluid, just as it would in a pipe of varying section. P and Q are particles moving in these lines with velocities v and v + 8v, which at the instant considered are so placed that PQ the line joining them is normal to the stream. Then if h be the total head at Q, ,_ p v 2 '^w + W which remains by Art. 243 always the same as the particle moves. The total head at P is given by a similar equation, and the difference 5h of the two is found by differentiation. Thus ,7 . t dp v .dv 0/1 = 02 H (-- . iv g Now dz is the elevation of P above Q, that is, where is as shown in the figure the angle PQ makes with the vertical. But if we imagine a small cylinder described round PQ as an axis, and consider its equilibrium in the direction of the normal, it is clear that w v 2 op , a = . . a . PQ -w.a. PQ . cos , y P where a is the sectional area of the cylinder and p the radius of curvature of the stream lines at P or Q. Combining this with the preceding equation we get xh v * PO + v ' Sv V ' P Q( V ... Sv Ofl = . -r(/H = ~ + "757 gp 1100) ^=3c^(v-800). The resistance of an ogival-headed shot is of course much smaller ; it depends on the angle of the ogive, but in the elongated shot experimented on by BASHFORTH was about two-thirds that of a sphere. In shot of recent type it is no doubt still less. If the hinder part of the shot were elongated instead of flat the resistance would be greatly reduced : in bullets this idea has been carried out by making them tubular with ends fined off both in front and rear. At these high speeds the resistance is mainly due to sound waves, which by the aid of photography have been rendered visible in bullets. 628 NOTES AND ADDENDA. In a paper read at Chicago in 1893, and reproduced in the Philosophical Magazine for May 1894, Professor Langley shows that wind, however steady and uniform it apparently may be, is in fact a motion of an extremely complex character, each small portion of the fluid being in a state of pulsation. The velocity at any point of a wind current therefore goes through periodic changes of great magnitude, although the motion of a large body floating in the air may be perfectly uniform. It is believed that birds have the power of utilizing the internal energy corresponding to these periodic changes for the purpose of sustaining themselves, and even rising without visible movement of the wings. Hence it is, most probably, that the pressure on small areas exposed to wind is much greater than that just given for motion through still air. The value commonly accepted for the pressure per square foot of a wind of V miles per hour is F 2 /200. This corresponds to k = 2, being 50 per cent, greater than before. According to the best authorities on hydraulics, as stated in the text, there is a corresponding increase in the case of water, but it is difficult to say how far this is due to irregularity in the stream and how far to errors in the experiments. The pressure on a large area of 300 square feet has been shown by Sir B. BAKER to be only two-thirds that on a small area ; that is, it is about the same as for motion in still air. The relation between the pressures on an oblique and a normal surface, so far as is known, is the same in air as in water, but it must be remembered that the exposure of the surface will have an enormous influence. Thus, the pressure on a sloping roof will be much less if it rests on walls than if it is carried on pillars so that the air has free passage below. PAGE 537 [1900]. The outward flow turbine was introduced by FOURNEYRON about 1828, and its theory given by PONCELET in 1838. The inward flow THOMSON turbine followed some 20 years later. A wheel sometimes less than 2 feet diameter in these machines replaces a slow-moving cumbrous water-wheel, and may be made to yield a very considerable power. The efficiency of a turbine is in general not precisely a maximum when the conditions of no whirl at exit and no shock at entrance are satisfied. To explain this and some other points of more actual importance in the working of turbines, an approximate investigation of their efficiency is now added. The losses considered are (1) the kinetic energy of flow on discharge from the wheel together with the loss by friction of pipes and passages, and (2) the kinetic energy of whirl on discharge together with the loss by "shock" on entrance. For simplicity Case I. of the text is taken in which the vanes are radial at entrance and the angles of the guide-blades and vanes so proportioned that, at a certain speed, the two conditions of no whirl and no shock are simultaneously satisfied. The notation is that of the text with a slight modi- fication to be mentioned presently, and in addition the ratio r 2 /r a is denoted by m. The kinetic energy of flow is u z /2y, and, as the velocity through the pipes and the passages of the wheel is proportional to u, in a turbine of this type where all the passages are always completely filled with water, the loss by surface friction will be F. u 2 /2g where F as usual is a suitable co-efficient of resistance. The losses (1) are therefore ( 1 + F)u 2 /2(j. Now u - v 1 cot 6 1 and it is convenient to express u in terms of v^. These losses then become pv-fftg where is a co-efficient which may be supposed known. HYDRAULICS AND HYDRAULIC MACHINES. 629 The whirl at exit is v 2 = F 2 - u . cot 6. 2 = m( V l - vj, while the loss by shock at entrance is ( V\ - v^jlg, the losses (2) are therefore (1 + m z )( FI- Vi) 2 /2#, and the total loss of head (h f ) is given by the equation 2gh' = ^ + ( 1 + m z ) ( V - v) 2 , in which formula the suffix 1 is suppressed, being no longer necessary. For the useful work we have gh= V lVl - V 2 v z = V& the suffix 1 being suppressed as before. Adding together the losses and the work we obtain the energy exerted repre- sented by the fall or actual head H, so that The case considered in the text is that in which v=V. This gives 2 h 2 Hence the efficiency at this speed is 2/(2 + /3), and as this may be taken at 80 per cent, it follows that the value of (3 in well-designed turbines is about *5. At other speeds H remains the same, while v and V vary, and by differentiation CLV W 2 - 1 As already remarked v is proportional to u the velocity of flow, and therefore to the delivery. Hence in an outward flow turbine the delivery increases with the speed, and in an inward flow, for which m is fractional, it diminishes. This important conclusion might have been foreseen ; for reasoning as in the case of the reaction wheel (p. 531) the virtual head at discharge is increased in the outward flow and diminished in the inward flow, by the " head due to centri- fugal force" (compare page 546) a quantity which increases rapidly with the speed. The influence on the working of the machine is most important, for, in consequence, the outward flow turbine is in practice unstable and the inward flow stable ; and hence the value of m, which in a Fourneyron turbine should not exceed 1"25, may with advantage in a Thomson turbine be reduced to '5. (See Unwin, "Hydraulics," EncycL Britannica, page 529.) Next let us see how the efficiency varies. Differentiating the equation for A,, and