:;< v 7 ; REESE LIBRARY UNIVERSITY OF CALIFORNIA. OF CALIFORNIA *y> THE RELATIVE PROPORTIONS OP THE STEAM-ENGINE : COURSE OF LECTURES ON THE STEAM-ENGINE. DELIVERED TO THE STUDENTS OF DYNAMICAL ENGINEERING IN THE UNIVERSITY OF PENNSYLVANIA. BY WILLIAM D. MARKS, WHITNEY PROFESSOR Of DYNAMICAL ENGINEERING. WITH NUMEROUS UKIVEESITT PHILADELPHIA J. B. LIPPINCOTT & CO. 1879. Entered according to Act of Congress, in the year 1878, by J. B. LIPPINCOTT A CO., In the Office of the Librarian of Congress, at Washington. LIPPINCOTT'S PRESS, Philadelphia. PREFACE. IT is a source of regret to the author of these Lectures that none of the distinguished writers upon Mechanics or the Steam-engine have undertaken to give, in a simple and practical form, rules and formulae for the determination of the relative proportions of the component parts of the steam-engine. The authors of the few wprks as yet published in the English language either entirely ignore the proportions of the steam-engine or content themselves with scanty and general rules Rankine excepted, who in the attempt to be brief is sometimes obscure, leaving many gaps in the im- mense field which he has attempted to cover. This defi- ciency in the literature of the steam-engine is remarkable, because the problem which the mechanical engineer is most frequently called upon to solve is the determination of the dimensions of its various parts. From time to time hand-books of the steam-engine have been published giving practical (?) rules, the result of obser- vation of successful construction ; and with these rules the practising engineer, who has little time for original investi- gation, has had to content himself. It is of course reasonable to limit the correctness of these rules to cases in which all 2063304 4 PREFACE. the conditions are the same, as in the case or cases from which these rules have been derived, thus placing a serious obstruction in the way of improvement or alteration of de- sign, and rendering the rules worse than useless even dan- gerous in many cases. " The usual resource of the merely practical man is pre- cedent, but the true way of benefiting by the experience of others is not by blindly following their practice, but by avoiding their errors, as well as extending and improving t what time and experience have proved successful. If one were asked, What is the difference between an engineer and a mere craftsman ? he would well reply that the one merely executes mechanically the designs of others, or copies some- thing which has been done before, without introducing any new application of scientific principles, while the other moulds matter into new forms suited for the special object to be attained, and lets his experience be guided and aided by theoretic knowledge, so as to arrange and proportion the various parts to the exact duty they are intended to fulfil. " ' For this is art's true indication, When skill is minister to thought, When types that are the mind's creation The hand to perfect form hath wrought.'" STOXEY'S Theoj-y of Strains. Zeuner, in his elegant Treatise on Valve Gears, transla- ted by M. Miiller, has laid the foundation for the treatment of slide-valve motions for all time, and in his Mechanische Warmeiheorie has carried the application of the mechan- PREFACE. 5 ical theory of heat to the steam-engine as far as the present state of the science of Thermo-dynamics will permit. Poncelet, in his Mecanique appliquee aux Machines, has most thoroughly treated some members of the steam-engine, neglecting others of as great practical importance, Hirn, in his Theorie mecanique de la Chaleur, gives us, besides a very able treatise on the science of Thermo- dynamics, a valuable series of experiments upon the steam- engine itself, confirming Joule's results. A translation of Der Constructeur, by F. Reuleaux, 'would, if made, add much to our knowledge of the proper proportions of the steam-engine, as well as of other ma- chines. A rational and practical method of determining the proper relative proportions of the steam-engine seems as yet to be a desideratum in the English literature of the steam-engine; and these Lectures have been written with that feeling, purposely omitting the consideration of such topics as have already in many cases been over-written, and considering only those which have not received the atten- tion which their importance demands. In the choice of a factor of safety a matter wherein opinions widely differ the author, guided by considerations set forth in Weyrauch's Structures of Iron and Steel, trans- lated by DuBois, has fixed upon 10 as being the most cor- rect value. If any of our readers should prefer a different factor, the formulae deduced will be correct if the actual steam-pressure per square inch is divided by 10 and multi- 1* 6 PREFACE. plied by the preferred factor of safety, and the result used in the place of the actual steam-pressure. In reducing all the required dimensions of parts of the steam-engine to functions of the boiler-pressure or mean steam-pressure in the cylinder per square inch, of the diam- eter of the steam-cylinder, length of stroke, number of strokes per minute, and horse-power, he trusts that he has put the formulae in the simplest possible form for immediate use. It is indeed in this transformation of the formulae for the strength of materials that the usefulness of the book lies ; for the practitioner, once satisfied of their correctness, has but to insert quantities fixed at the commencement of his design, and derive from the formulae the required dimen- sions, being relieved of many formulae and details connected with the applications of statics to the strength and elasticity of materials. The constant references to the fourth section of Weis- bach's Mechanics of Engineering are necessary, as it is no part of the author's plan to discuss the strength and elasticity of materials any further than it is necessary to do so in their application to the steam-engine. Those unac- quainted with this branch of mechanical engineering will nowhere find it treated with greater simplicity and thor- oughness. Other references have been made for the pur- pose of directing the reader to such sources as have been drawn upon in the consideration of topics discussed in this work. PREFACE. I "We who. write at this late day are all too much in- debted to our predecessors, whether we know it or not, to complain of those who borrow from us ;" and each of us is only able to make his relay, taking up his work where others have left it. The lack of accurate experimental data has, in many cases, forced the writer to make, perhaps, bold assumptions which may not prove entirely correct ; however, as in all cases the method of reasoning is given, the reader, where he is in possession of more accurate data, can modify by substitution. The accidental loss of all of the original manuscript and drawings of these Lectures, and the necessity of rapidly re- writing them for use in daily instruction, have caused the work to be more abbreviated than was originally intended. Deeply sensible of the many unavoidable deficiencies of this little work, even in the limited field covered, its author still hopes that it will aid in the diffusion and advancement of real knowledge, upon whose progress the prosperity of our civilization depends. W. D. M. UNIVERSITY OF PENNSYLVANIA, , Philadelphia, 1878. CONTENTS. ART. LECTURE I. P AGH! 1. INTRODUCTORY 13 2. The Steam-Cylinder 13 3. Indicated Horse-Power 15 4. Thickness of the Steam-Cylinder 17 5. Thickness of the Cylinder-Heads 19 6. Cylinder-Head Bolts 20 LECTURE II. 7. Standard Screw-Threads for Bolts 22 8. The Steam-Chest 25 9. The Steam-Ports 28 10. The Piston-Head 29 11. The Piston-Kod 29 12. Wrought-Iron Piston-Eod 30 13. Steel Piston-Rod 34 LECTURE III. 14. Comparison and Discussion of Wrought-Iron and Steel Piston-Rods 37 15. Keys and Gibs 37 16. Wrought-Iron Keys 40 17. Steel Keys 43 18. The Cross-Head . 44 19. Area of the Slides 46 9 10 CONTENTS. AKT LECTURE IV. 20. Stress on and Dimensions of the Guides 49 21. Distance between Guides 51 22. The Connecting-Rod 53 23. Wrought-Iron Connecting-Rod 55 LECTURE V. 24. Steel Connecting-Rod 59 25. General Remarks concerning Connecting-Rods 62 26. Connecting-Rod Straps 63 27. Wrought-Iron Straps 63 28. Steel Strap 65 LECTURE VI. 29. The Crank-Pin and Boxes 66 30. The Length of Crank-Pins 70 31. Locomotive Crank-Pins, Length and Diameter of 73 LECTURE VII. 32. Diameter of a Wrought-Iron Crank-Pin for a Single Crank. 74 33. Steel Crank-Pins 75 34. Diameters of Crank-Pins from a Consideration of the Pres- sure upon them 76 35. Of the Action of the Weight and Velocity of the Recipro- cating Parts > 76 LECTURE VIII. 36. The Single Crank 85 37. Wrought-Iron Single Crank 90 LECTURE IX. 38. Steel Single Crank 93 39. Cast-Iron Cranks 95 40. Keys for Shafts 95 41. Wrought-Iron Keys for Shafts 98 CONTENTS. 1 1 ART. PAGE 42. Steel Keys for Shafts 99 43. The Crank or Main Shaft 99 44. Shaft Subjected to Flexure only 100 45. Wrought-Iron Shaft, Flexure only 101 LECTURE X. 46. Steel Shaft, Flexure only 103 47. Shaft Subjected to Torsion only 103 48. Wrought-Iron Shaft, Torsion only 104 49. Steel Shaft, Torsion only 104 50. Shaft Submitted to Combined Torsion and Flexure 105 51. Flexure and Twisting of Shafts 108 52. Comparison of Wrought-Iron and Steel Crank-Shafts Ill 53. Journal-Bearings of the Crank-Shaft Ill LECTURE XI. 54. Double Cranks 115 55. Triple Cranks 118 56. The Fly-Wheel 120 57. Fly-Wheel, Single Crank 123 58. Fly-Wheel, Double Crank, Angle 90 127 LECTURE XII. 59. Fly-Wheel, Triple Cranks, Angles 120 129 60. Of the Influence of the Point of Cut-Off and the Length of the Connecting-Rod upon the Fly-Wheel 131 LECTURE XIII. 61. The Weight of the Rim of Fly- Wheels 137 62. Value of the Coefficient of Steadiness 138 63. Area of the Cross-Section of the Rim of a Fly-Wheel 139 64. Balancing the Fly-Wheel 140 65. Speed of the Rim of the Fly-Wheel 142 12 CONTENTS. ART. LECTURE XIV. PAGE 66. Centrifugal Stress on the Arms of a Fly-Wheel 143 67. Tangential Stress on the Arms of a Fly-Wheel for a Single Crank 147 68. Work Stored in the Arms of a Fly-Wheel 150 69. The Working-Beam 151 70. General Considerations 151 71. Conclusion... 154 TABLES. 