f 
 
m 
 
 OF THE 
 
 Name of Book and Volume, 
 
-S/ 
 
PRACTICAL ESSAY 
 
 ON 
 
 THE STEENGTH OF CAST IKON 
 
 AND OTHER METALS. 
 BY THOMAS TREDGOLD. 
 
 THE FIFTH EDITION, WITH NOTES, BY 
 EATON HODGKINSON, F.R.S. 
 
 TO WHICH ARE ADDED 
 
 EXPERIMENTAL RESEARCHES 
 
 ON THE 
 STRENGTH AND OTHER PROPERTIES OF CAST IRON. 
 
 BY THE EDITOR. 
 WITH ILLUSTRATIVE PLATES AND DIAGRAMS. 
 
PRACTICAL ESSAY 
 
 ON THE 
 
 STRENGTH OF CAST IRON 
 
 AND OTHER METALS : 
 
 CONTAINING 
 
 PRACTICAL RULES, TABLES, AND EXAMPLES, FOUNDED ON A 
 SERIES OF EXPERIMENTS; 
 
 WITH AN EXTENSIVE 
 
 TABLE OF THE PEOPEETIES OF MATEEIALS. 
 
 BY THE LATE '" 
 
 THOMAS TREDGOLD, C.E. 
 
 THE FIFTH EDITION, EDITED, WITH NOTES, BY 
 
 EATON HODGKINSON, F.R.S., 
 
 TO WHICH ARE ADDED 
 
 EXPEEIMENTAL EESEAECHES 
 
 ON THE 
 
 STEENGTH AND OTHEE PEOPEETIES OF CAST IEON ; 
 
 BY THE EDITOR. 
 
 <biti0u. [ ' v j \* p i> ^ 
 
 1 < h > I 1 \ O F 
 
 CALIF* 
 
 ^ 
 LONDON : 
 
 JOHN WEALE, 59, HIGH HOLBOEN. 
 1860-1. 
 
LONDON ! 
 BRADBURY AND EVANS, PRINTERS, WHITEFRTARS. 
 
ADVERTISEMENT. 
 
 A New Edition of the Work of the late Mr. TREDGOLD, 
 edited with further Experimental Researches on the Strength 
 of Cast Iron and other Metals by EATON HODGKINSON, 
 F.R.S., now being much called for, and holding the highest 
 reputation as a Standard Work of reference, for the use of 
 the Scientific and Practical Builder, is now presented together 
 as a fifth edition of Tredgold, and a second edition of Mr. 
 Eaton Hodgkinson's "Experimental Researches." 
 
 It is intended by Mr. Hodgkinson to publish in a col- 
 lected volume, all his valuable experiments scattered in the 
 publications of several Scientific Works, viz. : " Transactions 
 of the Royal Society of London," " The House of Commons' 
 Report on Iron/' and " The Memoirs of the Manchester 
 Society." 
 
 The Second portion of this Work may be had sepa- 
 rately for the convenience of those who desire to possess, 
 alone, Mr. Hodgkinson's " Researches." 
 
 t J. W. 
 
 November, 1860. 
 
CONTENTS OF PART I. 
 
 SECTION I. 
 
 PAGE 
 
 INTRODUCTION. Art. 1 7 . 1 
 
 Kemarks on Using the Tables . .' . . . &C- ; ; > ^ ; . 4 
 Table I. Of the sizes of Square Beams, of different lengths, to 
 
 sustain from 1 cwt. to 500 tons. Art. 5 . . . , >' f - . 9 
 Table II. Showing the Loads which Beams of various sizes will 
 
 support without producing permanent set. Art. 6 . . . 13 
 Table III. To show the Loads which Pillars and Columns of dif- 
 ferent diameters will support with safety. Art. 7 . ,16 
 
 SECTION II. 
 
 Explanation of the Tables, with Examples of their Use. Art. 8 23 
 
 first Table 17 
 
 second Table .... u . ' ., '. 17 
 
 third Table . ' . . . " . . . 18 
 
 Examples of the Use of the first and second Table . . . . 19 
 third Table . . . . . 29 
 
 SECTION III. 
 
 Of the Forms of greatest Strength for Beams to resist Cross 
 Strains. Art. 2435 31 
 
 SECTION IV. 
 
 Of the Strongest Form of Section for Beams to resist Cross Strains. 
 
 Art. 3645 36 
 
 Of the Strongest Form of Section for revolving Shafts. Art. 
 
 4647 ... 41 
 
viii CONTENTS OF PART I. 
 
 SECTION V. 
 
 PAGE 
 
 An Account of Experiments on Cast Iron. Art. 48 87 . . 44 
 
 Experiments on the Resistance to Flexure. Art. 49 70 . / ". 44 
 
 _ , Iron of different qualities. Art. 71 77 ; . 59 
 
 _ Resistance to Tension. Art. 7880 . .'.-.. 67 
 
 Resistance to Compression. Art. 81 84. . ..- 68 
 
 . Resistance to Torsion. Art. 85 . . , . 71 
 
 the Effect of Impulsive Force. Art. 86 . . . 74 
 
 To distinguish the Properties of Cast Iron by the Fracture. Art. 87 74 
 
 SECTION VI. 
 
 Experiments on Malleable Iron and other Metals. Art. 88 97 . 76 
 Experiments on the Resistance of Malleable Iron to Flexure. 
 
 Art. 8891 .- .". . . . *" > ;> '..''.' ; 76 
 Experiments on the Resistance of Malleable Iron to Tension. 
 
 Art. 92' '.' . . '".-" ';' . -. ' '' ; * V ! ' <v . ' '' ,- ? , 81 
 Experiments on the Resistance of Malleable Iron to Compression. 
 
 Art. 93 . . . . . . .-' ".'.; ''.' 3 " : V'7 U \ 81 
 Experiments on the Resistance of Malleable Iron to Torsion. 
 
 Art. 94 , . . .... . 83 
 
 Experiments on Various Metals. Art. 95 97 . . . . 85 
 
 Steel. Art. 95 . . ' ; ; v> ;' . ' . ir:: V : ^'. 85 
 
 Gun-metal. Art. 96 . - . ; "~; '. . . 85 
 
 . Brass. Art. 97 . 86 
 
 SECTION VII. 
 
 On the Strength and Deflexion of Cast Iron, &c. Art. 98247 . 88 
 Practical Rules and Examples for Resistance to Cross Strains. Art. 
 
 142210 ... '. x . / 113 
 
 Table of Cast Iron Joists for Fire-proof Floors, &c. . ' ." ; . 141 
 Practical Rules and Examples for Deflexion by Cross Strains. Art. 
 
 211247 . 147 
 
 SECTION VIII. 
 
 Of Lateral Stiffness, with Rules and Examples applied to Engine 
 Beams, Cranks, Wheels, Shafts, &c. Art. 248261 155 
 
CONTENTS OF PART I. ix 
 
 SECTION IX. 
 
 PAGE 
 
 Resistance to Torsion. Art. 262274 165 
 
 Practical Rules and Examples of Stiffness to resist Torsion. Art. 
 273275 . .173 
 
 SECTION X. 
 
 Of the Strength of Columns, Pillars, or other Supports com- 
 pressed, or extended in the direction of their Length. Art. 
 
 276293 175 
 
 To find the Area of Short Blocks to resist pressure. Art. 281282 180 
 
 Of the Strength of long Pillars or Columns. Art. 286292 . 182 
 
 Of the Strength of Bars and Rods to resist Tension. Art. 293 . 186 
 
 SECTION XL 
 
 Of the Resistance to Impulsive Force. Art. 294 312 . .191 
 Practical Rules for the Resistance to Impulsive Force. Art. 313 
 351 199 
 
 An ALPHABETICAL TABLE OF DATA, for applying the Rules to 
 other Materials, with other Numbers useful in various Calcula- 
 tions 210 
 
 On the CHEMICAL ACTION of certain Substances on Cast Iron . . 219 
 
 Explanation of the Plates 221 
 
 List of Authors referred to in the Table of Data, &c. . . . 224 
 
 Plates I., II., III., IV. . . to face 220 
 
 All the quantities, proportions, <frc., are stated in English weights and measures, th-e- 
 pound being the avoirdupois pound, except the contrary be stated. 
 
L I B It A K Y 
 
 UNIVKKS1TY OF 
 
 CALIFORNIA. 
 
 SECTION I. 
 
 MR. TREDGOLD'S ORIGINAL INTRODUCTION. 
 
 ART. 1. In consequence of the security which cast iron 
 gives, when it is properly employed, for supporting consider- 
 able weights, pressures, or moving forces, it has lately been 
 very much used ; and is likely to wholly supersede the use of 
 timber for many important purposes. Indeed, so considerable 
 are the improvements which have arisen out of its use, that 
 the period of its general introduction has been very justly 
 considered as forming a new era in the history of machines.* 
 " All other improvements," it has been remarked, " have been 
 limited ; confined to particular machines ; but this, having 
 increased the strength and durability of every machine, has 
 improved the whole. "f 
 
 Cast iron is a valuable material, because it gives safety 
 against fire ; it is not liable to sudden decay, nor soon 
 destroyed by wear and tear, and it can be easily moulded into 
 the form of greatest strength, or that which is best adapted 
 for our intended purpose. 
 
 The fatal consequences that might result from the use of 
 timber for supporting heavy buildings, either in case of fire 
 or of decay, have often been foreseen ; but in a few instances 
 
 * Essays on Mill Work, &c., by Robertson Buchanan, Essay II. p. 254, 2nd edit. ; 
 or 3rd edit, by G. Renuie, Esq., p. 177. 
 
 t Mr. Dunlop's Account of some Experiments on Cast Iron. Dr. Thomson's 
 Annals of Philosophy, vol. xiii. p. 200. 
 
 ' 
 
2 INTRODUCTION. [SECT. i. 
 
 it has happened that where iron has been used for greater 
 security against fire, the structure has failed from want of 
 strength. Such failures have not occurred from any defect 
 in the material itself; for it too often happens that such 
 works are conducted by persons of little experience, and less 
 scientific knowledge. Men of little experience too frequently 
 imagine that a large piece of iron is almost of infinite strength; 
 and they often have a like indistinct notion of pressure. They 
 design to please the eye, without regard to fitness, strength, 
 or durability ; instead of ornamenting a support, they make 
 the support itself the ornament, and sacrifice everything to 
 lightness of effect. The dimensions of the most important 
 parts of structures are too often fixed by guess or chance ; 
 and the person who calculates the value of materials to the 
 fraction of a penny, seldom if ever attempts to estimate their 
 power, or the stress to which they will be exposed. 
 
 The manner in which the resistance of materials has been 
 treated by most of our common mechanical writers, has also, 
 in some degree, misled such practical men as were desirous 
 of proceeding upon surer ground; and has given occasion 
 for the sarcastic remark, " that the stability of a building is 
 inversely proportional to the science of the builder/'* 
 
 When it is considered that it is absolutely necessary that 
 the parts of a building or a machine should preserve a certain 
 form or position, as well as that they should bear a certain 
 stress, it will become obvious that something more than the 
 mere resistance to fracture should be calculated. In cases 
 where the parts are short and bulky, it may do very well to 
 employ the rules for resistance to fracture, and make the parts 
 strong enough to sustain four times the load, but such cases 
 rarely occur ; and where long pieces are loaded to one-fourth 
 of their strength, we may expect much flexur^, vibration, and 
 instability. 
 
 If 'a material of any kind be loaded with more than a 
 
 H Ency. Method. Diet, Architecture, art. Equilibre. 
 
SECT, i.] INTRODUCTION. 3 
 
 certain quantity, it loses the power of recovering its natural 
 form, when the load is removed ; the arrangement of its 
 particles undergoes a permanent alteration ; and if it supports 
 the same load during a considerable time, the deflexion will 
 increase, and the more in proportion as the load is above the 
 elastic force of the material.* 
 
 On this part of the resistance of materials I have made 
 many experiments, both with metals of various kinds, and 
 with timber : I find that while the elastic force or power of 
 restoration remains perfect, the extension is always directly 
 proportional to the extending force, and that the deflexion 
 does not increase after the load has been on for a second or 
 two ; but when the strain exceeds the elastic force, the exten- 
 sion or deflexion becomes irregular, and increases with time. 
 I was led into this important inquiry by considering the 
 proportions for cannon, and the common method of proving 
 them. It appears from my experiments, that firing a certain 
 number of times with the same quantity of powder would 
 burst a cannon when the strain is above the elastic force of the 
 material, though the effect of the first charge might not be 
 sensible. The same remarks apply to the methods of proving 
 
 * This important fact appears to have been first noticed by Coulomb, while 
 making his experiments on torsion. (Some account has ^been given of Coulomb's 
 experiments by Dr. Young, Nat. Phil. vol. ii. p. 383, and also by Dr. Brewster in his 
 additions to Ferguson's Lectures, vol. ii. p. 234, third edit.) But, in a great number 
 of substances, we seem to have an instinctive knowledge of this property of matter : 
 a bent wire retains its curvature ; and it may be broken by repeated flexure, with 
 much less force than would break it at once : indeed, when we attempt to break any 
 flexible body, it is usually by bending and unbending it several times, and its 
 strength is only beyond the effort applied to break it when we have not power to 
 give it a permanent set at each bending. A permanent alteration is a partial frac- 
 ture, and hence it is the proper limit of strength. Dr. Young, with his usual profound 
 discrimination, pointed out the importance of this limit in applying the discoveries 
 in science to the useful arts. 
 
 While the previous edition was in preparation, a copy was received of the " Essai 
 Theorique et Experimental sur la Resistance du Fer Forge"," of M. Duleau, which is 
 founded on similar views of the strength of wrought iron. M. Duleau has ascertained, 
 with an apparatus much more imperfect than mine, the fact that iron cannot be con- 
 sidered a perfectly elastic body when the strain exceeds a certain force. I shall, in 
 the course of this edition, compare the results of his experiments with those I have 
 made, whenever the conditions are similar. 
 
 B* 
 
4 INTRODUCTION. [SECT. i. 
 
 the strength of steam engine boilers and pipes, by hydraulic 
 pressure : if the strain in proving exceeds that which produces 
 permanent alteration, an irreparable injury is done by the 
 trial. 
 
 In the moving parts of machines the strain should obviously 
 be under the elastic force of the material, and in the second 
 Table will be found the flexure and load a piece of a given 
 size will bear without destroying the elastic force. 
 
 I think every one, who carefully examines the subject, will 
 feel satisfied that the measure of the resistance of a material to 
 flexure is the only proper measure of its resistance, when it 
 is to be applied where perfect form or unalterable position is 
 desirable ; and the measure of its resistance to permanent 
 alteration, when it is used where flexure is not injurious nor 
 objectionable. 
 
 In order to supply practical men with a convenient and 
 ready means of assigning the dimensions of cast iron beams, 
 columns, &c., to support known pressures, or moving forces, 
 I have drawn up this volume. I am persuaded that its use- 
 fulness will find it a place among the common works of 
 reference, which are more or less necessary to every architect, 
 engineer, and builder. To bring it within as small a compass 
 as possible, I have arranged the Tables so as to include as 
 many distinct applications as the nature of the subjects 
 seemed capable of admitting. 
 
 SOME PARTICULARS TO BE OBSERVED IN USING THE TABLES. 
 
 2. The weight of the beam itself is always to be estimated, 
 and added to the load to be supported; or (because this 
 method renders it necessary to estimate the weight before the 
 bulk be determined) find the dimensions ftf the piece that 
 would support the load by one of the Tables, and increase 
 the breadth in the same proportion as the weight of the piece 
 increases the load. If the weight of the piece, for example, 
 
SECT, i.] INTRODUCTION. 5 
 
 be an eighth part of the load, then to the breadth, found by 
 the Table, add an eighth part of that breadth ; and so of any 
 other proportion. It is not an absolutely correct method, but 
 it is simple and correct enough for use. 
 
 3. The Tables and Rules are calculated for soft gray cast 
 iron. Metal of this kind yields easily to the file when the 
 external crust is removed, and is slightly malleable in a cold 
 state. Dr. C. Hutton has justly given the preference to such 
 iron, because it is " less liable to fracture by a blow, or shock, 
 than the hard metal. 3 '* 
 
 White cast iron is less subject to be destroyed by rusting 
 than the gray kind ; and it is also less soluble in acids ; there- 
 fore it may be usefully employed where hardness is necessary, 
 and where its brittleness is not a defect ; but it should not 
 be chosen for purposes where strength is necessary. When 
 it is cast smooth, it makes excellent bearings for gudgeons or 
 pivots to run upon, and is very durable, having very little 
 friction. 
 
 White cast iron, in a recent fracture, has a white and 
 radiated appearance, indicating a crystalline structure. It is 
 very brittle and hard. 
 
 Gray cast iron has a granulated fracture, of a gray colour, 
 with some metallic lustre ; it is much softer and tougher than 
 the white cast iron. 
 
 But between these kinds there are varieties of cast iron, 
 having various shades of these qualities ; those should be 
 esteemed the best which approach nearest to the gray cast 
 iron. 
 
 Gray cast iron is used for artillery, and is sometimes called 
 gun -metal. 
 
 The best and most certain test of the quality of a piece of 
 cast iron, is to try any of its edges with a hammer ; if the 
 blow of a hammer make a slight impression, denoting some 
 degree of malleability, the iron is of a good quality, provided 
 
 ' * Tracts, vol. i. p. HI. 
 
6 INTRODUCTION". [SECT. i. 
 
 it be uniform : if fragments fly off, and no sensible indenta- 
 tion be made, the iron will be hard and brittle.* 
 
 The utmost care should be employed to render the iron in 
 each casting of an uniform quality, because in iron of different 
 qualities the shrinkage is different, which causes an unequal 
 tension among the parts of the metal, impairs its strength, 
 and renders it liable to sudden and unexpected failures. 
 When the texture is not uniform, the surface of the casting is 
 usually uneven where it ought to have been even. This 
 unevenness, or the irregular swells and hollows on the surface 
 of a casting, is caused by the unequal shrinkage of the iron 
 of different qualities. A founder of much observation and 
 experience in his business pointed out to me this test of an 
 imperfect casting. 
 
 Now, when iron of a particular quality is obtained by 
 mixture of different kinds, it will be difficult to blend them 
 so thoroughly as to render the product perfectly uniform; 
 hence we easily perceive one reason of iron being improved 
 by annealing, for in passing slowly to the solid state, the 
 parts are more at liberty to adjust themselves, so as to 
 equalise, if not neutralise, the tension produced by shrinking. 
 But, it is clear that an annealing heat applied after the metal 
 has once acquired its solid state, must be sufficiently intense 
 to reduce the cohesive power in a very considerable degree, 
 otherwise it will not be sensibly beneficial, f These remarks 
 apply to glass, and to various metals as well as to cast iron. 
 
 It has been remarked that " iron varies in strength, and 
 not only from different furnaces, but also from the same 
 furnace and the same melting ; but this seems to be owing to 
 some imperfection in the casting, and in general iron is much 
 
 * For more information upon this subject, see Mr. Fairbairn's Experiments upon 
 the Transverse Strength, &c., of Bars of Cast Iron, from various parts of the United 
 Kingdom. (Manchester Memoirs, vol. vi. new series). EDITOE. 
 
 t Dr. Brewster has shown that the mechanical condition of unannealed glass is not 
 capable of being altered by the heat of boiling, water. Edin. Phil. Journal vol ii 
 p. 399. 
 
SECT, i.] INTRODUCTION. 7 
 
 more uniform than wood." * I am glad to find my own 
 experience supported by the opinion of a writer so well 
 known to practical men as Mr. Banks. But the very great 
 strain which large "masses of well mixed cast iron will bear, 
 when applied to resist the greatest stresses in mill and engine 
 work, is now extremely well known in this country. Its 
 value was foreseen by our celebrated Smeaton at an early 
 period of his practice. Upwards of fifty years ago he com- 
 bated the prejudices against it in the following language : 
 " If the length of time of the use of these (cast iron) utensils 
 is not thought sufficient, I must add, that in the year 1755, 
 for the first time, I applied them as totally new subjects, 
 and the cry then was, that if the strongest timbers are not 
 able for any great length of time to resist the action of 
 the powers, what must happen from the brittleness of cast 
 iron? It is sufficient to say, that not only those very 
 pieces of cast iron are still at work, but that the good effect 
 has in the North of England, where first applied, drawn 
 them into common use, and I never heard of one failing." f 
 These remarks were written in 1782, and the good opinion 
 of Smeaton has been fully justified by the experience of suc- 
 ceeding engineers ; the grand and varied works of Wilson, 
 Rennie, Boulton and Watt, Telford, &c., &c., abundantly 
 confirm it. J 
 
 * Banks on the Power of Machines, p. 73. See also p. 94 of the same work. 
 
 t Reports, vol. i. pp. 410, 411. 
 
 J One of the boldest attempts with a new material was the application of cast iron 
 to bridges : the idea appears to have originated, in the year 1773, with the late 
 Thomas Farnolls Pritchard, then of Eyton Turrett, Shropshire, architect, who, in 
 communication with the late Mr. John Wilkinson, of Brosely and Castlehead, iron- 
 master, suggested the practicability of constructing wide iron arches, capable of 
 admitting the passage of the water in a river, such as the Severn, which is much 
 subject to floods. This suggestion Mr. Wilkinson considered with great attention, 
 and at length carried into execution between Madely and Brosely, by erecting the 
 celebrated iron bridge at Colebrook Dale, which was the first construction of that 
 kind in England, and probably in the world. This bridge was executed by a Mr. 
 Daniel Onions, with some variations from Mr. Pritchard's plan, under the auspices 
 and at the expense of Mr. Darby and Mr. Reynolds, of the iron works of Colebrook 
 Dale. Mi. Pritchard died in October, 1777. He made several ingenious designs, to 
 
8 INTRODUCTION. [SECT. i. 
 
 Yet I must not omit to remark, that cast iron when it fails 
 gives no warning of its approaching fracture, which is its 
 chief defect when employed to sustain weights or moving 
 forces ; therefore care should be taken to give it sufficient 
 strength. And it will be obvious from the preceding remarks, 
 how much its strength depends upon the skill and experience 
 of the founder. 
 
 4. The parts of eacli casting should be kept as nearly of 
 the same bulk as possible, in order that they may all cool at 
 the same rate. 
 
 Great care should be taken to prevent air bubbles in 
 castings ; and the more time there can be allowed for cooling 
 the better, because the iron will be tougher than when 
 rapidly cooled; slow cooling answers the same purpose as 
 annealing. 
 
 In making patterns for cast iron, an allowance of about 
 one-eighth of an inch per foot must be made for the contrac- 
 tion of the metal in cooling. Also the patterns that require 
 it should be slightly bevelled to allow of their being drawn 
 out of the sand without injuring the impression; about 
 one-sixteenth of an inch in six inches is sufficient for this 
 purpose. 
 
 In notes at the foot of each Table, the mode of applying 
 these Tables to other materials is shown, which will be useful 
 in exhibiting the comparative strength of different bodies 
 when applied to the same purpose, as well as in giving the 
 proportions of these materials for supporting a given load. 
 
 show how stone or brick arches might be constructed with cast iron centres, so that 
 the centre should always form a permanent part of the arch. These designs were in 
 the possession of the late Mr. John White, of Devonshire Place, one of his grandsons, 
 to whom we are indebted for the preceding particulars of this note. 
 
SEJT. I.] 
 
 INTRODUCTION. 
 
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 INTRODUCTION. 
 
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 INTRODUCTION. 
 
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 INTRODUCTION. 
 
 [SECT. i. 
 
INTRODUCTION. 
 
 st iron bar wi 
 exure of the wrought ir 
 trength of good OAK. is o 
 size. And the flexure o 
 To find the weight a beam of 
 ltiply the flexure cast iron 
 In the same man the Table 
 wn. See the aletical Tab 
 
 the 
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16 
 
 INTRODUCTION. 
 
 [SECT. i. 
 
 TABLE III.* ART. 7. A Table to show the iveiyht or pressure a cylindrical pillar or 
 column of cast iron will sustain, with safety, in hundred weights. 
 
 Length 
 or height. 
 
 2ft. 
 
 4ft. 
 
 6ft 
 
 8ft. 
 
 10ft. 
 
 12ft 
 
 14ft. 
 
 16ft. 
 
 18ft. 
 
 20ft. 
 
 22ft 
 
 24ft. 
 
 
 
 -P ro* 
 H -t_< 
 
 -t> 00 
 
 5-3 
 
 5| 
 
 #4 
 
 543 
 
 4-3 GO* 
 
 ^ 
 
 .* 
 
 77 
 
 542 
 
 * 
 
 
 Diameter. 
 
 fl 
 
 .tapis 
 o> ^ 
 
 bo 
 
 
 
 3 o 
 
 || 
 
 II 
 
 fl 
 
 '3 u 
 
 '3 
 
 |S 
 
 Diameter. 
 
 
 
 
 
 5 
 
 3 
 
 2 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 
 1 in. 
 
 18 
 
 12 
 
 8 
 
 
 
 1 in. 
 
 14- 
 
 44 
 
 36 
 
 28 
 
 19 
 
 16 
 
 12 
 
 9 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 IK- 
 
 
 82 
 
 72 
 
 60 
 
 49 
 
 40 
 
 32 
 
 26 
 
 22 
 
 18 
 
 15 
 
 13 
 
 11 
 
 2 
 
 21 
 
 129 
 
 119 
 
 105 
 
 91 
 
 77 
 
 65 
 
 55 
 
 47 
 
 40 
 
 34 
 
 29 
 
 25 
 
 2i 
 
 3 
 
 188 
 
 178 
 
 163 
 
 145 
 
 128 
 
 111 
 
 97 
 
 84 
 
 73 
 
 64 
 
 56 
 
 49 
 
 3 
 
 31- 
 
 257 
 
 247 
 
 232 
 
 214 
 
 191 
 
 172 
 
 156 
 
 135 
 
 119 106 
 
 94 
 
 83 
 
 31- 
 
 4 
 
 337 
 
 326 
 
 310 
 
 288 
 
 266 
 
 242 
 
 220 
 
 198 
 
 178 160 
 
 144 
 
 130 
 
 4 
 
 4A 
 
 429 
 
 418 
 
 400 
 
 379 
 
 354 
 
 327 
 
 301 
 
 275 
 
 251! 229 
 
 208 
 
 189 
 
 41 
 
 5 
 
 530 
 
 522 
 
 501 
 
 479 
 
 452 
 
 427 
 
 394 
 
 365 
 
 3371 310 
 
 285 
 
 262 
 
 5* 
 
 6 
 
 616 
 
 607 
 
 592 
 
 573 
 
 550 
 
 525 
 
 497 
 
 469 
 
 440 
 
 413 
 
 386 
 
 360 
 
 G 
 
 7 - 
 
 1040 
 
 1032 
 
 1013 
 
 989 
 
 959 
 
 924 
 
 887 
 
 848 
 
 808 
 
 765 
 
 725 
 
 686 
 
 7 
 
 8 
 
 1344 
 
 1333 
 
 1315 
 
 1289 
 
 1259 
 
 1224 
 
 1185 
 
 1142 
 
 1097 
 
 1052)1005 
 
 959 
 
 - 
 
 9 
 
 1727 
 
 1716 
 
 1697 
 
 1672 
 
 1640 
 
 1603 
 
 1561 
 
 1515 
 
 1467 
 
 1416 
 
 1364 
 
 1311 
 
 9 
 
 10 
 
 2133 
 
 2119 
 
 2100 
 
 2077 
 
 2045 
 
 2007 
 
 1964 
 
 1916 
 
 1865 
 
 1811 
 
 1755 
 
 1697 
 
 10 
 
 n 
 
 2580 
 
 2570 
 
 2550 
 
 2520 
 
 2490 2450 
 
 2410 
 
 2358 
 
 2305:2248:2189 
 
 2127 
 
 11 
 
 12 
 
 3074 
 
 3050 
 
 3040 
 
 3020 
 
 2970J2930 
 
 2900 
 
 2830 
 
 2780i 
 
 2730 
 
 2670 
 
 2600 
 
 12 
 
 This Table was calculated by Equation xviii. art. 290. It is one of those cases 
 where a Table is most useful even to the quickest calculator, because the weight to 
 be supported and the length being given, a quadratic equation must be solved to find 
 the diameter : here it is found by inspection. This Table does not admit of accurate 
 application to other materials, on account of the form of the equation. It will be 
 nearly correct for wrought iron, but is not applicable to timber. 
 
 * This Table has no solid basis. The very ingenious reasoning, from which the 
 formula is deduced by which the Table was calculated, depends upon assumptions 
 which Mr. Tredgold was induced to adopt through want of experimental data. See 
 Mr. Barlow's Report on the. Strength of Materials, 2nd vol. of the British Association. 
 An abstract of an experimental research, to supply this deficiency, is given in the 
 " Additions." EDITOE. 
 
SECTION II. 
 
 EXPLANATION OF THE TABLES, WITH EXAMPLES OF THEIR USE. 
 EXPLANATION OF THE FIRST TABLE. 
 
 8. The first Table (page 9, art. 5), shows by inspection, 
 the dimensions of square beams to sustain weights or pressures 
 of from one hundred weight to 500 tons ; so as not to be 
 bent or deflected in the middle, more than one-fortieth of an 
 inch for each foot in length. 
 
 The length is the distance between the supports, as A B, 
 fig. 1, Plate I., and the stress, whether it be from weight or 
 pressure, is supposed to act at the middle of the length, as at 
 C in the figure. The breadth and depth are supposed to be 
 the same in every part of the length, and equal to one another. 
 
 The horizontal row of figures at the top of the Table 
 contains the lengths in feet. 
 
 The columns, at the outsides, contain the weights in cwts. 
 and tons, and the second column, on the left-hand side, 
 contains the weights in pounds avoirdupois. 
 
 The horizontal row of figures at the bottom shows the 
 deflexion for each length. The other columns show the 
 depths in inches. 
 
 EXPLANATION OF THE SECOND TABLE. 
 
 9. The second Table (page 13, art. 6), is intended to show 
 the greatest weight a beam of cast iron will bear in the 
 
18 EXPLANATION AND USE OF THE TABLES. [SECT. IT. 
 
 middle of its length, when it is loaded with as much as it 
 will bear, so as to recover its natural form when the load is 
 removed. If a beam be loaded beyond that point, the 
 equilibrium of its parts is destroyed, and it takes a per- 
 manent set. Also, in a beam so loaded beyond its strength, 
 the deflexion becomes irregular, increasing very rapidly in 
 proportion to the load. 
 
 The horizontal row of figures along the top of the Table 
 contains the lengths in feet, that is, the distances between 
 the points of support ; and the horizontal row at bottom, the 
 length of a beam supported or fixed at one end only, which 
 with the same load would have the same deflexion. 
 
 The columns on the outsides contain the depths in inches. 
 
 The other columns contain the weights in pounds avoir- 
 dupois, and the deflexions they would produce in inches and 
 decimal parts, when the beams will be only just capable of 
 restoring themselves. 
 
 The breadth of each beam is one inch, therefore the Table 
 shows the utmost weight a beam of one inch in breadth 
 should have to bear ; and a piece five inches in breadth will 
 bear five times as much, and so of any other breadth, 
 
 EXPLANATION OF THE THIRD TABLE,* 
 
 10. The third Table (page 16, art. 7), shows by inspection 
 the weight or pressure a cylindrical pillar or column of cast 
 iron will bear with safety. The pressure is expressed in 
 cwts., and is computed on the supposition that the pillar is 
 under the most unfavourable circumstances for resisting the 
 stress, which happens, when, from settlements, imperfect 
 fitting, or other causes, the direction of the stress is in the 
 surface of the pillar, as shown in fig. 31, Plate IV. 
 
 The horizontal row of figures along the top of the Table 
 contains the lengths or heights of the pillars in feet. 
 
 * See note to that Table. EDITOR. 
 
SECT. IL] EXPLANATION AND USE OF THE TABLES. 19 
 
 The outside vertical columns of the Table contain the 
 diameters of the pillars in inches. 
 
 The other vertical columns of the Table show the weight in 
 cwts. which a cast iron pillar, of the height at the top of the 
 column, and of the diameter in the side columns, will support 
 with safety. Consequently, of 'the height, the diameter, and 
 the weight to be supported, any two being given, the other 
 will be found by inspection. 
 
 EXAMPLES AND USE OF THE TABLES. 
 
 1 1 . Example 1 . To find the depth of a square bar of cast 
 iron, twenty feet in length, that would support ten tons, the 
 deflexion not exceeding half an inch. 
 
 Find the column in Table I. which has the length twenty 
 feet at the top, and in that column, and opposite to ten tons 
 in either of the side columns, will be found the proper depth 
 for the bar, which is 9*8 inches. 
 
 If the depth 9 8 be multiplied by 1*71, it will give the 
 depth of a square beam of fir that would support the same 
 load with the same deflexion. Thus, 1*71 x 9-8 = 16*76 
 inches nearly, the depth of the fir beam. 
 
 If the depth of an oak beam be required, multiply by 1'83 ; 
 thus 1*83 x 9'8 = 17'93 inches, the depth of an oak beam. 
 
 12. Example 2. Required the weight a cast iron beam 
 would support without impairing its elastic force, the length, 
 breadth, and depth being given ? 
 
 Let the length be twenty feet, and the breadth the same 
 as the depth, ten inches. In the second Table, under the 
 length twenty feet, and opposite the depth ten inches, we 
 find the weight 4250 fts. for the load a beam one inch in 
 breadth would bear ; and this multiplied by 10, gives 
 42,500 fts., or nearly nineteen tons; and the deflexion 
 would be 0*8 inches, but the weight of the beam itself would 
 be nearly fairee tons, and its effect the same as if half the 
 
 c2 
 
20 EXPLANATION AND USE OF THE TABLES. [SECT. n. 
 
 three tons were applied in the middle ; consequently the 
 greatest load that the beam should be liable to sustain should 
 not exceed seventeen tons and a half. 
 
 An oak beam of the same size would support only one- 
 fourth of 42,500 fts. or 10,625 fts. ; and its deflexion in 
 the middle would be 0*8 multiplied by 2*8 = 2*24 inches. 
 
 A fir beam of the same size would support three-tenths of 
 42,500 fts. = 12,750 fts. ; and its deflexion in the middle 
 would be 0-8 x 2-6 = 2'08 inches. 
 
 A wrought iron bar of the same size would support 1*12 
 times the weight of the cast iron one, that is, 42,500x1-12 
 = 47,600 ft s. ; and its deflexion in the middle would be 0*8 
 multiplied by 0-86 = 0*688 inches. But the reader will 
 remember that wrought iron possesses this great stiffness 
 only in consequence of the operations of forging or rolling, 
 and these operations have very little effect where the thick- 
 ness is considerable. 
 
 13. There are cases where a greater degree of flexure may 
 be allowed, and there are others where it ought to be less ; 
 but I consider that to which the first Table is calculated as 
 nearly the mean, and it is easy to make any variation in this 
 respect. 
 
 Example 3. Let it be required to find the depth of a 
 square cast iron bar to support ten tons without more 
 deflexion than one-tenth of an inch, the length being twenty 
 feet. 
 
 By examining the deflexion for twenty feet at the foot of 
 the column in Table I. it will be found five times one- tenth 
 of an inch; hence take the depth opposite five times the 
 weight or fifty tons, which is 14*6 inches, the depth 
 required. 
 
 14. Example 4. Find the depth of a square bar of cast 
 iron to support ten tons, the deflexion not to exceed one inch, 
 the length being twenty feet. 
 
 This degree of deflexion is double that at the foot of the 
 
SECT. IL] EXPLANATION AND USE OF THE TABLES. 21 
 
 column headed 20 feet in Table I. ; therefore look opposite 
 half the weight, or five tons, and the depth will be found to 
 be 8 '2 inches. 
 
 I have taken the same length and weight in each of these 
 examples for the purpose of showing how much the depth 
 must be increased to give stiffness. 
 
 15. When a bar or beam is employed to support a load in 
 the middle, or at any other point of the length, a great saving 
 of the material is made by making the bar thin and deep,* 
 provided it be not made so thin as to break sideways. 
 