then putting v= V which on substitution for dv/d V gives dh_ /3(7tt 2 -l) 9 ~ 2 The speed for which v= Fis that at which there is no shock at entrance and no whirl at exit ; and we see that this is not the speed of maximum efficiency ; in an outward flow turbine dh/dv is then negative and a lower speed is better, while in an inward flow it is positive, and the best speed is higher. The reason is that reduction of flow by a change of speed reduces the losses (1), thus more than compensating for the introduction of losses (2). If, however, Ah be the 630 NOTES AND ADDENDA. change in h consequent on a change A V in speed from the value \/gk which gives v= V _ h ~ ~ V ' "r+m a + j8' from which it appears that the change of efficiency is only a small fraction of the change of speed. Thus to reduce the efficiency from 80 to 79 per cent, or to increase it to 81 per cent, a change of at least 7 per cent, in the speed is necessary. This agrees well with common experience, since it is well known that a change of 10 per cent, in the speed has little influence on the efficiency. We might now proceed further and determine the actual speed of maximum efficiency from the equations just given, but the result would be of no practical importance, as the frictional resistances are not, and perhaps cannot be, deter- mined with sufficient accuracy. It is, however, probably safe to infer that the best speed of an inward flow turbine is decidedly greater than that of an outward flow. The external resistances referred to on page 535 have the effect of lowering the speed of maximum efficiency. The useful work (U) is wQh, and therefore proportional to the product vh. From the values of dv/d V and dh/d V given above it is easy to derive the value of A(vA), and hence at the speed for which vV, _- U -I 1 P) y The supply of available energy therefore increases with the speed in an outward, and diminishes in an inward, flow turbine ; unless by obstruction of the supply the efficiency becomes less than two-thirds when the converse will hold good. But the rate of increase is small, and varies greatly at different speeds, while, for the reason stated above, it cannot be determined accurately. A similar calculation may be made in the more general case (page 537) in which the vanes are not radial at entrance but inclined at a given angle. The results are similar though more complex, and may be expressed by replacing the con- stants j8 and m of the formulae by other values dependent on this angle. PAGE 538. The undisturbed motion of a perfect liquid within fixed boundaries is always reversible, that is, if every particle of liquid were imagined to be set in motion with the same velocity in the reverse direction, the motion would continue undisturbed. But if water be set in motion from rest this will generally not be the case. If, for example, we imagine a pipe connected with a tank by a mouthpiece in the form of the vena contracta, then, when water flows out of the tank, it will issue in a continuous stream with small loss of head ; but if the motion be reversed most of the energy of motion of the water in the pipe will be wasted in the internal motions soon after entering the tank. The loss is not unavoidable, as will be seen on reference to the case of a trumpet-shaped pipe (Fig. 175, page 477), but may be rendered small by enlarging the pipe very gradually. PAGE 540 [1895]. The second of the two approximate calculations of the efficiency of a centrifugal pump has been added to this article in the present (1895) edition, for the purpose of showing that when the chamber is properly designed radial vanes are not necessarily less efficient, and may be more efficient than curved back vanes. But it must not be supposed that the form is actually of little importance. The investigation is based on the supposition that the only loss during the passage through the fan is due to surface friction, as ELASTIC FLUIDS. 631 "would be the case if the motion were steady and continuous. But it is by no means certain that steady continuous flow is possible under the circum- stances, and if breaking up occurs the value of ft might be greatly increased, and might depend on the speed. In a centrifugal pump with high lift the changes of velocity imposed on the water during the passage through the fan are enormously great and rapid, and the form of vane may be of great importance in facilitating or otherwise the tendency to break up. On page 536 it has already been suggested that it might be advantageous to curve the vanes, so that IT should change (in this case increase) uniformly from entrance to exit. It is only by systematic experiment that the best form and number of vanes could possibly be determined ; any reasoning on this point must be very un- certain. The equation given in Art. 278, page 546, gives the stead}' flow through a pipe attached to a rotating casing, but it does not necessarily follow that steady flow is possible in a radiating current. In recent designs of fans for blowing air the vanes are curved forwards instead of backwards, and it is quite conceivable that the tendency to break up may be diminished in this way. The RATEAU fan described in the Engineer for May 24th, 1895, is an example. It is said to give excellent results. VI. ELASTIC FLUIDS. PAGE 562 [1900], Some important experiments by Professor Osborne Reynolds and Mr. Moorby on the mechanical equivalent of heat are described by them in the Philosophical Transactions for 1897. The result obtained, omitting decimals, is 777 ft.-lbs., the unit of heat being such that the mean specific heat of water between 32 and 180 F. is unity. It is probable that this number will be generally accepted as the true value, but for the purposes of this work it has not been thought necessary to alter the provisional value given in the text. PAGE 566 [1895]. If Aft be a small quantity of heat supplied at temperature T when raising the temperature of a Ib. of water, the mechanical value of that heat as explained in the text is &h(T - TJ/T, and the total mechanical value of the whole heat supplied in raising the temperature of the feed water from T to 7\ is T If the specific heat of water is taken as unity this becomes which is a formula very commonly used. The result is too small, because the specific heat of water increases as the temperature rises. If we adopt an approxi- mation suggested by RANKINE we may take and on substitution the formula given in the text is obtained. The result is greater than before, and the error of the approximation partially compensates for the neglect of the excess specific heat of water. The formula in the text may therefore be preferred unless special tables of the "entropy "of water are available. 