72. TABLE V. The Elasticity and Strength of Extension and Compression 158 73. TABLE VI. The Elasticity and Strength of Flexure or Bending 159 74. TABLE VII. The Elasticity and Strength of Torsion 160 75. TABLE VIII. The Proof-Strength of Long Columns 161 NOTE. These Tables will be found to contain almost all the formulae referred to in these Lectures. THE RELATIVE PEOPOETIOI^S STEAM-ENG UNIVERSITY LECTUR (1.) Introductory. In the present" cially stated, the single-cylinder, double-acting steam-engine only will be the subject of discussion. In making use of the following rules and formulae, if a non-condensing engine is under consideration, the pressures per square inch above the atmosphere, or as registered by an ordinary steam-gauge, must be used. If a condensing engine be considered, fifteen pounds per square inch must be added to the pressures above the atmosphere. While it is impossible to take up every form of steam- engine which has been invented, the formulae are sufficiently general to admit of adaptation to any form of engine which the engineer may wish to devise. (2.) The Steam-Cylinder. In many cases occurring in practical Mechanics other considerations than economy of steam determine either the stroke or the diameter of the steam-cylinder. When, however, these dimensions are not fixed by other considerations, that of economy of steam should have the precedence, as being a constant source of gain ; and it being demonstrable by the differential calculus that the surface of any cylinder closed at the ends and en- closing a given volume is a minimum when the diameter 13 14 THE RELATIVE PROPORTIONS of that cylinder is equal to its length,* it follows that any given volume of steam has a minimum of surface of con- densation, and consequently loses less by condensation than it would in any other cylinder of equal volume and differ- ent relative dimensions ; and therefore that the best relative proportions of the stroke and diameter of the steam-cylinder are attained when they are equal. The importance of the action of the walls of the steam- cylinder in condensing steam, and the inability of a steam- jacket to do more than keep the cylinder warm, without actually communicating any appreciable amount of heat to the enclosed steam, are becoming more clearly recognized among engineers, and are forcing them to adopt higher pis- ton-speeds, f to take means of reducing the surface of con- * Demonstration. Let the surface of the cylinder = S " " volume " " =F= ( TT From eq. (2) we have * + =(!) 3n. (2) 4F = -J- or jr*y=4Fy~ 1 . Fio. 1. -y Substituting in eq. (1), we have 8-4r r *+&. Differentiating, we have giving . , show- Substituting this value of y in eq. (2), we have also z = ing, since the second diff. coef. is positive, a minimum. t For a full discussion of the consumption of steam, see A Trea- tise on Steam, chap, iv., Graham ; Report of the Committee on Designs, Rankine; The Steam-Engine, 3. H. Cotterill ; The Steam-Enyine, Prof. Rankine. OF THE STEAM-ENGINE. 15 densation in the cylinder, and to superheat the steam to a limited extent before its introduction into the engine. If the cylinder and heads be of a uniform thickness, the quantity of metal required to form a cylinder of the de- sired volume is nearly a minimum : we say " nearly," because sufficient length must be added to the cylinder o provide for the thickness of the piston-head and the required clear- ance at the ends. The use of shorter cylinders than has hitherto been cus- tomary has the advantage of reducing the piston-speed, and consequently the wear upon the piston-packing, for any given number of revolutions, although the wear upon the interior of the steam-cylinder is constant for any constant number of strokes per minute. (3.) Indicated Horse-Power. In the present work the indicated horse-power only will be referred to or made use of. Let (-HP) = the indicated horse-power. " P =the mean pressure of steam on the piston- head in pounds .per square inch. " L = the length of stroke in feet. " A = the area of the piston-head in square inches. " N = the number of strokes ( = twice the number of revolutions of the crank) per minute. We have the well-known formula, PLAN = ~* If in formula (1) we make the length of stroke in inches equal to the diameter of the steam-cylinder, it becomes, letting d = the diameter of cylinder in inches, 12x33000 ' 16 THE RELATIVE PROPORTIONS and reducing, we have for the common diameter and stroke of a steam-cylinder of any assumed horse-power, This formula gives, for any assumed horse-power, mean pressure, and number of strokes per minute, the common diameter and stroke of the steam-cylinder in inches. The reduction in size, and the consequent economy in using steam, resulting from assuming the pressure per square inch, P, and the number of strokes per miuute, N, as large as circumstances will permit, in designing an engine of any desired horse-power (.HP), Avill at once be perceived upon inspection of formula (2). The advantages resulting from high pressures, early cut-off, and rapid piston-speed will be more thoroughly discussed in Art. 35 of this work. Example. In a given cylinder, Let L = 4 ft. - 48 inches. " d = 32 inches. " P = 40 Ibs. per square inch. " N= 40 per minute = 20 revolutions of the crank. Using formula (1), we have PLAN 40x4x804.25x40 = 156 > a PP rox - If now we assume the horse r power (HP*) = 156, and Let, as before, JV= 40 per minute, P=40 Ibs. per square inch, we have, using formula (2), 1 ^fi = 36.73 inches, for the required common stroke and diameter of a cylinder of equal power with the first. OF THE STEAM-ENGINE. 17 Thickness of the Steam-Cylinder. Steam-cylin- ders are usually made of cast iron ; and in order that the engine may be durable, this casting should be made of as hard iron as will admit of working in the shop. Steel lining- cylinders for ordinary cast-iron cylinders have sometimes been used, and have well repaid in durability their greater cost. Large steam-cylinders should always be bored in either a horizontal or vertical position, similar to that in which they are to be placed when in use. Weisbach, in Art. 443, vol. ii., of the Mechanics of En- gineering, gives the following formula for the thickness of steam-cylinders : Let t = the thickness in inches of cast-iron cylinder-walls. " P b = the boiler-pressure in pounds per square inch. " d = the diameter of the cylinder in inches. Then I = 0.00033P 4 <2 + 0.8 inch, (3) which makes 0.8 inch the least possible thickness of a steam- cylinder. Van Buren, in Strength of Iron Parts of Steam-Machinery, page 58, establishes the following formula by means of a discussion of a 72-inch English steam-cylinder which had been found to work well: Reuleaux, in Der Constructeur, page 561, gives the fol- lowing empirical formula for the completed thickness of steam-cylinders : t = 0.8 inch + ^. (5) 2* 18 THE RELATIVE PROPORTIONS Inspection of these differing formulae, all founded upon successful practice, would lead to the conclusion that it is best first to calculate the thickness necessary to withstand the pressure of the steam, and then to make an addendum sufficient to provide for boring* and re-boring, and also to give the cylinder perfect rigidity in position and form. Good cast iron has an average tensile strength of 18,000 pounds per square inch cross- section, and with a factor of safety of 10 gives 1800 pounds per square inch as a safe strain. That this factor is not too large will be conceded when we consider that with some forms of valve-motion the admission of steam to the cylinder partakes of the nature of a veritable explosion. From Weisbach's Mechanics of Engineering, vol. i., sec. vi., Art. 363, we have, if we take safe strain = 1800 pounds, Example. For a locomotive cylinder, Let P b = 150 pounds per square inch. " d = 20 inches. Trom formula (3) we have t = 1.8 inches. (4) " " < = 1.34 " (5) " " < = 1.00 ' (6) " " t= .83 " Any of the thicknesses given would probably serve success- fully, and about \\ inches is the best practice. It is not * The best means of securing an approximately true cylinder is to finish to size with a shallow broad cut, giving a rapid feed to the lathe or boring-mill tool. If the cylinder is bored with a fine feed and deep cutting-tool, the gradual heating and subsequent cooling are apt to make the interior tapering in form, as well as to require the running of the shop out of hours in order to avoid stopping, and thereby caus- ing a jog in the cylinder. OF THE STEAM-ENGINE. 19 advisable to make a steam-cylinder of less than 0.75 inch thickness under any circumstances. In deciding upon the thickness to be given to any cylin- der, the method of fastening it, as well as the distorting forces that are likely to occur, should be carefully con- sidered. (5.) Thickness of the Cylinder-Heads. It is demon- strated in Weisbach's Mechanics of Engineering, vol. i., sec. vi., Art. 363, that if the cylinder-heads were made of a hemispherical shape, they would need to be of only half the thickness of the cylinder-walls ; and in designing, the attempt is sometimes made to attain greater strength by giving to the cylinder-heads the form of a segment of a sphere. In considering rectangular plane surfaces subjected to fluid pressure in Weisbach's Mechanics of Engineering, vol. ii., sec. ii., Art. 412, the following formula is deduced for square plane surfaces, which will of course be, with greater safety, true for a circular inscribed surface : Let 100 pounds. A good practical rule for engines in which the pressure does not exceed 100 pounds per square inch is to make the thickness of the cylinder-heads one and one-fourth that of the steam-cylinder walls. (6.) Cylinder-Head Bolts. Having assumed a conve- nient width of flange upon the steam-cylinder, the diameter of the bolt should be assumed at one-half that width, and thoroughfare bolts used preferentially to stud bolts, as a stud bolt is likely to rust and stick in place, and be broken off in the attempt to remove it. The bolts fastening the cylinder-head to the cylinder should not be placed too far apart, as that would have a tendency to cause leaks. Taking 5000 pounds per square inch of the nominal area of a bolt as the safe strain,* in order to cover fully the strain upon the bolt due to screwing its nut firmly home, as well as the strain due to the steam-pressure, and dividing the total pressure of the steam upon the cylinder-head by it, we will obtain the area of all the bolts required ; and divid- * The Baldwin Locomotive Works use eleven J-inch stud bolts to secure the head of an 18-inch steam-cylinder. If we assume the greatest steam-pressure to be 150 pounds per square inch, we have for the stress per square inch of nominal area of the bolts about 5800 pounds, and we are therefore well within limits which have been found thoroughly practical by a tentative process. OF THE STEAM-ENGINE. 