 The depth of a beam is sometimes limited by circumstances, 
 and as no proportion could be given that would suit for every 
 purpose, it is left entirely to the judgment of the person who 
 may use the Table. But there is a limit to the depth, which, 
 if it be exceeded, renders the use of cast iron for bearing 
 purposes very objectionable and dangerous where the load is 
 likely to acquire some degree of momentum from any cause ; 
 for if the depth be increased, it renders a beam rigid or 
 nearly inflexible, and then a comparatively small impulsive 
 force will break it. A very rigid beam resembles a hard 
 body ; it will bear an immense pressure, but the stroke of a 
 small hammer will fracture it. 
 
 In order to mark the point where the depth has arrived 
 at that proportion of the length which makes it become 
 dangerously rigid, I have stopped the column of depths at 
 that point, and should it be required to sustain a greater 
 weight, the breadth must be increased instead of the 
 depth. 
 
 1G. Example 5. Find the depth of a rectangular bar of 
 cast iron to support a weight of 10 tons in the middle of its 
 length, the deflexion not to exceed one-fortieth of an inch per 
 foot in length, and the length 20 feet; also let the depth be 
 six times the breadth. 
 
 * The term depth is always employed for the dimension in the direction of the 
 pressure. 
 
22 EXPLANATION AND USE OF THE TABLES. [SECT. n. 
 
 Under the length 20 feet in Table I. and opposite six 
 times the weight, will be found the depth, which in this case 
 is 15 - 3 inches, and the breadth will be one-sixth of this 
 depth, or 2' 6 inches. 
 
 If a fir beam be proposed to support the same weight with 
 the same quantity of deflexion, multiply the depth 15 '3 
 inches by 1*71, which gives 26*2 inches for the depth of the 
 fir beam, and its breadth will be 
 
 26*2 
 
 - = 4-37 inches nearly. 
 
 The depth of an oak beam for the same purpose may also 
 be found by using the multiplier given for oak at the foot of 
 the Table. * 
 
 In the same manner, if the depth had been fixed to be 
 four times the breadth, look opposite four times the weight 
 for the depth, and make the breadth one-fourth of the depth, 
 and so of any other proportion. 
 
 17. Example 6. If the breadth and length of a beam be 
 given, and it be required to find the depth such that the 
 beam may sustain a given weight without impairing its elastic 
 force; then, in the second Table, the depth and deflexion 
 may be found thus : Divide the given weight by the 
 breadth ; the quotient will be the weight a beam of one inch 
 in breadth would sustain, which being found in the column 
 of weights under the given length, the depth required will be 
 opposite to it, and also the deflexion. 
 
 Let the given breadth be three inches, the weight to be 
 supported 10 tons or 22,400 fts., and the length 20 feet. 
 Then 
 
 and the weight nearest to 7466 fts., in the column for 20 feet 
 lengths in the second Table is 8330, and the depth 14 inches, 
 and- the deflexion would be 0*57 inches. 
 
 Example 7. The second Table may be usefully applied to 
 
SECT, ii.] EXPLANATION AND USE OF THE TABLES. 23 
 
 proportion the parts of a very simple weighing machine for 
 weighing very heavy weights. For the flexure of a beam 
 being directly proportional to the load upon it, while its 
 elastic force is perfect, this flexure may be made the measure 
 of the weight upon the beam. And a multiplying index may 
 be easily made to increase the extent of the divisions so as to 
 render them distinct enough for any useful purpose. 
 
 Suppose that 4 tons (8960 ibs.) is the greatest load to be 
 weighed, and that the distance between the supports is 16 
 feet ; and make the breadth of the bar 7 inches. Then, 
 
 and the nearest load above this under the length 16 feet in 
 Table II. is 1328fts. and the corresponding depth 5 inches, 
 which may be the depth of the bar. The flexure will be 
 T02 inches, but if the beam be formed as fig. 4, Plate I., the 
 flexure will be greater, being nearly 1'7. (The calculation 
 may be made by art. 232.) 
 
 By making the index move over 5 inches when the 
 deflexion is one inch, each cwt. will cause the index to move 
 over one-tenth of an inch ; but the scale should be graduated 
 by the actual application of ton weights. 
 
 Two such beams and an index would form a simple weigh- 
 bridge, which would be very little expense ; a correct enough 
 measure of weight for any practical use, not likely to get out 
 of order, and would require no attention in weighing except 
 examining the index. And this index might be enclosed, if 
 necessary, so as to be inaccessible to the keeper of the weigh- 
 bridge. 
 
 18. Example 8. To find the diameter for a mill shaft 
 which is to be a solid cylinder of cast iron, that will bear a 
 given pressure, the flexure in the middle not to exceed one- 
 fortieth of an inch for each foot in length. 
 
 Let us suppose the distance of the supported points of a 
 shaft tp be 20 feet, and the pressure to be equal to 10 tons. 
 
24 EXPLANATION AND USE OF THE TABLES. [SECT. n. 
 
 Then multiply the pressure* by the constant multiplier 1'7, 
 that is, 
 
 10 x 1-7=17, 
 
 and in this case, opposite 17 tons in the first Table, and 
 under 20 feet, we find 11 '2 inches for the diameter of the 
 cylinder or shaft. 
 
 But a mill shaft should have less flexure than one-fortieth 
 of an inch for each foot in length ; about half that degree of 
 flexure will be as much as should be allowed to take place. 
 Therefore opposite double the weight, or twice 17 tons, will 
 be found the diameter to give the shaft that degree of stiff- 
 ness, that is, 13' 3 inches. 
 
 If it be for a water wheel, for example, the stress should 
 include every force acting on the shaft ; that is, the weight of 
 the wheels on the shaft, and twice the weight of water in the 
 buckets of the water wheel ; and though it will exceed the 
 actual stress as much as the difference between the weight of 
 the water and its force to impel the wheel, the difference is 
 too small to render it necessary to adopt a more accurate 
 mode of computation. 
 
 Example 9. Large shafts are often made hollow in order 
 to acquire a greater degree of stiffness with a less weight of 
 metal, not only to lessen the first expense, but also to lessen 
 the pressure, and consequently friction on the gudgeons. If 
 the thickness of the metal be made one-fifth of the external 
 diameter, the stiffness of the hollow tube will be half that of 
 a square beam, of which the side is equal to the exterior 
 diameter of the tube. Therefore in Table I., opposite double 
 the stress on the shaft, will be found the diameter in inches 
 under the given length. 
 
 Tor instance, let the shaft be 25 feet long, and the stress 
 upon it when collected in the middle 18 tons; under 26 feet 
 in Table I. and opposite 2 x 18, or 36 tons, will be found 
 
 * See art, 258, or Elementary Principles of Carpentry, Sect. II. art. 96 ; or edition 
 by Mr. Barlow, 4 to, 1840, 
 
SECT, ii.] EXPLANATION AND USE OF THE TABLES. 25 
 
 15 '3 inches, the diameter of the shaft, provided it may bend 
 0*65 in., or a little more than half an inch at every revolution. 
 If it should bend only half this, then look opposite twice 36 
 tons ; the nearest in the Table is 75 tons, and the diameter is 
 18-| inches. The thickness of metal will' be one-fifth, or 
 nearly 3f inches. 
 
 19. Example 10. When the diameter of a solid cylinder 
 is given, and the length, to find the greatest load it will 
 sustain without injury to its elasticity, and the deflexion that 
 weight will cause. 
 
 Suppose the diameter to be 11 inches, and the length 20 
 feet, then in the second Table, opposite the depth 11 inches, 
 and under the length 20 feet, will be found 5142 fts. Let 
 this be multiplied by the diameter 11 inches, and divided by 
 the constant number T7; the result will be the weight 
 required in pounds. 
 
 In this case it is 33,271 fbs., 
 
 5142 x 11-^1-7 = 33,271. 
 
 The deflexion opposite 11 inches and under 20 feet is 
 73 in. 
 
 Any different degree of deflexion may be allowed for in the 
 same manner as shown in the third and fourth examples. 
 
 APPLICATION TO CASES WHERE THE LOAD IS TO BE UNI- 
 FORMLY DISTRIBUTED OVER THE LENGTH OF THE BEAM. 
 
 20. Whether a load be uniformly distributed over the 
 length from A to B, fig. 2, Plate L, or it be collected at 
 several equidistant points, as at 1, 2, 3, 4, 5, 6, and 7, in the 
 same figure, the same rule may be used, as it causes no 
 difference that need be regarded in practice. But the effect 
 of this load in producing flexure differs from its effect in 
 producing permanent alteration. 
 
 It is ^proved by writers on the resistance of solids, that the 
 
26 EXPLANATION AND USE OF THE TABLES. [SECT. n. 
 
 whole of a load upon a beam, when it is uniformly distributed 
 over it, will produce the same degree of deflexion as five- 
 eighths of the load applied in the middle * (see experiment, 
 art. 54, 61, and 62). Consequently, take five-eighths of the 
 w r hole load upon the beam, and with this reduced weight pro- 
 ceed as in the foregoing examples. 
 
 21. Example 11. Let it be required to find the dimen- 
 sions of a cast iron bar to support 10 tons uniformly dis- 
 tributed over its length, the depth of the bar to be four times 
 its breadth, and the deflexion to be not more than one- 
 eightieth part of an inch for each foot in length, or one-fourth 
 of an inch, the length being 20 feet. 
 
 Here the five-eighths of 10 tons is 6 tons and a quarter, 
 and as the depth is to be four times the breadth, multiplying 
 six and a quarter by four gives 25 tons ; but the deflexion is 
 to be only half that given in the Table; therefore the 25 
 must be doubled, which gives 50 for the number of tons 
 opposite which the depth is to be found. The depth opposite 
 50 tons, and under 20 feet, is 14'6 inches, and the breadth is 
 
 - or 3'65 inches ; 
 
 that is, a bar 14 6 inches deep, and 3*65 inches in breadth, 
 will bear a load of ten tons uniformly distributed over it when 
 the length of bearing is 20 feet, and the deflexion in the 
 middle a quarter of an inch. 
 
 Example 12. Let it be proposed to find the proper 
 dimensions for an open girder of cast iron, for supporting the 
 floor of a room, the girder being formed as described in fig. 
 11, Plate II. (See art. 41). 
 
 Suppose the distance between the walls to be 25 feet, and 
 the distance x between girder and girder to be 10 feet, then 
 there will be 
 
 10x25=250 
 
 * Dr. Young's Lectures on Nat. Phil. vol. ii. art. 325, 329. Mr. Barlow's Treatise 
 on the Strength of Timber, Cast Iron, &c., art. 55. 1837. 
 
SECT, ii.] EXPLANATION AND USE OF THE TABLES. 27 
 
 superficial feet of floor supported by each girder ; and the 
 load on each foot being IGOfbs., (see Alphabetical Table, art. 
 Moor), 
 
 160 X 250 = 40,000 fts. 
 
 is the whole load distributed over the girder. But five- 
 eighths of 40,000 is 25,0001fes., and multiplying* 25,000 by 
 6 '3, we ha,ve 157,000fts. ; the nearest number in Table I. is 
 156,800Ibs., and the mean between the depths for 24 and 26 
 feet is 17*8 inches, which is the depth for the girder; the 
 breadth should be one-fifth of the depth, or 
 
 - = 3'56 inches, 
 5 
 
 and the section at A B, and C D, square. 
 
 If the girder were actually loaded to the extent we have 
 calculated upon, the depression in the middle would be about 
 one-third more than is stated at the foot of the Table, in con- 
 sequence of the girder being diminished towards the ends ; 
 but the greatest variable load in practice is seldom more than 
 half that we have assumed, and it is the flexure from the 
 variable load which is most injurious to ceilings, &c. 
 
 Again, let the length of. bearing be 20 feet, and the 
 distance of the girders 8 feet, and the weight IGOfts. upon a 
 superficial foot of the floor, then 
 
 20 X 8 X 160 = 25,600 tts. 
 
 the whole load distributed over the girder. And five-eighths 
 of this load multiplied by 6 '3 is 
 
 The nearest number in Table I. is 103,040, and the depth 
 
 * It is shown in a note to art. 200, that where the breadth and depth of the section 
 of the beam at A B, or C D, fig. 11, is one-fifth of the entire depth of the beam in 
 the middle, the strength is to that of a square beam as 1 : 6-3, and the stiffness is in 
 the same,, proportion. 
 
28 EXPLANATION AND USE OF THE TABLES. [SECT. n. 
 
 corresponding to a 20 feet bearing is 14'3 inches, the depth 
 of the girder required : and 
 
 = 2-86 inches, 
 
 
 the breadth. 
 
 The examples here given of girders show the dimensions of 
 some that were executed several years ago. 
 
 Example 13. The same calculations apply to the form of 
 girder shown in fig. 24, Plate III. When the extreme 
 breadth at the upper or lower side is one-fifth of the depth, 
 divide this breadth into ten equal parts, and make the thick- 
 ness in the middle of the depth four of these parts ; the depth 
 of the projections should be three-fourths of the breadth.* 
 With these proportions, the depth at the middle of a girder 
 for a 25 feet bearing should be 17 "8 'inches, and the extreme 
 breadth 3' 56 inches, as in the preceding example. 
 
 And for a 20 feet bearing 14^ inches deep, and 2*86 
 inches in breadth. I have seen some of less dimensions 
 employed in several instances, but it is to be hoped such 
 examples are not very common. A review of my mode of 
 calculation will show that no more excess of strength is 
 allowed than ought to be in such a material. 
 
 When there is not any length and weight in the Table 
 exactly the same as those which are given, take the nearest ; 
 the dimensions thus obtained will always be sufficiently near 
 for practice. 
 
 22. In applying the second Table, the effect of a load uni- 
 formly distributed over the length is to be considered equal 
 to that of half the load collected at the middle point, (art. 
 139). Therefore considering this half load the weight to be 
 supported, proceed as in the other examples of the use of the 
 second Table. 
 
 * See note to art, 186, for the reason of this rule. 
 
SECT, ii.] EXPLANATION AND USE OF THE TABLES. 
 
 EXAMPLES OF THE USE OF THE THIRD TABLE. 
 
 23. Example 14. Let it be required to support the floor 
 of a warehouse by iron pillars, where the greatest load on 
 any pillar will be 70 tons, the height of the pillars being 14 
 feet. 
 
 Seventy tons is equal to 1400 cwt. ; and in the column 
 having 14 feet at the head, in the third Table, 1561 cwt. is 
 the nearest weight ; and the diameter opposite this weight in 
 the side column is 9 inches, the diameter required. 
 
 If it be wished to approach nearer to the proportion, take 
 the mean between the weight above and that below 1400 ; 
 that is, the mean between 1561 and 1185, which is 1373, or 
 nearly 1400 ; hence it appears that a little more than 8-J 
 inches would be a sufficient diameter, but it is seldom 
 necessary to calculate so near. 
 
 Example 15. If it be desired to fix on the diameter for 
 story posts of cast iron to support the front of a house ; such 
 a one for example as is commonly erected in London where 
 the ground story is to be occupied with shops; in such a case, 
 each foot in length of frontage may be estimated at 25 cwt. 
 for each floor, and 12 cwt. for the roof: hence in a house 
 with three stories over the shops, the extreme load will be 
 
 3 x 25 + 12 = 87 cwt. 
 
 on each foot of frontage. Now if the posts be 7 feet apart, 
 and 12 feet high, we have 7x87 = 609 cwt. the load upon 
 one post ; and hence we find by the Table, that a pillar 6 \ 
 inches in diameter would be sufficient; the load 525 cwt., 
 which corresponds to a diameter of 6 inches, being too small. 
 If there be only two stories above the pillars, and the 
 height of a pillar be 10 feet, the distance from pillar to pillar 
 7 feet ; then, 
 
 (2 X 25; + 12 X 7 = 4 >1 cwt, 
 
30 EXPLANATION AND USE OF THE TABLES. [SECT. ir. 
 
 the whole load for one pillar : and it appears by the Table, 
 that a pillar 5 inches in diameter would sustain 452 cwt. ; 
 consequently 5 inches will be a proper diameter for the 
 pillars. 
 
 When pillars are placed at irregular distances, that which 
 carries the greatest load should be calculated for, and if it 
 happen that such a pillar stands 10 feet from the next 
 support on one side, and 6 feet from the next support on the 
 other side, add these distances together, and take the mean 
 for the distance apart ; thus, 
 
 10+6 __ t6_ 
 ~~2~ ~~ ~2~ *' . 
 
 the mean distance of the supports. 
 
 The strain upon a pillar cannot be exactly in the direction 
 of the axis when the pillars are placed at unequal distances to 
 support an uniform load ; and since this unequal distribution 
 of supports is extremely common in story posts, the propriety 
 of adopting the mode of calculation I have followed is 
 evident. 
 
 The diameter of a story post is sometimes made so small 
 in respect to its height and the load upon it, that a very slight 
 lateral stroke would break it : while we hope that no serious 
 accident may occur through such hardihood, we cannot but 
 dread the consequences of trusting to these inadequate 
 supports. 
 
L 1 P K A 
 
 OF ; 
 
 LI F< > 
 
 SECTION III. 
 
 OF THE FORMS OF GREATEST STRENGTH FOR BEAMS. 
 
 24. In the Introduction, I have stated that one of the 
 most valuable properties of cast iron consists in our being 
 able to mould it into the strongest form for our intended 
 purpose ; and in order to apply this property with the most 
 advantage, it will be useful to consider the means of applying 
 our theoretical knowledge on this subject to practice. 
 
 There are two means of increasing the strength of a beam ; 
 the one consists in disposing the parts of the cross section in 
 the most advantageous form ; the other, in diminishing the 
 beam towards the parts that are least strained, so that the 
 strain may be equal in every part of the length. 
 
 OF FORMS OF EQUAL STRENGTH FOR BEAMS TO RESIST 
 CROSS STRAINS. 
 
 25. Before I point out the forms of equal strength corre- 
 sponding to different modes of applying the load or straining 
 force, let us consider the conditions that are essential in a 
 practical point of view. In the first place, supported parts 
 must have sufficient magnitude to insure stability ; for it is 
 much more important that every connection or joining should 
 be firm, and that the bearing parts should be secure against 
 crushing or indentation, than it is that a small portion of 
 material should be saved. When mathematicians investigate 
 a form of equal strength, the manner of connecting it or 
 
32 OF THE STRONGEST FORMS FOR BEAMS. [SECT. in. 
 
 supporting it is not considered. Girard lias shown that 
 whatever line generates a solid of equal resistance, the solid 
 always terminates in a simple point, or in an arris which is 
 either perpendicular or parallel to the direction of the strain- 
 ing force.* Therefore the forms given by this mode of 
 investigation do not answer in practice unless they be properly 
 modified. 
 
 26. It may be easily proved, that in a rectangular section, 
 when a weight is supported by a beam, the area of the 
 section at the point of greatest strain should be to the area 
 at the place of least strain, as six times the length is to the 
 depth at the point of greatest strain ;f and this is the least 
 proportion that ought to be given. Now when the length 
 and depth are equal, the area at the point of least strain 
 should be one -sixth of the area at the point of greatest strain, 
 instead of being a simple point or an arris. 
 
 27. If a beam be supported at the ends, and the load 
 applied at some one point between the supports, and always 
 acting in the same direction, the best plan appears to be to 
 keep the extended side perfectly straight, and to make the 
 breadth the same throughout the length ; then the mathe- 
 matical form of the compressed side is that formed by draw- 
 ing two semiparabolas A C D and BCD, fig. 3, C being the 
 point where the force acts, j Now the curve terminating at 
 A, it is necessary in applying it to use, to add some such 
 
 * Traits' Analytique de la Resistance des Solides, art. 129. 
 t For it is shown (art. 110) that 
 
 /&<*'- w 
 61 W ' 
 
 but the force to resist detrusion being as the area simply; therefore wo must have 
 / V d' ~ W at the weakest point. Consequently 
 
 or 
 
 6Z: d : :bd:V d'; 
 
 where I is the length, I d the area at the point of greatest strain, and U d' the area at 
 the point of least strain. 
 
 J Greg. Mechanics, i. art. 180. It was first shown by Galileo. 
 
SECT, in.] OF THE STRONGEST FORMS FOR BEAMS. 33 
 
 parts as are indicated by the dotted lines at the extremities. 
 The same form is proper for a beam supported in the middle, 
 as the beam of a balance. 
 
 28. Irregular additions of this kind, however, render it diffi- 
 cult to estimate the effect of the straining force ; therefore, 
 some simple straight-lined figure to include the parabolic form 
 is to be preferred : this may be easily effected as proposed by 
 Dr. Young, * by making the lines bounding the compressed 
 side tangents to the parabolas, as in fig. 4. If A E be equal to 
 half C D, then E C is a tangent to the point C of an inscribed 
 parabola A C, having its vertex at A. 
 
 By forming a beam in this manner, one-fourth of the 
 material is saved ; but the flexure will be somewhat more than 
 one-third greater, therefore there is a loss of stiffness in using 
 this form. 
 
 29. If the beam be strained sometimes from one side and 
 sometimes from the other, both sides should be of the same 
 figure, as in fig. 5. In the beam of a double acting steam 
 engine, the strain is of this kind. AE and BF should be 
 equal, and each equal to half C D as before. 
 
 30. It is sometimes desirable to preserve the same depth 
 throughout ; and in this case, the section through the 
 length of the beam made perpendicular to the direction of the 
 straining force should be a trapezium, described in the 
 manner shown in the 6th figure,! the force acting perpen- 
 dicularly at C, the points of support being at A and B. A 
 figure of this kind would obviously be without stability, but 
 modified as shown by fig. 7, the end being formed as shown 
 at B/ any degree of stability may be given, and with a less 
 quantity of material than when the depth is diminished as in 
 the parabolic form. Also, the deflexion is less, which gives 
 this form a considerable advantage for bearing purposes. In 
 a beam supported in the middle, the same form may be used 
 when the weights act at the ends, as in a balance. 
 
 * JSTafc Philos. vol. i. p. 767. t Gregory's Mechanics, i. art. 179. 
 
 D 
 
34 OP THE STRONGEST FORMS FOR BEAMS. [SECT. n. 
 
 31. When a beam or bar is regularly diminished towards 
 the points that are least strained, so that all the sections are 
 similar figures, whether it be supported at the ends and 
 loaded in the middle, or supported in the middle and loaded 
 at the ends, the outline should be a cubic parabola ; * and if 
 the section of the beam be a circle at the point of greatest 
 strain, the form of the beam should be that generated by the 
 revolution of the cubic parabola round its axis, the vertex 
 being at the point of least strain. 
 
 But in practice, a frustum of a cone or a pyramid will 
 generally answer better, the diameter of the point of greatest 
 strain being to that at the point of smallest strain as 3 : 2.f 
 
 The same figure is proper for a beam fixed at one end, and 
 the force acting at the other ; consequently, it is a proper 
 figure for a mast to carry a single sail. 
 
 32. If a weight be uniformly distributed over the length 
 of a beam supported at both ends, and the breadth be the 
 same throughout, the line bounding the compressed side 
 should be a semi-ellipse when the lower side is straight, J as 
 shown in fig. 8. 
 
 Instead of an ellipse, I usually make the compressed side a 
 portion of a circle, of which the radius is equal to the square 
 of half the length divided by the depth of the beam. The 
 dotted line in fig. 8 shows this form. 
 
 The same form of equal strength should be employed when 
 the beam is intended to resist the pressure of a load rolling 
 over it ; hence the beams of a bridge, the rails of a waggon- 
 way, and the like, should be of this figure. 
 
 33. If a beam has to bear a weight uniformly distributed 
 over its length, and its depth be everywhere the same, the 
 beam being supported at both ends, then the outline of the 
 breadth should be two parabolas A C B, A D B, set base to 
 
 * Gregory's Mechanics, i. art. 181, or Emerson's Mechanics, prop. Ixxiii. cor. 1. 
 f Such a cone or pyramid will include the figure of equal strength, the subtangcnt 
 of the curve being three times its abscissa. 
 
 J Gregory's Mechanics, i. art. 182, or Kmerson's Mechanics, prop. Isxiii. cor. 3. 
 
SECT. III.] 
 
 OF THE STRONGEST FORMS FOR BEAMS. 
 
 base, their vertices C and D being in the middle of the length, 
 as shown in the next perspective sketch.* In practical cases, 
 the arcs A C B, A D B, may be portions of circles. 
 
 When the ends are modified as in fig. 7, Plate I., this will 
 be the most advantageous form for a beam for supporting a 
 load uniformly distributed over its length, as lintels, bres- 
 sumrners, joists, and the like. 
 
 34. When a beam is fixed at one end only, and has to 
 support a weight uniformly distributed over its length ; if the 
 breadth of the beam be every where the same, the form of 
 equal strength is a triangle A C B,f fig. 21, Plate III. 
 
 35. If a beam be fixed at one end only, and the weight be 
 uniformly diffused over the length, the section being every- 
 where circular, then the form of equal strength would be that 
 generated by the revolution of a semi-cubic parabola round 
 its axis 4 
 
 It will be sufficient in practice to employ the frustum of a 
 cone of which the diameter at the unsupported end is one- 
 third of the diameter at the fixed end. 
 
 * Young's Nat. Phil. i. p. 767. 
 
 t Emerson's Mechanics, prop. Ixxiii. cor. 2. 
 
 For the equation of the curve is, 
 
 Ibid. 
 
 ax y 
 
 hence, 
 
 x the subtangent = l'5x; 
 
 and the length of the cone that would include the form of greatest strength is 1*5 
 times the length of the beam. 
 
SECTION IV. 
 
 OF THE STRONGEST FORM OF SECTION. 
 
 36. When a rectangular beam is supported at the ends, 
 and loaded in any manner between the supports, it may be 
 observed that the side against which the force acts is always 
 compressed, and that the opposite side is always extended ; 
 while at the middle of the depth there is a part which is 
 neither extended nor compressed ; or, in other words, it is 
 not strained at all. 
 
 Any one who chooses to make experiments may satisfy 
 himself that this is a correct statement of the fact, in any 
 material whatever ; whether it be hard and brittle as cast iron, 
 zinc, or glass ; or tough and ductile as wrought iron and soft 
 steel ; or flexible as wood and caoutchouc ; or soft and ductile 
 as lead and tin. In very flexible bodies it may be observed 
 by drawing fine parallel lines across the side of the bar before 
 the force is applied; when the piece is strained the lines 
 become inclined, retaining their original distance apart only 
 at the neutral axis. In almost all substances, the fracture 
 shows distinctly that a part has been extended, and the rest 
 compressed ; and in some substances, as wood, lead, tin, 
 wrought iron, &c., the place of the axis of motion may be 
 observed in the fracture. It was first noticed in experi- 
 ment, and applied to correct Galileo's theory by Marriotte.* 
 
 * Treatise on the Motion of Water. &c., translated by Desaguliers, p. 243, Svo. 
 London, 1718. 
 
SECT, iv.] OF THE STKONGEST FORM OF SECTION. 37 
 
 Coulomb * and Dr. Young have made it the basis of their 
 investigations, the latter showing the important fact that an 
 oblique force changes the position of this axis ; f as has been 
 investigated more in detail in this Essay. Lately the place 
 of the neutral axis in horizontal beams has been more closely 
 examined by Barlow in a numerous course of experiments ; J 
 and Duleau has ingeniously shown its place by experiment on 
 wrought iron . The same thing is exhibited in a refined and 
 beautiful contrivance of Dr. Brewster's, which he calls a 
 teinometer, and employs to measure the effect of force on 
 elastic bodies. || 
 
 The strains decrease from each side towards the middle, 
 and in the middle they are insensible. I will call the part at 
 the middle of the depth the neutral axis, or axis of motion. 
 See Sect. VII. art. 107. 
 
 37. In the case of equilibrium, between the straining force 
 and the resistance of a beam, it is a necessary condition that 
 the resistance on one side of the axis of motion should be 
 exactly equal to the resistance on the other side ; or, that the 
 force of compression should be equal to the force of extension. 
 Now, in practice, a body should never be strained beyond its 
 power of restoring itself ; and as it is known from experience, 
 that while their elastic force remains perfect, bodies resist the 
 same degree of extension or of compression with equal forces, 
 it is obvious that, in the section of a beam of the greatest 
 strength, the form on each side of the axis of motion should 
 be the same ; because whatever is the strongest form for one 
 side of the axis must be equally so for the other. Hence, 
 the axis of motion in beams of the greatest strength will 
 always be at the middle of the depth. 1F 
 
 * M^moires de 1'Acaddmie des Sciences. Paris, 1773. 
 f* Lectures on Natural Philosophy, vol. ii. p. 47, 4to. London, 1805. 
 Essay on the Strength of Timber, &c., 8vo. London, 1838. 
 
 Essai The'orique et Experimental sur la Resistance du Fer Forge, p. 26, 4to. 
 Paris, 1820. 
 
 || Additions to Ferguson's Lectures, vol. ii. p. 232, 8vo. Edinburgh, 1823. 
 
 U Thes*e remarks apply only to bodies subjected to very moderate strains, par- 
 
38 OF THE STRONGEST FORM OF SECTION. [SECT. IT. 
 
 38. And, as it is shown by writers on the resistance of 
 solids, that the power of any part in the same section is 
 directly as the square of its distance from the axis of motion, 
 (art. 108,) when the strain upon it is the same, it is obviously 
 an advantage to dispose the parts of the section at the greatest 
 possible distance from the axis of motion, provided that the 
 middle parts be kept sufficiently strong to prevent the strain- 
 ing force from crushing the extreme parts together, and that 
 the breadth be made sufficient to give stability. 
 
 39. It must also be observed that when the parts are not 
 of equal thickness, the metal cools unequally, and therefore is 
 partially strained by irregular contraction ; it is sometimes 
 even fractured by such irregular cooling : for this reason, the 
 parts of a beam should be nearly of the same size. A good 
 founder may generally reduce the danger of irregular cooling, 
 but it is always best to avoid it altogether. 
 
 40. The form of section which I usually adopt in order to 
 fulfil these conditions is represented in fig. 9.* AM is the 
 axis of motion ; the parts on each side of the axis of motion 
 are the same ; the metal is nearly of equal thickness, and the 
 parts necessary to give strength and stability are disposed at 
 the greatest distance from the axis of motion. 
 
 A section of this form is adapted for many purposes ; such, 
 for example, as the beam of a steam engine, as in fig. 26 ; or 
 for supporting arches, as in fig. 10, for girders, bearing beams, 
 and the like. 
 
 41.^When it is necessary to leave some part of the middle 
 of the beam quite open, or when the depth is considerable, 
 I have recourse to another method, which has, in such cases, 
 a decided advantage in point of economy. It consists in 
 
 ticularly in cast iron; since that metal requires, on the average, nearly seven times 
 as much force to crush it as to tear it asunder, and the breakiug strength of beams 
 depends upon these forces. EDITOR. 
 
 * This is not the form of greatest strength to resist fracture; and the beam pro- 
 posed in the next article (fig. 11, Plate II.) breaks irregularly, and is remarkably 
 weak. See Additions. EDITOR. 
 
HECT. iv.] OF THE STRONGEST FOEM OF SECTION. 39 
 
 making the compressed side of the beam, or that against 
 which the force acts, a series of arches, and the other side a 
 straight tie. (See fig. 11, Plate II.) If the tie be not 
 straight, there is a great loss of strength, and a greater loss 
 of stiffness. 
 
 In this figure, the thickness is supposed to be everywhere 
 the same, and the narrowest part of the curved side of the 
 same width as the straight side ; or, so that the area of the 
 section at A B may be the same as the area of the section 
 at CD. 
 
 The sketch in the figure is for the case in which the load is 
 uniformly distributed over the length, and then the upper 
 side should be the proper curve of equilibrium for an uniform 
 load. This curve is a common parabola, but a circular arc 
 will always be sufficiently near when the rise is so small. The 
 upper part of the beam forms an arch, of which the continued 
 tie forms the abutments, and the smaller arches are merely to 
 connect the two parts and give stability to the whole. 
 
 The connexion thus formed is necessary for supporting the 
 tie ; and in consequence of this connexion the effect of the 
 straining force will be similar to that on a solid beam. 
 Several girders and beams for floors have been formed on this 
 principle ; and a simple method of proportioning them will be 
 found in the second Section. 
 
 All the parts should be kept as nearly as convenient of the 
 same bulk, to prevent irregular contraction. 
 
 42. If the load be distributed in any other manner, the 
 curve should be the proper curve of equilibrium for that load.* 
 
 For if it be not the proper curve, partial strains will be 
 produced in the beam, which will impair its strength. The 
 curve of equilibrium should pass everywhere at the middle of 
 the depth of the curved part of the beam, and should meet 
 the axis of the straight tie in the centres of the supports 
 
 * The method of finding the curve of equilibrium is shown in my "Elementary 
 Principle's of Carpentry," Sect. T. art. 47-61. 
 
40 OF THE STRONGEST FORM OF SECTION. [SECT. IT. 
 
 upon which the beam rests. Thus A C being the curve of 
 equilibrium, A D the axis or centre line of the tie ; A B 
 should be the centre of the support on which the beam rests. 
 
 43. If the load be applied at one point, the upper side 
 should be formed of two straight lines, meeting in the point 
 where the load is to rest, as at A in fig. 12. 
 
 The openings should be disposed as may best answer the 
 purpose for which the beam is intended, but they may 
 generally be from 2 to 3 feet each. When such beams, as 
 fig. 11, are used as girders, the openings receive the binding 
 joists instead of mortises. 
 
 44. When a beam is to bear a load at one end, the other 
 being fixed ; or when a beam is loaded at both ends and the 
 support is in the middle ; then the tie should be the upper 
 part of the beam : it should obviously be straight for the 
 reasons already assigned; and the other parts should be 
 straight also, except the small degree of curvature which 
 would cause the weight of the part to be balanced by the 
 forces concerned. Indeed, the arrangement for this strain 
 should be the same as fig. 12 inverted, the support being at 
 A, and the load at B and D. 
 
 45. But when the load is uniformly distributed over the 
 length, the lower side A C, in the annexed figure, should be 
 curved ; the proper curve for an uniform load being a 
 common parabola with its vertex at A. By a combination of 
 
SECT, iv.] OF THE STRONGEST FORM OF SECTION. 41 
 
 such beams, a bridge might be formed which would have no 
 lateral pressure on its piers or abutments. C D being one of 
 the piers, the distance between the points C and D may very 
 easily be so arranged, that a given force at A or B would not 
 disturb the equilibrium of the frame. 
 
 A bridge of this kind would not be affected by contraction 
 and expansion ; because no connexion would be necessary at 
 
 the junction of the beams at A, but such as would allow of 
 the motion of contraction or expansion without injury. 
 
 In a design for a large bridge on this principle, which I 
 made some years ago, it was contrived to put together in 
 parts, without the assistance of centering ; the open work of 
 the spandrils being composed of vertical and diagonal supports 
 and braces. ^ LI B R A K i 
 
 UNIVERSITY OJ 
 
 OF THE STRONGEST FORM OF SECTION FOR REVOLVING 
 
 SHAFTS. IT CALIFORNIA. 
 
 46. When a beam revolves, while Ihe straining force 
 continues to act in the same direction upon it, that form is 
 obviously the best which is of the same strength to resist a 
 stress at any point of the perimeter of its section, and the 
 circle is the only form of section which has this property.* 
 
 * This conclusion Las been objected to by Navier (Application de la Me'cauique, 
 note to art. 494) iu the following terms : 
 
 " The most convenient figure for axes of rotation is made a subject of inquiry in 
 the Pi-actual Essays on Mill-work by Buchanan, with notes by T. Tredgold, and re- 
 
42 OF THE STRONGEST FORM OF SECTION. [SECT. TV. 
 
 If a shaft be of any other form than cylindrical, the flexure 
 will be different in different parts of the revolution, and 
 therefore the motion will be unsteady, and particularly in 
 new work. In a square shaft (and such shafts are chiefly 
 employed), the resistance to pressure at one point is to the 
 resistance to the same pressure at another point in the peri- 
 meter as ten is to seven nearly (art. 112). In feathered 
 shafts, that is, shafts of which the section is similar to fig. 13, 
 the resistance is more regular, but not perfectly so.* 
 
 For the same reasons, the sections of the masts of vessels 
 should be circular. 
 