632 NOTES AND ADDENDA. The second or mechanical formula for the available heat of steam M=P 9 V 9 .lQfr?i Po given in the text is based on the fact explained in the author's work on the Steam Engine that the saturation curve is approximately midway between an hyperbola and an adiabatic curve starting from the same initial pressure. If then an hyper- bola is traced starting from the lotcer pressure p Q on the saturation curve the area of the hyperbola must be very approximately the same as that of the adiabatic curve. The result given by this formula is too small : the deficiency increasing as the pressure-ratio increases ; but the error does not exceed 2 per cent. For pressure-ratio less than 5 it is insensible. PAGE 575 [1892]. The facts relating to the transmission of energy by com- pressed air are much better known now than when this book originally appeared. The remarks made on the subject have therefore been re-written and amplified. PAGE 578. If the fluid be supposed at rest, and elevation be taken into- account, we obtain KpT+z = Const,, or 3'5 PF + z = Const. This gives the distribution of pressure and temperature of the atmosphere for "convective equilibrium" (CLEEK MAXWELL'S Theory of Heat, 1st edition, p. 301). Energy is then uniformly distributed. PAGE 579 [1900]. Until recently little of any importance had been added to the ordinary theory of efflux described in the text, the sketch given in Art. 300 remaining unaltered since it was originally written fin 1884. Since 1895 the subject has attracted some attention, and it has been pointed out that the velocity through the section of maximum density when the critical ratio given by the table on page 580 is over-passed is the velocity of sound in the issuing fluid at the temperature of outflow. This observation, originally, it is believed, due to M. Hugoniot, can be easily verified by means of the formulae given in the text. More recently experiments by M. Parentey, Mr. W. Rosenhain, and others have raised many points of interest. Rosenhain's experiments, described in a paper published in a recent volume (vol. 140, p. 99) of the Proceedings of the Institution of Civil Engineers, are chiefly devoted to the comparison of different types of orifice. They show that, as might have been expected, in trumpet-shaped nozzles the expansion down to atmospheric pressure takes place within the nozzle. In forms of orifice where this is not the case the expansion is extremely sudden and the jet subsequently is for some distance cylindrical. The discharge is greater than the theoretical maximum, as calculated on the supposition of dry steam and adiabatic flow ; the excess in trumpet-shaped nozzles with rounded inlet being apparently about 14 per cent. How far this excess can be accounted for by the water mixed with the steam or by the error of the hypothesis of adiabatic flow is uncertain. PAGE 582. This formula for the flow of air in a long pipe was given by UN WIN (Min. Proc. Inst. C. E., Vol. XL1IL), and somewhat earlier by GRASHOF. It is a question of considerable practical interest. By comparison with experi- ment it has been shown that the co-efficient is given by a formula of the same form (DARCY'S), as in the flow of water through pipes, an important verification of theoretical principles. The equation for the case where the temperature varies can be obtained without difficulty, but has not as yet been practically applied. EESISTANCE AND PROPULSION OF SHIPS. 633' VII. RESISTANCE AND PROPULSION OF SHIPS. [1892.] The importance of this subject is so great that though outside the intended limits of this work some information relating to it may be useful. The brief summary here given of the leading facts relating to it would, however, require- expansion into two or three long chapters if anything like a full statement w r ere attempted. Submerged Bodies. If a uniform current be flowing through a straight pipe or cylindrical casing of indefinite length, which it completely fills, and a solid of any size or shape be fixed within it, the particles of water after passing the solid return ultimately to the original straight line paths in which they moved before reaching the solid, unless the current be disturbed by the causes discussed at length in Ch. XIX. of this book. Each particle, after passing, has ultimately the same velocity and pressure as it had before reaching the solid, no permanent change being possible except such as may be produced (1) by viscosity (page 479) or (2) by eddies due to surface friction or other causes (pages 501-7). Hence it follows that the longitudinal resultant pressure upon the solid must be zero- The grounds on which these statements are made, the qualifications to which- they are subject, are discussed on page 626. If the water in the casing be at rest and the body move uniformly through it in a straight line parallel to the sides, the relative motion of solid and water is the same as when the solid is at rest and the water moves. We therefore conclude that the water will offer no resistance to the motion except such as may be due to- hydraulic losses. And as the casing may be supposed of any size we please, this conclusion must be tiue for any case where a body is sufficiently deeply submerged. In bodies of very small size, such as particles of a finely divided solid, the direct action of viscosity is the principal cause of resistance, but in bodies of the size of a ship or even of a model of a ship the direct action of viscosity is so small as to be negligible, and the resistance of a submerged solid is therefore practically due to exactly the same causes as produce loss of head in a pipe or passage. Eddy resistance has already been discussed (page 502). On examination of Fig. 191 it will be seen that eddies are not formed immediately in front but behind the- corners and in the rear. A solid, therefore, may be blunt ended in front without giving rise to eddy resistance, provided the shoulders are rounded off sufficiently, and a tail of sufficient length be attached behind. Eddy resistance may thus be reduced to a very small amount, and surface friction then becomes by far the most important cause of resistance in a deeply submerged body moving uniformly in a straight line. Surface Friction has been discussed on page 485, and a table given of FROTJDE'S results, which is directly applicable when the surface is plain. When the surface is curved the question is in principle much more complex, because the water glides over the different parts of the surface with a different velocity. Let q be the ratio which the velocity of gliding over a small area 5S, bears to the speed of the solid (V), then /. q*V5S is the energy dissipated by surface friction per second, and /. V 2 I q*dS is the corresponding resistance, which is the same as that of a plane the area of which is 634 NOTES AND ADDENDA. If the velocity of gliding was not altered by friction the co-efficient q would be on the average greater than unity. In short [surfaces therefore S 1 is greater than the actual area of wetted surface, and is described as the Augmented Surface. The idea of an augmented surface is due to RANKING, who based upon it a well-known formula for the resistance of ships. It has long been recognized that this formula is not of practical value, and the reason for its failure is simply that the velocity of gliding over a surface of any considerable length is so much disturbed by friction as to be far less than the calculation value, and the friction is correspondingly reduced. Calculations of the surface friction of vessels are