21 ing this latter area by the area of one bolt of the assumed diameter, we have the number of bolts required. Let P b = the boiler-pressure in pounds per square inch. " d = the diameter of the steam-cylinder in inches. c = the area of a single bolt of the assumed diameter in square inches. " 6 = the number of bolts required. We have 078M*P. 5000c Example. In a given steam-cylinder, Let d = 32 inches. " P b = 81 pounds per square inch. " c = 0.442 square inches (diameter of bolt f inch). We have, from formula (9), 6 = 0.0001571 1024 * 81 -30, 0.442 showing about 30 three-quarter inch bolts to be required. With so wide a margin as is given by the assumption of 5000 pounds per square inch as a safe strain, considerable variation may be made from this number. Mr. Robert* Briggs, in a paper in the Journal of the Franklin Institute for February, 1865, page 118, says : " Ordinary wrought iron, such as is generally used in bolts, can be stated to be reliable for a maximum load under 20,000 pounds per square inch, and the absolute (ultimate ?) tensile strength of any bolt may be safely estimated on that basis." 22 RELATIVE PROPORTIONS LECTURE II. (7.) Standard Screw-Threads for Bolts. The stand- ard American pitch and dimensions of head and nut of bolts as now used in all the mechanical workshops of the United States was first proposed by Mr. Wm. Sellers. (See Report of Proceedings, Journal of the Franklin Institute for May, 1864 ; Report of Committee, Journal of the Franklin In- titute, January, 1865. For a full critique and comparison with other systems, see Journal of the Franklin Institute, February, 1865; On a Uniform System of Screw-Threads, Robert Briggs.) The advantages of uniformity of dimensions in an element of a machine so frequently occurring do not need discussion. The " Committee on a Uniform System of Screw-Threads " reported as follows : "Resolved, That screw-threads shall be formed with straight sides at an angle to each other of 60, having a flat surface at the top and bottom equal to one-eighth of the pitch. The pitches shall be as follows, viz. : Diameter of bolt.. No. threads pr. in. 20 Diameter of bolt.. No. threads pr. in. is 16 14 2J-2J 4 4 13 12 11 10 21 H U 4| 51 21 6 5f " The distance between the parallel sides of a bolt head and nut, for a rough bolt, shall be equal to one and a half diameters of the bolt plus one-eighth of an inch. The thick- ness of the heads for a rough bolt shall be equal to one-half the distance between its parallel sides. The thickness of the nut shall be equal to the diameter of the bolt. The thick- ness of the head for a finished bolt shall be equal to the OF THE STEAM-ENGINE. 23 thickness of the nut. The distance between the parallel sides of a bolt head and nut and the thickness of the nut shall be one-sixteenth of an inch less for finished work than for rough." FIG. 2. The required dimensions of bolts and nuts can be best expressed in a general way by means of formulae and Fig. 2. 24 THE RELATIVE PROPORTIONS Let D - the nominal diameter of any bolt in inches. Then p = the pitch - 0.24j//) + 0.625 - 0.175. n = the number of threads per inch = -. P d = the effective diameter of the bolt i. e., at root of thread -D-1.3p. S=the depth of thread = 0.65p. H= the depth of nut = D. d n = the short diameter of hexagonal or square nut = -Z>+0".125. A = the depth of the head of bolt = |Z) + T 1 T inch. 3 d k = the short diameter of the head of bolt = -D + 0.1 25. The VAlue of the pitch p, in terms of D, was derived from a graphical comparison of the then existing threads as used in the most prominent workshops in the United States. The depth of the thread S is deduced as follows : Since the angles of a complete V thread are each = 60, its sides and the pitch would form an equilateral triangle, and we would have for its depth p sin 60 = 0.866p ; but in the actual thread \ is taken off at the top and bottom, leav- ing only | of the depth of a complete F thread. Q Therefore, S = -0.866/j =*0.65p. 4 Were it possible or always convenient to have uniformly close work, such an accident as the stripping of the thread from a bolt with the dimensions stated would be impossible. The proportions established are the result and an average of the practical requirements of machine-shop practice, and are therefore to be preferred to proportions which might be established from a theoretical consideration of the strength OF THE STEAM-ENGINE. 25 of materials. The appended table of dimensions (pp. 26, 27) is furnished by Messrs. Wm. Sellers & Co. (8.) The Steam-Chest. In deciding upon the dimen- sions of the steam-chest, it must be borne in mind that it ought to be as small as the dimensions and travel of the valve will permit, in order to avoid loss by condensation of steam. The chest is subject to many modifications of form, but usually consists of the ends and two sides of a cast-iron box, resting upon a rim surrounding the valve-face, upon which a flat cover is placed, and the whole firmly secured to the steam-cylinder by means of stud-bolts passing through the cover outside of the sides of the box, in order to avoid rusting of the bolts as well as to diminish the contents of the box. The number of bolts required can be determined, as shown in Art. (6), from a consideration of the steam-pressure upon the steam-chest cover. It is customary to make the sides and cover of the steam- chest of the same material and thickness as the cylinder- walls, sometimes strengthening the cover by casting ribs upon it. To deduce the theoretical thickness, we have, from Weis- bach's Mechanics of Engineering, vol. ii., sec. ii., Art. 412, the following formula : Let / = the longest inside measurement of chest in inches. " b = the breadth of chest in inches. " P b = the boiler-pressure in pounds per square inch. " T = the safe tensile strain upon cast iron per square inch = 1800 pounds. " ty = the required thickness of the steam-chest cover in inches. 3 26 THE RELATIVE PROPORTIONS o fn CO H^ o! o PH qSnoa paqsjaij j..nin:j ji tJoq paqsnnj ssaa^oiqx qSnoa aavnbg qSnojj J,oniBip ?ao no tntujs J" 1OOJ j j,ai qont jad spvaaqx jo ia-a (N C ) >O lO OOOOOT^I^rtrH ^r-C^OlC^l^t-MTI O O < eo-**.T -t CO l-H CO >C f~ O CO CO O .OS^iTt< COOOi-iCO t^ CO 00 CO o o eo eo CO CO 1C O Tjl t^ C5 CM i-i i-i CM M CM CM eo eo co co * ** CM CM CM CM CM ^[Oq JO 1918 Q -UIBJp IBUJtUO^I ^ ^,- CQ K. . E~ 42000000 . / and K for steel, , we have = 8. OF THE STEAM-ENGINE. 37 LECTURE III. 14. Comparison and Discussion of Wrought-iron and Steel Piston-Rods. If, considering the formulae for rupture by crushing or tearing only for wrought iron and steel, we divide (21) by (16), we have For steel For wrought iron d^ 12.75-*/ (25) LN That is, the steel rod would have but 0.58 the diameter of the wrought-iron rod, and 0.34 the area. If we treat the formulae (24) and (19) for rupture by buckling in a similar manner, we have For steel d, 0.9394 For wrought iron d, 1.039 N =0.90. (26) That is, the steel rod has 0.9 the diameter and 0.81 the area of a wrought-iron rod working under the same conditions. In either case the use of steel is productive of economy in weight, varying from 34 to 81 per cent. Letting T^=the volume of the piston-rod in cubic inches, " y = the weight of a cubic inch of wrought iron or steel = 0.27 lb., approx., we have, for purposes of comparison, 4 38 THE RELATIVE PROPORTIONS V-* d *L. (27)* 4 Letting represent the numerical constant, we have, by substituting for d* its value, derived equation (16) or (21) lor crushing, r-jG^lp, (28) and we see from (28) that the volume, and consequently the weight, of a piston-rod are inversely as the number of strokes per minute for any assigned horse-power. Substituting formulae (19) and (24) for rupture by buck- ling in (27), we have If in this we assume _L = c?, we have from equation (2), letting Ci represent the constant in that equation, F_ n sw /n> ~A Ol giving an expression similar to (28), but affected by the steam-pressure, and showing an economy in weight to be derivable from high pressure as well as piston-speed for this particular case that is, when the stroke and diameter of the steam-cylinder are equal, and the diameter of the rod does not exceed for wrought iron ^ and for steel -J- of its length. If we multiply the results of formula (28), (29) or (30) by y = 0.27, and by the factor representing the ratio of the * This statement is not accurately true, as the piston-rod must be somewhat longer than the stroke. Multiplying (27) by a factor of from f to 2 would give an approximate result. OF THE STEAM-ENGINE. 39 total length of the piston-rod to the stroke, we obtain its weight in pounds for a uniform pressure. Referring to formula (15) or (20) for crushing or tearing, and to formula (17) or (22) for rupture by buckling, we see that the diameter of the piston-rod increases with the square root of the pressure, or its area increases directly as the pressure of the steam in the first case i. e., for crushing or tearing and that the diameter of the piston-rod increases with the fourth root of the steam-pressure, or its area with the square root of the pressure, in the second case. All of these formula alike show that a rapid increase in the boiler-pressure does not cause a correspondingly rapid increase in the diameter of the rod, and explain the success of empirical rules giving a constant ratio between the diam- eters of piston-rods and steam-cylinders regardless of the steam-pressure. For the purpose of showing how little variation in the diameters of piston-rods results from great changes in the steam-pressure, the following short table of second and fourth roots of the usual steam-pressures is given : TABLE II. Number. Square root. Fourth root. 25 5.00 2.236 50 7.07 2.659 75 8.66 2.949 100 10.00 3.163 125 11.18 3.344 Thus we see that for a pressure increased 5 times formula? (15) and (20) increase the diameter of the rod but a little over 2 times, and formulae (17) and (22) increase the diam- eter of the rod but a little less than one-half. Generally it will be found that the formula for crushing (15) will give the greatest diameter for a wrought-iron 40 THE RELATIVE PROPORTIONS piston-rod, and that formula (22) will give the greatest diameter of a steel piston-rod. (15.) Keys and Gibs. Reserving keys for shafts for discussion in Article (40), we will consider that form of key used in making the connection between the piston-rod, piston-head and cross-head, and also for the connecting-rod. If we wish to avoid weakening the piston-rod at the point where the key passes through it a precaution not found to be practically necessary in most cases we must increase the diameter of the rod at that point. It is customary to thicken the rod where it enters the piston-head, but for convenience not to do so where the piston-rod enters the cross-head. If it is desired to thicken the rod at both ends, it is best to make it of a uniformly enlarged diameter from end to end. Let d' = the diameter of -the enlarged end of rod in inches. " di = the diameter of the piston-rod, as derived in Art. (12), (13) or (14), in inches. The average dimensions of keys are assumed as follows : h = the breadth of the key = d'. ji i = the thickness of the key = . d'* Therefore, 2,ht = , which nearly equals the cross-section 4fi of the enlarged end - - = 0.5354! = the pressure upon the guide in pounds. / = the length of the connecting-rod in. inches. " r = the length of the crank = in inches. " n = - = the rates of the length of the connecting- T rod to the crank. " P 6 = the boiler-pressure in pounds per square inch. " d = the diameter of the steam-cylinder in inches. " L = the length of stroke in ii " (HP) = the indicated horse powe/f' ^ Referring to Fig. 5, FIG. 5. we have S : Si : : 1/nV - r 2 : r. o Therefore, 1/n 2 -! The value of n commonly varies between 4 and 8. For n = 4, formula (38) becomes 1 (38) 1/16-1 = 0.25825. 