 47. As the circle is the best form for the section of a shaft, 
 a hollow cylinder will be the strongest and stiffest form for a 
 shaft ; and the same form is also best calculated for resisting 
 a twisting strain to which all shafts are more or less exposed. 
 
 The idea of making hollow tubes for resisting forces that 
 often change their direction, has been undoubtedly borrowed 
 from nature ; but in art we cannot pursue the principle to so 
 much advantage, because it is difficult to make a perfect 
 casting of a thin tube; and in shafts, &c., of small diameter, 
 it is much greater economy to make them solid. 
 
 It is usual to make hollow tubes of uniform diameter with 
 
 edited by George Rennie, F.R.S., and in the Practical Essay on the Strength of Cast 
 Iron. Mr. Tredgold appears to think that the circle is the only figure which gives to 
 the axes the property of offering in every direction the same resistance to flexure. 
 This error proceeds from his considering the resistance to flexure as being measured 
 by the expression which measures the resistance to rupture. We have already re- 
 marked that a square section gave the same resistance to flexure in the direction of 
 the sides and of the diagonals. But moreover, this section gives an equal resistance 
 in every direction ; and the same is the case with regard to a great number of figures, 
 which may be formed by combining in a symmetrical manner the circle and the 
 square. It thence results that if the axes strengthened by salient sides, which the 
 English call feathered shafts, do not answer as well as square axes, or full cylindrical 
 ones, this arises probably from their not being as well disposed to resist torsion, and 
 not from the inequalities of flexure of these axes." EDITOR. 
 
 * In heavy astronomical instruments, and in all machines where steady and accu- 
 rate movements are necessary, every attention should be paid to the effect of flexure. 
 Irregularity may be diminished by excess of strength, but it cannot be wholly 
 removed. The reader who wishes to pui-sue this subject, as far as regards astrono- 
 mical instruments, may consult the Philosophical Magazine, vol. Ix. p. 338, and vol. 
 Ixi. p. 10. 
 
SECT, iv.] OF THE STRONGEST FORM OF SECTION. 43 
 
 gudgeons cast separate, to fix at the ends. The manner of 
 calculating the stiffness of hollow tubes for shafts is shown 
 in art. 259 and 260, and an easy popular mode at art. 18. 
 When they are applied to other purposes, consult art. 178, 
 and those following it in the same proposition. 
 
SECTION V. 
 
 AN ACCOUNT OF SOME EXPERIMENTS ON THE RESISTANCE 
 OF CAST IRON. 
 
 48. There have been very few experiments made on the 
 resistance of cast iron, in which the degree of flexure pro- 
 duced by a given weight has been measured; but the few 
 that have come to my knowledge, and that are sufficiently 
 described to admit of comparison, I purpose to compare with 
 the rules I made use of in calculating the Tables in this 
 work ; and to add several new experiments. 
 
 MR. BANKS'S EXPERIMENTS.* 
 
 49. Mr. Banks made some experiments on cast iron, and 
 noticed the deflexion, but only at the time of fracture. These 
 experiments were made at a foundry at Wakefield. The iron 
 was cast from the air-furnace ; the bars one inch square, and 
 the props exactly a yard distant. One yard in length 
 weighed exactly 9fts., excepting one, which was about half 
 an ounce less, and another a very little more. They all bent 
 about an inch before they broke. 
 
 1st bar broke with 963 Ibs. T 
 
 2d bar broke with 958 " I Mean 
 
 3d bar broke with 994 " J97l|lbs. 
 
 4th bar, made from the cupola, broke with 864 " 
 
 * From a treatise " On the Power of Machines," by John Banks. Kendall, 1803, 
 p. 96. 
 
SECT, v.j EXPERIMENTS ON CAST IRON. 45 
 
 50. Now the rule according to which the first Table was 
 calculated is expressed by the equation 
 
 001 W L 2 = B D 3 , 
 
 in which the weight in pounds is denoted by W, the length 
 in feet by L, the breadth in inches by B, the depth in inches 
 by D, and the number '001 is a constant multiplier, which I 
 shall sometimes denote by a. 
 
 The rule determines the dimensions for a deflexion of as 
 
 many fortieths of an inch as there are feet in length, or ^ ; 
 and if d be the deflexion in inches determined by experiment, 
 we have 
 
 which being substituted for the weight in the equation above 
 it, becomes 
 
 J^BJ)., 
 
 40 d 
 
 40 B D 3 d 
 
 ovooi- = - ^j^. 
 
 The equation, in this form, may be called a formula of 
 comparison, as when the value of a determined by it is the 
 same I have used, or nearly the same, it will be evident that 
 the Table is calculated from proper data. 
 
 51. Taking the mean of the first three of Mr. Banks's 
 experiments, we have 
 
 40 B D 3 d 40 
 
 And in the bar from the cupola, or fourth experiment, 
 
 40BD3,* 40 
 
 W L 3 864 x 27 
 
 
 The experiments of Mr. Banks indicate therefore that he 
 had employed iron of a more flexible quality, but they are 
 not sufficiently accurate for establishing the elements of a 
 practical rule, because the deflexion was not correctly 
 
EXPERIMENTS ON CAST IRON. 
 
 [SECT. v. 
 
 observed, nor observed at a proper stage of the experiment. 
 For when a bar is strained nearly to the point of fracture, the 
 deflexion becomes extremely irregular, and increases more 
 rapidly than in the simple proportion of the weight, (see art. 
 56, 63, 65, and 67,) and consequently must give a much 
 higher value to a than the true one, as we find to be the case 
 with these experiments. 
 
 M. RONDELET'S EXPERIMENTS.* 
 
 52. M. Rondelet has described some experiments on different 
 kinds of cast iron in his work on building, which were made 
 upon specimens of 1*066 inches square, supported at the ends, 
 and loaded in the middle of the length. 
 
 M. Rondelet' s First Experiments. Distance between the Supports 3*83 feet. 
 
 Weight in tt>s. 
 
 134 
 
 201 
 
 268 
 
 335 
 
 Remarks, &c. 
 
 Kind of iron. 
 
 Defl. 
 inch. 
 
 Defl. 
 inch. 
 
 Defl. 
 inch. 
 
 Defl. 
 inch. 
 
 
 1. Gray cast iron 
 2. Do. do. 
 
 089 
 156 
 
 .'2 
 
 313 
 
 357 
 38 
 
 49 
 
 49 
 
 Broke with 482 ffis. 
 Broke with 482 lt>s. 
 
 
 
 
 
 2) -98 
 
 49 mean of deflexions, with 
 335 K>s. 
 
 3. Soft cast iron 
 4. Do. do. 
 5. Do. do. 
 6. Do. do. 
 
 134 
 0223 
 089 
 089 
 
 313 
 
 067 
 156 
 178 
 
 466 
 134 
 245 
 29 
 
 62 
 
 2 
 38 
 445 
 
 Broke with 700Tbs. 
 Broke with 1140 KM. 
 Broke with 375 Ibs. 
 Broke with 605 Ibs. 
 
 
 
 
 
 4)1 '645 
 
 411 mean of deflexions, 
 
 with 335lbs. 
 I 
 
 Extracted from his Traitd Thdorique et Pratique de 1'Art de Batir, tome iv. 
 
SECT. V.] 
 
 EXPERIMENTS ON CAST IRON. 
 
 47 
 
 M. Rondelet's Second Experiments. Distance between the Supports 1' 91 5 feet. 
 
 Weight in fts. 
 
 322 
 
 483 
 
 644 
 
 805 
 
 Remarks, &c. 
 
 Kind of iron. 
 
 Den. 
 inch. 
 
 Defl. 
 inch. 
 
 Defl. 
 inch. 
 
 Defl. 
 inch. 
 
 
 1. Gray cast iron 
 2. Do. do. 
 
 067 
 0445 
 
 089 
 089 
 
 112 
 
 134 
 
 Broke with 580 Ibs. 
 Broke with 1063 Ibs. 
 Mean of deflexions, with 
 483 Ibs. is -089 inch. 
 
 3. Soft cast iron 
 4. Do. do. 
 
 0445 
 0445 
 
 089 
 067 
 
 2) -156 
 
 134 
 134 
 
 153 
 
 Broke with 1770 Ibs. 
 Broke with 1360 Ibs. 
 
 Mean of deflexions with 
 483 fcs. is -078 inch. 
 
 078 
 
 In order to compare these results with the formula used in 
 calculating the Tables, I have taken the mean deflexions 
 corresponding to the load of 33 5 fibs, in the long pieces, 
 and to 483 fibs, in the short ones ; and in the gray cast 
 iron. 
 
 For the long lengths a = '00134 
 
 For the short lengths a - '00135 .. 
 
 In the soft cast iron, 
 
 For the long lengths a = '00112 
 
 For the short lengths a = '0011S 
 
 These values of a were calculated by the formula of com- 
 parison given in art. 50, and the latter ones nearly agree with 
 that employed to calculate the Table. 
 
 MR. EBBELS S EXPERIMENT. 
 
 53. According to a trial communicated to me by Mr. R. 
 Ebbels, a bar of cast iron, 1 inch square, and supported at 
 the ends, the distance of the supports being 3 feet, the 
 deflexion in the middle was ^ths of an inch, with a weight of 
 308 fts. suspended from the middle. The iron was of a hard 
 
48 EXPERIMENTS ON CAST IRON. [SECT, v 
 
 kind, not yielding very easily to the file; it was cast at a 
 Welsh foundry. 
 
 In this trial we have 
 
 4Q * 3 m -000902 = a 
 L 3 W 27 x 308 x 16 
 
 Consequently, iron of this kind is about ^th stronger than 
 that which the Table is calculated from, or rather it would 
 bend ^th part less under the same strain. 
 
 Experiment 1. 
 
 54. A joist of cast iron of the form described in fig. 9, 
 Plate L, was submitted to the following trials. It was 
 supported at the ends only ; the distance between the 
 supports 19 feet, and placed on its edge. The deflexion 
 from its own weight was 4^ ths of an inch. 
 
 When it was laid flatwise, the deflexion from its own 
 weight was 3*5 inches, the distance of the supports remaining 
 19 feet. 
 
 The whole depth a d, fig. 9, was 9 inches, the breadth, a b, 
 was 2 inches ; the depth of the middle part, e /, was 7-Jr 
 inches ; and the breadth of the middle part f ths of an inch. 
 
 55. It may be easily shown that to derive the value of a, 
 from the experiment on the edge, we may use an equation of 
 this form, (see art. 192 and 215,) 
 
 = 40BD 3 d(l -p 3 g) = 64 B^D 3 d (1 -p*g) t 
 | W L 3 W L :j 
 
 hi which D is the whole depth, and p D the depth of the 
 middle part, and B the whole breadth, and q B the breadth 
 after deducting that of the middle part. 
 
 In our experiment D = 9 inches, and p D = 7*5, or 
 p = -833. Also, B = 2 inches, and deducting f ths, the 
 breadth of the middle, we have q B = 1*25, or q '625. 
 And' the weight of the part of the joist between the supports 
 being 540 fts., we find a = '00124. 
 
SECT, v.] EXPERIMENTS ON CAST IRON. 49 
 
 The equation for finding the value of a, in the experiment, 
 with the joist flatwise, is 
 
 Where 
 
 D = 2 inches, B = 9 7'5 = 1-5, p = -75 7'5 
 
 I consider the value of a derived from the experiment with 
 the joist flatwise as nearest the truth, because the deflexion 
 was so considerable, that a small error in measuring it would 
 not sensibly affect the result, while there must be ^ some un- 
 certainty in ascertaining so small a deflexion as / O ths of an 
 inch in 19 feet ; and a very small error in this measure would 
 cause the difference between the results. I have, however, 
 given it, as I determined it at the time, and the manner of 
 calculation may be useful in other cases. If the mean be 
 taken between the results, it is 
 
 00124 + -00092 _ 
 
 In the experiment flatwise, we obtain a constant multiplier 
 extremely near to that determined from a bar of the same iron 
 an inch square, and 34 inches long (art. 57), and it differs 
 only about j^th part from the one employed for calculating 
 the Table, page 12, art, 5. 
 
 Experiment 2. 
 
 56. I now purpose describing the direct experiments I have 
 made for obtaining the constant multipliers used in this work ; 
 I call experiments direct when known weights are applied as 
 the straining force, without the intervention of mechanical 
 powers, without loss of effect from friction, or a risk of error 
 in estimating the quantity of force, when the yielding of the 
 supports cannot affect the measure of the deflexion, and when 
 the deflexion can be accurately measured. -.[,, 
 
 The iron I used was soft gray cast iron ; it yielded easily 
 
50 
 
 EXPERIMENTS ON CAST IRON. 
 
 [SECT. v. 
 
 to the file, and extended a little under the hammer, before it 
 became brittle and short.* 
 
 The first experiment was made with a bar of an inch square, 
 cast by Messrs. Dowson, London, with the supports 34 inches 
 apart; the weights were placed in a scale suspended from the 
 middle of the length; the load was increased by 10 fts. at a 
 time, and the deflexion measured each time, the quantity of 
 deflexion being multiplied by means of a lever index. The 
 whole time of making the experiment was nearly four hours ; 
 the thermometer varying from 65 to 66 degrees. Only half 
 the number of observations is inserted here. 
 
 Weight 
 in tts. 
 
 Defl. in 
 inch. 
 
 Remarks. 
 
 Weight 
 inlbs. 
 
 Defl. iii 
 inch. 
 
 Remarks. 
 
 20f 
 
 02 
 
 
 
 240 
 
 13 
 
 
 40 
 
 03 
 
 
 
 260 
 
 14 
 
 
 60 
 
 80 
 100 
 
 04 
 05 
 06' 
 
 
 
 280 
 300 
 320 
 
 15 
 16 
 
 17 
 
 j unloaded, and it returned to its 
 \ natural state. 
 
 120 
 
 07 
 
 
 
 340 
 
 18 
 
 
 140 
 
 08 
 
 
 
 360 
 
 19 
 
 
 160 
 
 09 
 
 
 
 380 
 
 20 
 
 
 
 
 
 unloaded, and it 
 
 400 
 
 21 
 
 
 180 
 
 200 
 220 
 
 10 
 
 11 
 
 12 
 
 
 returned to its 
 natural state. 
 
 410 
 
 22 
 
 ( deflexion became irregular ; and 
 )when the load was removed, it 
 j had taken a permanent set, with 
 ( a curvature of '015 inch. 
 
 From this experiment we find that the deflexion of cast iron 
 is exactly proportional to the load, till the strain arrives at a 
 certain magnitude, and it then becomes irregular ; and at or 
 near the same strain a permanent alteration takes place in the 
 structure of the iron, and a part of its elastic force is lost. 
 The same thing occurs in experiments on other metals : I 
 have tried wrought iron, tin, zinc, lead, and alloys of tin and 
 lead, with a view to measure their elastic forces, and the 
 strains that produce permanent alteration. 
 
 * A considerable degree of malleability is a good quality in cast iron for bearing 
 purposes, because it lessens the risk of sudden failure. The iron was a mixture of 
 Butterly iron, two parts, with one part of old iron. 
 
 t The weight of the scale, 8 Ifcs., ought to have been added. 
 
SECT, v.] EXPERIMENTS ON CAST IRON. 51 
 
 57. According to this experiment, 
 
 40 B D 3 d _ 40 x -21 
 ~ 400 x 227 -' 00092 = a ' 
 
 Experiment 3. 
 
 58. The next experiments were made with an uniform bar 
 of iron, cast by Messrs. Dowson, 3 inches by 1 and 1^ inches, 
 and 6*5 feet between the supports. When this bar was 
 placed on its edge, and loaded in the middle with 
 
 150 Ibs. the deflexion in the middle was 1 fortieth of an inch. 
 290 Ibs. V .V . . . .2 do. 
 
 360 Ibs. ,0/Mi iff'/ ?, . . 24 do. 
 
 440 Ibs 3 do. 
 
 The same deflexions were observed in removing the load, 
 and it perfectly regained its natural state. Whence we have, 
 
 40BD 3 d 1-5 x 27 x 3 
 
 440 x 271-625 = ^S nearly = a. 
 
 Experiment 4. 
 
 59. The same piece, with the supports at the same distance, 
 placed flatwise, and loaded in the middle with 
 
 180 Ibs. the deflexion in the middle was 5 fortieths of an inch. 
 360 Ibs. . v f ..- r . . r V . 10 do. 
 
 The bar restored itself perfectly when the weights were 
 removed, and the trial was repeated with the same results ; 
 the load, of 3601bs., remained upon it ten hours without 
 impairing its elastic force, or increasing the deflexion in the 
 slightest degree. 
 
 60. From this and the preceding experiment, the ratio of 
 the breadth and depth to the quantity of deflexion may be 
 compared when the weight is the same. According to the 
 theory of the resistance to flexure (art. 256), 
 
 K2 
 
52 EXPERIMENTS ON CAST IRON. [SECT. v. 
 
 and for the weight of 360fts. we have 
 
 as it was found to be by experiment. 
 
 To find the constant multiplier from the last experiment, 
 we have 
 
 40BD 3 d 3 x 3-375 x 10 
 
 -WT3- = 360 x 274-625 == ' Q102 = a ' 
 
 This value of a does not exactly agree with the one 
 calculated from the first experiment on the same piece ; but 
 it is as near as can be expected in a case of this kind; and in 
 a practical point of view it is as near an approach to accuracy 
 as the nature of the subject requires. 
 
 Experiment 5. 
 
 61. I was desirous of trying the effect of an uniformly 
 distributed load, and my weights, which. are cubical pieces of 
 cast iron, all of the same size, and each weighing lOfts., are 
 very well adapted for the purpose. 
 
 The same piece that was used for the last experiment was 
 laid flatwise upon supports, the supports being 6 feet 6 inches 
 apart, and 18 weights (in all ISOfts.) were laid along the 
 upper side, just so as to be clear of one another, in the 
 manner shown in fig. 2, Plate I. The deflexion produced by 
 these weights was f- ih$ of an inch. 
 
 A second tier of weights being added, making the whole 
 weight upon the bar 360fts., the deflexion was ^ths of an 
 inch. 
 
 62. Hence it appears, that when the weight is uniformly 
 distributed over the length, the deflexion is directly as the 
 weight. 
 
 And comparing this with the preceding experiment, it 
 appears, that the deflexion from the weight uniformly dis- 
 tributed over the length, is to the deflexion from the same 
 weight applied in the middle of the length, as 6 is to 10. 
 
SECT. V.] 
 
 EXPERIMENTS ON CAST IRON. 
 
 53 
 
 The proportion obtained by theoretical investigation is as 5 
 is to 8 ; but as 6 : 10 : : 5 : 8^. This small difference arises 
 undoubtedly from error in measuring the deflexions in the 
 experiments. 
 
 To compare the value of the constant multiplier by this 
 experiment, the equation 
 
 1 A I 13 LV xx tV 
 
 = 
 
 UNIVERSITY 
 
 W L 3 
 
 must be used, whence we find a ='00098 
 
 I CAL1FORN 
 
 Experiment 6. 
 
 63. This experiment was made upon a piece of iron cast 
 by Messrs. Bramah, of Pimlico, London. It crumbled sooner 
 under the hammer than that used in the preceding experi- 
 ments, and did not yield quite so readily to the file ; it was 
 regular and fine-grained. 
 
 The piece was uniform, and ^ths of an inch square ; the 
 supports were 3 feet apart, and the weight was applied in the 
 middle of the distance between the supports. 
 
 Weight 
 in Ibs. 
 
 Defl. in 
 inch. 
 
 
 Remarks. 
 
 Weight 
 in Ibs. 
 
 Defl. in 
 inch. 
 
 Remarks. 
 
 20 
 
 02 
 
 
 
 220 
 
 225 
 
 
 40 
 
 04 
 
 
 
 240 
 
 245 
 
 
 60 
 
 00 
 
 
 ' When unloaded it returued to 
 
 260 
 
 27 
 
 When this load 
 
 80 
 
 08 
 
 
 its original form : loaded again, 
 
 280 
 
 293 
 
 had been on 20 
 
 100 
 
 10 
 
 
 the deflexion was the same, and 
 
 300 
 
 318 
 
 minutes, it be- 
 
 120 
 
 12 
 
 
 it remained loaded 1 2 hours with- 
 
 320 
 
 34 
 
 came '32 inch. 
 
 140 
 
 14 
 
 
 out sensible increase, when on 
 
 340 
 
 365 
 
 
 160 
 180 
 
 162 
 183 
 
 - 
 
 being unloaded it was found to 
 have acquired a permanent set of 
 
 360 
 380 
 
 392 
 42 
 
 
 200 
 
 21 
 
 
 02 in. The index was set to 
 
 400 
 
 445 
 
 
 
 
 
 nothing, and the weights pro- 
 
 420 
 
 475 
 
 
 
 
 
 duced the same deflexions as at 
 
 440 
 
 5 
 
 
 
 
 
 first ; and it was further loaded 
 
 460 
 
 532 
 
 j which became in 
 
 
 
 
 Las described. 
 
 480 
 
 57 
 
 ( an hour '58. 
 
 When the weights were removed, the piece retained a per- 
 manent deflexion of "075 inch ; but it was several hours 
 before it returned to that curvature. I did not break the 
 specimen, because I had not weight enough by me for that 
 
EXPERIMENTS ON CAST IRON. 
 
 [SECT. v. 
 
 purpose, neither would it have given a fair measure of the 
 strength of the iron after the trials I have described ; but I 
 hope the effect of these trials will make the reader sensible of 
 the necessity of limiting the strain within the range of the 
 elastic force of the material. 
 According to this experiment, 
 
 40BD 3 cJ 40 x -9 4 x -21 
 
 COMPARISON OF THE PRECEDING EXPERIMENTS. 
 
 64. If the mean value of the constant a be taken for the 
 experiments from art. 53 to 63, it is 0-0010446. The 
 number used in calculating the first Table (art. 5, p. 9) was 
 O'OOl, a sufficiently near approximation, with the advantage 
 of much simplicity. 
 
 Experiments 7, 8, and 9. 
 
 65. The next trials were made with specimens formed as 
 shown in fig. 4, Plate I., with the deepest part C D exactly 
 in the middle of the length, and the depth, at CD, 0'975 
 inch; the depth EA and BF were each half that at C D. 
 The distance of the supported points A B was 3 feet, and the 
 breadth of the bars 0'75 inch. The load was suspended from 
 the point C in the middle of the length, and the deflexion was 
 measured at the same point: the load was increased by lOfts. 
 at a time. 
 
 Weight acting on 
 the bar. 
 
 1st Specimen. 
 Deflex. produced. 
 
 2nd Specimen. 
 Deflex. produced. 
 
 3rd Specimen. 
 Deflex. produced. 
 
 ibs. 
 
 in. 
 
 in. 
 
 i 
 in. 
 
 40 
 
 052 
 
 065 
 
 052 
 
 80 
 
 104 
 
 13 
 
 105 
 
 120 
 
 16 
 
 19 
 
 16 
 
 160 
 
 215 
 
 25 
 
 21 
 
 180 
 
 245 
 
 28 
 
 24 
 
 200 
 
 272 
 
 32 
 
 265 
 
 500 
 
 84 
 
 
 
 540 
 
 Broke. 
 
 
 
OOF. v.] EXPERIMENTS ON CAST IROK 55 
 
 On the first specimen the load of ISOfts. remained twelve 
 hours ; the deflexion did not sensibly increase, and it returned 
 to its natural form when unloaded ; it was again loaded to 
 200fts., which remained upon it two hours ; it was then un- 
 loaded again, and was found to have taken a permanent set 
 with a deflexion of * 00 5 inch. The specimen was then loaded 
 again, and the deflexions observed at every 20fts. : the de- 
 flexion produced by the addition of 20fts. was at first '026, 
 became '03, "04, and towards the end of the experiment *05. 
 When the load had been increased to 360fts., in every 
 succeeding addition of lOfts. I observed that the deflexion 
 increased by starts of as much as jiToth of an inch each, 
 which appeared to be caused by the ends sliding on the sup- 
 ports, at the moment the weight was added ; the bar emitted 
 a slight crackling noise, like that produced by bending a 
 piece of tin. There was a small defect in the bar at the place 
 where it broke, which was 4 inches distant from the middle. 
 
 When the second specimen was unloaded, immediately, 
 from a weight of 200fos. it barely returned to its natural 
 form; but a load of ISOfts. produced a permanent deflexion 
 of '005 when it remained upon it fourteen hours. 
 
 The load of 200Sbs. remained twenty-one hours upon the 
 third specimen, and when it was unloaded the index returned 
 to zero ; therefore this strain was less than would produce a 
 permanent set. The set was nearly '01 when the load was 
 increased to 210fcs., and remained upon it ten hours. It 
 was a smoother and better casting than the other specimens. 
 
 There did not appear to be any sensible difference in the 
 quality of the iron in these specimens, except that the second 
 specimen was more brittle under the hammer than the other 
 two. They were all fine grained, and yielded easily to the 
 file. They were cast by Messrs. Bramah. 
 
 66. I was proceeding with a trial of a piece of the same 
 kind of iron, formed as described in fig. 4, Plate I., when it 
 broke uddenly, at about a foot from the end, at an air 
 
56 
 
 EXPERIMENTS ON CAST IRON. 
 
 [SECT. v. 
 
 bubble. The bubble was not apparent on the surface, and 
 yet so near it, that a slight stroke of a hammer would have 
 broken into it. Founders should be very careful to avoid 
 defects of this kind ; and beams to sustain great weights 
 should always be proved to a deflexion within their range of 
 elasticity before they are used. 
 
 Experiments 10, 11, and 12. 
 
 67. These trials were made on three pieces of uniform 
 breadth and depth, with the supports 3 feet apart, the load 
 being applied in the middle of the length. The depth *9 
 inch, and the breadth the same. 
 
 Weight acting on 
 the bar. 
 
 1st Specimen. 
 Deflex. produced. 
 
 2nd Specimen. 
 Deflex. produced. 
 
 3rd Specimen. 
 Deflex. produced. 
 
 Ibs. 
 
 in. 
 
 in. 
 
 in. 
 
 40 
 
 041 
 
 042 
 
 041 
 
 80 
 
 082 
 
 09 
 
 08 
 
 120 
 
 124 
 
 136 
 
 12 
 
 160 
 
 165 
 
 18 . 
 
 16 
 
 180 
 
 185 
 
 202 
 
 18 
 
 200 
 
 206 
 
 
 20 
 
 The load of 200 fts. remained twelve hours on the first 
 specimen, and when it was unloaded the quantity of per- 
 manent deflexion was barely sensible ; and it was loaded and 
 unloaded again with the same result. 
 
 The load of 180 fts. remained three hours on the second 
 specimen ; it had not increased the deflexion, but when the 
 load was removed, it was found that the bar had acquired a 
 permanent set of nearly y^oth of an inch. 
 
 In the third specimen the bar returned perfectly to its 
 natural form when the load was removed, after being upon 
 it three hours. 
 
 Of these specimens the third was the most brittle under 
 the hammer, and the hardest to the file; there was not a 
 sensible difference between the other two > both were soft 
 iron. These specimens were cast by Messrs. Bramah. 
 
SECT, v.] EXPERIMENTS ON CAST IRON. 57 
 
 68. The chief object in view in the experiments No. 2, 6, 
 7, 8, 9, 10, and 11, was to determine the strain a square 
 inch of cast iron would bear without permanent alteration, 
 and the extension corresponding to that strain. Calling f 
 this strain in pounds, the experiment 2 gives /= 15,300 ibs. 
 = 6*8303 tons, as calculated in art. 143 ; and the others 
 being calculated by the same formula, in experiments 6, 10, 
 and 12,/ = 14,814 ; in experiments 7, 8, and 9,/ = 15,160 ; 
 and in experiment 11, /= 13,333 fts. The greatest dif- 
 ference amounts to about ^th of the highest value of/; but, 
 in the experiment 2, the load was taken off after remaining 
 only about ten minutes on the bar ; in the others it remained 
 for several hours. The former I consider most strictly 
 applicable to practice ; and yet it was desirable to show that 
 a force acting a considerable time will produce a permanent 
 set, Avhen the same force could riot produce it in a few 
 minutes. 
 
 69. In art. 212, it is calculated that the extension pro- 
 duced by the strain of 15,300 ibs. in experiment 2, was j^- 4 
 of the length;* and by the same mode of calculation the 
 extension in experiment 6 is found to be ^- 3 , in experiment 
 
 10, jjjg, in experiment 11, ^, and in experiment 12, j^. 
 Also, by the equation, art. 127, the extension in experiment 
 7 is found to be y^, in experiment 8, ^y^, an( * experiment 
 
 ^ 136T 
 
 The difference between the extension in the 8th and 9th 
 experiments is the most considerable ; and the mean between 
 these is ^^, which differs very little from y^, the number 
 used in the rules. 
 
 70. A Table of the chief experiments that have been made 
 
 * The extension in experiment 2 has been re-calculated, and found to be the same 
 as here stated, by Professor Leslie, whose mode of calculation is different. See 
 Leslie's Elements of Natural Philosophy, vol. i. p. 240. Edinburgh, 1828. 
 
EXPERIMENTS ON CAST IRON. 
 
 [SECT. v. 
 
 on the absolute strength of cast-iron bars to resist a cross 
 strain, the bars supported at the end, and loaded in the 
 middle. 
 
 No. 
 
 Description. 
 
 Length | Dimensions 
 between. at the 
 the sup- : strained 
 ports in I point in 
 feet. inches. 
 
 | 
 
 Weight in 
 fts. that 
 broke it. 
 
 Calculated 
 weight that 
 would de- 
 stroy elastic 
 force in Ibs. 
 
 Ratio of the; 
 calculated 
 weight to 
 the breaking ; 
 weight. 
 
 
 
 ft. in. jbrdth. dpth. 
 
 
 
 
 1 
 
 Uniform bar 
 
 30!1 1 
 
 756 
 
 283 
 
 1 :2-7 
 
 ' 2 
 
 Ditto 
 
 3 I 1 1 
 
 735 
 
 233 
 
 1 :2'6 
 
 3 
 
 Ditto 
 
 2 6 i 1 1 
 
 1008 
 
 340 
 
 1 :2-96 
 
 4 
 
 Ditto 
 
 3 | 1 1 
 
 963 
 
 283 
 
 1 :3'4 
 
 5 
 
 Ditto 
 
 301 1 
 
 958 
 
 283 
 
 1 : 3-38 
 
 6 
 
 Ditto 
 
 3 j 1 1 
 
 994 
 
 283 
 
 1:3-5 
 
 7 
 
 Ditto cast from ) 
 the cupola. \ 
 
 30:1 1 
 
 864 
 
 283 
 
 1 : 3-05 
 
 8 
 
 Parabolic bar cast ) 
 from the cupola. } 
 
 30 11 
 
 874 
 
 283 
 
 1:3-08 
 
 9 
 
 Uniform bar 
 
 80:1 1 
 
 897 
 
 283 
 
 1:3-17 
 
 10 
 
 Ditto 
 
 2 8 i I 1 
 
 1086 
 
 318-75 
 
 1:3-4 
 
 11 
 
 Ditto 
 
 ] 4 ! 1 1 
 
 2320 
 
 637-5 
 
 1:3-6 
 
 12 
 
 Ditto 
 
 2 8 i 2 4 
 
 2185 
 
 637-5 
 
 1 : 3-42 
 
 13 
 
 Ditto 
 
 1 4 j 2 
 
 4508 
 
 1275 
 
 1 : 3-53 
 
 14 
 
 Ditto 
 
 2 8 ! 3 i 
 
 3588 
 
 956-25 
 
 1 : 3-63 
 
 15 
 
 Ditto 
 
 1 4 | 3 J 
 
 6854 
 
 1912-5 
 
 1 : 3-58 
 
 16 
 
 Ditto 
 
 28;4 i- 
 
 3979 
 
 1275 
 
 1 : 3-12 
 
 17 
 
 Semi-ellipse 
 
 2 8 : 4 | 
 
 4000 
 
 1275 
 
 1:3-14 
 
 18 
 
 Parabolic 
 
 284 
 
 3860 
 
 1275 
 
 1 : 3-03 
 
 
 ( Uniform strain in } 
 
 
 
 
 
 19 
 
 < the direction of > 
 
 2 8 JV2~ V2~ 
 
 851 
 
 224-5 
 
 1 : 3-79 
 
 
 (diagonal ) 
 
 
 
 
 
 
 
 j 
 
 
 
 
 The two columns on the right hand side are added to show 
 the relation between the load which permanently destroys a 
 part of the elastic force, and that which breaks the piece. It 
 will be seen that the load which would produce permanent 
 alteration, according to the formula as derived from my 
 experiments, is about ^rd of that which actually broke the 
 
 specimens ; in the worst kind tried, it is ^ of the breaking 
 
 weight. 
 
 In the preceding Table, the experiments 1, 2, and 3, were 
 made by Mr. Reynolds. No. 1 was twice repeated with the 
 same result. No. 2 is a mean of three experiments.* Hence 
 
 * Banks on the Power of Machines, p, 39. 
 
SECT, v.] EXPERIMENTS ON CAST IRON. 59 
 
 the mean ratio will be about 1 : 2*7. The experiments No. 4, 
 5, 6, 7, and 8, were made by Mr. Banks ; * the mean ratio 
 being 1 : 3'3. The rest were made by Mr. George Rennie, 
 and all of the bars of his experiments were cast from the 
 cupola; f the mean ratio being 1 : 3*4. 
 
 71. Allowing that the preceding experiments are sufficient 
 to fix with considerable certainty the utmost strain that ought 
 to exist in any structure of cast iron, still there is abundant 
 scope for new experimental research ; and that which perhaps 
 may be considered of most importance is the effect produced 
 by combining iron of different qualities. 
 
 Through the kindness of Mr. Francis Bramah, I am enabled 
 to begin this inquiry. He has furnished me twelve specimens, 
 of six different kinds of iron ; that is, two specimens of each 
 kind. Of these kinds three were run from pig iron from 
 different iron works ; one kind was run from old iron, usually 
 termed scrap iron ; another kind a mixture of old iron and 
 pig iron in equal parts, and the sixth kind pig iron with an 
 alloy of j^th of copper. 
 
 Before I begin to describe the experiments, it will be 
 proper to inform the reader what method I pursued in 
 making them. I knew, from previous trials, that the force 
 which produces a permanent set cannot be determined with 
 that precision which is necessary in comparing iron of 
 different kinds ; we can merely observe when it is, and when 
 it is not sensible ; and it is most likely that it becomes so by 
 gradations which we cannot trace. It was desirable to ascer- 
 tain whether a load equivalent to 15,300fts. upon a square 
 inch would produce a set or not ; and a load of 162 ibs., on 
 the middle of a bar of the size of the specimens, causes that 
 degree of strain : hence, in specimens of the same size, the 
 flexure by this load gives the comparative power of the 
 
 * Banks on the Power of Machines, p. 90. 
 
 t Philosophical Transactions for 1818, Part I,, or Philosophical Magazine, voL liii. 
 p. 173. . 
 
60 EXPERIMENTS ON CAST IRON. [SECT. v. 
 
 different kinds, particularly when compared with the quantity 
 of set produced by this or some additional load. But in 
 specimens of different sizes, the comparison is most easily 
 made by calculating the modulus of resilience, or resistance 
 to impulsion, which gives the toughness or relative power of 
 the material to resist a blow. Yet, even then it should be 
 tried what strain will produce permanent alteration, or that 
 which causes fracture, otherwise the comparative goodness of 
 the iron will not be known : I have tried both in each of the 
 varieties of iron. 
 
 For all purposes where strength is required, that iron is to 
 be esteemed the best which will bear the greatest degree of 
 flexure without set, and the greatest load. The worst and 
 most brittle pieces of iron have the greatest degree of stiff- 
 ness ; consequently the highest modulus of elasticity ; for 
 even the most flexible kind of iron is sufficiently stiff.* 
 
 In the iron which was taken as a good medium to calculate 
 from (see experiment, art. 56), we found 
 
 The force that it would bear without permanent alteration 15,300 tfes. 
 
 The extension in parts of the length extended . . . ^j 
 
 The modulus of elasticity for a base 1 inch square . . 18,400,000 fts. 
 