therefore made as if the surface were plain, due regard being had to the length and nature of the surface in estimating the probable value of the co-efficient. The area of wetted surface is calculated from the drawings of the vessel, but as the calculation is complex, and throws no light on the relation between the wetted surface, the displacement, and the draught of water, the formula may be used in which A is the displacement in salt water in tons, L the length, D the draught of water, both in feet. This formula in most cases gives a very fair approximation to the surface of the bare hull, as has recently been shown by Mr. Archibald Denny if the co-efficient 2 be replaced by 1'7. In this note, however, it will be used without this modification and then includes a certain margin for bilge keels, or similar appendages. If J3BDL be the cubic dis- placement where B is the beam and j8 a co-efficient of fineness the formula becomes Law of Comparison. When a careful estimate of the surface friction and eddy resistance of a solid is compared with the actual resistance the solid offers to uniform motion in a straight line, it will be found that in a submerged body of fair form the two are nearly the same, and this is also true for a vessel at low speeds. But in a floating body the difference at high speeds is very con- siderable, and increases rapidly with the speed. This difference is described as the Residuary Resistance of the vessel, and is mainly due to the formation of waves at the surface of the water, a cause which would operate even if there were no hydraulic resistances of any kind. It is no longer true as in a submerged body that the resultant pressure on the body in the direction of motion is zero. Hence so far as independent of hydraulic resistance the residuary resistance must be subject to the law of comparison stated in Art. 247 (page 483) of this book, so that in similar vessels at corresponding speeds under similar circum- stances the residuary resistance must be in the proportion of their displacements. The most convenient way of expressing this principle, is by taking F-c.VZ where V is the speed in knots, L the length in feet, and c a co-efficient of speed. The law of comparison may now be expressed by saying that the residuary resistance when expressed in Pounds per Ton of displacement must be a function of c the speed co-efficient, which function must be the same in similar vessels. As a general principle in hydrodynamics this law of comparison had long been known, but FROUDE made it his own in its application to vessels by showing (1) that it was not applicable to surface friction, the resistance due to which is much more important on a small scale than the law would imply, and (2) RESISTANCE AND PROPULSION OF SHIPS. 635 that it was applicable to the residuary resistance. The experimental verification consists partly in the famous experiments made on the Greyhound and her model, and partly in the fact that it is now possible to predict the power required to propel an entirely new type of vessel by means of experiments made on a model of the vessel, a method systematically employed by FROUDE'S successors. It must be distinctly understood that no particular law of speed is implied, but only a relation connecting the law of size with the law of speed. To illustrate this point, suppose the resistance to vary as the fourth power of the speed, then by the law of comparison the resistance is Ji = k 1 . A.c 4 where , is a numerical co-efficient. Replacing c by its value V/\ f L, and remembering that A must be proportional to the cube of the linear dimension R = k.l. V 4 where k is another co-efficient depending on the particular linear dimension (I) chosen which may be the length, beam, draught of water, or any linear combina- tion of these quantities. If then the resistance vary at V 4 it must also vary in direct proportion to the linear dimensions of the vessel. Similarly, if the resistance vary as the square of the speed, it must also vary as the square of the linear dimension, that is, as the transverse section, and consequently eddy resistance pure and simple (page 503) satisfies the law. On the other hand surface friction does not satisfy it, for, referring to pages 495, 496, it will be seen that the general form of the formula is when //, the co-efficient is taken from FROUDE'S table for fresh water, in which the standard speed is 600 feet per minute, or approximately 6 knots. This may be written using the second formula for S where/ the co-efficient is not constant as it should be if the law of comparison applied, but is given by Wave Resistance. Waves produced on the surface of* water by the action of a body moving through it are of two distinct kinds. The first is a solitary shallow water wave generated in front of a barge moving in a narrow canal. In such a wave the particles of water are lifted up, carried forward along with the wave through a short distance, and then set down at rest, while the wave travelling onward leaves them behind. A wave of this class is described as a Wave of Translation ; it possesses a certain definite amount of energy, which is transmitted with it from particle to particle as it moves, and hence when of any size it travels for great distances without external agency when once created. Such a wave is consequently only a cause of resistance to the generating body while it is being formed. The second class occur in series, and the particles of water oscillate backwards and forwards : the translation along with the wave is relatively small and for most purposes may be neglected : they are therefore described as Oscillating Waves. In a purely oscillating wave the particles 636 NOTES AND ADDENDA. describe closed curves resembling an ellipse which becomes a circle in deep- water. The speed of an oscillating wave in deep water depends on its length only, the speed in knots of a wave of length X feet being F 2 =1'8X. This formula shows that the length X of a wave travelling at the same speed in< the same direction as a vessel of length L is where c is the speed co-efficient. An oscillating wave possesses both kinetic and potential energy in nearly equal amounts, but as was first pointed out by OSBOKNE REYNOLDS, the kinetic energy is not transmitted along with the wave but remains behind, and therefore when such a wave travels onwards into still water its height necessarily diminishes unless it is kept up by external agency, such as a moving body which supplies it with energy. Waves then are a cause of resistance, not only when a new wave is continually being created as the vessel moves, but also when waves already existing are kept up to their full height. The energy of a complete wave is proportional to its length and the square of its height; and of this a certain definite fraction has to be supplied by the vessel as it moves through a wave length. Hence the resistance due to a wave system of a given type varies as the square of the height. If the type remained the same at different speeds the height of the waves would vary as the square of the speed, and the corresponding wave resistance as the fourth power of