48 THE RELATIVE PROPORTIONS For n = 8, it becomes S=0.126 1/64-1 If in this we substitute the value S= -d*P t , we have 4 (39) V w'-l Or, supposing the pressure per square inch, P t , to be uni- form throughout the stroke, we have 896000 (HP) (4Q) 1/V-l LN' We have, then, from either equation (39) or (40) the maximum pressure upon the slides. One hundred and twenty-five pounds per square inch is as high a pressure per square inch as should be used, and the most modern English locomotive practice takes forty pounds per square inch as the proper pressure, the prac- tical result being a great diminution in the wear upon both guides and slides. Let A = the area of a slide in square inches. " b = the pressure per square inch allowed. Then will we have for the area of a slide, A-%. (41) o Example. Let P b P = 40 pounds per square inch. d = 32 inches. " .ZV = 40 per minute. L =48 inches. " (HP) = 156 horse-power. b = 1 25 pounds per square inch. " n =5. OF THE STEAM-ENGINE. 49 Substituting in formula (40), we have 396000 156 i = = 6568 pounds. -j/24 48 x 40 Substituting in formula (41), we have 6568 A = = 52.5 square inches. 125 LECTURE IV. (20.) Stress on and Dimensions of the Guides.* The first requisite of a guide is that it shall be perfectly rigid under all circumstances. In many cases the guides are so attached to the bed-plate or frame-work of the engine as to require no calculation of their rigidity. Cast iron is used for guides where they are firmly fastened throughout their length to a bed-plate or framing. Under other circumstances wrought iron or steel is to be preferred as having greater moduli of elasticity, and conse- quent rigidity. When it is considered necessary to calculate the dimen- sions of a guide, the following method will give a result which is safe: * Parallel motions, which take the place of guides in some engines, are discussed in Willis's Principles of Mechanism, pp. 350-363. How to Draw a Straight Line, by A. B. Kempe, is a particularly interest- ing little book, giving all the later discoveries in parallel motions which have followed the invention of Peaucellier's perfect parallel motion. 5 50 THE RELATIVE PROPORTIONS Let Si = the stress upon the guide [see formulae (39), (40) Art. (19)] in Ibs. " I' = the length of guide in inches. " TF=the measure of the moment of flexure of the guide, " E = the modulus of f For wro't iron = 28000000 Ibs. elasticity, 1 For steel =30000000" " a = the deflection in inches. We have (Weisbach's Mechanics of Engineering, sec. iv., art. 217), for a beam supported at both ends and loaded in the middle, M0 , Since perfect rigidity is unattainable, let us concede a deflection, a = y^ of an inch, and formula (42) becomes for wrought iron, W= ^ (43) 13440000 Assuming a rectangular cross-section for the guide, Let b = the breadth in inches. " . h = the depth in inches. Then W= , and formula (43) becomes /t 3 = 12S/!_ (44) 134400006 Substituting the value of /$, derived from equation (39), and extracting the cube root, (44) becomes (45) OF THE STEAM-ENGINE. 5] and substituting the value of Si from equation (40), -. (46) Example. Let f = 60 inches. d =32 inches. L =48 inches. P 6 =P=40 Ibs. per square inch. 6=4 inches. " (-HP) = 156 indicated horse-powers. " N = 40 per minute. n =5. Using formula (45), we have 8 / OO2 ., Af\ h - 0.00889 x 60 J = = 6.82 inches. x Using formula (46), we have h = 0.7071 x 60 J 156 =6.82 inches. V 4x48x40]/25-l (21.) Distance between Guides. It is important to know at what angle of the crank the connecting-rod requires the greatest distance between the guides, if the plane of vibration of the connecting-rod intersects them. We can then determine the least possible distance be- tween the guides, or that position of the connecting-rod which, clearing all parts of the engine, will make it impos- sible for it to touch with the crank at a different angle. Solution. (Approximate.) Let r = radius of crank = CB. " I = nr = length of connecting-rod = AB. We have EF:BD:: [2r - r(l - cos a)] : j/T - r 1 sin 2 a. 52 THE RELATIVE PROPORTIONS FIG. 6. __ flr ~^ \ / / T \ \ ^L ulX-4 ! " 1 \ / tt \ / >^ ,, / Let .EF = a;, we have BD = r sin a, r* sin a (1 + cos a) then x = v , ri/ri 1 - sin* a and neglecting for the present sin* a in the denominator, we have x = - sin a (1 + cos a). n But since a = 2 sin - cos - and (1 + cos a) = 2 cos* -, which gives x = sin a cos* -. Differentiating, we have 71 2 dx 4rr a a . a~\ - 3 sin* - cos* - + cos 4 - , da n I 22 2J and placing this equal to and dividing by cos* -, 2i , d o o o' or tan 1 - = 4, tan |-i/i-.578, - - 30 and a = 60 approximately. 2* Showing that when the crank forms an angle of 60 with the centre line of the cylinder, we have the maximum dis- OF THE STEAM-ENGINE. 53 tance from that centre line required to make the centre line of the connecting-rod clear the guides, 1.5x0.866 1.3r 1.3r x = r or . 1/ri* - .75 V/V-.75 To get the whole distance between the guides we multi- ply by 2, giving n To this value must be added the thickness of the connect- ing-rod. For any point, as K, on the connecting-rod, we have, knowing the angle a = 60, the proportion, AL : AD : : KL : BD, ,, T to which we must add the half thickness of the connecting- rod and multiply by 2 to get the whole distance between the guides. It must be borne in mind, when arranging the guides, that room must be made for the introduction of the key into the stub end of the connecting-rod, if that end cannot be slid outside of the guides at one end of the stroke. (22.) The Connecting-Rod. The connecting-rod of a steam-engine is usually made from 4 to 8 times the length of the crank that is, 2 to 4 times the length of the stroke of the steam-cylinder. FIG. 5. Referring to Fig. 5, we see that when the crank is at 5* 64 THE RELATIVE PROPORTIONS right angles to the centre line of the piston-rod, the strain upon the connecting-rod is a maximum. Let $ = the total steam-pressure upon the piston-head in pounds. " S, = the stress upon the connecting-rod in pounds. " I = the length of the connecting-rod in inches. " r = the radius of the crank. " n = -= the ratio of the length of the connecting-rod to the crank. We have S : Therefore, S t = S . (47) y V? '. If in (47) we let n = 4, we have & = S 4 -1.0328& If we let n = 8, we have -1.0008& 1/64-1 These numerical results show the rapidity with which the value of the maximum stress on the connecting-rod ap- proaches the constant stress upon the piston-rod as the ratio of the connecting-rod to the crank is increased, and further by what a small percentage = .03 the stress upon the con- necting-rod is greater in the extreme case for the value of n = 4. A relatively short connecting-rod is productive of econ- omy of material, and the increased pressure upon the sides can be provided for by increased area. (See Art. 19.) The connecting-rod, being free to turn about its pins at OF THE STEAM-ENGINE. 55 either end, must be regarded as a solid column not fixed at either end, but neither end free to move sideways. To determine the point at which a column of this cha- racter has an equal tendency to rupture by crushing or buckling, we place the formulae for crushing and buckling equal to each other (Weisbach's Mechanics of Engineering, sec. iv., art. 266), and letting cZ 2 = the diameter of the con- necting-rod, or, substituting the values of F and W, I V V 4 ~7 "64 ' i HE which gives -7 = -7854* I . d t K. I 7ft , /1 128000000' For wrought iron, = J854 \ 31Q()0 = 23 i I 142000000 For steel, - - -7854^^ = 16, (49) and we see that a wrought-iron column will rupture by crushing when its diameter is greater than ^ of its length, and will rupture by buckling when its diameter is less than ^5- of its length, and that a steel column will rupture by crushing when its diameter is greater than y 1 ^ of its length, and will rupture by buckling when its diameter is less than Y 1 ^ of its length. (23.) Wrought-iron Connecting-Rod. Let F= the cross-section of the rod in square inches. LetjK"=the ultimate crushing strength of wrought iron per square inch. 56 THE RELATIVE PROPORTIONS Referring to formula (47), we have for crushing, fl- n S = FK. (50) V/n'-l Let P = the boiler-pressure in pounds per square inch. " d t = the diameter of the connecting-rod in inches. " d = the diameter of the steam-cylinder in inches. " the factor of safety be 10, as before. Then formula (50) becomes 31000. 4 Therefore, -0.0179di/P,/-2 . (51) >f s -l In this formula for n = 4 we have \ - =j..uiu, n 1 - 1 v 15 and as the value of n increases this quantity becomes more and more nearly equal to unity, and can therefore be neg- lected in all cases in which n = 4 or a greater number. Formula (51) then becomes Let (-HP) = the indicated horse-power. " L = the length of the stroke in inches. " N = the number of strokes per minute. Formula (52) becomes, in terms of the horse-power, (53) LN The formula for rupture by buckling for long solid OF THE STEAM-ENGINE. 