 The modulus of resilience '-'. " 127 
 
 These numbers being compared with the results of the 
 experiments now to be described will afford the means of 
 judging both of the qualities of the iron experimented upon, 
 and of the fairness of the mean data I have employed. 
 
 OLD PARK IRON. 
 
 72. Two specimens run from this kind of pig iron, each 
 
 * I have here followed the principles of comparing materials which were first given 
 in iny "Elementary Principles of Carpentry," art. 368373. The toughness is 
 measured by the same data as in that work, only here a general number of com- 
 parison is used instead of making one material a standard of comparison. The term 
 modulus of resilience, I have ventured to apply to the number which represents the 
 power of a material to resist an impulsive force ; and when I say that one material is 
 tougher than another, it is in consequence of finding this modulus higher for that 
 which is described as the toughest : see arts. 299 to 304, further on. 
 
SECT. V.] 
 
 EXPERIMENTS ON CAST IRON. 
 
 61 
 
 3 feet in length, and smooth, clean, and regular castings, 
 were first put in trial. The section of the bars rectangular ; 
 depth 0*65 inch; breadth 1*3 inches; the supports 2*9 feet 
 apart ; and the load suspended from the middle. 
 
 Weight applied. 
 
 Effect on 1st bar. 
 
 Effect on 2nd bar. 
 
 ibs. 
 
 in. 
 
 in. 
 
 60 
 
 bent 01 
 
 bent 01 
 
 120 
 
 0-2 
 
 0-203 
 
 162 
 
 0-265 
 
 0-275 
 
 182 
 
 0-305 small set. 
 
 n . q1 (set barely 
 ( (perceptible. 
 
 190 
 
 0-32 set -005 
 
 0-33 set -005 
 
 The iron was slightly malleable in a cold state ; yielded 
 easily to the file. The fracture dark gray with a little metallic 
 lustre ; fine grained and compact. 
 
 We may consider 162 fts. as the greatest load it would 
 bear without impairing its elastic force ; and 0*27 is the 
 mean between the flexures produced by this weight ; there- 
 fore, calculating on these data, we have 
 
 I 5 . 390 
 
 The strain it would bear on a square inch without permanent 
 alteration ......... . 
 
 Extension in length by this strain ..... 
 
 Modulus of elasticity for a base of an inch square . . 17,744,000 fts. 
 Modulus of resilience ........ 13'4 
 
 Specific gravity ......... r"OM 
 
 The absolute strength to resist fracture was tried by fixing 
 the bar at one end, the load being applied by fixing a scale 
 at the other end, and adding weights till the bar broke. The 
 second bar tried in this manner broke with 184 fts., the 
 leverage 2 feet ; fracture close to the fixed end, metal sound 
 and perfect at the place of the fracture.* 
 
 Hence, calculating by equation, art. 110, the absolute 
 cohesion of a square inch is 48,200 fts.,f or 3'15 times 
 
 * These are circumstances which must have place, otherwise the experiment does 
 not give a fair measure of the strength. 
 
 t Thia-' erroneous conclusion as to the great strength of cast iron, into which 
 
62 EXPERIMENTS ON CAST IRON. [SECT. v. 
 
 1 5,300 fts., the strain which has been found incapable of 
 causing permanent set. 
 
 Hence I infer, that this iron is superior in toughness, and 
 less stiff than the mean quality. 
 
 ADELPHI IRON. 
 
 73. The specimens of this iron were clean good castings of 
 the same dimensions as those of Old Park iron. That is, 
 depth 0*65 inch; breadth 1 '3 inches ; distance between the 
 supports 2 '9 feet. 
 
 Weight applied. 
 
 Effect on 1st bar. 
 
 Effect on 2nd bar. 
 
 fts. 
 60 
 120 
 162 
 182 
 
 in. 
 bent O'l 
 0-2 
 0'26 no set. 
 0-3 set -0075 
 
 in. 
 bent 0-1 
 0-205 
 0-27 no set. 
 0-305 set -005 
 
 Comparing this with the preceding experiments on Old 
 Park iron, it is sthTer, and sooner acquires a permanent set. 
 It is also somewhat harder to the file, and more brittle under 
 the hammer. The colour of the fracture was a lighter gray, 
 with less metallic lustre. 
 
 Its elasticity is not affected by the load of 1621bs. ; there- 
 fore 
 
 It will bear upon a squai'e inch without permanent alteration 15,390 Ibs. 
 And the mean of the two experiments gives the extension . n l ra 
 
 Modulus of elasticity for a base of 1 square inch . . . 18,067,000 fla. . 
 
 Modulus of resilience 13*1 
 
 SpeciEc gravity / . l - 7'07 
 
 The second bar, fixed at one end with a leverage of 2 feet, 
 broke with 173 fts. ; the fracture close to the fixed end, and 
 the place of fracture sound and perfect. 
 
 Navier, as well as Tredgold, has fallen, arises principally from a supposition that tlie 
 neutral line remains stationary during the flexure of the body. See " Additions," or 
 Notes to Arts. 68 and 143. -EDITOR. 
 
SECT, v.] EXPERIMENTS GIST CAST IRON. 63 
 
 According to this experiment, the absolute cohesion is 
 45,300 fts. for a square inch, or 2'96 times 15,300 tbs. 
 
 A comparison of these trials shows that the difference 
 between Adelphi and Old Park iron is not much, but that 
 the Old Park is superior, particularly in absolute strength ; 
 for it required 184 fbs. to break the one, and only 173 fts. to 
 break the other. 
 
 ALFRETON IRON. 
 
 74. There was not a sensible difference between the size of 
 these bars and the others. The depth 0'65 inch; breadth 
 1*3 inches ; distance between the supports 2*9 feet. 
 
 Weight applied. 
 
 Effect on 1st bar. Effect on 2nd bar. 
 
 Ibs. 
 60 
 120 
 162 
 183 
 
 
 in. 
 bent 01 
 0-2 
 0-27 no set. 
 0'31 small set. 
 
 in. 
 bent 01 
 0-195 
 0-28 no set. 
 0-325 small set. 
 
 This iron differs very little from Old Park, a little more 
 flexible, but very little. It seemed, if anything, somewhat 
 harder to the file, but of a less degree of malleability ; for 
 instead of extending, it crumbled under the hammer. Fracture 
 scarcely differing from that of Adelphi iron. 
 
 These bars bore 162 fts. without set, and the mean 
 deflexion was '275. Hence, 
 
 The iron would bear upon a square inch without permanent 
 
 alteration .-- ...) 15,390 fos. 
 
 Extension in length by this strain . . . . . 1I 1 T 
 
 Modulus of elasticity for a base of 1 inch square . . . 17,406,000 Ibs. 
 Modulus of resilience ;. . ;,. ~i . ,, r . ;.>.*,, r , ... . ,- 13'6 
 Specific gravity . '".'.' . .' T' . ... 7*04 
 
 The second bar, fixed at one end, broke with 153 Ibs., the 
 leverage being two feet, the fracture close to the fixed end, 
 and the metal sound and perfect at the place of fracture. 
 
 The absolute cohesion, according to this trial, is 40,0001bs. 
 for a square inch, or 2'63 times the force of 15, 300 Ibs. 
 
64 
 
 EXPERIMENTS ON CAST IRON. 
 
 [SECT. v. 
 
 This is a soft species of iron, and may answer extremely 
 well alone, for castings where strength is not required ; but 
 it is the weakest iron I have tried, and would most likely be 
 much improved by mixture. 
 
 SCRAP IRON. 
 
 75. These bars were run from old iron. They were 
 uneven on the surface, indicating that irregularity of shrink- 
 age which has been noticed in the Introduction (page 8). 
 The depth of the bars 0'65 inch; the breadth T3 inches;, 
 the distance between the supports 2*9 feet. 
 
 Weight applied. 
 
 Effect on 1st bar. 
 
 Effect on 2nd bar. 
 
 ibs. 
 
 in. 
 
 in. 
 
 60 
 
 bent 0-09 
 
 bent 0-09 
 
 120 
 
 018 
 
 0-18 
 
 162 
 
 0-25 no set. 
 
 0-255 no set. 
 
 180 
 
 0-28 no set. 
 
 0-285 no set. 
 
 190 
 
 0'3 small set. 
 
 0>q ( set not 
 " ' 1 certain. 
 
 210 
 
 34 set -005 
 
 0-34 set -004 
 
 This iron was very hard to the file, and very brittle, frag- 
 ments flying off when hammered on the edge, instead of 
 indenting as the preceding specimens. 
 
 The fracture dead or dull light gray ; no metallic lustre ; 
 not very uniform ; fine grained. 
 
 These bars showed no sign of permanent set with a load 
 of 1 SOfts.; but, to whatever cause this greater range of 
 elastic power may be owing, it certainly would be unsafe to 
 calculate upon it in practice. I shall therefore consider the 
 load of 162 fts., and a flexure of 0*25 inch, the data to 
 calculate from ; accordingly, the 
 
 Force of a square inch without permanent alteration . . 15,390 Ibs. 
 
 Extension in length by this strain . . . '. . ^ 
 
 Modulus of elasticity for a base of 1 square inch . . . 19,130,000 fts. 
 
 Modulus of resilience 12'4 
 
 Specific gravity .... 7-219 
 
SECT. V.] 
 
 EXPERIMENTS ON CAST IRON. 
 
 65 
 
 Fixed at one end, with a leverage of 2 feet, the second bar 
 broke with 168fts., with one fracture at the fixed end; but 
 the bar flew into several pieces. 
 
 This gives the absolute cohesion of a square inch 44,000 fos., 
 or nearly 2' 9 times that strain which I consider to be the 
 greatest cast iron should have to sustain. 
 
 From these elements we may conclude, that a casting of 
 scrap iron will be ^ stiffer than one from Old Park iron ; 
 that it has ^ less power to resist a body in motion, and that 
 it is less strong in the ratio of 168 to 184. 
 
 MIXTURE OF OLD PARK AND GOOD OLD IRON IN 
 EQUAL PARTS. 
 
 76. The castings run from this mixture were even and 
 clean ; such as indicate a perfect union of the materials. The 
 depth 0*65 inch; the breadth 1*3 inches; and the distance 
 between the supports 2'9 feet. 
 
 Weight applied. 
 
 Effect on 1st bar. 
 
 Effect on 2nd bar. 
 
 fts. 
 
 in. 
 
 in. 
 
 72 
 
 bent O'l 
 
 bent O'l 
 
 140 
 
 0-2 
 
 0-2 
 
 162 
 
 0-24 no set. 
 
 0-245 no set. 
 
 182 
 
 0-27 no set. 
 
 0-28 no set. 
 
 202 
 
 0-3 small set. 
 
 0-31 small set. 
 
 220 
 
 
 0-34 set -005 
 
 300 
 
 
 0-475 set -03 
 
 The iron was rather hard to the file ; it indented with the 
 hammer, but was rather short and crumbling. 
 
 Fracture a lighter gray, and more dull than Old Park iron ; 
 very compact, even, and fine grained. 
 
 The bars did not set with a load of 182fts., therefore the 
 load of 162fts. is sufficiently within the limit; the flexure 
 with that load is '245, consequently we may state its 
 properties thus : 
 
66 
 
 EXPERIMENTS ON CAST IKON. 
 
 [SECT. v. 
 
 Force on a square inch that does not produce permanent 
 
 alteration ."'... 15,390 K>3. 
 
 Extension under this strain T^W 
 
 Modulus of elasticity for a base of 1 inch square . . 19,514,000 R>s. 
 Modulus of resilience ... .... l2'l 
 
 Specific gravity . . . . . . . 7*104 
 
 When the second bar was fixed at one end, it broke with 
 174 fts., acting with a leverage of 2 feet ; fracture close to the 
 fixed end. Therefore the absolute cohesion of a square inch 
 is 45,600 fts., or very nearly three times the strain of 
 15,300 fts. 
 
 In this mixture there is clearly too great a proportion of 
 old iron ; it is rather inferior to the quality of our mean 
 specimen (art. 56). About one of old iron to two of the Old 
 Park pig iron would be a better proportion. It is worthy of 
 remark, that the absolute strength is nearly the mean of the 
 two kinds which form the mixture, and so is the specific 
 gravity. 
 
 ALLOT OF PIG IRON SIXTEEN PARTS, COPPER ONE PART. 
 
 7 7. It has been said that iron is much improved by a small 
 proportion of copper ; it was desirable, therefore, to ascertain 
 its effect, and the advantage, if any, of employing it. The 
 breadth of the specimens 1*25 inches; the depth '675 inch; 
 the distance between the supports 2'9 feet. The load which 
 ought not to produce permanent alteration, about 167 fts. 
 
 Weight applied. 
 
 Effect on 1st bar. 
 
 Effect on 2nd bar. 
 
 Ibs. 
 
 in. 
 
 in. 
 
 60 
 
 bent 01 
 
 bent 0-1 
 
 122 
 
 0-2 
 
 0-2 
 
 167 
 
 0-275 no set. 
 
 0-265 no set. 
 
 380 
 
 0-3 no set. 
 
 0-29 no set. 
 
 203 
 
 0-34 set '003 
 
 0-325 set -002 
 
 300 
 
 ,, 0-5 
 
 
 These bars yielded freely to the file, but were short 
 and crumbling under the hammer. I expected to have found 
 
SECT, v.] EXPERIMENTS ON CAST IRON. 67 
 
 them more ductile. The fracture dark gray, fine grained, 
 and more compact than Old Park iron; with less metallic 
 lustre. 
 
 The load of 167 fts. did not produce any degree of set, the 
 mean flexure by this load 0'27 ; and assuming this to be as 
 great a load as it should bear in practice, we have, 
 
 Force on a square inch that does not produce permanent 
 
 alteration .." '^ '..,. . . . ^ . 15,300 K>s. 
 
 Extension under this strain ...... Yiog 
 
 Modulus of elasticity for a base of 1 inch square ' \ '. 16,921,000 Ibs. 
 
 Modulus of resilience . , 'ft . . . \, . . , .- 13 - 8 
 
 Specific gravity . ., " ?i . ..... 7'13 
 
 To try the absolute strength, the second bar was fixed at 
 one end, and the scale suspended from the other end; 
 weights were then added till the bar broke : the fracture took 
 place close to the fixed end, and it required 194fts. to break 
 the bar. 
 
 According to this experiment, the cohesive force of a square 
 inch is 52,000 Bbs., or 3'4 times the strain that will not give 
 permanent alteration. 
 
 It appears that copper increases both the strength and 
 extensibility of iron. 
 
 EXPERIMENTS ON THE RESISTANCE TO TENSION. 
 
 78. According to an experiment made by Muschenbroek, 
 a parallelopipedon, of which the side was '17 of a Rhinland 
 inch, broke with 1930 fts. ; * and since the Rhinland foot is 
 1-03 English feet, and the pound contains 7038 grains, this 
 experiment gives 63,286 fts. for the weight that would tear 
 asunder a square inch, when reduced to English weights and 
 measures. 
 
 79. An experiment made by Capt. S. Brown is thus de- 
 scribed : " A bar of cast iron, Welsh pig, 1^ inch square, 3 
 feet 6 inches long, required a strain of 11 tons 7 cwt. 
 
 * Muschenbroek's Introd. ad Phil. Nat. vol. i. p. 417. 1762. 
 
68 EXPERIMENTS ON CAST IRON. [SECT. v. 
 
 (25,424 fts.) to tear it asunder: broke exactly transverse, 
 without being reduced in any part ; quite cold when broken ; 
 particles fine, dark bluish gray colour/'* 
 
 Capt. Brown's machine for trying such experiments being 
 constructed on the principle of a weigh-bridge, Mr. Barlow is 
 of opinion it may show less than its real force ; it also may be 
 remarked, that to obtain the real force of cohesion, the re- 
 sultant of the straining force should coincide exactly with the 
 axis of the piece, for so small a deviation in this respect as ^th 
 of the breadth would reduce the strength one half. 
 
 From this experiment it appears that 16,265 fts. will tear 
 asunder a square inch of cast iron. 
 
 80. In some experiments made by Mr. G. Rennie, it is 
 obvious, from the description of the apparatus, that the strain 
 on the section of fracture would not be equal ; and, therefore, 
 that the straining force would be less than the cohesion of the 
 section. The specimens were 6 inches long, and ^th of an 
 inch square at the section of fracture. A bar cast horizontally 
 required a force of 11 66 fts. to tear it asunder. A bar cast 
 vertically required a force of 1218 fts. to tear it asunder. f 
 
 Per square inch. 
 
 In the horizontal casting the force was equal to . . . .18,656 Ibs. 
 And in vertical casting 19,488 
 
 EXPERIMENTS ON THE RESISTANCE TO COMPRESSION IN 
 SHORT LENGTHS. 
 
 81. The power of cast iron to resist compression was 
 formerly much over-rated. Mr. Wilson estimated the power 
 necessary to crush a cubic inch of cast iron at 1000 tons = 
 2,240,000 fts. ; and in describing an experiment by Mr. 
 William Reynolds, of Ketley, in Shropshire, a cube of ^th of 
 an inch of cast iron, of the quality called gun-metal, was said 
 
 * Essay on the Strength of Timber, &c., by Mr. Barlow. 
 
 t Philosophical Transactions for 1818, Part I., or Philosophical Magazine, vol. liii. 
 p. 167. 
 
SECT, v.] EXPERIMENTS ON CAST IRON. 69 
 
 to require 448,000 fts. to crush it.* But Mr. Telford, for 
 whom the experiments were made, was so kind as to com- 
 municate the correct results of the experiments made by Mr. 
 Reynolds ; and it appears that 
 
 Per square inch. 
 A cube of th of an inch of soft gray metal was crushed by 
 
 SOcwt = 143,360 Ibs. 
 
 Ditto of the kind of cast iron called gun-metal was crushed 
 
 by200cwt , =350,400 Ibs. 
 
 82. Such was the state of our knowledge on this important 
 subject, when Mr. G. Rennie communicated a valuable series 
 of experiments to the Royal Society, which were published in 
 the first part of their Transactions for 1818. 
 
 Mr. Rennie's Experiments on cubes from the middle of a large UocJc ; specific 
 gravity 7'033 : 
 
 Force per sq. in. 
 
 in. Ibs. in Ibs. 
 
 Side of cube | was crushed by 1,454, highest result = 93,056 
 
 Ditto & ditto 1,416, lowest ditto = 74,624 
 
 Ditto 4- ditto 10,561, highest ditto = 168,976 
 
 Ditto ditto 9,020, lowest ditto = 144,320 
 
 On cubes from horizontal castings, specific gravity 7 '11 3 
 
 in. Ibs. Ibs. per. sq. in. 
 
 Side of cube was crushed by 10,720, highest result = 171,520 
 Ditto i ditto 8,699, lowest ditto = 139,184 
 
 On cubes from vertical castings, specific gravity t 7 '074 
 
 in. R>s. fl. per. sq. in. 
 
 Side of cube was crushed by 12,665, highest result = 202,640 
 Ditto ditto 9,844, lowest ditto = 157,540 
 
 On pieces of different lengths. 
 
 in . fos. fcs.persq.in. 
 
 Area J x I length f was crushed by 1,743 = 111,552 
 
 Ditto i x i 1 do. 1,439 = 92,096 
 
 Ditto i x k \ do. 9,374 = 149,984 
 
 Ditto i x i 1 do. 6,321 = 101,136 
 
 These experiments were on too small a scale to allow of 
 
 * Edin. Encyclo. art. Bridge, p. 544 ; or Nicholson's Journal, vol. xxxv. p. 4. 1813. 
 t It is singular that the specific gravity of the vertical castings should be less than 
 that of the horizontal ones. 
 
70 EXPERIMENTS ON CAST IEON. [SECT. v. 
 
 that precision in adjustment which theory shows to be 
 essential in such experiments; therefore there still remains 
 much to be done by future experimentalists. It does not 
 appear, within the limits of these experiments, that an increase 
 of length had any sensible effect on the result. 
 
 I have selected the highest and lowest results, and such of 
 the single trials that were made under the greatest difference 
 of length ; in all Mr. Rennie made thirty-nine trials on the 
 resistance of cast iron to compression.* 
 
 EXPERIMENTS Otf THE RESISTANCE TO COMPRESSION OF 
 PIECES OF CONSIDERABLE LENGTH. 
 
 83. The only experiments of this kind that I know of 
 were made by Mr. Reynolds, and are described as follows in 
 Mr. Banks's work on the ' Power of Machines/ p. 89. 
 
 " Experiments on the strength of cast iron, tried at Ketley, 
 in March, 1795. The different bars were all cast at one time 
 out of the same air furnace, and the iron was very soft, so as 
 to cut or file easily. 
 
 "Exp. 1. Two bars of iron, 1 inch square, and exactly 3 
 feet long, were placed upon a horizontal bar, so as to meet 
 in a cap at the top, from which was suspended a scale ; these 
 bars made each an angle of 45 with the base plate, and of 
 consequence formed an angle of 90 at the top : from this cap 
 was suspended a weight of 7 tons (15,680!bs.), which was 
 left for sixteen hours, when the bars were a little bent, and 
 but very little. 
 
 " Exp. 2. Two more bars of the same length and thickness 
 were placed in a similar manner, making an angle of 22^ 
 with the base plate ; these bore 4 tons (8960 fts.) upon the 
 scale : a little more broke one of them which was observed 
 to be a little crooked when first put up." 
 
 * Philosophical Transactions for 1818, Paii I., or Philosophical Magazine, vol. liii. 
 pp. 164, 165. 
 
SECT. V.] 
 
 EXPERIMENTS ON CAST IRON. 
 
 71 
 
 84. By the principles of statics,* 
 
 2 sin. 45 : Had. : : 15,680 fts. : 11,087 fts. 
 
 equal the pressure in the direction of either bar in the first 
 experiment. And, 
 
 2 sin. 22i : Rad. : : 8960 K>s. : 11,709 Bbs. 
 
 the pressure in the direction of either bar in the second 
 experiment. 
 
 If we consider the direction of the force to have been 
 exactly in the axis in these trials, then, according to the 
 equation, art. 288, the greatest strain in the direction of one 
 of these bars should not have exceeded 5840 fts. ; but if the 
 direction of the pressure was at the distance of half the 
 depth from the axis, which it is very probable it would be, 
 the greatest strain in actual construction should not have 
 exceeded 2720 fts. See art. 287. 
 
 EXPERIMENTS ON THE RESISTANCE TO TWISTING. 
 
 85. Table of the principal experiments of the strength of 
 cast iron to resist a twisting strain. 
 
 No. 
 
 Description. 
 
 Leverage. 
 
 Length. 
 
 Side or 
 diameter 
 in 
 inches. 
 
 Weight in 
 ibs. that 
 broke the 
 piece. 
 
 Calculated 
 resistance 
 without 
 destroying 
 the elastic 
 
 Katio of 
 the calcu- 
 lated re- 
 sistance to 
 the break- 
 
 
 
 
 
 
 
 
 force. 
 
 ing weight 
 
 
 C Bar placed verti- ^ 
 cally, fast at one 
 
 ft. in. 
 
 
 
 
 
 
 
 1 
 
 -| end and twisted I 
 ] by a wheel at 
 
 1 
 
 not given 
 
 1x1 
 
 631 
 
 150 
 
 1:4-2 
 
 
 I the other ...J 
 
 
 
 
 
 
 
 
 2 
 
 Cylinder fixed at 
 one end, twisted 
 by a lever at the 
 
 14 2 
 
 inches. 
 2| 
 
 inches. 
 2 
 
 250 
 
 73-7 
 
 1 : 3-39 
 
 
 other ... 
 
 
 
 
 
 
 
 
 3 
 
 Ditto 
 
 14 2 
 
 3i 
 
 2 
 
 
 384 
 
 111 
 
 1 : 3-46 
 
 4 
 
 Ditto 
 
 14 2 
 
 3 
 
 2! 
 
 
 408 
 
 140 
 
 1 :2-9 
 
 5 
 
 Ditto 
 
 14 2 
 
 3 
 
 2' 
 
 
 700 
 
 184 
 
 1 :3'8 
 
 6 
 
 Ditto ... ' M 
 
 14 2 
 
 4 
 
 3; 
 
 
 1170 
 
 309 
 
 1 : 3-78 
 
 7 
 
 Ditto 
 
 14 2 
 
 5 
 
 3, 
 
 
 1240 
 
 402 
 
 1 : 3-08 
 
 8 
 
 Ditto 
 
 14 2 
 
 5 
 
 3; 
 
 
 1662 
 
 481 
 
 1 : 3-45 
 
 9 
 
 Ditto 
 
 14 2 
 
 5 
 
 4 
 
 
 1938 
 
 580 
 
 1 : 3-34 
 
 10 
 
 Ditto 
 
 14 2 
 
 6 
 
 4< 
 
 h' 
 
 2158 
 
 713 
 
 1 : 3-02 
 
 * Gregory's Mechanics, vol. i. art. 48. 
 
72 EXPERIMENTS ON CAST IRON. [SECT. v. 
 
 The experiment No. 1 was made by Mr. Banks.* The 
 others were made by Mr. Dunlop, of Glasgow. Nos. 4 and 
 7 were faulty specimens.! Some experiments on a very 
 small scale were made by Mr. George. Rennie, but they are 
 not inserted here, because they were not sufficiently described 
 to admit of comparison.^ 
 
 I am indebted to Messrs. Bramah for a description of some 
 new and interesting experiments on torsion, which they had 
 made in order to ascertain what degree of confidence they 
 might place in the theoretical and experimental deductions of 
 writers on this subject. They were also desirous of knowing 
 the effect of a small portion of copper on the quality of cast 
 iron. 
 
 I have given a tabular form to the results of these experi- 
 ments, in order that they may be more easily compared ; and 
 I have added two columns to the Table, to show how these 
 experiments agree with the rules of this work. 
 
 The bars were firmly fixed at one end, in a horizontal 
 position, and to the other end the straining force was applied, 
 acting with a leverage of 3 feet. To prevent the effect of 
 lateral stress, the bar rested loosely upon a support at the 
 end to which the straining force was applied. 
 
 * " Power of Machines." 
 
 t Dr. Thomson's Annals of Philosophy, vol. xiii. p. 200-203. 
 
 I Philosophical Magazine, vol. liii. p. 168. 
 
 
 LI BUA It Y 
 
 jj UXIVKKSITV OP 
 
 CALIFORY 
 
SECT. V.] 
 
 EXPERIMENTS ON CAST IRON. 
 
 73 
 
 "Sfl 
 
 II 
 
 S3 
 s ex 
 
 * 
 
 Description of iron. 
 
 Length of 
 bar. 
 
 Side of 
 bar. 
 
 Weight 
 applied 
 acting 
 with three 
 feet 
 leverage. 
 
 Effect of 
 weight or 
 angle of 
 torsion. 
 
 Calculated 
 angle 
 of torsion. 
 
 Ratio of the 
 force which 
 would not 
 produce set 
 to the break- 
 ing weight. 
 
 
 Square bar of an 
 
 feet. 
 
 inches. 
 
 ft>S. 
 
 degrees. 
 
 degrees. 
 
 
 I 
 
 alloy of 16 parts 
 
 
 
 
 
 
 
 J. 
 
 iron to one of 
 
 1 
 
 4 
 
 166 
 
 H 
 
 4-25 
 
 
 
 copper ... 
 
 
 
 215 
 
 broke 
 
 
 1 :3'6 
 
 2 
 
 Square bar of the 
 same kind as No. 1 
 
 2 
 
 H 
 
 111 
 213 
 213 
 
 $ 
 
 broke 
 
 5 V 7 
 10-9 
 
 1 :3'5 
 
 
 Square bar a mix- 
 
 
 
 
 
 
 
 3 
 
 ture of equal parts 
 of Adelphi, Alfre- 
 ton, and old iron... 
 
 . 1 
 
 i* 
 
 217 
 330 
 
 14 
 broke 
 
 5-6 
 
 1 :5-5 
 
 
 Bar same kind as 
 
 1 
 
 IA 
 
 166 
 
 n 
 
 4-25 
 
 
 
 No. 3 
 
 
 
 310 
 
 broke 
 
 
 1 : 5-16 
 
 
 
 Bar same kind as 
 
 XT O 
 
 2 
 
 *A 
 
 164 
 213 
 
 4 
 
 18 
 
 8-4 
 10-9 
 
 
 ( 
 
 No. 3 
 
 
 
 280 
 
 28 
 
 14-3 
 
 
 
 
 
 
 
 Broke by 
 
 
 
 
 
 
 
 
 slipping 
 one of the 
 
 
 
 
 
 
 
 
 weights. 
 
 
 
 6 
 
 Square bar of cast 
 iron 
 
 1 
 
 i 
 
 237 
 
 broke 
 
 
 1 : 4-72 
 
 7 
 
 Square bar same 
 kind as No. 6... 
 
 i 2 
 
 i 
 
 218 
 
 broke 
 
 ... 
 
 1 : 4-35 
 
 The comparison between our rule (Equation iv. art. 265), 
 and the force that broke these specimens, which is given in 
 the last column, is very satisfactory, and very nearly agrees 
 with former experiments on this strain as shown in the first 
 Table of this article. 
 
 The observed angle of torsion is very irregular, and in all 
 these experiments it greatly exceeds the angle calculated by 
 Equation xiv. art. 272. But it will be remarked, that the 
 angle was measured after the strain was far beyond that 
 degree where it is known that flexure increases more rapidly 
 than the load and no allowance was made for the compres- 
 sion at the fixed points. M. Duleau, in his experiments on 
 wrought iron, (see Sect. VI. art. 94 of this Essay,) allowed 
 for the latter source of error by taking, as the measure of 
 torsion, the angle through which the bar returned when the 
 weight was taken off ; * and the formula applied to his 
 
 * Essai sur la Resistance du Fer Forge", p. 49. 
 
74 EXPERIMENTS ON CAST IRON. [SECT. v. 
 
 experiment gives an error in excess, here it is in defect : I 
 shall, therefore, not endeavour to make the rules agree with 
 either set of experiments, because I know that the flexure will 
 be too great in Messrs. Bramah's experiment ; and assuming 
 this to be so, the rules will be nearly true ; whereas, if M. 
 Duleau's turn out to be most correct, it will only cause shafts 
 to be made a small degree stronger than necessary. 
 
 EXPERIMENTS ON THE EFFECT OF IMPULSIVE FORCE. 
 
 86. The height from which a weight might fall upon a 
 piece of cast iron without destroying its elastic force was 
 calculated by Equation v. art. 306, for the specimens of *9 
 inch square, used in the preceding experiments (art. 67). 
 Repeated trials with that height of fall were made without 
 producing a sensible effect. I then let the weight fall from 
 double the calculated height, and every repetition of the blow 
 added about itroth f an i ncn ^ the curvature of the bar. 
 I could not measure the effect of each trial very correctly, but 
 a few trials rendered the bar so much curved as to be easily 
 seen. I hope, at some future time, to be able to resume 
 these experiments with an apparatus for measuring correctly 
 the degree of permanent set.* See art. 313 350, where 
 practical rules will be found. 
 
 TO DISTINGUISH THE PROPERTIES OF CAST IRON BY 
 THE FRACTURE. 
 
 87. I shall close this section with a few remarks on the 
 aspect of cast iron recently fractured, with a view to distin- 
 guish its properties. 
 
 There are two characters by which some judgment may be 
 formed ; these are the colour and the lustre of the fractured 
 surface. 
 
 * This hope of the Author had not been realised when Practical Science was 
 unfortunately deprived by his death, of one of its most able supporters. EDITOR. 
 
SECT, v.] EXPERIMENTS ON CAST IRON. 75 
 
 The colour of cast iron is various shades of gray ; some- 
 times approaching to dull white, sometimes dark iron gray 
 with specks of black gray. 
 
 The lustre of cast iron differs in kind and in degree. It 
 is sometimes metallic, for example, like minute particles of 
 fresh cut lead distributed over the fracture ; and its degree, 
 in this case, depends on the number and size of the bright 
 parts. But in some kinds, the lustre seems to be given 
 by facets of crystals disposed in rays. I will call this lustre, 
 crystalline. 
 
 In very tough iron the colour of the fracture is uniform 
 dark iron gray, the texture fibrous, with an abundance of 
 metallic lustre. If the colour be the same, but with less 
 lustre, the iron will be soft but more crumbling, and break 
 with less force. If the surface be without lustre, and the 
 colour dark and mottled, the iron will be found the weakest 
 of the soft kinds of iron. 
 
 Again, if the colour be of a lighter gray with abundance 
 of metallic lustre, the iron will be hard and tenacious ; such 
 iron is always very stiff. But if there be little metallic lustre 
 with a light colour, the iron will be hard and brittle ; it is 
 very much so when the fracture is dull white ; but in the 
 extreme degrees of hardness, the surface of the fracture is 
 grayish white and radiated with a crystalline lustre. 
 
 There may be some exceptions to these maxims, but I 
 hope they will nevertheless be of great use to those engaged 
 in a business which is every day becoming more important. 
 
SECTION VI. 
 
 EXPERIMENTS ON MALLEABLE IRON AND OTHER METALS. 
 
 EXPERIMENTS ON THE RESISTANCE OF MALLEABLE IRON 
 TO FLEXURE. 
 
 88. There have been a greater number of experiments 
 made on malleable iron than on any other metal ; but those 
 on the lateral strength are chiefly by foreign experimentalists. 
 Prom those of Duleau I shall select a few for the purpose of 
 comparison ; but in the first place I propose to describe some 
 of my own trials. 
 
 The following experiments were made on bars of English 
 and of Swedish iron ; the bars were supported at the ends, 
 and the weight applied in the middle between the supports ; 
 the length of each bar was exactly 6 feet, and the distance 
 between the supports 66^ inches. 
 
 English, Iron. 
 
 Kind of bar and 
 dimensions. 
 
 Weight 
 of 6 
 feet in 
 length. 
 
 Deflexion with 
 
 Weight of 
 modulus of elas- 
 ticity for a base 
 of 1 sq. inch. 
 
 58 Ibs. 
 
 114 fbs. 
 
 iro H>S. 
 
 Bar 11 inch square . . 
 Bar li . . 
 Barl . . 
 Round bar 1 inch diamtr. 
 Round bar 1 
 
 fts. 
 33 
 25 
 20 
 
 24 
 17 
 
 inch. 
 0625 
 125 
 15 
 125 
 25 
 
 inch. 
 1 
 25 
 32 
 25 
 5 
 
 inch. 
 1875 
 375 
 5 
 375 
 8 
 
 Ibs. 
 27,240,000 
 20,830,000 
 24,990,000 
 23,154,000 
 26,500,000 
 
 Mean weight of modulus 24,542,800 fts. 
 
SECT. VI.] 
 
 EXPEEIMENTS ON MALLEABLE IKON. 
 
 77 
 
 Swedish Iron. 
 
 Kind of bar and 
 dimensions. 
 
 Weight 
 of 6 
 feet in 
 length. 
 
 Deflexion with 
 
 Weight of 
 modulus of elas- 
 ticity for a base 
 of 1 sq. inch. 
 
 5Sibs. 
 
 114 ibs. 
 
 170 ibs. 
 
 Bar 1'2 inch square . . 
 Bar 11 ".'; . 
 Barl -n;..V 
 
 ibs. 
 32 
 27 
 33 
 
 inch. 
 0625 
 08 
 125 
 
 inch. 
 125 
 161 
 25 
 
 inch. 
 19 
 25 
 375 
 
 Ibs. 
 32,000,000 
 31,245,000 
 33,328,000 
 
 Mean weight of modulus 32,191,000 fos. 
 
 The bars of Swedish iron varied in dimensions con- 
 siderably; the dimension in the first column was taken at 
 the point of greatest strain in each bar. The apparently 
 superior stiffness of the Swedish iron is partly to be attributed 
 to this cause ; but it is in a greater degree owing to the mode 
 of manufacture which gives more density as well as elastic 
 force to the iron. If the English iron had been formed 
 under the hammer, in the same manner, it would have been 
 perhaps equally dense and strong, and as fit for the nicer 
 purposes of smiths' work as the Swedish. All these 
 specimens were tried in the same state as the bars are sent 
 from the iron works ; the trials were made in July, 1814. 
 