the speed, a law of resistance already mentioned. A third class of waves not necessary here to consider are the "capillary" waves, so called because their motion is in great measure governed by capillary action, that is, by what is known to physicists as "surface tension." They are of very minute size, and are also known as "ripples." The wave resist- ance of a model vessel would be affected by surface tension if the model was small enough. No effect of this kind appears to have been noticed at present. Interference. If the residuary resistance of a vessel as determined by a set of speed trials upon a model be divided by the fourth power of the speed the quotients are in general neither constant nor continuously increasing or diminishing. On the contrary they show very distinctly a periodic change, being alternately greater and less than a certain mean value. The cause of this remarkable result was conclusively shown by FROUDE to be the interference of two distinct wave systems one created at the bow, the other at the stern of the vessel. The experimental demonstration of this consisted in comparing the residuary resistances at a given speed of a set of models, the fore body and after body of which were the same in all, but which had different lengths of middle body. If for simplicity we suppose that the bow and stern generate simple waves of heights h-Ji z at points distant s from each other, measured along the side of the vessel, the result of the combination by a principle well known in physical science will be a simple wave of height h given by the formula &3 = ^2 + h * + 2^ . C os 27T -, X X being the wave length which is the same for all, being connected as before- explained with the speed of the vessel. Hence when s is changed by varying the RESISTANCE AND PROPULSION OF SHIPS. 637 length of middle body the residuary resistance suffers a. periodic change, and this conclusion was exactly verified by the experiment. When a given model is tried at various speeds * remains nearly the same, but X as well as hfi* varies as the square of the speed, the formula then shows that the resistance, while increasing rapidly on the whole, suffers a periodic change whereby the rate of increase is alternately excessive and moderate. The fact that the residuary resistance is a periodic function of the speed is shown graphically by the " humps" and "hollows" which are found in curves of resistance, and is mainly accounted for by interference. But the interference is of a more complex kind than in the simple case supposed, and it would not be safe to conclude that interference is the sole cause of variation in type of the wave system especially at certain critical speeds. The form of water surface has been investigated by LORD KELVIN (Sir W. Thomson), but a rational formula for wave resistance is probably unattainable, varying as it must according to the lines of the vessel. Approximate Formula. From what has been said it appears that omitting (1) the resistance of the air, (2) the resistance of various appendages to the vessel her- self and her propelling apparatus, an item which may be considerable, and which must be separately estimated the resistance of a vessel in Pounds per Ton is given by the general formula r = ac 2 + xc 4 , where the first term gives the surface friction in terms of a co-efficient , which can be calculated with a fair degree of approximation, while the second term gives the wave resistance in terms of x, a periodic function of c. The character of the resistance will depend on the value of c, the second term being relatively small at low speeds. The values of c which occur in actual vessels may be grouped as follows : (1.) In steamers employed for the transport of merchandise c ranges from '5 to '7, and by the formula already given the length of waves travelling at the same speed as the vessel rarely exceeds one-fourth the length of the vessel and is usually much less. The wave resistance is in this case one-fourth or less of the whole, and the single constant formula R=K .A3 .F 2 may conveniently be employed. The value of K for displacements in tons and speeds in knots ranges from "55 to '66 in full-sized sea-going vessels, excluding any resistance due to the nature or condition of the wetted surface and to (2.) In recent ironclads and in mail steamers the value of c ranges from "7 to '95, and the length of waves travelling at the same speed as the vessel increases to nearly one-half her length. The wave resistance now becomes nearly one- half the whole, and the term representing it cannot be merged into the term representing the surface friction. Already, before FROUDE'S researches, this had been recognized, and formulae had been given showing that the. resist- ance of a ship increased faster than the square of the speed. By far the most important of these is the formula given by BOCRGOIS in a treatise referred to further on. The first two terms of this formula represent surface friction and eddy resistance which may be better effected in the way already explained. The physical mean- ing of the third term was only partially understood by BOURGOIS, but it is now 638 NOTES AND ADDENDA. evident that it amounts to taking an average value of the periodic function x and assuming that the beam B of the vessel is the linear dimension which is principally effective in the production of wave resistance. On substituting we find the average value of x to be where b is a numerical co-efficient. The value of k employed by BOURGOIS was -14 in French units for all cases except where the beam is as much as one-fourth the length, when it is increased to '16. The value of 6, which corresponds to k= '14, is '23, and in some types gives good results, but it may be doubted whether it applies so universally as BOURGOIS assumed. The values of both x and a necessarily depend on the lines of the vessel, so that no fixed relation can exist between the two, but the author has found that the formula gives good results in a great variety of types, though in heavy ironclads the number 8 should be reduced to 7, or in some cases perhaps still further. The same restrictions must be understood as in the preceding case. The resistance of vessels of small draught of water is much greater, and may be approximately estimated by the formulae already given. The resistance of the air and of append- ages may be included by a suitable addition to the constant a. (3.) In cruisers and torpedo gunboats, by the use of engine power amounting to from 2 to 5 H.P. per ton, values of c are obtained exceeding unity, sometimes even reaching 1 '4, the waves are now about the same length as the vessel, and at this critical point the periodic variation of x is so great that no formula is of much value. (4.) Beyond this speed no full-sized vessel can be propelled from the impossi- bility of putting sufficient engine power on board, but in torpedo boats a power of 15 H.P. to the ton can be employed, and we find values of c ranging from 1'8 to 2*3. The character of the wave resistance has now altogether changed, as might be expected since the waves are now two or three times the size of the vessel. It increases much more slowly, probably nearly as the square of the speed. The total resistance of a torpedo boat appears to be about 30c 2 pounds per ton. Effective Horse Power. From the formulae for the resistance of a vessel we may immediately deduce formulae for the effective horse power required to propel her. The first of these gives the absolute power A-TP where C is a constant at low speeds, which under the restrictions already men- tioned may be taken as 500 or 600. If applied to high speeds the value of C is much reduced as it diminishes rapidly with the speed. The second gives the effective horse power per ton e.h.p.