57 columns, not fixed at either end, is Weisbach's Mechanics of Engineering, sec. iv., art. 266. Letting W= the measure of the moment of flexure, " E = the modulus of elasticity - 28000000 pounds per square inch, " I = the length of the rod in inches = nr = n , i) Substituting in this the values of S, = 8, W= - j/V-1 64 and E, we have, with a factor of safety, 10, \/ Substituting in this the value of eZ 2 from equation (68), we have, letting C= constant, f. (75) Reducing V^ W^j^j^-. (76) Equation (76) shows that the volume of the connecting- rod is affected by the varying conditions in a similar man- ner to a piston-rod, excepting that the square of the ratio n enters in and affects the volume of the rod. * It is customary to make round connecting-rods with a taper of about one-eighth of an inch per foot from the centre to the necks, which should be of the calculated diameter. Experiment does not show an increased strength from a tapering form. OF THE STEAM-ENGINE. 63 Thus, if we first take n = 4, and then n = 8, we see that the volume in the first case is but of the volume in the latter case. The increase of the area of the slides to provide for. shorter connecting-rods is quite slow (see Art. 19). It is customary to make connecting and parallel rods of a rectan- gular cross-section in locomotive practice. When this is done it will be safe to make the smaller of the rectangular dimensions equal to the diameter of a round rod suitable to withstand the strains to which it will be subjected. The resistance of a long rectangular column to rupture varies as the cube of its smallest dimension of cross-section. (See Weisbach's Mechanics of Engineering, sec. iv., art. 266.) (26.) Connecting-Rod Straps. By means of gibs and keys the straps draw the brasses solidly against the stub-end of the connecting-rod. If necessary the uniform cross-section of the straps can be preserved by thickening them at the point where they are slotted to receive the gib and key. (See Art. 15.) Fig. 7 a shows the usual form of strap for locomotives, and Fig. 7 b a solid stub-end in which the keys are used only to set the brasses without moving the strap. Fig. 7 c represents the ordinary form of strap in which the brasses and strap are held by means of the gib and key. (27.) Wrought-Iron Strap. Let FI = the area of one leg of the strap in square inches. " the safe strain per square inch = 5000 pounds. " d = the diameter of the steam-cylinder in inches. " P & = the pressure per square inch of the steam. We have 2 x 5000^ = 0.7854P 4 d' ; therefore F l = 0.000078P 6 d* square inches. (77) 64 THE RELATIVE PROPORTIONS or if we assume the pressure to be uniform, and Let L = the length of the stroke in inches, JV=the number of strokes per minute, " (-HP) = the horse-power (indicated), square inches. CSJ,- (78) OF THE STEAM-ENGINE. 65 Example. Let d = 32 inches. L = 48 inches. JV=40 per minute. Pb = P= 40 pounds per square inch. " (-HP) = 156 indicated horse-powers. Substituting in formula (77), we have F 1 = 0.000078 x 40 x 1024 = 3.19 square inches, or substituting in formula (78), we have "t E\R FI 39.6 = 3.21 square inches. 48x40 (28.) Steel Strap. Let 9000 pounds per square inch equal the safe working-strain in tension. Let F! = the area of one leg of the strap in sq. inches. d = the diameter of the steam-cylinder in inches. P 4 = the steam-pressure per square inch. L = the length of stroke in inches. N= the number of strokes per minute. " (-HP) = the indicated horse- power. We have, as in the preceding article, 2x9000^=0.7854^, therefore F l = 0.0000437d 2 P 4 square inches, (79) or assuming the stream-pressure to be uniform throughout the stroke, and substituting for cfPj its value in terms of the horse-power, FI = 22.034^ ^ square inches. (80) Dividing formula (80) by formula (78), we find that the cross-section of a steel strap is but f of the cross-section of a wrought-iron strap of equal strength. 6* 66 THE RELATIVE PROPORTIONS Example. Let the data be the same as in the example appended to the preceding article. Substituting in formula (79), we have F! = 0.0000437 x 1024 x 40 = 1.77 square inches. Substituting in formula (80), we have 1 ^fi *\ = 22.034 - = 1.78 square inches. 48x40 LECTUEE VI. (29.) The Crank-Pin and Boxes. The crank-pin has ever been one of the most troublesome parts of the steam- engine to the mechanical engineer. The mere determination of its proportions, so that it will not break under the strain put upon it by the pressure of the steam upon the piston- head, does not suffice, and often results in trouble from heating when the engine is at work. It therefore becomes a first consideration to so proportion crank-pins as to pre- vent heating, their strength being a matter of secondary importance, to be afterward investigated if it is deemed necessary to do it Before taking up the mathematical part of our consid- eration, it will be of practical value to quote, from the writings of General Morin, the following remarks : " But it is proper to observe that from the form itself of the rubbing body (cylindrical) the pressure is exerted upon a less extent of surface according to the smallness of the diameter of the journal, and that* unguents are more easily expelled with small than with large journals. This circum- stance has a great influence upon the intensity of friction, and upon the value of its ratio to the pressure. OF THE STEAM-ENGINE. 67 " The motion of rotation tends of itself to expel certain unguents and to bring the surfaces to a simply unctuous state. The old mode of greasing, still used in many cases, consisted simply in turning on the oil or spreading the lard or tallow upon the surface of the rubbing, and in renewing the operation several times in a day. " We may thus, with care, prevent the rapid wear of journals and their boxes ; but with an imperfect renewal of the unguent, the friction may attain .07, .08, or even .10, of the pressure. " If, on the other hand, we use contrivances which renew the unguent, without cessation, in sufficient quantities, the rubbing surfaces are maintained in a perfect and constant state of lubrication, and the friction falls as low as .05 or .03 of the pressure, and probably still lower. " The polished surfaces operated in these favorable con- ditions became more and more perfect, and it is not sur- prising that the friction should fall far below the limits above indicated." (Bennett's Morin, pp. 307, 308.) If the unguents are expelled by extreme pressure, so that the "surfaces are simply unctuous, the friction increases rap- idly, and the surfaces begin to heat and wear immediately. These statements apply with equal force to cast iron and cast iron ; cast iron and wrought iron ; cast iron and brass or Babbitt's metal; or with steel or wrought iron in the place of cast iron. The supposed superiority of brass or Babbitt's metal lined boxes over iron boxes in positions very liable to heating lies in their greater softness and conductivity for heat. Brass will conduct heat away from two to four times as rapidly as iron. However, the film of unguent interposed may render the conductivity of brass of less avail than is generally supposed, and the advantage lies only in the fact that, being a softer metal, in case of heating, the surface of 68 THE RELATIVE PROPORTIONS the softer metal receives the principal damage. Phosphor bronze, which is a patented alloy, being nothing more than brass or gun-metal in which the formation of oxide has been prevented by the introduction of phosphorus, is coming into general use for positions in which the wear is very great. It is, perhaps, good practice to use brass or soft metal wherever the pressure exceeds 125 pounds per square incli of projected area. At lower pressures a good lubricating oil may be relied upon to form a film and run without breaking at ordinary speeds. (The continuity of the film of lubricant is affected by so many different conditions that it is impos- sible to fix any exact limit of pressure.) With soft metal or brass bearings good results can be ob- tained at pressures of 1000 pounds or more per square inch of projected area. (See Hand-Book of the Steam-Engine, Bourne, page 183, where 1400 pounds per square inch is given as the greatest pressure per square inch of projected area allowable on crank-pins. Arthur Rigg, A Practical Treatise on the Steam-Engine, page 147.) If, however, the film of unguent does break at these higher pressures, heat- ing begins almost instantly ; and if the surfaces in contact are both of hard metal, as iron and iron, injury to both at once results, while, if the boxes are brass or some of the softer metals, the continuity of the surface film may be re- stored by increased lubrication or by stopping and cooling as soon as heating is observed. Several expedients are used to keep bearings which have a tendency to heat cool until they have worn smooth. The introduction of rotten stone or sulphur with oil is perhaps the best. Quicksilver or lead-filings, introduced with oil, coat the rubbing surfaces and diminish the heat- ing where the rubbing surfaces are very much scored. (See The Working Engineer's Practical Giiide, pages 48 and 49, Joseph Hopkinson.) OF THE STEAM-ENGINE. 69 A great increase of the velocity of the rubbing surfaces renders bearings more liable to heat than a great increase in pressure, although the total amount of work done by friction is the same in both cases, and is probably accounted for by the more rapid expulsion of the lubricant. As the cause of the heating of bearings, when they are of tolerably good workmanship, is the transformation of the work of friction into heat, we see that it is necessary to reduce the friction as much as possible by the perfect smoothness of surfaces in contact, the interposition of lubri- cants, and the reduction of the speed and pressure upon the rubbing surfaces. In all machines there is a limit below which we cannot reduce the speed and pressure of the rubbing surfaces, and we must, therefore, so proportion journal-bearings as to cause no more work due to friction i. e., heat to be pro- duced than can be conveyed away by the unguents, the atmosphere and the conductivity of the metals without raising the temperature of the bearing appreciably. From the statistics of the working of the crank-pins of four screw propellers in the United States Navy (Van Buren, Strength of Iron Parts of Steam Machinery, page 24) we take the following statement and table : " The crank-pins of these vessels worked cool, giving but little trouble, which is the ex- ception rather than the rule for screw-engines." The projected area of crank- pin journal, given in column 7, is that rectangular area formed by a central section of the crank-pin journal in the direc- tion of its length. Shown cross-hatched in Fig. 8. Columns 1, 2, 4, 5 and 6 are given. Columns 3, 7, 8, 9, 10 and 11 are calculated from them. 70 THE RELATIVE PROPORTIONS In calculating column 9 from columns 3 and 8 we have assumed the coefficient of friction at .05, which is the high- est value given by General Morin for constant lubrication, and probably greater in the present cases.