 89. The objects of my next experiments on malleable iron 
 were, to determine the force that would produce permanent 
 alteration ; the effect of heating iron so as to give it uniform 
 density ; and the effect of temperature on its cohesive power. 
 For this purpose, Mr. Barrow, of East Street, selected for me 
 a bar of what he esteemed good iron, bearing the mark 
 Penydarra. A piece 38 inches long, weighing 10*4fts., was 
 cut off this bar : its section did not sensibly differ from 1 inch 
 square. With the supports 3 feet apart, and the weight 
 applied in the middle, the following results were obtained. 
 
 Weight. 
 
 Deflexion in the middle with 
 the bar as obtained from the 
 iron works. 
 
 Deflexion in the middle after 
 the bar had been uniformly 
 heated and slowly cooled. 
 
 ibs. 
 126 
 252 
 310 
 330 
 
 inch. 
 05 
 10 
 12 
 13 
 
 inch. 
 059 
 
 117 
 145 
 154 
 
78 EXPERIMENTS ON MALLEABLE IRON. [SECT* vi. 
 
 In both states it bore the weight of 330 fts. without 
 sensible effect, though it was let down upon it, and relieved 
 several times ; but in either state an addition of 20 fts. ren- 
 dered the set perceptible ; in the softened bar it appeared to 
 be sensible when only 10 fts. had been added. 
 
 Hence, by art. 110 we have the force that could be re- 
 sisted ; without permanent alteration 17, 820 fts. per square 
 inch : by art. 121 the extension in the softened state, is Wo-o 
 of its length ; and by art. 105 the modulus of elasticity is 
 24,920,000fts. for a base of an inch square. The modulus 
 before being softened is 29,500,000 ft s. 
 
 90. To try the effect of heat in decreasing the cohesion of 
 malleable iron, I heated it to 212 of Fahrenheit, having 
 previously got the machine ready, so that a weight of 300 ft s. 
 could be instantly let down upon the bar as soon as it was 
 put in, and the index adjusted to one of the divisions of the 
 scale. These operations having been effected in a close and 
 warm room, with as little loss of heat as possible, the window 
 was thrown open, and the effect of cooling observed. The 
 deflexion decreased as the bar cooled, but it was allowed to 
 remain nearly two hours, in order to be perfectly cooled down 
 to the temperature of the room, or 60. Each division on the 
 scale of the index is Troth of an inch, and as nearly as I could 
 determine, with the assistance of a magnifier, the deflexion 
 had decreased three-fourths of one of the divisions ; and it 
 returned through fourteen divisions when the load was 
 removed ; therefore we may conclude, that by an elevation of 
 temperature equal to 212 60 = 152 degrees, iron loses 
 about a 20th part of its cohesive force, or a 3040th part for 
 each degree. 
 
 M. DULEATj's EXPERIMENTS.* 
 
 91.- The most part of the experiments of M. Duleau were 
 
 * Taken from his Essai Thdorique et Experimental sur la Resistance du Fer Forgd 
 4 to. Paris, 1820. 
 
SECT. VI.] 
 
 EXPERIMENTS ON MALLEABLE IRON. 
 
 79 
 
 made with malleable iron of Perigord ; some of the specimens 
 were hammered to make them regular, others were put in 
 trial in the state they are sent from the iron works; the 
 former are distinguished from the latter by an h added to the 
 number of the experiment ; and these numbers are the same 
 as in M. Duleau's work. The experiments are divided into 
 two classes : in the first the elasticity was observed to be im- 
 paired by the action of the load ; in the second it was not. 
 The specimens were supported at the ends, and the load 
 suspended from the middle of the length. The dimensions 
 of the pieces are in the original measures, as well as the 
 weights : but the deductions are in our own measures and 
 weights. All the experiments I have selected are on Perigord 
 
 iron. 
 
 Number 
 of experiments. 
 
 Distance 
 between the 
 supports. 
 
 Breadth. 
 
 Depth. 
 
 Depression 
 in the 
 middle. 
 
 Weight 
 producing it. 
 
 Extension 
 in parts of 
 length. 
 
 
 Millim. 
 
 Millim. 
 
 Millim. 
 
 Millim. 
 
 Kilogramm. 
 
 
 1 of 
 
 15 
 
 2000' 
 
 45' 
 
 12- 
 
 54- 
 
 45 
 
 000972 
 
 
 I7 h 
 
 2000- 
 
 40" 
 
 11-5 
 
 52-5 
 
 25 
 
 000906 
 
 
 36* 
 
 3000- 
 
 60' 
 
 20- 
 
 33' 
 
 50 
 
 000441 
 
 9nrl 
 
 21 ft 
 
 2000- 
 
 11-5 
 
 40- 
 
 15-03 
 
 90 
 
 000902 
 
 
 22 
 
 3000" 
 
 77- 
 
 14- 
 
 72- 
 
 50 
 
 000672 
 
 class 
 
 OQ/l 
 
 3000- 
 
 15- 
 
 25- 
 
 70- 
 
 50 
 
 001167 
 
 The last column shows the extension of an unit of length 
 by the strain as calculated by M. Duleau ; my formula, art. 
 121, gives the same results. The extension that malleable 
 iron will bear without permanent alteration is -nrVo- = "000714, 
 according to my experiment ; but in M. Duleau's experiment, 
 No. 36^, the extension of '000441 produced a permanent set, 
 while in No. 29 fe the extension was "001167 without pro- 
 ducing a set : this is a considerable irregularity, but such as 
 may be expected in experiments on such long heavy 
 specimens of small depth. In all such experiments, the 
 effect of the weight of the piece should be observed. It is 
 also essential that the points of support should be perfectly 
 solid and firm, or that the flexure should be measured from a 
 
80 EXPERIMENTS ON MALLEABLE IRON. [SECT. vi. 
 
 point, of which the position is invariable in respect to the 
 points of support. 
 
 The mean weight of the modulus of elasticity, as de- 
 termined by the above 'experiments, is, 28,000,000fts. for a 
 base of 1 inch square. Experiment No. 22 gives the highest, 
 being 31,864,000 fts. ; and No. 17 the lowest, being 
 22,974,000 fts. ; therefore it appears that the elastic force of 
 Perigord iron is not greatly different from English iron. 
 
 M. Duleau concludes that a bar of malleable iron may be 
 safely strained till the extension at the point of greatest strain 
 is equal to -g-sW of its original length without losing its 
 elasticity ; and that the load upon a square inch which pro- 
 duces this extension is 8 540 fts. In many of his own 
 experiments the extension was three times this without 
 permanent loss of elasticity. 
 
 It has been my object to fix the limit which will produce 
 permanent alteration of elasticity in a good material ; to say, 
 that beyond this strain you must not go, but approach it as 
 nearly as your own judgment shall direct, when you are 
 certain that you have assigned the greatest possible load it 
 will be exposed to. Where a great strain is to be sustained, 
 a good material is most suitable and most economical ; to a 
 defective material no rules whatever will apply ; for who can 
 measure the effect of a flaw in malleable iron, an air bubble 
 in cast iron, a vent in a stone, or of knots and rottenness in 
 timber ? But the presence of most of these defects can be 
 ascertained by inspection of the material itself ; and since the 
 greatest strain is at the surface of a beam or bar, the defects 
 which impair the strength in the greatest degree are always 
 most apparent. 
 
 Experiments on the flexure of malleable iron have also 
 been made by Rondelet,* Aubry, and Navier,f which 
 accord with the theoretical principles developed in this Essay. 
 
 * Trait^ de 1'Art de Bfttir, tome iv. p. 509 and 514. 4to. 1814. 
 t Gauthey's Construction des Fonts, tome ii. p. 151. 4to. 1813. 
 
SECT, vi.] EXPERIMENTS ON MALLEABLE IRON. 81 
 
 EXPERIMENTS ON THE RESISTANCE TO TENSION. 
 
 92. The experiments on the absolute resistance of mal- 
 leable iron to tension are very numerous : in many experiments 
 it has been found above 80,000fts., per square inch, and in 
 very few under 50,000fts,, indeed in none where the iron was 
 not defective. About 60,000fts. seems to be the average 
 force of good iron : and according to this estimate, the force 
 that would produce permanent alteration is to that which 
 would pull a bar asunder as 17,800 : 60,000, or nearly as 
 1 : 3 '37. Hence we see, that on whatever principle it was 
 that Emerson* concluded a material should not be put 
 to bear more than a third or a fourth of the weight that 
 would break it, the maxim is agreeable with the laws of 
 resistance. 
 
 Experiments on the absolute strength of malleable iron 
 have been made by Muschenbroek,t Buffon,J Emerson, 
 PerronetJ Soufflot,^ Sickingen,** Rondelet,ft Telford,}} 
 Brown, and Rennie. |||| Those by Messrs. Telford and 
 Brown were made on the largest scale ; and are minutely de- 
 scribed in Professor Barlow's Essay, to which I must refer 
 the reader. 
 
 EXPERIMENTS ON THE RESISTANCE TO COMPRESSION. 
 
 93. Very few experiments have been made on this species 
 of resistance, and from some circumstances in such 
 experiments requiring attention which the authors of them 
 do not appear to have been aware of, we can make no use of 
 
 * Mechanics, 4to edit. p. 116. 1758. 
 
 t Introd. ad Phil. Nat. i. p. 426. 4to. 1762. 
 
 Gauthey's Construction des Ponts, ii. pp. 153, 154. 
 
 Mechanics, p. 116. 4to edit. 1758. 
 
 || Gauthey'a Construction des Ponts, ii. pp. 153, 154. 
 
 m Rondelet's L'Art de Batir, iv. pp. 499, 500. 4to. 1814. 
 
 ** Annales de Chimie, xxv. p. 9. 
 
 ft Rondelet's L'Art de Batir, iv. pp. 499, 500. 4to. 1814. 
 
 it Barlow's Essay on Strength of Timber, &c. Ibid. 
 
 Illl Philosophical Magazine, liii. p. 167. 1819. 
 
82 EXPERIMENTS ON MALLEABLE IROK [SECT. vi. 
 
 them in illustrating our theoretical principles, unless it be to 
 show that when we consider the direction of the force to 
 nearly coincide with one of the surfaces of the bar, we shall 
 always be calculating on safe data ; and from the nature of 
 practical cases in general, we can scarcely think of employing 
 a less excess of force than is given by this rule. 
 
 On pieces of considerable length experiments have been 
 made by Navier, Rondelet, and Duleau; and the force 
 necessary to crush short specimens has also been ascertained 
 by Rondelet. 
 
 Rondelet employed cubical specimens, the sides of the cubes 
 varying from 6 to 10 J and 12 lines; and cylinders of 6, 8, 
 and 12 lines in diameter, the height being the same as the 
 diameter in each cylinder. The mean resistance of the cubes 
 was equivalent to 512 livres on a square line; the mean 
 resistance of the cylinders 515 livres per square line: 512 
 livres on a square line is 70,000fts. on a square inch in our 
 weights and measures. The force necessary to crush the 
 specimens was in proportion to the area ; when the area was 
 increased four times, this ratio did not differ from the result 
 of the experiment so much as a fiftieth part.* 
 
 He observed in experiments on bars of different lengths, 
 that when the height exceeded three times the diameter, the 
 iron yielded by bending in the manner of a 4ong column. 
 Rondelet's experiments on longer specimens are not suffi- 
 ciently detailed.! 
 
 Navier's experiments were made on long bars, and show the 
 force that broke them ; whether the flexure was sudden or 
 gradual is not stated. J 
 
 * There does not appear to be an abrupt change in the crushing of wrought iron to 
 enable an experimenter to draw any very definite conclusions of this kind. According 
 to my observations, wrought iron becomes slightly flattened or shortened with from 
 9 to 1 tons per square inch ; with double that weight it is permanently reduced in 
 length about ^, and with three times that weight about j^th of its length. (Philo- 
 sophical Transactions, Part II., 1840, p. 422.) EDITOR. 
 
 t Traite de 1'Art de Batir, iv. pp. 521, 522. 
 
 J Gauthey's Construction des Ponts, ii. p. 152. 
 
SECT. VI. J 
 
 EXPERIMENTS ON MALLEABLE IRON. 
 
 A bar of any material, in which the stress is very ac- 
 curately adjusted in the direction of the axis, will bear a 
 considerable load without apparent flexure, but the load is in 
 unstable equilibrium, so much so indeed, that in a bar where 
 the least dimension of the section is small in respect to the 
 length, the slightest lateral force would cause the bar to bend 
 suddenly and break under the load. . In such a case it is not 
 so much owing to the magnitude of the force that fracture is 
 produced, as the momentum it acquires before the bar attains 
 that degree of flexure which is necessary to oppose it. The 
 reader will find this view of the subject to be agreeable to 
 experience, particularly in flexible materials ; in fact, I do not 
 think any one can be aware of the danger of over-loading a 
 column who has never observed an experiment of this kind. 
 
 M. Duleau found that a bar of malleable iron 11*8 feet 
 long, and 1*21 inches square (31 millimetres), doubled under 
 a load of 4400 fts. (2000 kilogrammes). Another specimen 
 about 11*8 feet long, the breadth 2*38 inches, and the depth 
 0'8 inch, doubled under a load of 2640 fts. : this piece did 
 not become sensibly bent before it doubled.* In the 
 last experiment, our rule (Equa, xv. art. 288,) gives 876 fts. 
 as the greatest load the bar ought to sustain in practice ; 
 which is about one-third of the weight that doubled the 
 piece ; a similar result obtains in other cases. 
 
 EXPERIMENTS ON THE RESISTANCE TO TORSION. 
 
 94. Mr. Rennie made some experiments on the resistance 
 of malleable iron to torsion. The weight acted with a lever 
 of 2 feet, and the specimens were Jth of an inch square ; the 
 strain was applied close to the fixed end : 
 
 English iron, wrought, was wrenched asunder by 
 Swedish iron, wrought, . .''.'.; . 
 
 tbs. oz. 
 10 2 
 9 8 f 
 
 * Essai The*orique et Experimental sur la Resistance du Fer Forge*, p. 26-37. 
 t Philosophical Magazine, vol. liii. p. 168. 
 
 G 2 
 
EXPERIMENTS ON MALLEABLE IRON". 
 
 [SECT. vi. 
 
 If we could suppose the pieces so fitted that the distance 
 between the centres of action, of the force and the fixing 
 apparatus, was equal to the diameter of the specimen ; then 
 our formula gives 1*31 5 fts. as the force that such a bar 
 would resist without permanent change : this is only about 
 -J-th of the force that produced fracture. A like irregularity 
 occurs in his experiments on the torsion of cast iron, which 
 may very likely be in consequence of the strain not being 
 applied exactly as I have supposed it to be. 
 
 The experiments on the resistance of malleable iron to 
 torsion made by M. Duleau were all directed to determiniDg 
 its stiffness. The bars were fixed at one end in a horizontal 
 position, and the force was applied to a wheel or large pulley 
 fixed on the other end. In order to prevent lateral strain, 
 the end to which the wheel was fixed reposed freely upon a 
 support. It was found that the bars yielded a little at the 
 fixed points ; the permanent alteration produced by this 
 yielding was allowed for by deducting the angle of set from 
 the angle observed.* 
 
 
 
 
 
 Angle of torsion 
 
 
 
 
 
 with a wt. of 
 
 
 Nature of the specimens. 
 
 Length of 
 the part 
 
 Sid of diameter. 
 
 10 kilogrammes 
 (22 Ibs.) with 
 
 Angle of 
 torsion as 
 
 
 twisted. 
 
 
 leverage of 320 
 
 calculated. 
 
 
 
 
 millimetres (1-22 
 
 
 
 
 
 feet). 
 
 
 
 Millimetres. 
 
 Millimetres. 
 
 degrees. 
 
 degrees. 
 
 Round iron, English, \ 
 
 
 
 
 
 marked DOWLAIS, as f 
 
 2400 
 
 19-83 
 
 4 
 
 10-4 
 
 from the iron works ; f 
 
 (7-9 ft.) 
 
 (-78 in.) 
 
 
 
 hot short. ) 
 
 
 
 
 
 Round iron,Perigord, as ) 
 
 2890 
 
 23-03 
 
 3 
 
 7 
 
 from the iron works. ) 
 
 (9- 5 ft.) 
 
 (91 in.) 
 
 
 
 Square iron, English ) 
 
 4120 
 
 20x20 
 
 6| 
 
 10 
 
 marked C 2, hot short. \ 
 
 (13-5 ft.) 
 
 (79 in.) 
 
 
 
 Square iron, Perigord, ) 
 
 2520 
 
 20-35x20-35 
 
 3-08 
 
 5-8 
 
 as from the iron works. \ 
 
 (8'3 ft.) 
 
 (-8 in.) 
 
 
 Flat iron, English. j 
 
 2910 
 (9-6 ft.) 
 
 34x8-56 
 (1-32 x -337 in.) 
 
 11-4 
 
 13-9 
 
 Essai sur la Resistance du Fer Forgd, p. 50-53. 
 
 
SECT. VI.] 
 
 EXPERIMENTS ON VARIOUS METALS. 
 
 85 
 
 The last column shows the angle calculated by the formula 
 (Equa. xiii. and xiv. art. 272). There is a considerable error 
 in excess according to these experiments ; see art. 85. 
 
 EXPERIMENTS ON VARIOUS METALS. 
 EXPERIMENTS ON STEEL. 
 
 95. The modulus of elasticity of steel was first determined 
 by Dr. Young .from the vibration of a tuning-fork; the 
 height of the modulus found by this method was 8,530,000 
 feet ; * hence the weight of the modulus for a base of an inch 
 square will be 29,000,000 fts. 
 
 M. Duleau has made some experiments on the flexure of 
 steel bars when loaded in the middle and supported at the 
 ends; in all he has described twelve experiments;! from 
 these I will take four at random. 
 
 Description of specimens. 
 
 Distance 
 between 
 the 
 
 Breadth. 
 
 Depth. 
 
 Depression 
 with 10 kilo- 
 
 Weight of mo- 
 dulus of elasticity 
 in fts. for a base 
 
 
 supports. 
 
 
 
 grammes. 
 
 an inch square. 
 
 
 Millioi. 
 
 Millim. 
 
 Millim. 
 
 Millim. 
 
 English fts. 
 
 English cast steel,"] 
 
 
 
 
 
 
 marked HUNTSMAN, 1 
 perfectly regular, un- j 
 
 930 
 
 13-3 
 
 5-9 
 
 32-05 
 
 34,000,000 
 
 tempered, but brittle.J 
 
 
 
 
 
 
 German steel (of ce-^j 
 
 
 
 
 
 
 mentation), marked 
 
 
 
 
 
 
 FORTSMAN, and 3 deer }- 
 
 680 
 
 14'5 
 
 7'8 
 
 8 
 
 20,263,000 
 
 heads, used for razors, 
 
 
 
 
 
 
 dimensions irregular. J 
 
 
 
 
 
 
 Same kind of steel. 
 
 1845 
 
 28-5 
 
 21-9 
 
 2-6 
 
 29,000,000 
 
 Ditto. 
 
 1350 
 
 52 
 
 26-6 
 
 0-5 
 
 17,880,000 
 
 Mean for German steel 22,381,000 fts. 
 
 EXPERIMENTS ON GUN-METAL. 
 
 96. A cast bar of the alloy of copper and tin, commonly 
 called gun-metal, of the specific gravity 8 '152, was filed true 
 and regular ; its depth was 0'5 inch, and its breadth 0*7 
 
 * Lectures on Natural Philosophy, vol. ii. p. 86. 
 t Essai sur la Resistance du Fer Forge, p. 38. 
 
86 EXPERIMENTS ON VAKIOUS METALS. [SECT. vi. 
 
 inch ; it was supported at the ends, the distance between the 
 supports being 12 inches ; and the scale was suspended from 
 the middle. 
 
 19 fts. bent the bar O'Ol inch. 
 38 ... 0-02 
 56 ... 0-03 
 78 ... 0-04 
 . ( This load was raised from the bar several times, but 
 
 * " | permanent set was not sensible. 
 
 } 20 0-06 S Every time the bar was relieved of this load, a set of 
 
 I about '005 was observed. 
 200 . ,. . 0-17 
 230 . ;. . 0-34 
 
 32Q j slipped through between the supports, bent nearly 
 ( 3 inches, but not broken. 
 
 We may therefore consider 100 fts. as the utmost that the 
 bar would support without permanent alteration, which is 
 equivalent to a strain of 10,285 fts. upon a square inch ; and 
 an extension of 9^0 th part of its length (see art. 110 and 
 121). Absolute cohesion greater than 34,000 fts. for a 
 square inch. 
 
 Calculating from this experiment, we find the weight of 
 the modulus of elasticity for a base 1 inch square, 9,873,000 
 fts. ; and the specific gravity of gun-metal is 8*152; there- 
 fore the height of the modulus in feet is 2,790,000 feet. 
 
 The deflexion increases much more rapidly than in pro- 
 portion to the weight, as soon as the strain exceeds the elastic 
 force ; a weight of 200 fts. more than trebled the deflexion 
 produced by 100, instead of only doubling it. 
 
 EXPERIMENTS ON BRASS. 
 
 97. Dr. Young made some experiments on brass, from 
 which he calculated the height of the modulus of elasticity of 
 brass plate to be 4,940,000 feet, or 18,000,000 fts. for its 
 weight to a base of 1 square inch. For wire of inferior 
 brass he found the height to be 4,700,000 feet.* 
 
 As cast brass had not been submitted to experiment, I 
 
 * Natural Philosophy, vol. ii. p. 86. 
 
SECT, vi.] EXPERIMENTS ON VARIOUS METALS. 87 
 
 procured a cast bar of good brass, and made the following 
 experiment : 
 
 The bar was filed true and regular ; its depth was 45 
 inch, and breadth 0*7 inch. The distance between the 
 supports 12 inches, and the scale suspended from the middle. 
 
 12 fes. bent the bar O'Ol inch. 
 
 23 ... 0.02 
 
 38 ... 0'03 j The bar was relieved several times, but it took no 
 
 52 ... 0-04 ( perceptible set. 
 
 65 . .' O'Oo relieved, the set was -01. 
 
 110 ... 0-18 
 
 , go \ slipped between the supports, bent more than 2 
 
 ( inches, but not broken. 
 
 Hence 521bs. seems to be about the limit which could 
 not be much exceeded without permanent change of struc- 
 ture. It is equivalent to a strain of 6700 flbs. upon a square 
 inch, and the corresponding extension is rstt f its length, 
 (see art. 110 and 121). Absolute cohesion greater than 
 21,000 fts. per square inch. The modulus of elasticity 
 according to this experiment is 8,930,000 fts. for a base of 
 an inch square. The specific gravity of the brass is 8*37, 
 whence we have 2,460,000 feet for the height of the modulus. 
 
 L I B R A R 
 
 U N I V EH S I TT 
 
 CALIFORNIA. 
 
SECTION VII. 
 
 OF THE STRENGTH AND DEFLEXION OF CAST IRON WHEN IT 
 RESISTS PRESSURE OR WEIGHT. 
 
 98. The doctrine of the Strength of Materials, as given in 
 this Work, rests upon three first principles, and these are 
 abundantly proved by experience. 
 
 The First is, that the strength of a bar or rod to resist a 
 given strain, when drawn in the direction of its length, 
 is directly proportional to the area of its cross section ; 
 while its elastic power remains perfect, and the 
 direction of the force coincides with the axis. 
 
 99. The Second is, that the extension of a bar or rod by a 
 force acting in the direction of its length, is directly 
 proportional to the straining force, when the area of the 
 section is the same ; while the strain does not exceed 
 the elastic power.* 
 
 100. The Third is, that while the force is within the elastic 
 power of the material, bodies resist extension and 
 compression with equal forces. 
 
 101. It is further supposed that every part of the same 
 piece of the material is of the same quality, and that there are 
 no defects in it. If there be any material defect in a piece of 
 
 * This limit should be carefully attended to, for as soon as the strain exceeds the 
 elastic. power, the ductility of the material becomes sensible. The degrees of ductility 
 are extremely variable in different bodies, and even in different states of the same 
 body. A fluid possesses this property in the greatest degree, for every change in the 
 relative position of its parts is permanent. 
 
SECT. VIL] RESISTANCE TO PRESSURE. 89 
 
 cast iron, it may often be discovered, either by inspection, or 
 by the sound the piece omits when struck ; except it be air 
 bubbles, which cannot be known by these means. 
 
 The manner of examining the quality of a piece of cast iron 
 has been given in the Introduction, p. 5 ; and such as will 
 bear the test of hammering with the same apparent degree of 
 malleability, will be found sufficiently near of the same strength 
 and extensibility for any practical deductions to be correct. 
 
 The truth of these premises being admitted, every rule that 
 is herein grounded upon them may be considered as firmly 
 established as the properties of geometrical figures. 
 
 102. A free weight or mass of matter is always to be 
 considered to act in the direction of a vertical line passing 
 through its centre of gravity ; and its whole effect as if 
 collected at the point where this vertical line intersects the 
 beam or the pillar, which is to support it. But if the weight 
 or mass of matter be partially sustained, independently of the 
 beam or pillar, in any manner, then the direction and intensity 
 of the force must be found that would sustain the mass in 
 equilibrium,* and this will be the direction and intensity of 
 the pressure on the beam or pillar. 
 
 103. Let/ denote the weight in pounds which would be 
 borne by a rod of iron, or other matter, of an inch square, 
 when the strain is as great as it will bear without destroying 
 a part of its elastic force, f Also, let W be any other weight 
 to be supported, and b = the breadth and t = the thickness 
 of the piece to support it, in inches. Then, by our first- 
 principles, art. 98, we have 
 
 / : W : : 1 : b t 
 
 or, ^ = 6 t (i.) 
 
 * The method of finding this force and its direction is explained in my Elementary 
 Principles of Carpentry, art. 24-29. 
 
 f- " A permanent alteration of form," Dr. Young has remarked, " limits the strength 
 of materials with regard to practical purposes, almost as much as fracture, since in. 
 general the force which is capable of producing this effect is suffici< nt, with a small 
 addition, to increase it till fracture takes place." Natural Philosophy, vol. i. p. 141. 
 
90 , RESISTANCE TO PRESSURE. [SECT. vn. 
 
 That is, the area should be directly as weight to be sup- 
 ported, and inversely as the force which would impair the 
 elastic power of the material. 
 
 104. If e be the quantity a bar of iron, or other matter, 
 an inch square and a foot in length, would be extended by 
 the force/; and / be any other length in feet ; then 
 
 !:::: A, or Ze=A = 
 
 the extension in the length /. (ii.) 
 
 For, when the force is the same, the extension is obviously 
 
 proportional to the length. 
 
 And, since by our principle, art. 99, the extension is as 
 
 the force ; we have / '. W : : e : extension produced by the 
 
 weight W = -~, and we obtain from Equation ii. ~ 
 = A. (iii.) 
 
 In which A is the extension that would be produced in 
 the length /, by the weight W. 
 
 105. Where a comparison of elastic forces is to be made, 
 it is sometimes convenient^ have a single measure which is 
 called the modulus of elasticity.* It is found by this ana- 
 logy : as the length of a substance is to the diminution of 
 its length, so is the modulus of elasticity to the force pro- 
 ducing that diminution. Or, denoting the weight of the 
 modulus in fts. for a base of an inch square by m 9 
 
 (iv.) 
 
 And if p be the weight of a bar of the substance 1 foot 
 in length, and 1 inch square ; then if M be the height of the 
 modulus of elasticity in feet,f 
 
 = M. (v.) 
 
 * The term was first used by Dr. Young. Lectures on Nat. Phil. vol. ii. art. 319. 
 t By this and the preceding equation, were calculated the height and weight of the 
 modulus of elasticity of the different bodies in the Alphabetical Table. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 91 
 
 106. Let the rectangular beam A A', fig. 14, be supported 
 upon a fulcrum D, in equilibrio, and for the present consi- 
 dering the beam to be acted upon by no other forces than 
 the weights W W ; which are supposed to have produced 
 their full effect in deflecting the beam, and the vertical 
 section at B D to be divided into equal, and very thin fila- 
 ments, as shown in fig. 15. 
 
 Consider B, fig. 14, to be the situation of one of the small 
 filaments in the upper part of the beam, and a a a tangent 
 to the curvature of the filament B, at the point B. Now, it 
 is clearly a necessary consequence of equilibrium that the 
 forces tending to separate the filament at B should be equal, 
 and in the direction of the tangent a a ; and the strain is 
 obviously a tensile one. 
 
 But since F A is the direction of the weight, we have, by 
 the principles of statics, 
 
 B a : A a : : S ( = the resistance of the filament B) : a 
 = its effect in sustaining the weight W. 
 
 These forces, we know both from reasoning and experi- 
 ence, will compress the lower part of the beam ; and let D 
 be a compressed filament, of the same area as the filament B, 
 and in the same position, and at the same distance from the 
 under surface as the filament B is in respect to the upper 
 surface. Also, let e e be a tangent to the filament at D, and 
 parallel to a a ; and representing one of the equal and oppo- 
 site strains on the filament D by e D ; we have, e D : e A. '.'.&' 
 (the resistance to compression of D) : e e ' I> = the effect of 
 
 the filament D in sustaining the weight W. 
 
 The effect of both the filaments, B D, in supporting the 
 weight will therefore be, 
 
 A. s e A . s' 
 
 Ba ~~eD ' 
 
 or since B a cT), and as portions of the same matter of 
 equal area resist extension or compression with equal forces 
 
92 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 (art. 100) S = S' ; therefore, ^ x (A a + e A) = the effect 
 of the filaments D and B.* But 
 
 A a + e A ="B D, 
 
 the vertical distance between the filaments, f Consequently 
 s B B g D = this effect in supporting the weight W. (vi.) 
 
 107. As one side of the beam suffers extension, and the 
 other side compression, there will be a filament at some point 
 of the depth, which will neither be extended nor compressed ; 
 the situation of this filament may be called the neutral axis, 
 or axis of motion. 
 
 The extension or compression of a filament will obviously 
 be as its distance from the neutral axis ; and when the 
 neutral axis divides the section into two equal and similar 
 parts, its place will be at the middle of the depth. 
 
 And since the effect of two equal filaments is as the dis- 
 tance between them, the effect of either will be as its distance 
 from the neutral axis ; for the filaments being equal, and 
 the strain on them equal, the axis will be at the middle of 
 the distance between them ; and the effect of both being 
 measured by the whole depth, that of one of them will be 
 
 * But when the strain exceeds the elastic force of a body, the resistance to com- 
 pression exceeds the resistance to tension ; consequently, the effect of the filaments 
 must be 
 
 A a . S + e A . S' 
 Ba 
 
 Now the difference between S and S' will be constantly increasing till fracture takes 
 place, the area of the compressed part being constantly increasing, and that of the 
 extended part diminishing. The variation will depend on the ductility of the ma- 
 terial, but it cannot be ascertained, unless some very careful experiments were made 
 for the purpose ; and fortunately it is an inquiry not required in the practical appli- 
 cation of theory. 
 
 f When the flexure becomes considerable, the curve is flattened in consequence 
 of the forces compressing the beam, and A a + e A will exceed the vertical distance 
 between the filaments ; and the point of greatest strain will be found to change to 
 the place where the line A B intersects the filament, This change of the point of 
 greatest strain is very apparent in experiment. 
 
SECT, vii.] RESISTANCE TO PRESSURE 93 
 
 measured by half the depth. Therefore the effect of a fila- 
 ment is 
 
 S.BD S.BcZ 
 Bl 
 
 108. When a beam is sustained in any position,* not 
 greatly differing from a horizontal one, by a fulcrum, as in 
 fig. 14, the power of a fibre or filament to support a weight 
 at A or A' is directly as its force, its area, and the square of 
 its distance from the neutral axis ; and inversely as its dis- 
 tance, F B, of the straining force from the point of support. 
 
 For the strain being as the extension, and the extension 
 of any filament being directly as the distance of that filament 
 from the axis of motion, therefore, the force of a filament is 
 as its distance from the axis of motion. But it has been 
 shown (art. 103,) that the force is also as the area ; and the 
 power in sustaining a weight has been shown (art, 107,) to 
 be directly as the vertical distance from the neutral axis, and 
 
 inversely as the length B a, that is, as 2^ ; and, since the 
 triangles F B a y B D/, are similar, 
 
 (/ d p x the force of filament x by its area = ^ ^.^ y. wiu mMn (vffi ( 
 
 F B 
 
 109. Let d be the depth, divided into filaments, each equal 
 to x the roth part of \ ; also put F B = /, the breadth of the 
 beam = b, and / the weight that a fibre of a given magni- 
 tude would bear when drawn in the direction of its length, 
 without destroying its elastic force. 
 
 Now, if we calculate the mean strain upon each filament 
 
 * It does not sensibly differ from the correct law of resistance till the beam be 
 so much inclined as to slide on its support ; but the general investigation will be 
 found in art. 276. 
 
94 
 
 RESISTANCE TO PRESSURE. 
 
 [SECT. vn. 
 
 by Equation viii. art. 108, we obtain the following progres- 
 sion, and its sum is the weight the beam will support.* 
 
 4/Ea? 
 id 
 
 x (1 + 2 2 + 3 2 m. 
 
 ~ ) = W . 
 
 110. If the beam be rectangular, the value of 
 
 w-/ &d2 
 
 (ix.) 
 
 (x.) 
 
 Therefore the lateral strength of a rectangular beam is 
 directly as its breadth, and the square of its depth, and in- 
 versely as its length. 
 
 And when the beam is square, its lateral strength is as the 
 cube of its side. 
 
 111. If a plate be fixed along one of its sides A B, and 
 the load be applied at the angle C ; 
 then if the distance A B be greater 
 than B C, the plate will break in the 
 direction of some line EB. To find 
 B this line, put F C = /, the leverage, and 
 E B = b, the breadth ; also t = the 
 tangent of the angle E B C. Then, by 
 similar triangles, 
 
 V 1 + 
 
 C : I = 
 
 EC x t 
 
 and 
 
 therefore 
 
 1 : V * + ? - ' B C : 6 = B C x V 1 + 
 
 6 I 
 
 But this equation is a minimum when t 1 ; that is, when 
 the angle E B C is 45 degrees ; consequently 
 
 * The first term of the progression is equivalent to the quantity called a fluxion, 
 
 and is usually written thu?, 3 The same remark applies to the other 
 
 progressions. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 95 
 
 fd 2 (l + t 2 ) fd? 
 
 -b- I = V = w - <*> 
 
 112. If the beam be rectangular, and the strain be in a 
 perpendicular direction to one of its diagonals AC, making 
 that diagonal = b, and the depth E P = a, the progression 
 becomes (because the breadth is successively 
 
 or ' W = < xii -) 
 
 If the beam be square, the direction of the straining force 
 coincides with the vertical diagonal, and in that case 
 
 in = w - < xiii -> 
 
 But the diagonal of a square beam is equal to its side multi- 
 plied by y/ 2; hence, if d be the side, we have 
 
 fd? 
 
 -^= = W. (xiv.) 
 
 6 V2J 
 
 Consequently the strength of a square beam, when the force 
 is parallel to its side, is to the strength of the same beam 
 when the force is in the direction of its diagonal as 
 
 .or as 10 is to 7 nearly. 
 
96 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 113. If the beam be a cylinder, and r the radius, then b is 
 successively 
 
 2 V r2 ~ x \ 2 -s/r 2 - ( 2 *)* &c -> 
 And -~ x j V^ 2 a* + 2 2 V i 2 ( 2 ^ + & c - { - W 
 
 If <^ be the diameter, then 
 
 The lateral strength of a cylinder is directly as the cube of 
 its diameter, and inversely as the length. 
 
 The strength of a square beam is to that of an inscribed 
 cylinder as 
 
 8 : 6 x -7854, that is, as 1 : -589, or as 1'7 : 1. 
 
 114. If the section of the beam be an ellipse, when the 
 strain is in the direction of the conjugate axis, we have by 
 the same process 
 
 where t is the semi-transverse, and c the semi-conjugate 
 axis. 
 