^-^-V, where the constant C will, subject to the remarks already made, usually be from 40 to 45, but may sometimes be increased to 50 or even more. It must be distinctly understood, however, that as formulae of this class take no account of the periodic variation of the resistance indicated by the " humps and hollows " of the resistance curve, no certain and close estimate of the E.H.P. KESISTANCE AND PROPULSION OF SHIPS. can be made except by the method of comparison. If a full-sized vessel of the same type exists, of which the E.H.P. is known, the principle may be applied with- out much error to the total resistance ; but if the type be new, a model must be tried and the principle employed to determine the residuary resistance alone : the surface friction, being relatively much greater in the model, as already explained must be separately calculated. Propellers in general. Let us next consider briefly the means by which the vessel is propelled through the water. Every propeller operates by driving astern some or all of the water passing through it, the reaction of which furnishes a propelling force equal and opposite to the "thrust" of the propeller. Since the resistance is directly astern, the velocity impressed on the water must be sternward as far as it is of any utility for the purpose of driving the vessel. Some forms of propeller as, for example, the screw give lateral motions to the water, but the energy thus employed is wasted. An ideally perfect propeller, then, impresses upon every particle of water passing through it a reaction astern which for simplicity we may suppose the same for all. This water may for the present be supposed to be initially at rest, and therefore to be passing the ship with the velocity V, which of course is the speed of the ship. After passing through the propeller this velocity is increased to v t v-V being the absolute velocity of the current or "race" produced by its action. For convenience we write v-V=), and A Q the area occupied before reaching the screw may therefore be taken as about irZ) 2 /4, while Q = A V will be the quantity of water acted on. Assuming as before cr V as the change of velocity produced on passing through the screw and applying the roughly approximate formula previouslv given, BD B C.M. 2 S 642 NOTES AND ADDENDA. Assuming D= "4B, this gives for the slip-ratios 1 1 * = n : ' = i2' corresponding to a slip of 8 per cent. This calculation is of interest as giving a theoretical minimum value for the slip of a screw ; the actual average must be much greater, because the whole of the water passing through the screw disc is not moved astern, and the other assump- tions made are all of a nature to reduce the calculated result. Efficiency of Screws. Though a screw, like every other propeller, operates by impressing a sternward velocity upon the water, yet the manner in which it does this is so entirely different from the action of a paddle that it is desirable to con- sider the question from a different point of view. Imagine a tube of uniform transverse section to be formed into a cylindrical spiral of uniform pitch z , and let the axis of the spiral be a shaft projecting astern exactly parallel to the direc- tion of motion of the vessel. The tube being fixed to the shaft rotates with it at N revolutions per second where F=A%. Neglecting the disturbance of the water by the passage of the vessel, the effect of this arrangement is that the spiral tube screws its way through the water without disturbing it in any way in the absence of friction, the velocity U with which the water moves through the tube being V . cosec a where a is the pitch angle. Under these circumstances the tube has no propelling effect ; but now, suppose that a portion of the tube is taken and its curvature altered so that the pitch, while remaining equal to z at the end where the water enters, gradually increases to 2^ at the end where the water issiies, the radius of the spiral being unchanged. The effect of this is that the stream flowing through the tube, while retaining the same mean velocity and pressure, has its direction altered by the small angle 0, by which the pitch angle at entrance differs from that at exit. By reasoning as on pages 489, 519, it is now easy to find the resultant action upon the tube of the water inside which will be given by the formula P = a . Q . U(f> = aSU 2 (f>, in which S is the sectional area of the tube and a a co-efficient which might be exactly calculated. The reasoning here given may be compared with that in Art. 260, p. 505, in which an equivalent result is arrived at. It was pointed out by FROUDE that a screw blade might be considered as a body moving nearly edgewise through the water, the small angle of obliquity (depend- ing as it does on the slip) being described as the " slip angle." This small angle is different at each point of the blade, but does not exceed 10 in practical cases. A particle of water in contact with the blade traces out upon it a spiral curve, and each of the spiral elements into which the blade may be thus divided behaves nearly as the spiral tube just described, deflecting through the small angle a stream the breadth and therefore the sectional area of which is proportional to the length of the element. Hence the propelling reaction is a force P normal to the blade given by the same formula as in the tube, S being now the area of the element. In ideal cases the co-efficient a can be calculated, but in an actual screw blade must be determined experimentally. In addition to the normal force there will be a tangential force fSU 2 due to friction, / being a co-efficient much less than a. Calling this F and v///a, 7, F = L = t P CHf) ' EESISTANOE AND PROPULSION OF SHIPS. 