* Column 11 is derived from columns 3 and 7. Column 10 is derived from columns 7 and 9. TABLE m. NAME or VEMBL. SwaUn.. Saco _ Wampanoag.. Wabash.... 72 1 3. 4. 5. 6. 7. 8. 9. 10. 11. Hi | a M 3 n-gjj s . e |i a 0. f. i. = 3 g5 8 fi. t 9 * a e a S 1 ? If ||| fi t h o 5 s ? II ei J3 ill || I f- 1| 2| 1? a. t* 85 s a B. a f H* pt B Ibs. Ibs. in. in. sq. in. 40 40716 160 12 8.5 102. 178. 362372 3552.6 399.5 40 28274 180 9 ^J5 67.5 176.7 249801 3700.7 419. 40 314160 62 27 16. 432. 129.8 2038898 4719.7 727.2 28 114002 100 16 15. 240. 196.3 1118910 4662.1 475. Lveraces 4159. 505. From column 10 of the table we find the average amount of work per square inch of projected area of crank-pin journal, which, in the cases cited, has been borne without heating, to be 4159 foot-pounds per minute; and in making use of this quantity in our subsequent calculations, we are on the safe side if the coefficient of friction (assumed at .05) has not been taken too small. (30.) The Length of Crank-Pins. Let d = the diam- eter of the piston-head in inches, P=the mean pressure in the steam-cylinder in pounds per square inch. * It is probable that the coefficient of friction, for crank-pins of marine propellor-engines under ordinary conditions, is 9 or 10 times greater than the assumed 0.05. OF THE STEAM-ENGINE. 71 Then 7854dlP = the mean pressure on the piston-head in pounds. Let / = the coefficient of friction. " la = the length of the crank-pin journal in inches. " d 3 = the diameter of the crank-pin journal in inches. The mean/orce of friction at the rubbing surfaces of any crank-pin journal per square inch of projected area is Let -ZV= the number of strokes per minute (equal twice the number of revolutions). " w = the work of friction per minute. The space passed over by the force due to friction in one minute 1.5708 Nd 3 inches, and we have for the work of friction i. e., heat per minute, w = 1.5708 x .7854/^-^ inch-pounds. From this formula the diameter of the crank-pin journal (c? 3 ) has vanished. Why it has vanished will be understood when we observe that the force per square inch of project- ed area due to friction is inversely as the diameter of the journal ; while the space passed over by this force is directly as its diameter. Replacing w in the last formula by the mean value de- rived from column 10 of the table, equal 4159 foot-pounds equal 49908 inch-pounds, we have 49908 = Therefore, 4 = .0000247/P^cf = 12.454/. (81) 72 THE RELATIVE PROPORTIONS Considering formula (81), we see that the length of the crank-pin increases and decreases with the coefficient of friction, the mean steam-pressure per square inch the num- ber of strokes per minute, and with the square of the diam- eter of the steam-cylinder. A consideration of the component formulae of (81) shows that as the crank-pin journal decreases in size the pressure per square inch becomes greater ; but if this reduction in size is obtained by a diminution of the diameter (rf s ) of the crank-pin journal, the work per square inch of projected area is not increased, for the velocity of the rubbing sur- faces by this means is decreased in the same ratio as the pressure is increased. Within reasonable limits as to pressure and speed of rub- bing surfaces, the general law may be enunciated : The longer any bearing which has a given number of revolutions and a given pressure to sustain is made, the cooler it will work, and its diameter has no effect upon its heating. Example. Let d = 30", N =180, and P=40 pounds per square inch. We have, by substitution, in formula (81), k = .0000247 x/x 40 x 180 x 900 = 160. /. If in this we take/=.03 to .05 for perfect lubrication, we have k = 4.8" to 8". If we take/=.08 to .10 for imperfect lubrication, we have I, - 12.8" to 16". The results show the great advantages arising from con- stant oiling of bearings and smoothness of surfaces. NOTE. If 0.05 be taken as the coefficient of the force of friction, we obtain the average length of the crank-pins quoted in Table III., OF THE STEAM-ENGINE. 73 Art. (29). About one-quarter of the length required for propellor crank-pins will serve for the pins of side-wheel engines with good results, and one-tenth for locomotive or stationary engines. (31.) Locomotive Crank-Pins, Length and Diameter. If for locomotive crank-pin journals we assume N, the number of strokes per minute = 600. P the pressure per square inch in pounds = 150, the form- ula (81) being changed to by removing the decimal point one place to the left, will give the length of journal commonly assumed in successful practice, if we assume the coefficient of friction at .06. The above formula then becomes /3 = .013d 3 . (82) This formula would prove the amount of heat per square inch of projected area conveyed away from the crank-pins of locomotives to be ten times greater than in the case of marine engines, did not the variations of speed and frequent stoppages of a locomotive prevent comparison. Example. Let d = 18 inches. "We have I, = .013x324 = 4.21 inches. The diameters of locomotive crank-pins are usually taken equal to their length. 7 74 THE RELATIVE PROPORTIONS LECTURE VII. (32.) Diameter of a Wrought-Iron Crank-Pin for a Single Crank.* It is necessary, first, to determine its length by formula (81), and with this length to determine the proper diameter. Let a = the deflection of pin under stress in inches. " S= the stress on pin in pounds. " E - the modulus of elasticity of wrought iron = 28000000 pounds. " W= the measure of the moment of flexure of the pin. " ly = the length of journal in inches. The deflection of a beam fixed at one end and loaded at the other (Weisbach's Mechanics of Engineering, sec. iv., art. 217) is SP 3 WE' If, again, the beam be supposed to be uniformly loaded and fixed at one end, its deflection will be (Weisbach's Mechanics of Engineering, sec. iv., art. 223) , S? and if for the load at the end we concede a deflection of Y^J of an inch, we have for the same load under the two cases above mentioned Oi = .01 inch, a, = .0038 inch. * In Arts. 54 and 55 will be found a discussion of the stresses on crank-pins for double and triple cranks. OF THE STEAM-ENGINE. 75 Then, taking the most unfavorable case i. e., the load at the end we have, letting 4 letting P d = the greatest pressure of steam in cylinder equal the boiler-pressure, letting 64' " ds '- 28000000 64 1600 ', (84) for a constant steam-pressure. Example. Let P b = 60 pounds per square inch. " 1 3 = 8 inches. " d = 30 inches. Substituting in formula (84), we have d 3 = .066^60x512x900- 4.79 inches. There is no need of an investigation of the strength of a crank-pin, as the condition of rigidity gives a great excess of strength. (33.) Steel Crank-Pins. The length of a steel crank- pin is just the same as that of a wrought-iron pin, and the modulus of elasticity of steel is so nearly equal to that of wrought iron as to make formula (84) serviceable for both steel and wrought iron alike. 76 THE RELATIVE PROPORTIONS The advantages of steel crank-pins over wrought iron are their greater strength, and the possibility of obtaining a much smoother surface because of the homogeneous struc- ture of steel. Their disadvantage is their liability to sud- den fracture when not working truly for any reason, as inac- curacy of workmanship or wrenching as hi a marine-engine. (34.) Diameters of Crank-Pins from a Considera- tion of the Pressure upon them. Referring to the table, we find the average pressure per square inch of projected area to be about 500 pounds. If now we divide the whole pressure upon the crank-pin by 500, we obtain the projected area required when this limit is not to be exceeded. TT d* P d?P The equation d^ - - gives d 3 = .00157 - . (85) 4 x 500 4 Example. Let P= 40 pounds, 4 = 8", d = 30", d a = .00157- ^-^ = 7.06 inches. This latter method is perhaps the most practical, and has the advantage of limiting the pressure. It will almost always give larger results than the preceding method. If we assume the pressure to be uniform throughout, the stroke (85) becomes, in terms of the horse-power, (86) (35.) Of the Action of the Weight and Velocity of the Reciprocating Parts. By the reciprocating parts we mean the piston-head, piston-rod, cross-head or motion- block, and the connecting-rod ; also, in a vertical engine, the working-beam if one is used. OF THE STEAM-ENGINE. 77 We are obliged to neglect the action of friction from the impossibility of determining it, and we will also at first neglect the influence of gravity and the angular position of the connecting-rod i. e., suppose it to be of infinite length. In order to clearly comprehend the motion of the recip- rocating parts in a horizontal direction for a horizontal en- gine, Fig. 9 (A), lay down a horizontal line, O X, and divide it into 12 equal spaces, O, 1, 2, 3, etc., to 12; at these points draw ordinates at right angles to O X. With any centre upon the line O X, as 2, and any radius as 2 C, describe the circle O C B 4 D O, and beginning at O divide the circumference of this circle likewise into 12 equal parts. Let 2 C represent the position of the centre line of the crank, let 2 be the centre of the crank-shaft, and let C be the centre of the crank-pin. Let the angle a = C 2 O be the variable angle formed by the centre line of the crank with the horizontal O X. 7* 78 THE RELATIVE PROPORTIONS Let r = the radius of the crank 2C. " $=the space passed over by the piston-head (neces- sarily in a horizontal direction OX) in the time t. " V=- the angular velocity of revolution of the crank (as- sumed constant). " T= the time of one revolution of the crank. 2tf "We have F= , = r(l-cosa). (87) Differentiating (87), we have eft? = r sin a da. (88) Letting v represent the velocity of the piston-head in a horizontal direction, and dividing (88) by dt, we have dS , da 2r and since = v and = v = , dt dt T v - - sin a. (90) Differentiating equation (90), we have o dv = -^ cos a d a, (91) and dividing by cfr, as before, we have for the acceleration dv 2nr da /27rV COS a. (92) If we assume the angular velocity = V= = 1, we can graphically compare the curves of these equations, Fig. 9 (A). OF THE STEAM-ENGINE. 