 115. If the beam be a hollow cylinder or tube, and r be 
 the exterior radius, n r being that of the hollow part, then by 
 the same process * we find 
 
 The radius of a solid cylinder that will contain the same 
 quantity of matter as the tube, is easily found by geometrical 
 construction in this manner : make B D perpendicular to 
 B C ; then C D being the radius of the tube, and B C that of 
 
 * Dr. Young gives a rule which is essentially the same, of which I was not aware 
 when my " Principles of Carpentry " was written. See Natural Philosophy, vol. ii. 
 art. 339, B. scholium. In a recent work on the Elements of Natural Philosophy, by 
 Professor Leslie, vol. i. p. 242, the learned author has neglected to consider the effect 
 of extension in bis investigation of this equation. 
 
SECT. VIL] RESISTANCE TO PRESSURE. 97 
 
 the hollow part, B D will be the radius of a solid cylinder 
 which will contain the same quantity of matter as the tube. 
 Because 
 
 BD 2 =CD 2 -BC 2 . 
 
 By comparing Equation xv. and xviii. we find that when a 
 solid cylinder is expanded into a tube, retaining the same 
 quantity of matter, the strength of the solid cylinder being 1, 
 that of the tube will be ( ^^- 3 - When the thickness F E 
 
 is yth of the diameter A E, the strength will be increased in 
 the proportion of T7 to 1. And when EE is - ihs of the 
 diameter, the strength will be doubled by expanding the 
 matter into a tube. But a greater excess of strength cannot 
 be safely obtained than the latter, because the tube would 
 not be capable of retaining its circular form with a less 
 thickness of matter. Erom ^th to ^th seems to be the 
 most common ratio in natural bodies, such as the stems of 
 plants, &c. 
 
 116. If a beam be of the form shown in Plate I., fig. 9 
 (see art. 38, 39, and 40), and d be the extreme depth, and b 
 the extreme breadth ; q b the difference between the breadth 
 in the middle and the extreme breadth, and p d the depth of 
 the narrow part in the middle ; then by the process employed 
 by calculating Equation x. we find 
 
 117. If the middle part of the beam be entirely left out, 
 with the exception of cross parts to prevent the upper and 
 
98 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 lower sides coming together, as in figs. 11 and 12, Plate TI. 
 (see art. 41), and d be the whole depth, p d the depth of the 
 part left out in the middle, and^ the breadth, then 
 
 118. Hitherto we have only considered those forms where 
 the neutral axis divides the section into identical figures ; but 
 there are some interesting cases * where this does not 
 happen, such, for example, as the triangular section. 
 
 Taking the case of a triangular section with a part removed 
 at the vertex, we shah 1 have a general case which will include 
 that of the entire triangle. Let d be the depth of the com- 
 plete triangle, and m d the depth of the part cut off at the 
 vertex ; and n d the depth of the neutral axis, M N, from the 
 upper side a b of the beam ; then the distance of the neutral 
 axis from the base will be (1 n m) d. If both sides of 
 the neutral axis were the same as the upper one, the strength 
 
 would be equal to that of a parallelogram a b p q, added to a 
 triangle a op ; hence from Equation x. and xii. we have 
 
 
 But to find the place of the neutral axis, we must compare 
 the strength of the lower side with the upper one ; and the 
 
 * They are interesting, because the early theorists fell into some serious errors 
 respecting them, and consequently have led practical engineers into erroneous 
 
 opinions. 
 
SECT, vii.j RESISTANCE TO PRESSURE. 99 
 
 strength of the lower side is equal to that of a rectangle 
 M N B C minus a triangle q r C, or 
 
 J-; \ 4 6 d? (1 - m n? b d? (1 - TO - n) 3 I = W. 
 o I ( ) 
 
 And consequently, 
 
 n 3 (4 m + n) = 4. (1 m n) 2 (1 m n) 3 . 
 
 Whence, 
 
 _ J7E*: 
 
 2 (1 - m) v V 2 (1 - m) / " 1 m 
 
 When m = Q'l, then ^zz'592, and 
 
 348/6^ = w< 
 
 But if m = o, or the triangle ABC be entire, then n ' 
 nearly.* And 
 
 .339 fbd* 
 
 (xxi.) 
 
 If m = 9 we have n=. '58166, and 
 
 347/6 <22 
 
 = W.f (xxn.) 
 
 (xxm.) 
 
 Where m = Q'I, the strength is about the greatest possible, 
 a triangular prism being about ^ 7 th part stronger when the 
 angle is taken off to n>th of its depth, as shown by the 
 shaded part of the figure. Emerson first announced this 
 seeming paradox, j but it is easily shown that his solution 
 only applies to the imaginary case where the neutral axis is 
 an incompressible arris, at the base of the section. 
 
 A triangular prism is equally strong, whether the base or 
 the vertex of the section be compressed : and by comparing 
 
 * Duleau obtains a result equivalent to n = '57 in this case, but the result only is 
 given. Essai The"orique, &c., p. 77. 
 
 t This rule was first published in the Philosophical Magazine, vol. xlvii. p. 22. 
 1816. 
 
 t Mechanics, Sect. VIIL, p. 114. 5th edit. 1800. 
 
 Duleau has proved the truth of this by his experiments on the flexure of trian- 
 gular bars. Essai sur la Resistance, &c., p. 26. 
 
 H-2 
 
100 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 Equations x. and xxii. it appears that its strength is to that 
 of a circumscribed rectangular prism as 339 : 1000, or nearly 
 as 1 : 3. But let it be remembered that this ratio only 
 applies to strains which do not produce permanent alteration 
 in the materials, and "where the arris is not injured by the 
 action of the straining force : if the strain be increased so as 
 to produce fracture, the triangle will be found still weaker 
 than in this proportion, when the arris is extended, and 
 somewhat stronger when the arris is compressed ; in the 
 former case, from the imperfection of castings, where there is 
 much surface in proportion to the quantity of matter, as in 
 all acute arrises ; in the latter case, from the saddle or other 
 thing used to support the weight reducing the quantity of 
 actual leverage. 
 
 It may be useful to remark, that a triangle contains half 
 the quantity of matter that there is in the circumscribing 
 rectangle, but its strength is only one-third ; hence it is not 
 economical to adopt triangular sections, and a like remark 
 applies to the T formed sections so commonly used. 
 
 119. If the whole depth of a T formed section be d, its 
 greatest breadth b, and its least breadth (1 q) b. Then, 
 supposing the depth of the neutral axis M N from the narrow 
 
 edge AE to be -d, the strength of the bar will be 
 
 Gin* 
 
 For -^r would be the square of the whole depth were both 
 sides of the neutral axis the same ; and the strength would 
 be equal to a rectangle of that depth with the breadth 
 
 (1 ?) & 
 ' But the strength of the other side of the axis by Equa- 
 
 tion xix. is 
 
 - q P 3 ) (n - I) 2 
 6 I n? 
 
SECT. VII.] 
 
 RESISTANCE TO PRESSURE. 
 
 hence we have the equation (n I) 2 (1 
 determine the place of the neutral axis ; or 
 
 101 
 
 = 1 ^ to 
 
 V 
 
 consequently, 
 
 l-p*q/ 
 
 This formula is complicated, but it affords some curious 
 results. If we make p = o, we have the strength of a bar 
 with its neutral axis at C, and the depth A C is 
 
 \i + vi-?/ ; 
 
 where d is the whole depth. 
 
 And if AE = J D B, then AC = f rf, when 
 the neutral axis is at C. 
 
 The neutral axis may be at any point that 
 may be chosen between the point C and half 
 the depth, by varying the values of q or p for F 
 that purpose. 
 
 If we make 
 
 also 
 
 The strength is 
 
 q = -75, and.p = -5, then A M = , and D F = C M; 
 
 A E = | D B. 
 
 fbd? 
 
 6 x 2-4 I 
 
 W. 
 
 xxv 
 
 The figure is drawn in these proportions, b is the whole 
 breadth D B, and d the whole depth. Its strength is to that 
 of the circumscribing rectangle shown by the dotted lines as 
 ~ : 1 or as 5 : 12. 
 
 These equations show the relation between the strength of 
 beams, and the weight to be supported in some of the most 
 
102 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 useful cases when the load is applied as in fig. 14 but 
 previous to considering how these equations will be effected 
 by varying the mode of supporting the beam, it will be 
 desirable to give some rules /or estimating the deflexion of 
 beams. 
 
 1*20. The deflexion of a beam supported as in fig. 16, 
 Plate II., is caused by the extension of the fibres of the 
 upper side, and the compression of those on the under side ; 
 the neutral line ABA' retains the same length. 
 
 If we conceive the length of a beam to be divided into a 
 great number of equal parts, and that the extension, at the 
 upper side of the beam, of one of these parts is represented 
 by a d, then the deflexion produced 'by this extension will 
 be represented by d e, and the angles a c b, dee, being 
 equal, we shall have 6 c : dc : : a b : de ; the smallness of 
 the angles rendering the deviation from strict similarity 
 insensible. 
 
 Now, however small we may consider the parts to be, into 
 which the length is divided, still the strain will vary in differ- 
 ent parts of it, and consequently the deflexion ; but if we 
 consider the deflexion produced by the extension of any 
 part, to be that which is due to an arithmetical mean between 
 the greatest and least strains in that part, we shall then be 
 extremely near the truth. 
 
 We have seen that the strain is as the weight and leverage 
 directly, and as the breadth and square of the depth inversely 
 (see art. 108). Our investigation will be more general by 
 considering the weight, breadth, and depth variable, by taking 
 /, <5, and d for the length, breadth, and depth of the middle 
 or supported point, and W for the whole weight, and %, y, 
 and w for the depth, breadth, and weight on any other point. 
 Then, the deflexion from the strain at any point c is as 
 
 Wlx w x (dcf m t m 21>d?fw(d c) 2 
 2 6 d* '' yy? '' '' * '' W Ly & 
 
SECT. VIL] RESISTANCE TO PRESSURE. 103 
 
 And, if z be the length of one of the parts into which we 
 suppose the whole length divided, then the deflexion from the 
 mean force on the length of z situate at c will be 
 
 2 b cP 6 w z /d 
 Vilya? X V 
 
 Since the whole deflexion D A is the sum of the deflexions 
 of the parts, we have 
 
 2 6 d 2 e w z 3 X m 2 \ 
 
 x( I 2 + 2 2 + &c. m 1 2 + 1 =D A. (i.) 
 
 W I y ic 3 \ 2 / 
 
 121. Case 1. When a beam is rectangular, the depth and 
 breadth uniform, and the load applied at one end. Then, 
 
 Therefore the progression becomes 
 
 UNITE KS ITT Of 
 
 of which the sum is 
 
 o - 72 
 
 = the deflexion DA.* (il) 
 
 122. Case 2. When the section of the beam is rectan- 
 gular, and the load acts at one end, the depth being uniform, 
 but the breadth varying as the length. 
 
 In this case the progression is 
 
 2 6 2 2 Z 2 
 
 T- x (1 + 2, &c.) = - = the deflexion D A. (iii.) 
 
 a a 
 
 This is the beam of uniform strength, described in art. 30, 
 fig. 6, and the deflexion is -grd more than that of a beam of 
 equal breadth throughout its length. The deflexion of the 
 beam of equal strength described in art. 33 is the same ; the 
 neutral axis becomes a circle in both these beams. f 
 
 * The same relation is otherwise determined in Dr. Young's Natural Philosophy, 
 vol. ii. art. 325. 
 
 + M. Girard arrives at the erroneous conclusion, that all the solids of equal 
 resistance curve into circular arcs, (Traite* Analytique, p. 82,) in consequence of 
 neglecting the effect of the depth of the solid on the radius of curvature. 
 
104 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 In this case it is easily shown by other reasoning that the 
 curve of the neutral axis is a portion of a circle, and it is well 
 known that in an arc of very small curvature (one of such 
 as are formed by the deflexions of beams in practical cases), 
 the versed sine is sensibly proportioned to the square of the 
 sine. This will enable the reader to form an estimate of the 
 accuracy of the method I here follow. I am perfectly satis- 
 fied that it is correct enough for use in the construction of 
 machines or buildings, and that it is a useless refinement to 
 embarrass the subject with intricate rules ; but this explana- 
 tion may be necessary to some nice theorists, who aim rather 
 at imaginary perfection than useful application. 
 
 123. Case 3. When the section of the beam is rectangu- 
 lar, the load acting at the end, the breadth uniform, and the 
 depth varying as the square root of the length ; which is the 
 parabolic beam of equal strength. (See art. 27, fig. 3, 
 Plate I.) 
 
 In this case the progression is 
 
 the deflexion D A. 
 
 The deflexion is double that of a uniform beam, while the 
 quantity of matter is only lessened Jrd. 
 
 124. Case 4. When the section of the beam decreases 
 from the supported point to the end where the load acts, so 
 that the sections are similar figures, then the curve bounding 
 the sides of the beam will be a cubical parabola ; that is, the 
 depth will be every where proportional to the cube root of the 
 length. 
 
 In this case the progression is 
 
 - the deflexion D A. (v.) 
 
 5 d 
 
 The deflexion is to that of a uniform beam as 1*8 : 1. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 105 
 
 125. Case 5. When a beam is of the same breadth 
 throughout, and the vertical section is an ellipse (see fig. 8, 
 art. 32), the deflexion from a weight at the vertex may be 
 exhibited in a progression as below : 
 
 P e z 3 x f 2 8 &c m 2 > 
 
 d *~ 2i - 22 
 
 xf 2 &c \_ DA 
 
 By actually summing this progression when m = 10, we 
 have 
 
 - - = the deflexion D A. (vi.) 
 
 126. Case 6. If a rectangular beam of uniform breadth 
 and depth be so loaded that the strain be upon any point c, 
 then 
 
 d c x W x (2 Z dc) 
 I 2 : dcx(2 I dc): :W:w= ~ - 
 
 This value of w being substituted in Equation i., we have 
 
 (4 - 1) = = the deflexion DA.* (vii.) 
 
 This is the deflexion of a beam uniformly loaded when it 
 is supported at the ends, / being half the length. 
 
 127. Case 7. If the section of a beam be rectangular, 
 and the breadth uniform, but a portion of the depth varying 
 as the length, and the rest of it uniform : then the depth at 
 any point c will be 
 
 the depth at the point where the weight acts being the 
 1 - n th part of the depth at the point of support. 
 
 * This relation is otherwise determined by Dr. Young, Nat. Philos. vol. i. art. 329. 
 
106 KESISTANCE TO PKESSURE. [SECT. vn. 
 
 This value of x being substituted in Equation i., art. 120, 
 it becomes 
 
 iii x { * + g ^ z)3+&c .} -D A. 
 
 And the general expression for the sum of this progression is 
 
 2 Z 2 e f 2 (l-w) 3 (l-w) 2 31, 1 
 
 When n = *5, as in the beams figs. 4 and 5, Plate I., we 
 have 
 
 - ==D A the deflexion. (viii.) 
 
 Hence the dcfl^ion of a uniform beam being denoted by 1, 
 this beam will be deflected 1*635 by the same force; the 
 middle sections being the same. 
 
 If the beam be diminished at the end to two-thirds of the 
 depth at the middle, then 
 
 -! 
 
 and 
 
 0-895 Z 2 6 = D A ^ deflexion> (ix } 
 
 128. Case 8. If a rectangular beam be supported in the 
 middle, and uniformly loaded over its length, then 
 
 Hence, when the beam is of uniform breadth and depth, 
 we have, by substituting this value of w in Equation i., 
 art. 120, 
 
 2-L? x (1 + 2 3 + 33 + &c.) = ^j = D A the deflexion. (x.) 
 
 t j Z Ct> 
 
 In this case the deflexion is fths of the deflexion of 
 the same beam having the whole weight collected at the 
 extremities. 
 
 129. Case 9. If a beam be generated by the revolution 
 of a semi-cubical parabola round its axis, which is the figure 
 
SECT, viz.] RESISTANCE TO PRESSURE. 107 
 
 of equal strength for a beam supported in the middle when 
 the weight is uniformly diffused over its length, then 
 
 (dc)W 
 and y=x\ alsow = - L -- 
 
 It 
 
 These quantities, substituted in Equation i., art. 120, give 
 
 2 4 
 
 LlJ^ x / 1 + 2* + 33 + &c. "I =^=D A the deflexion. (xi.) 
 
 d i j % a 
 
 Here the deflexion is f ths of that of a uniform beam with 
 the load at the extremities. 
 
 130. Case 10. If a beam to support a uniformly dis- 
 tributed load be of equable breadth, but the depth varying 
 directly as the distance from the extremity, as in fig. 21, 
 Plate III., then 
 
 b is constant, and 10 = 
 
 therefore, by Equation i., art. 120, 
 
 2 
 
 (dc)W 
 
 d 
 
 =D A the deflexion. ( x ii.) 
 
 If the beam had been uniform, and the loads at the extremi- 
 ties, the deflexion would have been only ^rd of the deflexion 
 in this case. 
 
 The cases I have considered are perhaps sufficient for the 
 ordinary purposes of business ; the next object is to show 
 how these calculations are affected by changing the position 
 and manner of supporting the beam, or the nature of the 
 straining force ; and to compare them with experiments, and 
 draw them into practical rules. For this purpose the mosb 
 clear and the most useful plan seems to be that of taking 
 known practical cases for illustration. 
 
108 RESISTANCE TO PRESSURE. [SECT. VIT. 
 
 BEAMS SUPPORTED IN THE MIDDLE, AND STRAINED AT THE 
 ENDS, AS IN THE BEAM OF A STEAM-ENGINE. 
 
 131. The distance I 1 B, fig., 14, of the direction of the 
 straining force from the centre of motion being constantly 
 the same, the strain will be the same in any position of the 
 beam (art. 108). Also, the deflexion from its natural form 
 will be the same in every position, because the strain is the 
 same ; and the length does not vary with the position. 
 
 Now the force acting upon the beam of a steam-engine 
 being impulsive, the practical rules for its strength will be 
 found in the eleventh section ; the formula calculated in this 
 section being used to establish those rules. 
 
 BEAMS FIXED AT ONE END ; AS CANTILEVERS, CRANKS, &C. 
 
 132. The strain upon a beam supported upon a fulcrum, 
 as in fig. 14, is obviously the same as when one of the ends 
 is fixed in a wall, or other like manner ; for fixing the end 
 merely supplies the place of the weight otherwise required to 
 balance the straining force. But though the strain upon the 
 beam be the same, the deflexion of the point where the strain 
 is applied will vary according to the mode of fixing the end ; 
 because the deflexion of the strained point will be that pro- 
 duced by the curvature of both the parts A B and B A'. 
 
 133. Let the dotted lines in fig. 17, Plate III., represent 
 the natural position of a beam fixed at one end in a wall : 
 when this beam is strained by a load at A, the compression 
 at C will always be enough to allow the beam to curve 
 between A' and B, and the strain at the point A' will obviously 
 be the same as if a weight were suspended there that would 
 balance the weight at A. Let ABA' be the curvature of 
 the beam by the load W, and a a a tangent to the point B. 
 Then A' a is proportional to the deflexion produced by the 
 strain at A', and 
 
SECT. VIL] RESISTANCE TO PRESSURE. 109 
 
 3x A' 
 
 ^TIT 
 the deflexion from the curving of the part A' B ; therefore 
 
 + A a=the whole deflexion D A. 
 A B 
 
 Now, since the deflexion is as the square of the length 
 (see Equation i.-xii., art. 120-130), we have 
 
 Therefore, 
 
 If the angle D B A be represented by c, then 
 
 BD = B Axcos. c; 
 
 and putting 
 
 Jt5 A. 
 
 we have 
 
 A a x (1 + r. cos. c) =D A. (ii.) 
 
 But since the deflexion is always very small, in practical 
 cases, we may always consider cos. c = 1, or equal to the 
 radius, and then we have 
 
 A a x (1 +r)=D A. (iii.) 
 
 134. In this equation r is the ratio of the length out of the 
 wall to the length within the wall ; that is, 
 
 B A : B A' : : 1 : r. 
 
 If the beam be either supported in the middle on a fulcrum, 
 or fixed so that the length of the fixed part be equal to that 
 of the projecting part, then 
 
 r=l,and2(Aa)=DA. (iv.) 
 
 135. If the fixed part be of greater bulk than the project- 
 ing part, or it be so fixed that the extension of the fixed part 
 would be very small, then the effect of such extension may be 
 
110 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 neglected, and the deflection D A and A a will be the 
 same ; particularly in the cranks of machinery, as in fig. 18, 
 because by employing this value of D A in calculating the 
 resistance to impulsion, we err on the safe side. See art. 327. 
 
 BEAMS SUPPORTED AT BOTH ENDS, AS BEAMS FOR 
 SUPPORTING WEIGHTS, &C. 
 
 136. When the same beam is supported at the ends, as in 
 fig. 19, instead of being loaded at the ends, and supported 
 in the middle, as in fig. 14, and the inclination and sum of 
 the load be the same in both positions, the strains will be the 
 same. 
 In either position of the beam we have 
 
 WxFB=W'xFB, 
 
 or as 
 
 W: W'::FB :FB; 
 
 and therefore, 
 
 W + W': W : :FF':F'B.* 
 
 Consequently, _ 
 
 If the beam be a rectangle, and the whole length F F = /, 
 and W the whole weight, then by art. 110, Equation x. 
 
 fbcP WxFBxFB 
 
 ~T~ ~T~ W 
 
 137. And the strain is as the rectangle of the segments 
 into which the point B divides the beam ; and therefore the 
 greatest when the point B is in the middle, as has been other- 
 wise shown by writers on mechanics. f 
 
 If the weight be applied in the middle, then 
 
 W+W'xFBxFBW + W' 
 
 FF 
 
 * Euclid's Elements, Prop. xviu. Book v. 
 t Gregory's Mechanics, vol. i. art. 178, cor. 2. 
 
SECT, vii.] RESISTANCE TO PRESSURE. Ill 
 
 In a rectangular beam, the whole length being /, and W 
 the whole weight, then 
 
 138. When a weight is distributed over the length of a 
 beam A B, fig. 20, in any manner, the strain at any point C 
 may be found. For let G be the centre of gravity of that 
 part of the load upon A C, and g that of the load upon B C. 
 Then by the property of the lever, 
 
 W AC "^the stress at C from the weight w of the load upon 
 AC. 
 
 Also, 
 
 * 
 
 ^ =the stress at C from the weight w of the load upon 
 
 CB. 
 
 Therefore the whole stress is 
 
 ; / i J^ +"/ 
 
 wxCBxAG + itfxACx^B 
 ACxCB 
 
 And by Equation v. art. 136, the strain will be 
 
 AG+^x (vi . } 
 
 139. Case 1. When the weight is uniformly distributed 
 over the length, then 
 
 the whole weight upon the beam ; these values being substi- 
 tuted in Equation vii. it becomes 
 
 Wx ACxCB 
 
 2 AB 
 
 = strain at C. 
 
 (viii.) 
 
 The strain is greatest at the middle of the length, for then 
 A C x C B is a maximum, and it is evidently the same as if 
 half the weight were collected there ; for in that case A C 
 being equal C B, and either of these equal to half A B, we 
 have in the case of a rectangular beam 
 
112 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 ff*!H a*-u^-w. , 
 
 140. Case 2. When the load increases from A to B in 
 proportion to the distance from A ; then 
 
 3 AB-2CB 
 
 A G = I A C, and g B = \ CB x 
 
 2AB- CB 
 
 Now since 
 
 w + -ui the whole weight, 
 
 and 
 
 \ AC 2 x W 
 w = , 
 
 also 
 
 rf = i CB x W 2AB ~ CB 
 A B 
 
 if these values be inserted in Equation vii. we have 
 W.AC 
 
 6 AB 
 
 (A B 2 A C 2 ) = the strain at C. (x.) 
 
 By the principles of maxima and minima of quantities, we 
 readily find that the strain is the greatest at the distance of 
 V-^A B from A. And the strain will be nearly 
 
 A "R2 "VV "\\r A R 
 
 ' at the point of greatest strain = ' when W is the whole weight, (xi.) 
 
 7'7o 
 
 This distribution of pressure applies to the pressure of a 
 fluid against a vertical sheet of iron ; as in lock-gates, reser- 
 voirs, sluices, cisterns, piles for wharfs, &c. 
 
 141. Case 3. When the load increases as the square of 
 the distance from A, we find by a similar process that the 
 strain at any point C is 
 
 The point of maximum strain in this case is at the distance 
 of i 1 A B from A. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 113 
 
 PRACTICAL RULES AND EXAMPLES. 
 RESISTANCE TO CROSS STRAINS. 
 
 142. Prop. i. To determine a rule for the breadth and 
 depth of a beam, to support a given weight or pressure, when 
 the distance between the supported or strained points is 
 given ; when the breadth and depth are both uniformly the 
 same throughout the length, and the strain does not exceed 
 the elastic force of cast iron. 
 
 143. Case 1. When a beam is supported at the ends, and 
 loaded in the middle, as in fig. 19. From Equation vi. art. 
 137, taking W for the weight, we have 
 
 W I = o , where I = F F ; fig. 19, 
 3 
 
 and the value of /is the only part required from experiment - 3 
 and 
 
 3ZW 
 
 Now, in the experiment described in art. 56, Sect. V., the bar 
 returned to its natural state when the load was 300 fts., and 
 I was perfectly satisfied that it would bear more than that 
 weight without destroying its elastic force. Therefore, from 
 this experiment, 
 
 That is, cast iron of the quality described in art. 56, will bear 
 15,300 fts. upon a square inch, when drawn in the direction 
 
 * Mr. Tredgold finds here that cast iron will bear a direct tensile force of 15,300 Its. 
 per square inch without injury to its elasticity, and concludes (arts. 70 76) that its 
 utmost tensile force is nearly three times as great as this, or upwards of 20 tons. 
 But it will be shown in the " Additions," art. 3, that a less weight per inch than 
 15,300 Its. was sufficient to tear asunder bars of several sorts of cast iron ; and the 
 mean strength of that metal, from experiments on irons obtained from various parts 
 of the United Kingdom, did not exceed 16,505 fts. per square inch. Mr. Tredgold 
 was mistaken in supposing the bar to have borne 300 fos. without injury to its 
 elasticity, as will be seen under the head " Transverse Strength " in the Additions. 
 EDITOR. 
 
 i 
 
114 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 of its length, without producing permanent alteration in its 
 structure. If this value of / be employed, our equation 
 becomes 
 
 2 x 15300 = l ^' 
 
 or, as it is convenient to take / in feet, 
 
 3 x 12 x I x W ZW 
 2 x 15300 ^SSO^ 
 
 144. 'Rule 1. To find the breadth of a uniform cast iron 
 beam, to bear a given weight in the middle. 
 
 Multiply the length of bearing in feet by the weight to be 
 supported in pounds; and divide the product by 850 times 
 the square of the depth in inches ; the quotient will be the 
 breadth in inches required.* 
 
 145. Rule 2. To find the depth of a uniform cast iron 
 beam, to bear a given weight in the middle. 
 
 Multiply the length of bearing in feet by the weight to be 
 supported in pounds, and divide this product by 850 times 
 the breadth in inches; and the square root of the quotient 
 will be the depth in inches. 
 
 When no particular breadth or depth is determined by the 
 nature of the situation for which the beam is intended, it will 
 be found sometimes convenient to assign some proportion ; 
 as, for example, let the breadth be the ^th part of the depth, 
 n representing any number at will.. Then the rule becomes 
 
 146. Rule 3. Multiply n times the length in feet by the 
 weight in pounds ; divide this product by 850, and the cube 
 root of the quotient will be the depth required: and the 
 breadth will be the nth part of the depth. 
 
 It may be remarked here, that the rules are the same for 
 inclined as for horizontal beams, when the horizontal distance, 
 "F I" fig. 19, is taken for the length of bearing. 
 
 * If the bar is to be of wrought iron, divide by 952 instead of 850. 
 If the beam be of oak, divide by 212 instead of 850. 
 If it be of yellow fir, divide by 255 instead of 850. 
 
SECT. VIL] RESISTANCE TO PRESSURE. 115 
 
 147. Example 1. In a situation where the flexure of a 
 beam is not a material defect, I wish to support a load which 
 cannot exceed 33,600fts. (or 15 tons) in the middle of a cast 
 iron beam, the distance of the supports being 20 feet ; and 
 making the breadth a fourth part of the depth. 
 
 In this case 
 
 , 4 x 20 x 33600 
 
 n = 4 and - - == 3162-35. 
 
 850 
 
 The cube root of 3162'35 is nearly 14-68 inches, the depth 
 required ; the breadth is 
 
 1^?= 3-87 inches. 
 
 In practice therefore I would use whole numbers, and make 
 the beam 15 inches deep, and 4 inches in breadth. 
 
 148. Case 2. When a beam is supported at the ends, but 
 the load is not in the middle between the supports. In this 
 case 
 
 W.FB x FB_/6d 2 
 
 (Equation, v. art. 136,) consequently 
 
 4FBxF'BxW 
 
 850 I 
 
 = bd*. 
 
 149. Rule. Multiply the distance FB in feet (see fig. 19) 
 by the distance Y B in feet, and 4 times this product, divided 
 by the whole length FF' in feet, will give the effective 
 leverage of the load, which being used instead of the length in 
 any of the rules to Case 1, Prob. i., the breadth and depth 
 may be found by them. 
 
 150. Example. Taking the same example as the last, 
 except that, instead of placing the 15 tons in the middle, it 
 is to be applied at 5 feet from one end ; therefore we have 
 F B = 5 feet, and consequently 
 
 12 
 
116 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 the number to be employed instead of the whole length in 
 Rule 3. That is, 
 
 4 x 15 x 33600 
 
 850 
 
 = 2372 nearly ; 
 
 and the cube root of 2372 is nearly 13'34 inches, the depth 
 for the beam, and 
 
 = 3-33 inches 
 
 for the breadth, or nearly 13^ inches by 3^ inches. 
 In the former case it was 1 5 inches by 4 inches. 
 
 151. Case 3. When the load is uniformly distributed 
 over the length of a beam, which is supported at both ends. 
 
 In this case 
 
 wi _f bd? 
 
 ~8~ '' 6 ' 
 
 (see Equation ix. art. 139,) hence 
 
 iw _ 
 
 X x 850 
 
 The same rules apply as in Case 1, art. 144, 145, and 146, 
 by making the divisor twice 850, or 1700. 
 
 152. Example. In a situation where I cannot make use of 
 an arch for want of abutments, it is necessary to leave an 
 opening 15 feet wide, in an 18 -inch brick wall ; required the 
 depth of two cast iron beams to support the wall over the 
 opening ; each beam to be 2 inches thick, and the height of 
 the wall intended to rest upon the beam being 30 feet ? 
 
 The wall contains 
 
 30 x 15 x \\ = 675 cubic feet ; 
 
 and as a cubic foot of brick-work weighs about lOOfts., the 
 weight of the wall will be about 67,500 fts. ; and half this 
 weight, or 33,750 tbs. will be the load upon one of the beams. 
 Since the breadth is supposed to be given, the depth will be 
 
 
SECT. VIL] RESISTANCE TO PRESSURE. ' 117 
 
 found by Rule 2, art. 145, if 1700 be used as the constant 
 divisor; thus 
 
 15 x 33750 
 1700x2 = 149 * e *rly. 
 
 The square root of 149 is 12^ nearly ; therefore each beam 
 should be 12 J inches deep, and 2 inches in thickness. This 
 operation gives the actual strength necessary to support the 
 wall ; but I have usually taken double the calculated weight 
 in practice, to allow for accidents. 
 
 In this manner the strength proper for bressummers, 
 lintels, and the like, may be determined. But if there be 
 openings in the wall so placed that a pier rests upon the 
 middle of the length of the beam, then the strength must be 
 found by the rule, art. 145. A rule for a more economical 
 form is given in art. 193. 
 
 153. Case 4. When a beam is fixed at one end, and the 
 load applied at 'the other ; also when a beam is supported 
 upon a centre of motion. By Equation x. art. 110, 
 
 and taking / in feet, and/ = 15,300fts., we obtain 
 
 wz 
 
 but the divisor 212 will be always sufficiently near for practice. 
 154. Rule 1. In a beam fixed at one end, take BD for 
 the length, fig. 17, Plate III., or if the beam be supported in 
 the middle, as in fig. 14, Plate II., take B F or B F' for the 
 length, observing to use the weight which is to act on that 
 end in the calculation. Then calculate the strength by the 
 rules to Case 1, art. 144, 145, and 146, using 
 
 ^f = 212 
 instead of 850 as a divisor. 
 
118 RESISTANCE TO PRESSURE. [SECT. VIT. 
 
 Example. By this rule the proportions for the arms of a 
 balance may be determined. Let the length of the arm, from 
 the centre of suspension to the centre of motion, be 1-^ feet ; 
 and the extreme weight to be weighed 3 cwt., or 336 fts., and 
 let the thickness be ^th part of the depth. Then by the rule 
 
 10 x 1-5 x 336 
 
 The cube root of 24 is 2 '88 inches, the depth of the beam 
 at the centre; and the breadth will be 0'28S inch. 
 
 Eor wrought iron the divisor is, in this case, 238, and 
 taking the same example, 
 
 10 x 1-5 x 336 = 21 . 2 
 238 
 
 The cube root of 21'2 is 2'77 inches, the depth required; 
 and the breadth is 0'277 inch. 
 
 155. Rule 2. If the weight be uniformly distributed over 
 the length of the beam, employ 425 as a divisor, instead of 
 850 in the rules to Case 1, art. 144, 145, 146. 
 
 156. Example. Required the depth for the cantilevers of 
 a balcony to project 4 feet, and to be placed 5 feet apart, the 
 weight of the stone part being 1000 fts., the breadth of each 
 cantilever 2 inches, and the greatest possible load that can 
 be collected upon 5 feet in length of the balcony 2200 fes. ? 
 
 Here the weight is 
 
 1000 + 2200 = 3200 fts. ; 
 
 and by Rule 2, Case 4, 
 
 3200 * 4 = 15-1 nearly 
 2 x 425 
 
 and the square root of 15'1 is 3' 80 nearly, the depth 
 required. 
 
 157. Remark. The depth thus determined should be the 
 depth at the wall, as A B, fig. 21, Plate III. ; and if the 
 breadth be the same throughout the length, the cantilever 
 
SECT. VIL] RESISTANCE TO PRESSURE. 119 
 
 will be equally strong in every part, if the under side be 
 bounded by the straight line B C ; * therefore, whatever 
 ornamental form may be given to it, it should not be reduced 
 in any part to a less depth than is shown by that line. 
 
 158. The strength of the teeth of wheels depends on this 
 case. But since in consequence of irregular action, or any 
 substance getting between the teeth, the whole stress may be 
 thrown upon one corner of a tooth ; and it has been shown 
 in art. Ill, that the resistance is much less in that case, for 
 then the strength of a tooth of the thickness d would only be 
 
 if it were everywhere of equal thickness ; and to make 
 allowance for the diminution of thickness, we ought to make 
 
 -f rf 2 
 
 Sfrf* = W. We have also to make an allowance for wear.f 
 
 5 ' ' 
 
 which will be ample enough at the rate of ^rd of the thick- 
 ness ; therefore, 
 
 In cast iron/ = 15,300; whence we have, with sufficient 
 accuracy, 
 
 Rule. Divide the stress at the pitch circle in fts. by 1500, 
 and the square root of the product is the thickness of the 
 teeth in inches. 
 
 * Emerson's Mechanics, 4to. edit. prop. Ixxiii. cor. 2. It was first demonstrated by 
 Galileo, the earliest writer on the resistance of solids. Opere del Galileo, Discorsi, 
 &c., p. 104, tome ii. Bonon, 1655. 
 
 f The allowance for wear should be for a velocity of 3 feet per second ; and in 
 proportion to the velocity, that is, 
 
 asS :!<:::<_". 
 
 Hence t & < *> i (1 h v ) *> or 12 t $ v )> should be deducted from the 
 thickness in the Table, for velocities differing from 3 feet per second. 
 
120 
 
 RESISTANCE TO PRESSURE. 
 
 [SECT. vii. 
 
 Example 1. Let the greatest power acting at the pitch 
 circle of a wheel be 6000 fts. Then 
 
 6000 _ 
 150*0 ; 
 
 and the square root of 4 is 2 inches, the thickness required. 
 
 The breadth of teeth should be proportioned to the stress 
 upon them, and this stress should not exceed 400 fts. for 
 each inch in breadth, when the pitch * is 2-J inches, because 
 the surface of contact is always small, and teeth work irregu- 
 larly when much worn. 
 