643 thus for a given slip angle the ratio F/P is given just as in the case of the friction of a solid screw in its nut, discussed on page 241. Introducing a " friction angle " which we may now call 0' to distinguish it from and proceeding as in the article cited it will be found that tan {a - d>\ Efficiency = -&, tan {a + 0'} In determining the maximum value of this we must remember that and are independent, but that 0' is connected with 0. When 0' is small 00' = y-. Hence the maximum efficiency is when = 0' = 7 and a = 45, the value being /I -y\ 2 Maximum Efficiency = ( = - J Thus the friction and efficiency of screw propellers as determined by this calcula- tion, which is due to FROUDE, are governed by laws closely analogous to those which govern an ordinary screw and its nut. The value ascribed to the co-efficient 7 for a simple element by FROUDE was 0685, corresponding to a maximum efficiency of 76 per cent, at a slip of about 13 per cent. To make a similar calculation applicable exactly to an ordinary screw blade it would be necessary to suppose that 7 had the same, or at any rate some known, value for all elements of the blade, but although quantitative results are unattainable, the principle of the calculation is undoubtedly correct. There must always be a slip of maximum efficiency which cannot be very small, and at small slips the waste by friction is enormously great. Experiment on model screws bears out this conclusion. Such experiments have been systematically made by Mr. R. E. Froude and others in great numbers, with the result of showing that in good examples the efficiency varies little at slips between 15 and 30 per cent., being then about '66 rising to nearly 70 per cent, at a slip of about 20 per cent. Disturbed Water. The conclusions we have arrived at appear at first sight con- trary to experience, for we know that the slip of screw propellers is commonly less than 15 per cent., and often less than the theoretical minimum of 8& per cent, obtained above. The reason of this is that the screw works in water which is not at rest, but travels onwards along with the ship with a mean velocity u, which probably often reaches 10 per cent, of the speed of the ship. Hence the water enters the screw not with velocity V, but with velocity V- u, and the real slip is correspondingly increased, being probably seldom less than 20 per cent, in good examples. The effect on the efficiency of the screw is complicated. In the first place, the useful work done in propelling the ship is greater for the same real slip, and therefore for the same turning moment and speed, so that prima facie the efficiency is increased. But on the other hand, a screw of ordinary dimensions sucks more water through it than would naturally flow there, an action which augments the resistance of the ship unless the screw is placed further astern than is possible for constructive reasons. In a screw with many blades of considerable length there would be little if any suction, but too great blade area is a cause of great loss by friction. FROUDE stated that the augmentation was often as much as 40 or 50 per cent., but it was afterwards explained by his son that this estimate included the resistance of thick square stern posts and appendages to the pro- peller, the augmentation proper varying from 8 to 18 per cent. The lower value applies to twin screws and vessels with fine lines. Experiment appears to show that in models the loss by augmentation on the average about compensates for the 644 NOTES AND ADDENDA. direct gain by working in disturbed water, the efficiency of a model screw being about the same when a corresponding vessel is run ahead of it as when the vessel is removed. The best results are doubtless obtained by an exact adaptation of the dimen- sions, number, and form of the screw blades to the type of vessel. At present such adaptation can only be effected by the principle of comparison from some example known to give good results. The method is fully explained by Mr. Sydney Barnaby in his work on Marine Propellers. Indicated Power. From what has been said it appears that the power required to drive a propeller will be (1+e^E.H.P. where e-^ is a fraction, which in the best examples of jet, paddle, or screw will seldom be much less than '5. This addition of 50 per cent, to the effective power is due to waste of energy in giving various motions to the water acted on by the propeller, including the production of eddies, by surface friction of blades and otherwise. To obtain the indicated power we must now consider the friction of the engines and other resistances, such as air-pumps, feed-pumps, and the like. These consist (as described on page 258) of two parts, a variable part proportional to the mean effective pressure, and a constant part most conveniently expressed as a fraction of the effective pressure at full speed. Thus the formula /. H. P. =( 1 + ei + e,) . E. H. P.+e s . E^H^ . -^ gives with sufficient accuracy for the present purpose the indicated power at the given speed of vessel and r solutions (JV) of the engines in a set of speed trials, where e. 2 , e 3 are fractions i the suffix 1 refers to full speed. The counter- efficiency at full speed is e-^ + e^ + e.^, which in screw propulsion in the best examples is about 1'8, and aking e l as '5 we find e. 2 + e 3 = '3, of which at least one- half is due to the constant .friction. At lower speeds the efficiency of propulsion is much less, because the el -t of the constant friction is relatively great. The ratio Af F 3 //. H. P. i: .escribed as the "displacement constant." It has long been known that it is i )t the same at different speeds in a set of progressive speed trials, but that it has a maximum value at a certain speed (about c= *7 in full sized vessels), diminifr ng considerably both at high speeds and at low speeds. The explanation of this is sufficiently clear from what has been said. It can, however, be used as a means of comparison if care be taken to compare only vessels at corresponding speeds with engines working at the same fraction of their full power. REFERENCES. A full account of the earlier experiments on the resistance of ships, including all that had been done on the subject before the time of FROUDE, will be found in a treatise published in 1857 entitled Memoire sur la Resistance de VEau, by Captain (afterwards Admiral) Bourgois of the French Navy. FROUDE'S researches are contained in two Reports on Surface Friction presented to the British Association in 1874, and in various papers for the most part pub- lished in the Transactions of the Institution of Naval Architects. It is to be hoped these papers will be published in a collected form. Sir W. H. White's well-known work on Naval Architecture will also be found invaluable. APPENDIX. B. ORGANIZATION OF THE CLASSES IN ENGINEERING AND NAVAL ARCHITECTURE IN THE ROYAL NAVAL COLLEGE. A SCHOOL of naval architecture was founded in Portsmouth dockyard so long ago as 1810, but, after existing for more than twenty years, was abolished in 1833. In 1844 it was re-established, but only to be once more abolished in 1853. In 1860 the Institution of Naval Architects was founded, and by its influence a third school was commenced under the direction of the Science and Art Department at South Kensington. For particulars respecting the two earlier of these schools the reader is referred to a paper by SCOTT RUSSELL in ^bhe Transactions I.X.A. for 1863. The third was afterwards incorporated with the Royal Naval College, of which it now forms a department. This department is divided into two classes, of v ; iich the junior serves as the final stage in the training of the engineer officers o r '\e navy, the majority of whom spend nine months at Greenwich immediately on eu ring the service, after several years spent in the dockyard. (See p. 647.) The semor is an advanced class, con- sisting partly of a small number of engineer officers Delected