79 The ordinates to the curve O B E F 12 show the distance of the piston-head from its starting-point during one revo- lution, which can be calculated also from equation (87). The ordinates to the curve B 6 G 12 show the velocities of the piston-head during one revolution, which can also be calculated from equation (90). The ordiuates to the curve H 3 K 9 L show the accelerations of the velocity of the piston-head during one revolution, which can also be calcu- lated from equation (92). All of the reciprocating parts are supposed to move in conjunction with the piston-head. If now we wish to determine the acceleration of the pis- ton-head for every position, Fig. 9 (B), on the line X, we lay off from O toward X the ordinates to the curve of dis- tances for six points, and at these points erect ordinates taken from the same positions and equal to the ordinates to the curve of accelerations. The extremities of these ordi- nates can be joined by a straight line, A B. For, if we sub- stitute in equation (87) the value of cos a derived from UV equation (92) we have, letting y = - = the acceleration, at Therefore 8~r- -- , (93) which is the equation of a straight line cutting O X at a distance r from the origin. Referring to equation (92), we observe that the accelera- tion varies with the cosine of , and therefore is a maxi- mum for cos a. = 1. This gives y-7V, (94) 80 THE RELATIVE PROPORTIONS which is the expression for the acceleration due to centrif- ugal force. If, therefore, we wish to balance a horizontal engine at its dead points, we must use a counter-weight so placed that its statical moment is equal to the statical mo- ment of the reciprocating parts supposed to be concentrated at the centre of the crank-pin. The engine cannot be bal- anced for any other than its dead points ; and when the crank is at right angles to the centre line of the cylinder, nearly the full centrifugal force of the counter-weight is felt. In engines driven at widely different speeds as, for in- stance, a locomotive the use for counter-weights seems to be the only practical method, and therefore the recipro- cating parts should be made as light as is consistent with sufficient strength to resist the stresses coming upon them. In the case of engines running at a constant speed, the weight and velocity of the reciprocating parts affect the stress upon the crank-pin in a manner which can be deter- mined, and are therefore worthy of consideration. Referring to Fig. 9 (B), we see that the piston resists, leaving each end of the steam-cylinder with a force equal to its centrifugal force (equation 94), Fia.9(B). and fisher, that the intensity of this force diminishes uniformly from the end to the centre of stroke 3, ^__ where it is zero. It will, therefore, ot 2 j\ F" ITx , be at once recognized that an amount of work represented by the area of the triangle A 3 O is sub- tracted from the work impressed upon the piston by the steam during the first half stroke, and that an equal amount, represented by the area of the triangle 3 X B, is added to the work impressed upon the piston during the last half of the stroke. OF THE STEAM-ENGINE. 81 In an ordinary indicator diagram, Fig. 10, we have the means of measuring the force acting upon the piston-head FIG. 10. at every point of the stroke. (For a thorough discussion and explanation of the steam-engine indicator refer to The Richards Steam-engine Indicator, Porter, or The Engine- Room, and who should be in it.) Let the line O X represent the atmospheric line of an indicator diagram taken from a non-condensing engine cut- ting off at one-half stroke. Let P b - the initial pressure of the steam upon the piston- head in pounds per square inch. " P f = the final pressure of the steam upon the piston-head in pounds per square inch. " y = the resistance due to the inertia of the reciprocat- ing parts in pounds per square inch. If now we impose the condition that the initial and final pressures upon the crank-pin be equal, we must have Therefore (95) (96) Let A = the area of the piston-head in square inches. " G = the weight of the reciprocating parts. 82 THE RELATIVE PROPORTIONS We have, equating equations (94) and (96), Transposing and substituting the values, A = 0.7854

X= ' [ 320 27' 35" 148 THE RELATIVE PROPORTIONS For a = 90 = 270 we have Sr X=+. 36338^. For a = = 180 we have X=-. 636618^. (199) This last value of X represents the maximum value of X, being its value for the two dead points ; and if we divide this by the number of arms, we obtain the force at the ex- tremity of each arm which tends to bend or to break it at its junction with the hub of the wheel, or the rim. We have (Weisbach's Mechanics of Engineering, sec. iv., art. 272) the following equation for a beam fixed at one end and loaded at the other: F \M) W in which M= the number of arms ; F= cross-section of an arm in square inches ; W= its measure of the moment of flexure ; / = the length of arm in inches ; T= the proof (or safe) stress per square inch ; Y= the radial stress on each arm ; X= the tangential stress on each arm ; and e = the half diameter of the arm in the plane of the fly-wheel; or inversely, In this formula, for round and elliptical arms w e For rectangular arms, Fe ^3 w e (Weisbach's Mechanics of Engineering, art. 236, sec. iv.) In formula (200) the value of e remains to be determined OF THE STEAM-ENGINE. 149 approximately. This can be done by substituting in either of the two following formulae, and taking the greater value, (Weisbach's Mechanics of Engineering, art 235, sec. iv.) ^=f. (202) Example. Let us assume the shape of the arm of the fly- wheel already discussed to be elliptical. Y Y= 282 pounds, - = 141 pounds. M We have, equation (202), F Therefore, e = in which b = the smaller ^-diameter of the arm assumed, and, equation (201), since TF= - - for an ellipse, (Weisbach's Mechanics of Engineering, art. 231, sec. iv.) Let b = 1", T= 1800, and I = 60 inches. 282x7 22x1800 .05 inch, /141x60x7 rt 592.2 n ... , or e --*/ - = 2\ = 2.44 inches. \ 22x1800 -Y 396 22x1800 13* 150 THE RELATIVE PROPORTIONS The second value of e = 2.44 inches must be substituted in formula (200), and we have T e\M 1800 2.44 = 7.862 square inches. We have assumed b = 1 inch ; and since F= *eb, we have e = x 7.86 = 2.5 inches. 22 We thus see that each arm of a fly-wheel of the dimen- sions indicated should be of an elliptical form, whose major and minor axes are respectively 5 and 2 inches.* It is customary to give the arms a slight taper from the hub to the rim. (68.) Work Stored in the Arms of the Fly- Wheel. If we wish to take into account the weight of the arms in estimating the work stored in the fly-wheel, we have, let- ting u = velocity of rim, W a = the total weight of the arms, and iv = work stored, W W= 0.325 w 2 , approximately, 2*7 which can be added to the work stored in the rim. For a more general and less practical analytical discus- sion of fly-wheels, reference may be made to the works of Morin, Dulos, Poncelet and Resal. Dr. R. Proel, in his Versuch einer Graphischen Dynamik, gives very clear and elegant graphical methods of represent- * If the power of the engine is conveyed by means of a band or geared fly-wheel, we must calculate the tangential stress upon the arms by means of the theorem of moments, regarding the crank an (he short lever at whose extremity the whole steam-pressure acts. OF THE STEAM-ENGINE. 151 ing the work lost and gained by a fly-wheel under various conditions. It perhaps appears superfluous to some of our readers to enter into detail to so great an extent as has here been done, but the danger and loss resulting from the accidental breakage of a fly-wheel demand the most painstaking care in estab- lishing its dimensions. (69.) The Working-Beam. The working-beam is be- coming less used as the speed of the steam-engine is increased ; it is preferably constructed of wrought iron or steel, or, if made of cast iron, is in many instances bound around with wrought iron. Its form, if solid, should be parabolic, with the vertex at the point where the connecting-rod joins it, and the load at that point is the total pressure of the steam upon the piston-head. (Weisbach's Mechanics of En- gineering, sec. iv., arts. 251-52-53.) The working-beam is supposed to be fixed at its central bearing, and thus becomes a beam fixed at one end and loaded at the other. See Table VI. Where web-bracing is used in working-beams, the graphi- cal method will afford the simplest solution. (See Graphical Statics, Du Bois.) (70.) General Considerations. The recent improve- ments in parallel motions will probably lead to their more general use in the place of guides and slides. A most in- teresting and instructive little work, How to Draw a Straight Line, by A. B. Kempe, suggests to the mechanician many forms which can be adapted to the steam-engine with little trouble. In the foundation and framework of engines every pre- caution must be taken to obtain RIGIDITY and immova- bility. Too much stress cannot be laid upon this point ; an in- Io2 THE RELATIVE PROPORTIONS secure foundation inevitably injures, and perhaps ruins, the engine upon it. The centrifugal governor has not been considered, because it forms one of the principal topics in almost every work on the steam-engine. The practical defects of the centrifugal governor are in- surmountable when a perfectly regular speed is desired of the engine. They are as follows : (1.) The engine must go fast in order to go slow, or the reverse, since the balls cannot move without a change of speed in the engine and themselves. (2.) The opening of the steam-valve is dependent upon the angle which the arms attached to the balls form with the central spindle around which they revolve. Thus, an engine having its full amount of work, and gov- erned by an ordinary ball-governor, will be kept at a uni- form speed by the governor so long as the average resistance to be overcome by the engine remains constant ; but when- ever any of the work is taken off, the speed of the engine will be increased to a higher rate, corresponding to the diminished work, and at this faster speed the engine will then run uniformly under the mastery of the governor so long as the work continues without further alteration. This arises from the fact that the degree of opening of the steam- valve is directly controlled by the angle to which the gov- ernor-balls are raised by their velocity of revolution, the steam-valve being moved only by a change of speed, and consequently by a change of the angle of suspension of the governor-balls ; whence it follows that a larger supply of steam for overcoming any increase of work can be obtained only in conjunction with a smaller angle of the suspension- rods of the governor-balls, and consequently with a slower speed, and that a larger angle of the ball-rods, and con- sequently a higher speed, must be attained in order to reduce OF THE STEAM-ENGINE. 