 The length of teeth ought not to exceed their thickness, 
 but the strength is not affected by the greater or less length 
 of the teeth, f 
 
 A Table of the Thickness, Breadth, and Pitch of Teeth for Wheel Work. 
 
 Stress in Ibs. at the 
 pitch circle. 
 
 Thickness of 
 teeth. 
 
 Breadth of 
 teeth. 
 
 Pitch J in inches. 
 
 Ibs. 
 
 inches. 
 
 inches. 
 
 inches. 
 
 400 
 
 0-52 
 
 1 
 
 11 
 
 800 
 
 0-73 
 
 2 
 
 1-5 
 
 1200 
 
 0-90 
 
 3 
 
 1-9 
 
 1600 
 
 1-03 
 
 4 
 
 2-2 
 
 2000 
 
 1-15 
 
 5 
 
 2-4 
 
 2400 
 
 1-26 
 
 6 
 
 2-7 
 
 2800 
 
 1-36 
 
 7 
 
 2-9 
 
 3200 
 
 1-43 
 
 8 
 
 3-0 
 
 3600 
 
 1'56 
 
 9 
 
 3-3 
 
 4000 
 
 1-64 
 
 10 
 
 3-4 
 
 4400 
 
 1-70 
 
 11 
 
 3-6 
 
 4800 
 
 1-78 
 
 12 
 
 37 
 
 5200 
 
 1-86 
 
 13 
 
 3-9 
 
 5600 
 
 1-93 
 
 14 
 
 4-0 
 
 6000 
 
 2-00 
 
 15 
 
 4-2 
 
 159. As good proportions for the teeth of wheels are of 
 much importance in the construction of machinery, I shall 
 
 * The surface of contact is nearly in the direct ratio of the pitch, and therefore the 
 breadth for a 24-inch pitch being given, the breadth for any other teeth will be 
 directly as the stress, and inversely as the pitch. 
 
 f On the length and form of teeth for wheels, the reader may consult the Additions 
 to Buchanan's Essays on Mill- work, vol. i. p. 39, edited by Mr. Rennie, 1842 ; or the 
 Paper on the Teeth of Wheels, by Professor Willis, given at p. 139 of that work, and 
 in the second volume of the Institution of Civil Engineers. 
 
 J The pitch is the distance from middle to middle of the teeth, and is here made 
 21 times the thickness of the teeth. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 121 
 
 illustrate the mode of applying this Table by examples of 
 different kinds. 
 
 Case 1. It is a common mode to compute the stress on 
 the teeth of a machine by the power of the first mover, 
 expressed in horses' power, and the velocity of the pitch 
 circle in feet per second. Now, though I have given the 
 stress in pounds in the Table, I have still kept this popular 
 measure in view ; and assuming a horse's power to be 200 
 fts. with a velocity of 3 feet per second, which we ought to 
 do in calculating the strength of machines, 
 
 Then, the breadth in inches will be equal to the horses' 
 power, to which the teeth are equal, when the velocity of the 
 pitch circle is 1 J feet per second ; twice the breadth will be 
 the horses' power when the velocity is 3 feet per second ; three 
 times the breadth will be the horses' power when the velocity is 
 4^- feet per second ; four times the breadth will be the horses' 
 power when the velocity is 6 feet per second ; five times the 
 breadth will be the horses' power when the velocity is 1\ feet 
 per second ; and generally n times the horses' power when 
 the velocity of the pitch circle is n times 1^ feet per second. 
 
 Example. Let a steam engine of 10 horses' power be 
 applied to move a machine, and it is required to find the 
 strength for the teeth of a wheel in it, which will move at 
 the rate of 3 feet per second at the pitch circle. 
 
 Here then the horses' power should be double the breadth ; 
 consequently the breadth will be 5 inches, and according to 
 the Table, the thickness of the teeth 1*15 inches, and pitch 
 2 '4 inches. 
 
 And the same strength of teeth will do for any wheel 
 where the horses' power of the first mover, divided by the 
 velocity in feet per second, produces the same quotient. In 
 this example it is 10 divided by 3 ; and the same strength 
 of teeth will do for 20 divided by 6 ; 30 divided by 9 ; and 
 so on. This will be of some advantage in the arrangement 
 of collections of patterns. 
 
122 KESISTANCE TO PKESSUKE. [SECT, vn, 
 
 Case. 2. When a machine is to be moved by horses, the 
 horses' power should be estimated higher, on account of the 
 jerks and irregular action of horses. We shall not estimate 
 above the strain which often* takes place in horse machines, 
 if we rate the horse power at 400 fts. with a velocity of 
 3 feet per second, and make the strength of the teeth 
 accordingly. 
 
 But the breadth of the teeth should be made in the same 
 proportion as in the preceding case. 
 
 Example. When the horse power is taken at 400 fts. 
 with a velocity of 3 feet per second, the stress on the teeth 
 is given for this case in the Table. Thus, in a machine to 
 be moved by four horses, the stress on all the wheels of 
 which the pitch circles move at the rate of 3 feet per second, 
 will be 1600 ibs., and the pitch should be 22 inches, and 
 thickness of teeth 1-03 inches ; the breadth half the breadth 
 in the Table, or 2 inches. 
 
 Then, for any other velocity, as suppose 6 feet per second, 
 it will be, as 
 
 6:3:: 1600 : 8(JO. 
 
 That is, the stress on the teeth from a first mover of four 
 horses is 800 ft s. when the velocity is 6 feet per second ; 
 and the thickness of teeth by the Table is 0*73 inch, and 
 pitch 1*5 inches. 
 
 Case 3. It remains now to show the general rule which 
 includes the preceding cases, and appears to me to be a more 
 direct and simple mode of proceeding. 
 
 If P be the power of the first mover in pounds, and V 
 the velocity of that power in feet per second, the stress on 
 the teeth of a wheel of which the velocity of the pitch circle 
 
 is v, will be 
 
 p v 
 
 = W, the stress on the teeth. 
 
 v 
 
 But we cannot always know the velocity of the pitch 
 circle, because it is not in general possible to vary the number 
 
SECT. VIL] RESISTANCE TO PRESSURE. 123 
 
 of teeth after the pitch is determined, so as to give it the 
 velocity we have assigned to it before the pitch was known. 
 
 The calculation may therefore be made with advantage in 
 this manner : Let N be the number of revolutions the axis 
 is to make per minute, on which the wheel is to be placed : 
 and r the radius the wheel should have if the pitch were two 
 inches, then _ 2-1 dx r _ d N r 
 
 V 
 
 19-09x24 218-16 
 
 Consequently, P v^ 218-16 P v = w 
 
 Hence 
 
 218-16 P V _ x-14544 P V\| 
 
 1500 N r ~ ' r V S> / 
 
 The equation affords this rule. 
 
 Rule. Multiply 0'146 times the power of the first mover 
 in pounds by its velocity in feet per second, and divide the 
 product by the number of revolutions the wheel is proposed 
 to make per minute, and by the radius the wheel should 
 have in inches if its pitch were two inches ; the cube root of 
 the quotient will be the thickness of the teeth in inches. 
 
 Example 1. Suppose the effective force acting at the 
 circumference of a water-wheel to be 300 fts. and its velo- 
 city 10 feet per second,* it is proposed to find the thickness 
 for the teeth of a wheel which is to make twelve revolutions 
 per minute, and have thirty teeth. 
 
 Here, . 146 x 300 x 10 = 433. 
 
 And since the radius of a wheel with thirty teeth and a pitch 
 of 2 inches is 9*567 inches ;f we have 
 
 438 
 
 12 x 9-567 
 
 3-815. 
 
 * The manner of estimating the effective force, and determining the best velocity 
 for water-wheels, is shown in the Additions to Buchanan's Essays on Mill-work, vol. ii. 
 p. 512-526, second edition ; or p. 326-333, in the edit, by Mr. Rennie, 1842. On the 
 subject of water-wheels the reader may consult Mr. Rennie's Preface, p. 22, for a 
 notice of the labours of Poncelet, Morin, &c., and the valuable experiments of the 
 Franklin Institute. 
 
 t This is easily ascertained by Donkin's Table of the radii of wheels. See 
 Buchanan's Essays, vol. i. p. 206, second edition; or p. 114, Rennie's edition. 
 
124 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 The cube root of 3*815 is very nearly 1*563, the thickness of 
 the teeth required in inches. 
 
 Example 2. Let the effective force of the piston of a 
 steam engine be 6875 fts.* and its velocity 3^ feet per 
 second; it is required to determine the strength for the 
 teeth of a wheel to be driven by this engine, which is to 
 have 152 teeth, and make 17 revolutions per minute. 
 
 In this case the radius for 152 teeth with a 2-inch pitch 
 is 48*387 inches ; therefore, 
 
 0-146 x 6875 x 3-5 
 17-5 x 48-387 
 
 And the cube root of 4*15 is very nearly 1*6 inches, the 
 thickness of the teeth required. 
 
 By referring to the Table it will be found that teeth of this 
 thickness should have a breadth of about 9 inches. 
 
 These rules will be found to give proportions extremely 
 near to those adopted by Boulton and Watt, of Soho; 
 Rothwell, Hick, and Rothwell, of Bolton, in Lancashire ; 
 and other esteemed manufacturers ; which is one of the most 
 gratifying proofs of the confidence that may be placed in 
 the principles of calculation I have followed. The diffe- 
 rence is chiefly in the greater breadths I have assigned for 
 the greater strains, and which being a consequence of the 
 principle adopted for proportioning these breadths, I cannot 
 agree to change till it can be shown that the principle is 
 erroneous. 
 
 160. Case 5. When the pressure upon a beam increases 
 as the distance from one of its points of support. Since the 
 point of greatest strain is at ^/ ^ I from the point A, where 
 the strain begins at (see fig. 20), we have by art. 140 and 
 
 110, 
 
 w i _ fbd? 
 
 775 ~ 6 
 
 or when / is in feet, and 
 
 w^ / 
 
 /= 15300fts.; =&cF; 
 
SECT. VIL] RESISTANCE TO PRESSURE. 125 
 
 a result which differs so little from Case 3, that the same 
 rule may serve for both cases. 
 
 161. Prop. ii. To determine a rule for the diagonal of 
 a uniform square beam to support a given strain in the 
 direction of that diagonal ; when the strain does not exceed 
 the elastic force of cast iron. 
 
 162. Case 1. When a beam is supported at the ends 
 and loaded in the middle, 
 
 Wl _/a 3 
 4 24 ' 
 
 art. 137 and 112 ; or when / is in feet, and 
 
 = 15300 m, 
 
 163. Rule. Multiply the length in feet by the weight in 
 pounds, and divide the product by 212; the cube root of 
 the quotient is the diagonal of the beam in inches. 
 
 164. Case 2. When a beam is supported at the ends, 
 and the strain is not in the middle of the length, 
 
 W x F B x V B _ f_a? 
 
 ~~r~ ~~ 24 ' 
 
 art. 112 and 136 ; or when/ = 15,300 fts. and the length 
 and distances F B, F B from the ends are in feet, 
 
 (W x FB x F' B\i 
 -53 -/ = " 
 
 165. Rule. Multiply the weight in pounds by the dis- 
 tance F B in feet, and multiply this product by the distance 
 from the other end, or FB in feet (see fig. 19). Divide the 
 last product by 53 times the length, and the cube root of 
 the quotient will be the diagonal of the beam in inches. 
 
 I limit the rules to these cases only, because a beam is 
 seldom placed in the position described in this proposition. 
 Examples are omitted for the same reason. 
 
 166. Prop. m. To determine a rule to find the diameter 
 
126 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 of a solid cylinder, to support a given strain, when the strain 
 does not exceed the elastic force of cast iron. 
 
 If the diameter be not uniform, the diameter determined 
 by the rule will be that at the point of greatest strain, and 
 the diameter at any other point should never be less than 
 corresponds to the form of equal strength. 
 
 167. Case 1. When a solid cylinder is supported at the 
 ends, and the weight acts at the middle of the length, 
 
 art. 113 and 137 ; or when / is in feet, 
 
 /= 15300 fts. and d = the diameter in inches. 
 
 we have 
 
 _ 
 
 168. Rule. Multiply the weight in pounds by the length 
 in feet ; divide this product by 500, and the cube root of 
 the quotient will be the diameter in inches.* 
 
 The figure of equal strength for a solid, of which the cross 
 section is everywhere circular, is that generated by two cubic 
 parabolas, set base to base,f the bases being equal, and 
 joining at the section where the strain is the greatest. 
 
 169. Example. Required the diameter of a horizontal 
 shaft of cast iron to sustain a pressure of 2000 fts. in the 
 middle of its length ; the length being 20 feet ? In this 
 case we have 
 
 2000 x 20 
 
 500 
 
 _ 
 
 and the cube root of 80 is 4*31 inches nearly, which is the 
 diameter required. 
 
 This is supposed to be a case where the flexure is of no 
 importance, otherwise the diameter must be determined by 
 the rules for flexure. 
 
 * For wrought iron divide by 560 instead of 500. For oak divide by 125 instead 
 of 500. 
 
 f Emerson's Mechanics, 4to. edit., prop. Ixxiii., cor. 4. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 127 
 
 170. Case 2. When a cylinder is supported at the ends, 
 but the strain is not in the middle of the length. By art. 113 
 and 136, 
 
 Wx FB xF'B_'7854/(ff. 
 
 or when the lengths are in feet, d is the diameter in inches, 
 and/~15,300, the equation becomes 
 
 /4 Wx FB x F'BNq_ 
 V 500 I ) = 
 
 171. Rule. Multiply the rectangle of the segments, into 
 which the strained point divides the beam, in feet, by 4 times 
 the weight in pounds ; when this product is divided by 500 
 times the length in feet, the cube root of the quotient will be 
 the diameter of the cylinder in inches. 
 
 The figure of equal strength is the same as in Case 1, 
 art. 168. 
 
 172. Example. Required the diameter of a shaft of cast 
 iron to resist a pressure of 4000 fts. at 3 feet from the end, 
 the whole length of the shaft being 14 feet ? In this 
 example 
 
 3 x 11 x 4 x 4000 
 
 = 75-43. 
 
 500 x 14 
 
 The cube root of 75*43 is nearly 4*23 inches, the diameter 
 required. 
 
 173. Case 3. When a load is uniformly distributed over 
 the length of a solid cylinder supported at the ends only. 
 By art. 113 and 139, 
 
 ^=7854 fd*; 
 
 therefore, when / is in feet, d the diameter in inches, and 
 /= 15,300, we have 
 
 174. Rule. Multiply the length in feet by the weight in 
 
128 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 pounds, and ^th of the cube root of the product will be the 
 diameter in inches.* 
 
 The figure of equal strength for a uniform load, the 
 section being everywhere circular, f is that generated by the 
 revolution of a curve of which the equation is 
 
 175. Example. A load of 6 tons (or 13,440 fts.) is to be 
 uniformly distributed over the length of a solid cylinder of 
 cast iron, of which the length is 12 feet ; required its diameter, 
 so that the load shall not exceed its elastic force ? 
 
 In this case 
 
 12 x 13440 = 161280; 
 
 and the cube root of 161,280 is 54 '44, and n>th of this is 
 5 '444 inches, the diameter required. 
 
 176. Case 4. When a cylinder is fixed at one end, and 
 the load applied at the other; also, when a cylinder is 
 supported on a centre of motion. By art. 113, 
 
 therefore, when d is the diameter, / is in feet, and / 
 15,300 fbs., we have 
 
 The figure of equal strength is the same as in Case 1, 
 art. 168. 
 
 177. Rule. Multiply the leverage the weight acts with, 
 in feet, by the weight in pounds ; the fifth part of the cube 
 root of this product will be the diameter required in inches. 
 
 The most important application of this case is to determine 
 the proportions for gudgeons and axles ; and this application 
 will be best illustrated by an example. I 
 
 The greatest stress upon a gudgeon or axle takes place 
 when, from any accident, that stress is thrown upon the 
 
 * For wrought iron divide by 10'88. t Emerson's Mechanics, prop. Ixxiii., cor. 3. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 129 
 
 extreme point of its bearing. But besides the greatest 
 possible stress we have to provide for wear ; perhaps yth of 
 the diameter may be allowed for this purpose. 
 
 Now taking the length / for the length from the shoulder 
 to the extreme point of bearing in inches, we have 
 
 i ( A i w)*= d (l-D; or i (I W)*= d. 
 
 Whence we have this practical rule : Multiply the stress in 
 pounds by the length in inches, and the cube root of the 
 product divided by 9 is the diameter of the gudgeon in 
 inches.* 
 
 Example. Let the stress on the gudgeon be 10 tons, or 
 22,400 fts., and its length 7 inches. Then 
 
 7 x 22400 = 156800; 
 
 and the cube root of this number is 54 nearly ; and 
 
 54 
 
 = 6 inches. 
 
 the diameter required. 
 
 But the stress of a gudgeon on its bearings ought to be 
 limited, otherwise they will wear away very quickly : let us 
 suppose this stress to be confined to a portion of the circum- 
 ference, which is equal to fths of the diameter of the 
 gudgeon; and that the pressure is limited to 1500 fts. upon 
 a square inch, which is about as great a pressure as we ought 
 to put on the rubbing surfaces when one of them is of gun- 
 metal. In this case we shall have 
 
 i= 4W w 
 
 3 x 15000 d ' 01 ~ 1125 d ' 
 
 and to allow for a small portion of freedom we make 
 
 ;_ W 
 
 1000 d 
 
 * For wrought iron divide by 9 '3 4. For wheel carriages less than 3-inch axles the 
 length may be 5 times the diameter ; then for wrought iron, 
 
 Above 3 inches it may be 4 times the diameter. 
 
130 
 
 RESISTANCE TO PKESSUKE. 
 
 [SECT. vn. 
 
 If this value of I be introduced in the preceding equatio n 
 we have T /W2 x , 
 
 ( y=d', or W= 854 d?; and I = '854 d. 
 
 According to these principles the following Table has been 
 calculated, and I hope it will be useful. 
 
 Table of the Proportions of Gudgeons and Axles for different degrees of Stress. 
 
 Diameter of 
 gudgeons. 
 
 Length of gudgeons. 
 
 Stress they may 
 sustain. 
 
 inches. 
 
 inches. 
 
 fts. 
 
 A 
 
 43 
 
 213 
 
 1 
 
 64 
 
 480 
 
 i 
 
 85 
 
 854 
 
 tft 
 
 1-25 
 
 1,921 
 
 2 
 
 17 
 
 3,416 
 
 3 
 
 2-5 
 
 7,686 
 
 4 
 
 3-4 
 
 13,664 
 
 5 
 
 4-3 
 
 21,350 
 
 6 
 
 51 
 
 30,744 
 
 7 
 
 5-9 
 
 41,846 
 
 8 
 
 6-8 
 
 54,656 
 
 9 
 
 7-7 
 
 69,174 
 
 10 
 
 8-5 
 
 85,400 
 
 Gudgeons exposed to the action of gritty matter may be 
 made larger in diameter about ^th part. 
 
 178. Prop. iv. To determine a rule for the exterior 
 diameter of a uniform tube or hollow cylinder * to resist a 
 given force where the strain does not exceed the elastic force 
 of cast iron. 
 
 179. Case 1. When a tube is supported at the ends, and 
 the load acts at the middle of the length. By art. 115 
 and 137, w . 
 
 hence, when d is the diameter in inches, / the length in feet, 
 = 15,300 fts., we have 
 
 / 
 
 V 
 
 500 (1 - N 4 ) 
 
 180. General Rule. Fix on some proportion between the 
 diameters; so that the exterior diameter is to the interior 
 
 * A considerable accession of strength and stiffness is gained by making shafts 
 hollow, which has been illustrated in art. 115; but it is difficult to get them cast 
 sound, therefore shafts of this kind require to be carefully proved. 
 
SECT. VIL] RESISTANCE TO PRESSURE. 131 
 
 diameter as 1 is to N ; the number N will always be a 
 decimal, and ought not to exceed 0*8.* 
 
 Then multiply the length in feet by the weight to be 
 supported in pounds. Also, multiply 500 by the difference 
 between 1 and the fourth power of N, and divide the product 
 of the length and the weight by the last product, and the 
 cube root of the quotient will be the diameter in inches. 
 
 The interior diameter will be the number N multiplied by 
 the exterior diameter, and half the difference of the diameters < 
 will be the thickness of metal. 
 
 If the proportion between the exterior and interior diameter 
 be fixed, so that the thickness of metal may be always ^th of 
 the exterior diameter of the tube; then N = '6; and the 
 rule is 
 
 And there being no difference between this equation and 
 that for a solid cylinder, except the constant divisor, we have 
 this rule : 
 
 Particular Rule. When the thickness of metal is to be 
 -^th of the diameter of the tube, let the diameter be calcu- 
 lated by the rule for a solid cylinder, art. 168, except that 
 435 is to be used as a divisor instead of 500. 
 
 181. Example. Let the weight of a water-wheel, including 
 the weight of the water in the buckets, be 44,800 fts., and 
 the whole length of the shaft 8 feet ; from which deducting 5 
 feet,f the width of the wheel, leaves 3 feet for the length of 
 bearing ; required the diameter of a hollow shaft for it ? 
 
 * In a large shaft there should be a tolerable bulk of metal to secure a perfect 
 casting. Mr. Buchanan, in his " Essay on the Shafts of Mills," vol. i., p. 305, second 
 edition (or page 202-3 in the edition of 1841), describes a hollow shaft of which the 
 exterior diameter was 16 inches, and the interim one 12 inches, therefore 
 
 16:12::! : N = ^ = '75. 
 
 This shaft was for an over-shot water-wheel of 16 feet in diameter. 
 
 t The wheel being so framed that the part of the length of the shaft it occupies 
 may be considered perfectly strong. 
 
 K2 
 
132 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 Making N= '7, its fourth power is '2401 ; and 
 
 1 _ -240 = -76. 
 
 Therefore, by the general rule we have 
 
 3 x 44800 
 
 ^~ 354 in the nearest whole numbers ; 
 
 and the cube root of 354 is 7 inches, the exterior diameter; 
 and 
 
 7 x '7 = 4*9 inches, the interior diameter. 
 
 By the particular rule the computation is easier, for it is 
 
 8x44800 _o o. 
 ~435~~ 9 ' 
 
 and the cube root of 309 is 6' 76 inches, the exterior 
 diameter ; and the thickness of metal -J-th of this, or If 
 inches nearly. 
 
 The particular rule will be found to give a good proportion 
 for the thickness of metal for considerable strains ; but in 
 lighter work, where stiffness is the chief object, recourse 
 should be had to the general rule. 
 
 182. Case 2. When a tube is supported at the ends, but 
 the strain is not in the middle of the length. When the 
 necessary substitutions are made, we have, by art. 115 
 and 136 
 
 /4WxFBxF'B\i 
 
 \ 500 I x (1-N 4 ) J 3== d ' 
 
 183. Rule. Multiply the rectangle of the segments into 
 which the strained point divides the beam, in feet, by four 
 times the weight in pounds ; call this the first product. 
 
 Multiply 500 times the length, in feet, by the difference 
 between 1 and the fourth power of JS T (N being the interior 
 diameter when the exterior diameter is unity) ; call this the 
 second product. 
 
 Divide the first product by the second, and the cube root 
 of the quotient will be the exterior diameter of the tube in 
 inches. 
 
SECT, vii.] RESISTANCE TO PRESSURE, 133 
 
 Or, making the thickness of metal yth of the diameter, 
 calculate by the rule art. 171, using 435 instead of 500 as a 
 divisor. 
 
 184. Example. Let the weight of a wheel and other 
 pressure upon a shaft be equal to 36,000 fbs., the distance of 
 the point of stress from the bearing at one end being 3 feet, 
 and the distance from the other bearing 1*5 feet ; N being *8 ; 
 required the exterior and interior diameter of the shaft ? 
 
 The fourth power of '8 is "409, and 
 
 1 _ -409 -591. 
 
 Therefore by the rule 
 
 3 x 1-5 x 4 x 36000 
 
 500 x 4-5 x -591 
 
 = 485; 
 
 and the cube root of 485 is 7 '86 inches, the exterior diame- 
 ter, and 
 
 7'86 x "8 = 6*3 inches, the interior diameter. 
 
 Cases 3 and 4 are not likely to occur iti the practical 
 application of tubes, but they may be supplied by Cases 
 3 and 4 for solid cylinders, by dividing the diameter of the 
 solid cylinder by the cube root of tlte difference between 
 1 and the fourth power of N ; or when the thickness of 
 metal is to be yth of the diameter, divide by 435 instead of 
 500. 
 
 185. Prop. v. To determine a rule for finding the depth 
 of a beam of the form of section shown in fig. 9, Plate L, 
 to resist a given force when the strain does not exceed the 
 elastic force of cast iron. 
 
 186. Case 1. When the beam is supported at the ends, 
 and the load acts in the middle of the length. By art. 116 
 and 137, 
 
134 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 or making / = the length in feet, and/ = 15,300 fts., 
 
 W7 
 
 . = ** <i -->. 
 
 187. Rule. Assume a breadth a b, fig. 9, that will answer 
 the purpose the beam is intended for ; and let this breadth, 
 multiplied by some decimal q, be equal to the sum of the 
 projecting parts, or, which is the same thing, equal to the 
 difference between the breadth of the middle part and the 
 whole breadth. 
 
 Also, let p be some decimal which multiplied by the whole 
 depth will give the depth of the middle or thinner part ef in 
 the figure. 
 
 Multiply the length in feet by the weight in pounds, and 
 divide this product by 850 times the breadth multiplied into 
 the difference between unity and the cube of p multiplied 
 by q ; the square root of the quotient will be the depth in 
 inches. 
 
 The figure of equal strength for this case is formed by two 
 common parabolas put base to base, as shown by the dotted 
 lines in fig. 22; for/: d 2 a property of the parabola, the 
 other being constant quantities. Tig. 22 shows how it may 
 be modified to answer in practice. When a figure of equal 
 strength is used, the depth determined by the rule is that at 
 the point of greatest strain, as C D in the figure. 
 
 188. Example 1. Required the depth of a beam of cast 
 iron of the form of section shown in fig. 9, Plate I., to bear a 
 
 * If we make p 7, and q = ' 6 ; then, 
 
 850 (1- q p s ) = 675; 
 and the rule is 
 
 WZ 
 
 675 = 6 ^ ; 
 
 and the breadth of the middle part = '4 b, and the depth of the middle part 7 d. 
 
 When the parts are in these proportionate strength is to that of the circum- 
 scribed rectangular section as 1 : 1'26. 
 
 If, with the same proportions, we make the breadth a 6 always one-fifth of the 
 depth b d, fig. 9, the strength will be to that of a square beam of the same depth as 
 1 : 6'3 ; and the stiffness will be in the same proportion. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 135 
 
 load of 33,600 fts. in the middle of the length, the length 
 being 20 feet, and the breadth, a b, 3 inches ? 
 
 Take *625 for the decimal q, and *7 for the decimal p, 
 which are proportions that will be found to answer very well 
 in practice.* 
 
 Then 
 
 20 x 33600 20 x 33600 
 
 850 x 3 x (1 -625 x -7 3 ) 3 x 667 
 
 335-4 nearly ; 
 
 and the square root of 335*4 is 18*4 inches, the depth 
 required. 
 
 The depth b d being 18'4, the depth ef will be 
 
 i 
 
 18'4 x -7 = 12-88 inches ; also, 
 
 3 x -625 = 1-875, and 
 3 -1-875 =1-125 inches, 
 
 the breadth of the middle part of the section. 
 
 Comparing this with the example, art. 147, it will be 
 found that the same weight requires only about f rds of the 
 quantity of iron to support it, when the beam is formed in 
 this manner. 
 
 Example 2. The same rule applies to determining the size 
 of the rails for an iron railway, where economy with strength 
 and durability is of much importance. As the weight has to 
 move over the length of the rail, the figure of equal strength 
 is that shown in Plate III., fig. 24, only it should be placed 
 with the straight side upwards. 
 
 Suppose the weight of a coal waggon to be about 4 tons, 
 8960 ft s. ; when the rails are shorter than twice the dis- 
 tance between the wheels, the utmost strain on a rail cannot 
 exceed half this weight, or 4480 fts., which will be allowing 
 half the strength nearly for accidents. The usual length of 
 
 * Since 
 
 850 (1 - '625 x -7 3 ) = 677 nearly; 
 
 whenever the same proportions are used, the divisor 677 may be employed instead of 
 repeating the calculation. 
 
= 9-96; 
 
 136 RESISTANCE TO PRESSURE.' [SECT. vn. 
 
 one rail is 3 feet,* and supposing the breadth to be 2 inches, 
 then, by the manner of calculation shown in the note to 
 art. 186, 
 
 W I 4480 x 3 
 675 x 6 = 675 x 2 
 
 and the square root of 
 
 9-96 = 3-1 6 inches, 
 
 the depth in the middle of the length. 
 
 * It is worthy of consideration whether this be the most economical length, or not, 
 for rails. This may be done as follows : 
 
 The weight of a bar of iron, an inch square and 700 feet long, is 1 ton ; therefore, 
 for a length of 700 feet, the area of the bar in inches multiplied by the price of a ton 
 
 of iron will be the amount of 700 feet of rail. Make the length of a single rail ; 
 
 3C 
 
 then, supposing the rail all of the same thickness, 
 
 W x 700 
 V 850 x b x ~ ' 
 
 the depth, and when it is reduced at the ends, 
 
 . 7& /Wx700 =tbearea: 
 V 850 x b x 
 
 and calling A the price of a ton of iron ; and B the price of fixing, and materials for 
 one block ; then the price of 700 feet will be 
 
 , A , 
 850 b x 
 
 Hence by the rules of maxima and minima it appears that the price will be the 
 least when the number of supports for 700 feet is 
 
 32 A 
 
 wherein W is half the weight of a waggon and its load in R>9. 
 
 The same equation will apply to the new railway invented by Mr. Palmer, when W 
 is made the whole weight of the waggon in Iks. 
 
 An example will illustrate the application : Let A the price of a ton of iron be 8 ; 
 B the price of one support 0'5 ; the weight of a waggon 8960 Iks. ; and the breadth 
 of the rail 3 inches. Then 
 
 32 x 8 x V&9tiO x 3\ I 
 
 = 89 ; 
 
 that is, there should be 89 supports in 700 feet, in order that the expense may be the 
 least possible at these prices, and for these proportions ; which makes the distance of 
 the supports nearly 8 feet. But it should be understood that these prices are only 
 what 1 have inserted for illustration ; they are not from actual estimate. 
 
SECT, vii.j RESISTANCE TO PRESSURE. 137 
 
 Also, 
 
 3-16 x -7 = 2-212 inches, 
 
 the depth of the thin part in the section at the middle of the 
 length, and 
 
 2 x -4 = 0'8 inch, 
 
 the thickness of the middle part of the section. 
 
 The depth of a rail, all of the same thickness, would be 
 2*83 inches in the middle, calculated by Rule 2, art. 145. 
 
 Example 3. In Palmer's railway a single rail carries the 
 waggon ; * and let its weight be 8960 fts., and the length of 
 the rail 8 feet, its breadth 3 inches. By the rule 
 
 WZ 8960 x 8 
 
 6756 675x3 
 
 35-4. 
 
 The square root of 35*4 is very nearly 6 inches, the depth 
 required ; and the depth of the middle part 
 
 6 x -7=4-2 inches. 
 
 The breadth 3 inches, and breadth of middle part 
 
 3 x -4 = 1-2 inches. 
 
 These are the dimensions for the middle of the length ; but 
 the under edge should be the figure of equal strength, 
 Plate III., fig. 24, with the straight side upwards. 
 
 189. Case 2. When the beam is supported at the ends, 
 but the load not applied in the middle between the supports. 
 When /is the length in feet, and/= 15,300 fts., 
 
 4FBxFBxW 2 
 
 '' 
 
 by art. 116 and 136. 
 
 190. Rule. Multiply the rectangle of the segments into 
 which the strained point divides the beam, in feet, by 4, and 
 divide this product by the length in feet ; use this quotient 
 instead of the length of the beam, and proceed by the last 
 rule. 
 
 * Description of a Railway on a New Principle, by H. R. Palmer, 8vo. London, 
 1323. 
 
138 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 191. Example. Let the load to be supported be 33,600 fbs. 
 at 5 feet from one end, the whole length being 20 feet. Also, 
 let the breadth of the widest part a b, fig. 9, be 4 inches. 
 
 Here F B = 5 feet, therefore F B = 15 feet, and 
 
 4x5 x 15 
 ~~20~ 
 
 the multiplier to be used instead of the whole length in the 
 rule. 
 
 Let p = '7, and q = '625 ; then 
 
 15 x 33600 15 x 33600 
 
 850 x 4 x (1 - -625 x V) = 4 x 677 
 
 ' * 9 
 
 of which the square root is 13*5 inches, the depth required. 
 The depth ef will be 
 
 7 x 13-5 = 9-45 inches, 
 
 and the breadth of the middle part of the section will be 
 
 4 - 4 x -625 = 4-2-5 = 1-5 inches. 
 
 . 192. Case 3. When the load is uniformly distributed 
 over the length of a beam. In this case 
 
 1700 (I_ 
 
 by art. 116 and 139. 
 
 193. Rule. Use half the weight instead of the whole 
 weight upon the beam, and proceed by the rule to Case 1, 
 art. 187. 
 
 The form of equal strength for this case, when the breadth 
 is uniform, is an ellipse, but in practical cases it will require 
 to be altered to the form shown in fig. 24. 
 
 194.' I propose to give as an example of this rule, its 
 application to the construction of fire-proof buildings ; but it 
 also applies to rafters, girders, bressummers, and all cases 
 where the load is uniformly distributed over the length. 
 
 A fire-proof floor is usually formed by placing parallel 
 beams of cast iron across the area in the shortest direction, 
 
SECT, vii.] RESISTANCE TO PRESSURE. 139 
 
 and arching between the beams as shown by fig. 10, Plate I., 
 with brick or other suitable material. Or they may be done 
 by flat plates of iron resting on the ledges, with one or two 
 courses of bricks paved upon the iron plates ; and when the 
 distance of the joists is considerable, the iron plates may be 
 strengthened by ribs on the upper side as the floor plates of 
 iron bridges are made. 
 
 When arches are employed, floors of this kind are least 
 expensive when the arches are of considerable span ; but then 
 it is necessary to provide against the lateral thrust of the 
 arches by tie bars. Also, since the arches ought to be flat, 
 we can only extend them to a limited span, otherwise they 
 would be too weak to answer the purpose. For instance, 
 when an arch is to rise only ^th of the span, and to be half 
 a brick (or 4^ inches) thick,* the greatest span that can be 
 given to the arch with safety in a floor for ordinary purposes 
 is 5 feet. If the arch rise only ^th of the span, the span 
 must be limited to 4 feet ; and if it rise only ^th of the 
 span, it must be limited to 3 feet. 
 
 Again, for arches of one brick (or 9 inches), to bear the 
 same load, and the rise ^th of the span, the greatest span 
 that can be given with safety is 8 feet ; f when the rise is 
 i^th of the span, 7 feet ; and when the rise is only ^th of 
 the span, the greatest span should not exceed 5 feet. 
 
 These limits were calculated from the ordinary strength of 
 brick, and on the supposition that the load upon the floor 
 will never be greater than 170fts. upon a superficial foot, in 
 addition to the weight of the floor itself. If the load be 
 greater, the span must be less, or the rise greater.! 
 
 For half-brick arches the breadth of the beam c d, fig. 9, 
 should be about 2 inches ; and for 9-inch arches, from 2^ to 
 3 inches. 
 
 * Rad. of curv. 6'75 feet 
 t Rad. of curv. 15 '6 feet. 
 
 See also Elementary Principles of Carpentry, art. 2*49 and 2VO; edition by Mr, 
 Barlow. 
 
140 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 Example. It is proposed to form a fire-proof room, but 
 from its situation it cannot be vaulted in the ordinary way 
 on account of the strong abutments required for common 
 vaulting, and also common vaulting is objectionable, because 
 so much space is lost in a low room. The shortest direction 
 across the room is 12 feet, and if iron beams of 3 inches 
 breadth be laid across at 5 feet apart, and arched between 
 with 9 -inch brick arches, it is required to find the depth for 
 the beams ? See fig. 10, Plate I. 
 