by competition from the preceding, and partly of students in naval rr" cecture originally selected by competition from the dockyard apprentices to j< \ the junior class. The full course in the advanced class lasts three years, of wtflch one is spent in the junior, and two in the senior class. There are also private students who generally go through the full course. The programme of these cJ'sses differs in some important respects from that of most other technical colleges, and it may be useful to describe it briefly here. The three principal branches of study are : I. Pure and Applied Mathematics ; II. Applied Mechanics ; jjy /Naval Architecture, I Marine Engineering ; to each of which the time allotted is about the same. In addition, there is a course in Physics and Chemistry. The mathematical course includes the theory of electricity, while the technical applications to electric lighting and torpedo work are included in the laboratory course on physics. The following remarks will be confined to the second and third of the principal subjects. In APPLIED MECHANICS the subjects are A. Elementary Subjects. j> /Stability and Oscillation of Ships. I. Theory of the Steam Engine. C. Wave motion. Resistance and Propulsion of Ships. 646 APPENDIX. Subjects A. are pretty closely represented by the present treatise, but there are a few omissions and some additions, especially in graphical statics and elementary theory of the steam engine. The course lasts two years, each subject being com- menced in the junior class and completed in the senior. Subjects B. are commenced in the second year and completed in the third. The first is studied by students in naval architecture only, and the lectures on it are at present given by the Instructor in Naval Architecture. The second is studied by students in engineering only. Subjects C. occupy the greater part of the third year. In NAVAL ARCHITECTURE the course followed is very fully explained in a paper by Mr. (now Sir) W. H. White in the Transactions I.N.A. for 1877 (Vol. XVIII. , p. 361), and it need not therefore be further considered here. In MARINE ENGINEERING the course for the junior class occupies nine hours a week. Each of the principal parts of the marine engine, including the boiler and propeller, are taken in detail, the dimensions proper for that part determined, and the other practical questions considered which are involved in its design. An example is set, and the student is expected to work out a design from the data proposed, and to produce working drawings. About 30 of these drawings are pre- pared in the session, the subjects being : Details of principal parts of Engine Piston, Piston Rod, Connecting Rod, Cross-head and Guides, Thrust-block,. Crank-shaft, Cylinders and Fittings. Propeller Shafting and Couplings, Boss, Blades. Slide Valves Zeuner's Diagrams for Solid and Open Bar Links, Valve-ellipse, Construc- tion and setting of Slide Valves, Link Motion. Boilers Dimensions and Structural Details, Fittings. Condensers and Air Pumps Fittings and General Arrangement. The foregoing course is gone through by all students in engineering. Those who are selected to enter the advanced class devote eight hours a week on the average to the subject in two following sessions. In the second year detailed drawings are made of the parts, and three views of the general arrangements, of a set of marine engines of large power suitable to propel a given ship at a given speed. The drawings of the details and propeller are completed, and the general drawings pencilled. In the third year the boilers are designed, and drawings made showing the disposition of the pipes and auxiliary engines. The general drawings are completed, the whole design being represented by a set of about 20 drawings. The practical training of students both in naval architecture and in engineering takes place in the dockyards for a period of at least 4 years before entering the College and during the three summer months in which the College is closed. This is a point of great importance, for, quite irrespectively of the absolute necessity of such training for its own sake, no theoretical course can be thoroughly understood without some preliminary knowledge of a practical kind. A college workshop is a very imperfect substitute and occupies time which is better spent elsewhere. The author, however, must not be understood to depreciate the importance of a TECHNICAL EDUCATION. 647 " mechanical laboratory," provided with testing machines, hydraulic apparatus, steam engines, and the like, for the purpose of studying mechanics experimentally. Such a laboratory, when properly organized, is capable of rendering great service, but it in no way replaces training in a large workshop carried on for commercial purposes. Nor are these remarks intended to apply to the lower grades of technical education, in which the workshop to a great extent plays the part of a laboratory. In the author's opinion, much the same may be said as to the use of models in teaching mechanics. An engineer does not use models ; he employs drawings almost exclusively ; and so, in the instruction of professional students, models are of little value for descriptive purposes. Nor should they be used to demonstrate the laws of motion. But in explaining a mechanical principle, a model is some- times of service ; it plays the same part as the figure in a proposition of Euclid in aiding the conception of the learner. And, as before, in the lower grades of technical education, models may properly be used for demonstrative purposes, while, in the case of non-professional students, they are often indispensable for descriptive purposes. In the " steam " department of the Royal Naval College, organized for the purpose of imparting to the executive officers of the navy a knowledge of the mechanism and working of a marine engine, models are freely used in this way. On the subject of technical education in naval architecture, the reader is referred to two valuable papers by Mr. John and Mr. W. Denny in the Transactions of the Institution of Naval Architects, the first in Vol. XIX., p. 120 ; the second in Vol. XXII., page 144. [1892.] In the year 1877 a college for the training of engineer students was established first at Portsmouth, on board the Marlborough, afterwards at Keyham. In recent years a certain proportion of the engineer officers on entering the Navy have been sent afloat at once instead of passing through Greenwich, and the majority of the students of naval architecture have been selected for study at Greenwich from the Keyham students. A large part of the time at Keyham is spent in practical work in the dockyard and engine factory ; but of late a part of the instruction in applied mechanics and marine engineering has been carried out there, leaving more time for the development of the subjects at Greenwich. GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLKHOSE AND CO. LTD. 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. .oi-D ~~~ v . 0^ * A - fc. -. MM General Library , University of Calif ornra Berkeley LD 21A-50m-3,'62 (C7097slO)476B YC r/f THE, UNIVERSITY OF CALIFORNIA LIBRARY