153 the supply of steam for meeting any reduction of work to be done by the engine. (3.) The governor must be sensitive i. e., quick to act. This result is usually attained in the centrifugal governor by giving to the balls a speed much greater than that of the engine, so that a slight variation of speed in the engine is multiplied in the governor many times. A high speed, however, is attended with the disadvantage of rapid wear, and, in the case of an ordinary governor, wear such as to admit of any lost motion is attended with much trouble to the engineer and sudden variations of speed in the engine. (4.) The governor must have power, which means an even and sure motion of the valve notwithstanding the almost unavoidable defects of workmanship, such as the sticking of the valve or the binding of the valve-stem through careless packing of the stuffing-box. In the ordi- nary governor this power is sought to be obtained either by a high speed, the defects of which have already been pointed out, or by means of very heavy balls, which results in a very cumbersome and large machine, besides adding largely to the expense. Thus we see that not only is the speed of the steam-engine entirely different with different loads, but also that with a constant load the speed varies between limits which are de- termined by the sensitiveness of the governor and is at no time regular. The necessity of a very sensitive governor is done away with by the use of a properly proportioned fly-wheel. The use of the governor to determine the point of cut-off, as shown in the Corliss engines, if the fly-wheel be of the proper weight and size, is attended with great regularity of speed and economy of steam. Siemens' chronometric gov- ernor, in which first the inertia of a pendulum and after- 154 THE RELATIVE PROPORTIONS ward hydraulic resistance were used as a point d'appui to move the valve from, produces a very regular speed of en- gine (Proceedings of the Institution of Mechanical Engineers, January, 1866), but is too costly for general use. Marks' isochronous governor (patented), in which the motion of the valve precedes any change of speed in the governor-balls, or. as since altered, in the hydraulic cup, subserves the same purpose, and is much cheaper than the former. (Journal of the Franklin Institute, May, 1876.) A vast number of forms of governor of varying merit have been invented, this portion of the steam-engine appear- ing to be the most attractive to, and the most considered by, mechanics and engineers. The only test of beauty and elegance of design in an en- gine is fitness and perfect proportion to the stresses placed upon the various parts. The severest simplicity of design should be adhered to. Every pound of metal, where it does not subserve some use- ful purpose, every attempt at mere ornament, is a defect, and should be avoided. The well-educated engineer should combine the qualities of the practical man and the physicist ; and the more he blends these together, making each mould and soften what the other would seem to dictate if allowed to act alone, the more will his works be successful and attain the exact object for which they are designed. The steam-boiler and its construction will be found to be very thoroughly treated in Professor Trowbridge's Heat and Heat Engines, in Wilson's Steam-boilers, and in The Steam- Engine, by Professor Rankine. (71.) Conclusion. Notwithstanding the ceaseless efforts of inventive genius to discover some more economical agent for the production of work than the steam-engine, it bids OF THE STEAM-ENGINE. 155 fair for many years, if not always, to retain its supremacy, and every invention adding to its perfection or economy of performance, every discovery of a new principle laying more clearly before us its proper arrangement and method of working, is a positive nay, almost incalculable benefit to mankind. In the marvelous tales of the The Arabian Nights we read of a poor fisherman who, casting his net into the sea, drew forth a sealed vessel ; breaking the seal, there issued forth a cloud of vapor, which finally assumed the form of a giant geui. Having subdued him to his will, he forced the geni to transport him from place to place with the speed of thought, to cause a stately city to spring up from the desert, and finally to provide for him boundless wealth. Incredible as such a tale the offspring of the romantic and magnificent Eastern fancy may seem, the steam-engine has performed for us a greater marvel. If accomplishment of work be our standard, if that is the longest life which has accomplished the most, it has length- ened the life of man an hundred-fold ; for now, with its aid, we can do in a few weeks or mouths what without it would have taken years or centuries. To-day the locomotive is drawing its trains with the speed of the wind from the Atlantic to the Pacific Ocean. The black geni is now toiling beneath the decks of our ocean steamers and carrying to us the products of the uttermost parts of the earth ; while in our shops and factories it is providing for us wealth, comfort and luxury greater than the story-teller of The Arabian Nights could have con- ceived. Stately cities spring up where but a few years ago was a ravage wilderness. Without the steam-engine, the greater part, of the continent of America would yet be inhabited by savage tribes, and our now busy and happy cities, vil- 156 PROPORTIONS OF THE STEAM-ENGINE. lages and farms a desolate wilderness. Truly, the steam- engine is the greatest factor in our civilization. Much thought and labor has been expended on the steam-engine by many different persons, " Yet all experience is as an arch wherethro' Gleams that untraveled world whose margin fades For ever and for ever when I move." TENNYSON'S Ulysses. And yet many improvements remain to be made. There are none in the history of mankind to whom the present generation owes a greater debt of gratitude than to the dis- coverers and inventors of the present form of the steam-en- gine, and there can be no nobler ambition, no greater ser- vice done to mankind, fraught only with benefit for all, with injury to none, than to add to the giant power of this magnificent servant. TABLES. THE following four tables, condensed from tbe fourth section of Weisbach's Mechanics of Engineering, which has formed the basis of this work, are inserted as a means of ready reference for ordinary problems in the strength and elasticity of materials. The same notation as that used by Weisbach is retained, in order to avoid confusion in refer- ring to his work. 14 157 158 THE RELATIVE PROPORTIONS TABLE V. (72.) Elasticity and Strength of Extension and Com- pression. (Arts. 201-214.) (Art. 204.) To find increase or decrease in length under a strain of extension or compression. Where the weight of the body under strain is not considered A = the amount of the extension in inches. P= the weight acting in pounds. J = the length of the body acted upon in inches. .F=the area of the cross-section in square inches. l?=the modulus of elasticity in pounds per square inch. Jl-i FE Let G = the weight of the body under strain. (Art. 207.) AVhere the weight of the body under the strain is also taken into account, ?. = - w^ -. (Art. 205.) To find the proof-strength of a body to be submitted to strain. T = the prdof-strength for extension per square inch in pounds. T! the proof-strength for compression per square inch in pounds. K =the ultimate strength for extension per square inch in pounds. JT, = the ultimate strength for compression per square inch in pounds. For a pull, P= FT. For a thrust, P, = FT V To find the ultimate strength of a body to be submitted to strain. To tear the body asunder, P= FK. To cnish it, P l = FJT,. OF THE STEAM-ENGINE. 159 3 3 3 3 c CO *J 3 P. 3, Ii. fr a X a 1 o 1 i S" a S 2. a^. S- s P P ^- o a" cr a a o o C" S 3 o o P P B B 2 CO CO CO s O B CD CD (B ft> a TJ ^ a? &'"- >z >> 00 a 5 ? ^ a? > u o *> g o " *' P i' i' p P 8 "- 5 5* 1 H * 1 I CO i CO 8 % H o o s* p a- a T ><*, J*.' 1 ? a if to s s5- o B t < e.-i 0- S* * o- i i ii y p B 1 1 II B 3-S- 3 p B E - g 5 5" |l|l B* 5* TO 2 c? a- 3 .8 * *^o 5 5 "I B 2^ *<$ Z?y " O 5To H >1 c K" "" ** *^ ^** P P a o. cs ? * * 3 o 3* -^" c" sS i ft ^f p A a, S g." Hi* B Jf 3" " p 1" 2, B" B N r Nil ,7 : *"' ? "< *. i For a hollow girder circular cross-sici' IT i ^ n ii 4 ^- ^ ^- 11" n s'| For a solid girder circular cross-sect w,-l TT./4 n u For a hollow or a webbed girder wit tangular cross-seel hA h,A, 1^ m >; For a solid girder wl tangular cross-sect ^ 3 "=" 10' Values of H' and M v | ga .' ** "* rf ..N " S* sr " 13 'E. p P ? ? i 160 THE RELATIVE PROPORTIONS TABLE (74.) The Elasticity and Strength of Torsion. (Arts. 262-265.) Notation. P= the load in pounds, a = the lever-arm of the load in inches. T=the modulus of proof-strength for shearing in pounds per square inch. TP=the measure of moment of torsion. e = the greatest distance in inches of any element of the cross-section from the Beutral axis. J=the length in inches submitted to stress. d = the diameter of round shafts in inches. b = the length of one side of a square shaft in inches. o = the angle through which the body is twisted in degrees. Form of body. Proof-strength. Angle of torsion. For any form of cross- \ p^TW See Art 264. Cast iron (Art. 263). For a solid round shaft... P= 0.1963.- a See Art 264. Wro't iron (Art. 263). a = 0.00006485-'. a* ' Cast iron (Art. 263). a = 0.0001 211^. For a solid square shaft.. 5 s T P= 0.2357 .- a See Art 264. Wro't iron (Art. 263). a = 0.0000382^. 6* OF THE STEAM-ENGINE. 161 TABLE VIII. (75.) The Proof-Strength of Long Columns. (Arts. 265 to 270.) I = the length of the column in inches. d = the diameter of the column in inches. When the length of columns is so increased as to cause rupture by first bending and then breaking across (buckling). The following formulae will apply approximately : Method of adjustment. Force necessary to rupture by buck- liug. Remarks. Column fast at the lower end, ") load applied at the upper 1 end, which is free to move f p-()V* See Art. 265 W and E are the same as in the case of flexure. Column not fixed at either 1 end, but neither end free X to move sideways J 4P. See Art. 266. Column fixed at both ends, "j and not free to move side- > ways J 16P. See Art. 266. According to Hodg- kinson's experi- ments, we have only 12P. (Art. 2G6.) For a solid cylindrical pillar not fixed at either end, but neither end free to move sideways, whose diameter is d and whose length is /, we must take the formulae for buckling, in preference to that for compression. -, = 9 or a larger number. 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