 The quantity of brickwork resting upon 1 foot in length of 
 joist will be 
 
 5 x -75 =3-75 cubic feet; 
 
 and the weight of a cubic foot being nearly lOOfts., the 
 weight of the brickwork will be 375 fts. 
 
 But since the space above is to be used, and the greatest 
 probable extraneous weight that will be in the room will arise 
 from its being filled with people, we may take that weight at 
 120Ibs. per superficial foot, and we have 
 
 5 x 120 = 600 fes. 
 
 for the weight on 1 foot in length. And supposing the 
 paving and iron to be 350fts. for each foot in length, the 
 whole load on a foot in length will be 
 
 375 + 600 + 350 = 1325 fts. or 
 12 x 1325 =*= 15900 fts. 
 
 the whole weight upon one joist. And as half this weight 
 multiplied by the length, and divided by the breadth and 
 constant number,* is equal to the square of the depth, we 
 have 
 
 7950 x 12 
 
 675 x 3 
 
 - = 47-11, 
 
 of which the square root is nearly 7 inches, the depth re- 
 quired. And 
 
 7 x '7= 4'9 inches 
 * See note to Rule, art. 186. 
 
SECT. VII.] 
 
 RESISTANCE TO PRESSURE. 
 
 141 
 
 the depth of the middle part, and 
 
 3 x -4 = 1-2 
 
 the breadth of the middle part. 
 
 By fixing the breadth, you avoid the risk of calculating for 
 a thinner beam than is sufficient to support firmly the 
 abutting course of bricks. 
 
 By means of this example we may easily form a small 
 Table of the depth of beams for fire-proof floors, which will 
 be often useful : in so doing, I shall not regard the difference 
 between the weight of a 9-inch and a 4^-inch floor ; because 
 the lighter floor will be more liable to accidents from percus- 
 sion, and therefore should have excess of strength. 
 
 Table of Cast Iron Joists for fire-proof Floors, when the extraneous load is not greater 
 than 12Gife8. on a superficial foot (see FLOORS, Alphabetical Table.) 
 
 Length of 
 
 Half-brick arches, breadth of beams 
 2 inches. 
 
 Nine-inch arches, breadth of beams 
 3 inches. 
 
 joists 
 
 
 
 
 
 3 feet span. 
 
 4 feet span. 
 
 5 feet span. 
 
 6 feet span. 
 
 7 feet span. 
 
 8 feet span. 
 
 feet. 
 
 Depth in 
 inches. 
 
 Depth in 
 inches. 
 
 Depth in 
 inches. 
 
 Depth in 
 inches. 
 
 Depth in 
 inches. 
 
 Di3pth in 
 inches. 
 
 8 
 
 4i 
 
 
 & 
 
 5| 
 
 H 
 
 5| 
 
 6 
 
 10 
 12 
 
 5| 
 
 6 4 
 
 
 64 
 
 7f 
 
 7 
 84 
 
 64 
 7f 
 
 9 
 
 n 
 
 9 
 
 14 
 
 7^ 
 
 
 9 
 
 10 
 
 *I 
 
 10 
 
 104 
 
 16 
 
 9 
 
 
 104 
 
 "1 
 
 104 
 
 114 
 
 12 
 
 18 
 
 10 
 
 
 HI 
 
 122 
 
 iif 
 
 13 
 
 134 
 
 20 
 22 
 24 
 
 11; 
 
 12; 
 13: 
 
 
 
 '. 
 
 13 
 Ml 
 
 154 
 
 14 
 ]5| 
 
 17 
 
 13 
 
 144 
 
 154 
 
 Ml 
 
 i? 1 
 
 15 
 164 
 18 
 
 For half-brick arches the breadth a d, fig. 9, Plate I., is 
 to be 2 inches, and the thickness of the middle part ^ths of 
 an inch ; the depth ef being j^ths of the whole depth; and the 
 whole depth is given in inches in the Table for each length 
 and span. 
 
 For 9-inch arches the breadth a 6, fig. 9, is to be 3 inches, 
 and the breadth of the middle part 1 inch and ^ O ths. The 
 depth n)ths of the whole depth, as in the 4^-inch arches. 
 
 If the floor be for a room of greater span than about 16 
 feet, let the beams be put 8 feet apart; and put the beams for 
 
142 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 8 feet bearing across at right angles to the other, in the 
 manner of binding joists, and arch between the shorter beams. 
 By casting the shorter beams with flanches at the ends, they 
 can be bolted to the other, and a complete firm floor be 
 made. This method has also the advantage of rendering it 
 extremely easy to fix either a wooden floor or a ceiling. 
 
 The construction of these floors renders a place secure from 
 fire without loss of space, and with very little extra expense ; 
 it may be of infinite use in the preservation of deeds, libraries, 
 and indeed every other species of property. In a public 
 museum, devoted to the collection and preservation of the 
 scattered fragments of literature and art, it is extremely 
 desirable that they should be guarded against fire ; otherwise 
 they may be involved in one common ruin, more dreadful to 
 contemplate than their widest dispersion. 
 
 195. Case 4. When a beam is fixed at one end, and the 
 load applied at the other. Also, when a beam is supported 
 upon a centre of motion. By art. 116, 
 
 or when / is in feet, and/=15,300 fbs., 
 
 wz 
 
 212(1-^) 
 
 bd?. 
 
 196. Ride 1. Calculate by the rule to Case 1, art. 187, 
 using 212 instead of 850 for a divisor. 
 
 Or when the breadth of the middle is made f^ths of 
 the extreme breadth, and the depth e f in fig. 9 is i^ths of 
 the whole depth; then, calculate by the rules art. 144, 145, 
 or 146, using 168 instead of 850 as a divisor. 
 
 The figure of equal strength is a parabola; see figs. 25 
 and 26. 
 
 197. Rule 2. If the weight be uniformly distributed over 
 the length, take the whole load upon the beam for the weight, 
 
SECT. VIL] RESISTANCE TO PRESSURE. 143 
 
 and calculate by the rule to Case 1, art. 187, except using 
 425 instead of 850 as a divisor. 
 
 198. Prop. vi. To determine a rule for finding the depth 
 of a beam when part of the middle is left open, as in figs. 11, 
 12, and 27, so that it will resist a given force; the strain not 
 exceeding the elastic force of the material. 
 
 199. When the depth is more than 12 or 14 inches, 
 angular parts in the middle become necessary, as in fig. 27 ; 
 the disposition of the middle part may in a great measure be 
 regulated by fancy, provided it allows of sufficient diagonal 
 and cross ties to bind the upper and lower parts together. 
 The middle parts should be made of the same size as the 
 other, in order that they may not be rendered useless by 
 irregular contraction. 
 
 If the beams be required so long as not to be made in a 
 single casting, and it is not a good plan to cast in very long 
 lengths, then they may be joined in the middle, as in fig. 27. 
 The connexion is made at the lower side only ; at the upper 
 side let the parts abut against one another, with only some 
 contrivance to steady them while they become fixed in their 
 places and loaded. Fig. 28 is a plan of the under side, 
 showing how the connexion may be made. 
 
 200. Case 1. When the beam is supported at the ends 
 and the load acts at the middle of the length. By art. 117 
 and 137, 
 
 or making /=the length in feet, and/=15,^Q r Ofts.^r >, x 
 
 Jr \ > ~* y s 
 
 Now, in general, we may make jo = *7, and then, 
 
144 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 or, in practice,* 
 
 wi 
 
 1 W 7 
 
 -S rf, then -^r = 
 
 y 62 
 
 201. Rule. Multiply the length in feet by the weight to 
 be supported in pounds ; and divide this product by 560 
 times the breadth in inches ; the square root of the quotient 
 will be the depth required in inches. Consult art. 41 and 43 
 respecting the form of beams of this kind. The depth 
 between the upper and lower part of the beam will be '1 d 
 inch, where d is the depth found by the rule. 
 
 202. Example. A beam for a 30-feet bearing is intended 
 to sustain a load of 6 tons (13,440 fts.) in the middle, the 
 breadth to be 4 inches ; required the depth ? 
 
 By the rule 
 
 30 x 13440 
 4x560 =18 - 
 
 the square root of 180 is nearly 13*5 inches, the whole 
 depth. 
 
 The depth between the upper and lower part is 
 
 7 x 13-5 = 9-45 inches. 
 
 If the depth be given, suppose 16 inches, and the breadth be 
 required, then 
 
 = 2-82, the breadth in inches; 
 
 16 x 16 x 560 
 
 when the depth is 16 inches, and the depth between the 
 upper and lower parts is 
 
 7 x 16 = 11-2 inches. 
 
 * If we make p 0'6, then 
 
 W / 
 
 p=d 3 ; and 6 = 0'2 d, 
 
 135 
 
 and the depth of the section at A B or C D, fig. 11, Plate II., will be the same as the 
 breadth of the beam. And as the equation for a square beam of the same depth ia 
 
 WJ_ 
 
 850~' 
 
 the strength of this beam will be to that of the square beam, of the same depth, 
 as 1 ; 6-3. 
 
SECT. VIL] RESISTANCE TO PRESSURE. 145 
 
 203. Case 2. When a beam is supported at the ends, but 
 the load is not applied at the middle. 
 
 When / is the length in feet, jo = *7, and /=! 5,300 H., 
 
 4BG x CD x W 
 
 558 1 ' 
 
 (see fig. 12, Plate II.;) or 
 
 BC x CD x W 
 
 139 I 
 
 bd?. 
 
 204. Rule. Multiply the rectangle of the segments into 
 which the strained point divides the beam, in feet, by the 
 weight in pounds, and divide this product by 139 times* the 
 length in feet multiplied by the breadth in inches ; the 
 square root of the quotient will be the depth required in 
 inches. 
 
 The depth between the upper and lower side will be '1 xby 
 the whole depth. Consult art. 41 and 43 respecting the 
 form, &c. of beams of this kind. 
 
 205. Example. Let C B, fig. 12, be 10 feet, and D C, 6 
 feet : and therefore B D the length, 16 feet; and the weight 
 to be supported at A, 20,000 fts., the breadth of the beam 
 being 2 inches ; required the depth ? 
 
 By the rule 
 
 10 x 6 x 20000 _ 
 "139 x 16 x 2 = 
 
 and the square root of 270 is 16^ inches nearly. 
 Also, 
 
 7 x 16-5 = 11 -55 inches 
 
 = the depth from a to b in fig. 12. 
 
 206. Case 3. When a load is distributed uniformly over 
 the length of a beam. When the length is in feet, p = '7, and 
 /=! 5,300 H)s., 
 
 by art. 117 and 139. 
 
 * In practice it will be sufficiently accurate to use 140 for ft divisor, 
 
146 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 207. Rule. Multiply the whole weight in pounds by the 
 length in feet; divide this product by 1116 times the breadth 
 in inckes, and the square root of the quotient will be the depth 
 in inches. 
 
 Multiply this depth by ?, which will give the depth 
 between the upper and lower parts. Respecting the form of 
 the beam, see art. 41. 
 
 208. Example. It is required to support a wall, 20 feet 
 in height, and 18 inches in thickness, over an opening 26 feet 
 wide, by means of two beams of cast iron, each three inches in 
 thickness ; required the depth ? 
 
 Suppose a cubic foot of brick-work to weigh lOOlfes.; 
 then 
 
 20 x 1-5 x 26 x 100 = 78000 ft>3. 
 
 the weight of the wall. 
 Therefore by the rule 
 
 78000 x 26 
 
 and the square root of 303 is 17^ inches, the depth 
 required. 
 
 The depth between the upper and lower parts is 
 
 7 x 17-5 = 12-25 inches. 
 
 209. Case 4. When a beam is fixed at one end, and the 
 load applied at the other. Also, when the load acts at one 
 end of a beam supported on a centre of motion. By art. 117 
 we have, when the length is in feet, jp='7, and/=15,300fts., 
 
 210. Ride. Calculate by the rule to Case 1, art. 201, 
 using 140 instead of 560 for a divisor. 
 
 If the weight be uniformly distributed over the length of 
 a beam fixed at one end, divide the weight by 2, and proceed 
 as above directed. 
 
SECT, vn.] RESISTANCE TO PRESSURE. 147 
 
 DEFLEXION FROM CROSS STRAINS. 
 
 211. Prop~ vn. To determine a rule for finding the 
 deflexion of a cast iron beam, when the section is rectangular, 
 and uniform throughout the length ; the strain being 
 15,300 fts. upon a square inch. 
 
 The same rules will apply to solid and hollow cylinders, to 
 beams formed as figs. 9, 11, 12, and 26, when they are 
 uniform throughout their length, and the depth used as a 
 divisor is the greatest depth. 
 
 212. Case 1. When a beam is supported at the ends, 
 and loaded in the middle, as in fig. 1. 
 
 By art. 121, 
 
 . = the deflexion, 
 
 O (It 
 
 when /=half the length ; therefore, 
 
 3 d x DA 
 
 ~272~~ 
 
 the greatest extension of an inch in length while the elastic 
 force remains perfect. According to the experiment described 
 in art. 56, the elastic force was perfect when the bar was 
 loaded with 300 fts. ; hence we have 
 
 3^vDA 3 x 1 x -16 1 
 
 = 00083 inch 
 
 2 I 2 2 x 17 2 1204 
 
 = e the extension of an inch in length, by a force equal to 
 15,300fts. upon a square inch; or generally, cast iron is 
 extended 1^4 P ar ^ f its length by a force equal to 
 15,300 fts. upon a square inch. 
 
 If this value of e be substituted in the equation, and /be 
 made the whole length in feet, we have 
 
 2 x -00083 x 122 x V 
 
 3x4xd = DA ' W 
 
 01992 F 
 d - DA ? 
 
 L 2 
 
148 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 hence it appears that the equation 
 
 may be used without sensible error. 
 
 Consequently, the deflexion of an uniform rectangular beam 
 supported at the ends may be determined by the following 
 rule : 
 
 213. Rule. Multiply the square of the length in feet by 
 02 ; and this product divided by the depth in inches is equal 
 to the deflexion in inches. 
 
 214. Example. Required the deflexion in the middle of 
 a beam 20 feet long, and 15 inches deep, when strained to 
 the extent of its elastic force ? 
 
 By the rule 
 
 ^ = '533 inch ; 
 
 therefore a beam loaded as in example (art. 147), will bend 
 more than half an inch in the middle. If it be wished to 
 reduce it to a quarter of an inch, double the breadth. 
 
 The deflexion of an uniform, beam may also be found by 
 Table II. art. 6. 
 
 215. Case 2. When a uniform rectangular beam is 
 supported at the ends, and the load is equally distributed 
 over the length. It has been shown in art. 139, Equation 
 viii., that in this case the strain at any point is as the 
 rectangle of the segments into which that point divides the 
 beam ; and the deflexion for that case is calculated by art. 
 126, Equation vii. And by comparing Equation ii. and vii. 
 
 JL JL. .' 02?2 ' 25 1 ' 2 
 
 3 : 6 : l ~~d~ ~T~ 
 
 Therefore the deflexion D A in the middle of a beam 
 
 P Till- '025 I 2 
 
 uniformly loaded is = d . 
 
 216. Rule. Multiply the square of the length in feet by 
 025 ; and the quotient, from dividing this product by the 
 depth in inches, will be the deflexion in the middle in inches. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 149 
 
 217. Example. Let it be required to determine the 
 deflexion that may be expected to take place in the example 
 to Case 3, Prop. i. art. 152, where the length is 15 feet and 
 the depth 12^ inches? 
 
 By the rule 
 
 15 x 15 x -025 ... , 
 
 126 ' 
 
 the deflexion required. 
 
 218. This mode of calculation may often remove ground- 
 less alarm, as well as inform us when a structure is dangerous ; 
 for if a beam be loaded so as to bend more than is determined 
 by the rule which applies to it, the structure may be justly 
 deemed insecure. We also, by this mode of calculation, 
 have an easy method of trying the goodness of a beam : for 
 let it be loaded with any part, as for example ^th of the 
 weight it should bear, then the deflexion ought to be Jth of 
 the calculated deflexion. When a beam is tried by loading 
 it with more than the weight it is intended to bear, it may be 
 so strained as to break with the lesser weight, besides the 
 difficulty and danger in trying such an experiment. 
 
 219. Case 3. When a beam is fixed upon a centre of 
 motion, and the force applied at the other end, the flexure of 
 the fixed part being insensible. The cranks of engines are in 
 this case. 
 
 The flexure will be the same as in Case 1, art. 212, but 
 the length of the beam being only half the length in that 
 
 case, we have 
 
 *os /" 
 
 = D A the deflexion. 
 d 
 
 220. Case 4. If a uniform rectangular beam be fixed at 
 one end, and the force be applied at the other, the deflexion 
 of the end where the force is applied will be 
 
 08 P 
 
 -j- x (1 + r). 
 
 For the deflexion from the extension of the projecting part 
 of the beam is ^~ } where I is the length of that part in feet ; 
 
150 RESISTANCE TO PRESSURE. [SECT, vn 
 
 and if r be equal the ]ength of f xed part ' then, by Equation, iii. 
 art. 133, 
 
 7 x (1 + ?) = the deflexion. 
 
 CL 
 
 221. Eule. Divide the length of the fixed part of the 
 beam by the length of the part which yields to the force, and 
 add 1 to the quotient; then multiply the square of the 
 length in feet by the quotient so increased, and also by *08 ; 
 this product divided by the depth in inches will give the 
 deflexion in inches. 
 
 222. Example. Conceive a beam, A B, fig. 26, to be 
 uniform, and to be the beam of a pumping engine, the end B 
 working the pumps, and the end A where the power acts 
 10 feet from the centre of motion, the end B 7 feet from the 
 centre of motion, and the strain at B equal to the elastic force 
 of the beam ; through how much space will the point A move 
 before the beam transmits the whole power to B, the depth of 
 the beam being 12 inches? 
 
 In this case, 
 
 therefore, 
 
 1-7 x 10 x 10 x -08 
 
 = 1'33 inches. 
 
 223. Prop. viii. To determine a rule for finding the 
 deflexion of a cast iron beam, of uniform breadth, when the 
 outline of the depth is a parabola, the strain being equal to 
 15,3001bs. per square inch. 
 
 The same rules will apply to beams of the form of section 
 shown in figs. 9 and 11, when the breadth is uniform. 
 
 224. Case 1. When a beam is supported at the ends, and 
 the load is applied in the middle. 
 
 The deflexion for this case is calculated in art. 123, 
 Equation iv. ; and comparing it with the deflexion of a 
 uniform beam we have 
 
 2 4 -02 P -04 Z 2 
 
 : o : : -r- : 7 = the deflexion. 
 
 o o a c& 
 
SECT, vii.] RESISTANCE TO PRESSURE. 151 
 
 225. Rule. Multiply the square of the whole length of the 
 beam in feet by '04 ; divide the product by the middle depth 
 in inches, and the quotient will be the deflexion in inches. 
 
 226. Example. Let the depth of a beam be 18 '4 inches, 
 and its length 20 feet, which is on the supposition that the 
 beam, of which the depth is found by example to Case 1, 
 Prop. v. art. 188, is parabolic. By the rule 
 
 20 x 20 x -04 
 
 -iw~ =' 87mch ' 
 
 the deflexion required. 
 
 If the beam were of uniform depth, the deflexion would be 
 only half this quantity, or '435. 
 
 227. Case 2. If a parabolic beam of uniform breadth be 
 fixed at one end, and the force be applied at the other, the 
 deflexion of the end where the force is applied will be 
 
 where / is the length of the part the force acts on in feet, and 
 r = the quotient arising from dividing the length of the fixed 
 part by the length /. 
 
 228. Rule. Divide the length in feet of the fixed part of 
 the beam by the length in feet of the part which yields to the 
 force, and add 1 to the quotient. Then multiply the square 
 of the length in feet by the quotient so increased, and also by 
 *16 ; divide this product by the middle depth in inches, and 
 the quotient will, be the deflexion in inches. 
 
 229. Example. Let A B, fig. 26, be the beam of a steam 
 engine, the moving force acting at A, and the resistance at 
 B, C being the centre of motion ; when A C = 12 feet, and 
 C B = 10, and the depth in the middle 30 inches; it is 
 required to determine the space the point A bends through 
 before the full action is exerted on B, the strain being equal 
 to the elastic force of the material ? 
 
152 RESISTANCE TO PRESSURE. [SECT. vn. 
 
 In this case the length of the part C B, which may be 
 considered as fixed, is 10 feet, and 
 
 ^ = 533, and 1 + -833 = 1-833 ; 
 
 therefore, 
 
 12 x 12 x 1-833 x -1(5 12 x 22 x '16 ' ... . 
 
 -so- -so- l ' m 1RChes ' 
 
 the deflexion of the point A. 
 
 Pew people are aware of the extent of flexure in the parts 
 of engines, and particularly when they are executed in a 
 material which has been considered as nearly inflexible. In 
 a well contrived machine, the importance of making the parts 
 capable of transmitting motion and power with precision 
 and regularity must be so obvious, that it appears almost 
 incredible how much the laws of resistance have been 
 neglected. 
 
 230. Prop. ix. To determine a rule for finding the 
 deflexion of a cast iron beam of uniform breadth, when the 
 depth at the end is only half the depth at the middle, the 
 strain being equal to 15,300 flbs. on a square inch. 
 
 231. Case 1. When a beam is supported at the ends, 
 and the load is applied in the middle. By art. 127, 
 Equation viii., 
 
 - DA the deflexion ; 
 
 when this is compared with Equation ii. art. 121, we have 
 
 2 -02 J 2 -0327 P . 
 
 : T09 : : T : ; =I ) A the deflexion. 
 
 3 da 
 
 232. Rule. Multiply the square of the length in feet by 
 '0327, and the product divided by the depth in the middle in 
 inches will give the deflexion in inches. 
 
 233. Case 2. When a beam is fixed at one end, and the 
 force is applied at the other. In this case 
 
 j (1 + 7') = the deflexion. 
 
SECT, vii.] RESISTANCE TO PRESSURE. 153 
 
 234. Rule. Calculate the deflexion by the rule, art. 228, 
 except changing the multiplier to '13 instead of *16. 
 
 235. Prop. x. To determine a rule for finding the 
 deflexion of a beam, generated by the revolution of a cubic 
 parabola, the strain being equal to 15,300fts. on a square 
 inch. 
 
 The same rules will apply to any cases where the sections 
 are similar figures, and the cube of the depth every where 
 proportional to the leverage the force acts with. 
 
 236. Case 1. When a beam is supported at the ends, and 
 the load is applied in the middle. 
 
 By art. 124, Equation v., 
 
 r =D A the deflexion ; 
 5 d 
 
 and comparing this with Equation ii. we have 
 
 2 6 -02 Z 2 -036 Z 2 , 
 
 - : = : : r : ; = the deflexion. 
 
 3 5 d d 
 
 237. Rule. Substitute '036 in the place of '04 in the rule 
 to Prop. vni. art. 225, and then calculate the deflexion by 
 that rule. 
 
 23 . Case 2. When a beam is fixed at one end, and the 
 force acts at the other. 
 In this case 
 
 - =the deflexion. 
 
 d 
 
 239. Rule. In the rule to Prop. vni. art. 228, use '144 
 instead of *1 6 as a multiplier, and calculate the deflexion by 
 that rule, so altered. 
 
 240. Prop xi. To determine a rule for finding the 
 deflexion of a cast iron beam, of uniform breadth, the depth 
 being bounded by an ellipse ; the strain being equal to 
 15,300 fts on a square inch. 
 
 If the Equations ii. and vi. be compared, it will be found 
 that 
 
 2 -02 Z 2 -0257 Z 2 
 
 s : '807 : : , : -j =the deflexion. 
 
 '' d 
 
154 RESISTANCE TO PRESSURE. [SECT. vii. 
 
 241. Rule. The deflexion may be calculated by the rule to 
 Prop. vin. art. 225, if the multiplier *0257 be employed 
 instead of "04. 
 
 242. Prop. xii. To determine a rule for the deflexion of 
 a beam of uniform depth, when the breadth is bounded by a 
 triangle, the strain upon a square inch being 15,300 fts. 
 
 From Equations ii. and iii. art. 121 and 122, we have 
 
 2 -02 I* -03 J 2 
 
 : 1 : : : r = the deflexion. 
 
 3 a d 
 
 243. Case 1. When a beam is supported at the ends, 
 and loaded in the middle. 
 
 244. Rule. Calculate by the rule to Prop. vin. art. 225, 
 using '03 instead of *04 as a multiplier. 
 
 245. Case 2. When a beam is supported at one end, and 
 fixed at the other. 
 
 In this case 
 
 12 I 2 
 
 r~ =t ke deflexion. 
 
 246. Rule. Calculate the deflexion by the rule to Prop, 
 vin. art. 228, using '12 as a multiplier instead of -16. 
 
 247. The rules derived from the twelve preceding pro- 
 positions are applicable to any kind of material. For example, 
 let it be required to adapt any one of the rules for oak : in 
 the Alphabetical Table at the end of this Essay, art. OAK, it 
 appears that oak is 0*25 as strong as cast iron; therefore, in 
 a rule for strength, multiply the constant number by 0'25. 
 Thus in the rules to Prop. i. Case 1, 
 
 850 x 0-25 = 212-5, 
 
 the number to be used in these rules when the material is oak. 
 Again, oak is 2*8 times as extensible as cast iron ; conse- 
 quently the deflexion being found for cast iron, 2*8 times that 
 deflexion will be the deflexion of oak, when it is strained to 
 the extent of its elastic power. 
 
SECTION VIII. 
 
 OF LATERAL STIFFNESS. 
 
 248. Definitions. The stiffness of a body is its resistance 
 at a given deflexion. And the lateral stiffness is the stiffness 
 to resist cross pressure. 
 
 249. Prop. xui. To determine the stiffness of a uniform 
 bar or beam, of which the section is a rectangle, when fixed at 
 one end, to resist a weight at the other ; or supported in the 
 middle on a centre to support a stress at each end. 
 
 When a beam is strained to the extent of its elastic force, 
 we have the weight it will bear, or 
 
 - Ql > . 
 
 (by art. 110,) and the deflexion under that strain will be 
 
 (by art. 121 and 133, Equation iii.) Then, since the de- 
 flexion is proportional to the strain, if a be the given de- 
 flexion, and w the weight which produces it, we have 
 
 fbd* fbd^a 
 
 and because i=**m, (art. 105,) we have 
 
 4^3(l+r)_^3 
 
 a m (i.) 
 
 If the length be in feet, then 
 
 6912 ML 3 (1 + r) _ &rf3 
 
 a m (ii.) 
 
156 OF LATERAL STIFFNESS. [SECT. vm. 
 
 Now in cast iron 7#=18,400,000fts. ; therefore 
 
 2662 a (ill} 
 
 Where L=the length in feet, a the deflexion in inches, b and 
 d the breadth and depth in inches, and w the weight in 
 pounds ; and r=the length of the fixed part divided by L. 
 When r=l, the lengths are equal, and (l+r)=2. 
 
 250. If the fixed part be of considerable bulk in respect 
 to the other, we may neglect its effect on the deflexion, and in 
 that case 
 
 wL 3 , 7 , 
 - = J d 3 . 
 
 2662 a (iv.) 
 
 If in any of the preceding equations the breadth be 
 diminished while the depth is uniform, the flexure will be 
 increased ; and when the outline of the breadths becomes a 
 triangle, this increase is half the deflexion of a beam of 
 uniform, breath ; or the deflexions with the same strain are, 
 as 2 : 3 (art. 122). 
 
 If the breadth be everywhere the same, but the beam be 
 made the parabolic one of equal strength, then the deflexion 
 will be twice as great as that of a beam of uniform depth 
 (art. 123), and the general Equation iii. becomes 
 
 . 
 1331 a (v.) 
 
 If the breadth be everywhere the same, but the outlines 
 of the depth be straight lines, and the depth at either of the 
 extremities half the depth at the point of greatest strain, 
 then the deflexion is to that of a beam of uniform depth as 
 T635 : 1 (art. 127) : and Equation iii. becomes 
 
 ,^ 73 
 
 1628 a . (vi.) 
 
 I shall illustrate this proposition by examples of its appli- 
 cation to beams of pumping engines, cranks, and wheels. 
 
SECT. VIIL] OP LATERAL STIFFNESS. 157 
 
 BEAMS FOR PUMPING ENGINES. 
 
 251. Example. Let it be required to determine the 
 breadth and depth of a beam for a pumping engine, its 
 whole length being 24 feet, and the parts on each side of 
 the centre of motion equal ; and the straining force 30,OOOJbs., 
 thee deflexion not to exceed 0*25 inch. 
 
 First, on the supposition that the beam is to be uniform, 
 then, by Equation iii., art. 249, 
 
 to L (1 + r)_ 30000 x!2x (1 + 1) ^ & ^ 3 = 155790 
 2662 a 2662 x -25 
 
 If the breadth be made 5 inches, the depth should be 3T5 
 inches ; for 
 
 31-5 3 x 5 = 156279, 
 
 which very little exceeds 155,790. 
 
 But if the depth at the middle be double the depth at 
 either end, use 1628 as a divisor instead of 2662 ; and 
 calculating by Equation vi. we find b ^ 3 =254,742, and if 
 the breadth be 5 inches, the depth should be 37 inches. 
 
 CRANKS. 
 
 252. Example. If the force acting upon a crank be 
 6000fts., and its length be 3 feet, to determine its breadth 
 and depth so that the deflexion may not exceed ^th of an 
 inch. 
 
 To this case, Equation iv., art. 250, applies, and 
 
 W L 3 6000 x3 3 
 
 2662 a 2662 x -1 
 
 .= 653 - 
 
 If the breadth be made 3 inches, the depth should be 6 
 inches, for the cube of 6 x 3=648. 
 
 When the depth at the end where the force acts is half the 
 depth at the axis, use 1G2S instead of 2662 for a divisor. 
 
158 OF LATERAL STIFFNESS. [ [SECT. vm. 
 
 WHEELS. 
 
 253, For wheels, if N be the number of arms, or radii, 
 our equation should be 
 
 2662 N a 
 
 254. Example 1. Let the greatest force acting at the 
 circumference, of a spur-weel be 1600fts., the radius of the 
 wheel 6 feet, and number of arms 8 ; and let the deflexion 
 not exceed ^ of an inch. 
 
 Then by the Equation, art. 253, 
 
 WL 3 1600x6 3 , , 3 ,,, 
 
 - = - = 6 a 3 = 163. 
 
 2062 JN a 2662 x 8 x -1 
 
 If we make the breadth 2*5 inches, then 
 
 *.= 66-2=d-'; 
 
 and the cube root of65'2=4'03 inches, nearly, for the depth 
 or dimension of each arm, in the direction of the force. 
 
 When the depth at the rim is intended to be half that at 
 the axis, use 1628 as a divisor instead of 2662 for a divisor. 
 
 If a wheel be strained till the arms break, the fractures 
 take place close to the axis ; there is a sensible strain at the 
 part of the arm near the rim, but it is so small in respect to 
 that at the axis, that its effect is neglected in our rule. 
 
 Example 2. When the stress on the teeth is 1 090ft s. 
 Suppose the wheel to be 4 feet radius, with 6 arms ; and 
 that a flexure of n>ths of an inch will not sensibly affect the 
 action of the wheel- work. Also, let the arms be diminished 
 in depth so as to be only half the depth at the rim of the 
 wheel ; the breadth being fixed at 2 inches. 
 
 By the Equation, art. 253, we have 
 
 W L 3 1090 x 64 
 
 1628 6 a N 2 x 1628 x 6 x -2 
 
 = 18 nearly = d 3 . 
 
SECT. VIIL] OF LATERAL STIFFNESS. 159 
 
 But the cube root of 18 is 2'62 inches; consequently next 
 the axis the arms should be 2 by 2 '62, and at the rim 
 2 by 1 -31, in order that the play in applying the power may 
 not exceed n)ths of an inch. This rule gives the quantity 
 of iron, with the rectangular section, but let it be disposed in 
 the form of greatest strength consistent with that required 
 for casting. 
 
 Again, let the pinion to be moved by the preceding wheel 
 have a radius of 0'75 foot, with four arms, and the breadth 
 of the arm 2 inches ; the angular motion produced by the 
 flexure being the same as above ; that is, if 
 
 4 : -2 : : -75 : '0375 = a. 
 
 Then, 
 
 WL3 4 WL2 . or 
 
 1628 baN 1628 6 NX -2' ' 
 
 4 x 1090 x -56 . 
 
 1628x2x4x-2 
 
 The cube root of '94 is *98 nearly, for the thickness of 
 the arm at the axis. 
 
 255. I think we may in most cases allow a flexure of ^ths 
 of an inch for a wheel of 4 feet radius for the effect of the 
 arms, and other njths for the flexure of the shafts. In con- 
 sequence, therefore, of such an arrangement, the strength of 
 the arms will be expressed by a more simple equation ; as 
 well as the strength of the shafts, to be treated in the section 
 on Torsion. 
 
 When the flexure is 0*2 for a radius of 4 feet, it is very 
 nearly a quarter of a degree ; and with this degree of 
 flexure, the arms of equal breadth, and the depth at the rim 
 half the depth at the axis, we have 
 
 ^il-Wa 
 
 81 N (vii.) 
 
 Hence we have this practical rule. Multiply the stress at the 
 pitch line in S)s. by the square of the radius in feet ; and 
 
160 OF LATERAL STIFFNESS. [SECT. vm. 
 
 divide the product by 81 times the breadth multiplied by 
 the number of arms ; and* the cube root of the quotient will 
 be the depth of the arm at the axis, and half this depth will 
 be the depth at the rim. 
 
 If the thickness of the rim be made equal to the thickness 
 of the teeth, and the breadth be proportioned by the Table, 
 art. 158, then the number of arms should be Ii times the 
 radius of the wheel in feet, divided by the square of the 
 thickness of the teeth in inches, taken in the nearest whole 
 numbers : it is usual to make an even number of arms ; but 
 there does not appear to be any reason for adhering to this 
 practice. Wheels are often broken in the rim by wedging 
 them on to the shaft ; but the practice of fixing the wheels on 
 by wedges has now given way to a much superior one, which 
 consists in boring the eye truly cylindrical, and the shaft 
 being turned to fit the eye, the wheel is retained in its place 
 by a slip of iron, fitted into corresponding grooves in the 
 shaft and in the eye of the wheel. 
 
 256. Prop. xiv. To determine the stiffness of a uniform 
 bar, or beam, supported at the ends, to resist a cross strain in 
 the middle. 
 
 If a beam be rectangular and uniform ; then, making a 
 the greatest deflexion that it ought to assume, we have by 
 Equation ii., art. 121, and art. 137, 
 
 And as 
 
 By some error of computation, Professor Leslie makes this equation 
 8 m b cZ 3 a 
 
 (Elements of Nat. Phil. vol. i. p. 237;) and, consequently, draws an erroneous morx- 
 sure of the modulus of elasticity from the experiments in art. 53 and 56. The 
 equation I have arrived at is the same as Dr. Young had previously determined, (sec 
 his Nat. Philos. vol. ii. art. 326,) 2 e in hia equation being Zin mine, and 2f=w. 
 
SECT, viii.] OF LATERAL STIFFNESS. 161 
 
 When L = the length in feet, then, 
 
 432 L 3 (ix.) 
 
 This equation answers for any material of which the weight 
 of the modulus of elasticity is known ; and this will be 
 found in the Alphabetical Table at the end of this Work, for 
 almost every kind of material in use. Its application to cast 
 iron will be sufficient for an example. 
 
 The weight of the modulus for cast iron is 18,400,000fts., 
 and, dividing this number by 432, we have for cast iron 
 
 _ 42600 I d 3 a 
 
 ~^~~ (*) 
 
 257. If a - Q of an inch, or the deflexion be as many 
 
 fortieths of an inch as there are feet in the length of the 
 beam ; then the equation will be 
 
 = 1065 5 ds . 
 
 L 2 
 
 which was made 
 
 001 10 L " = b d 
 
 to calculate the Table, art. 5. () 
 
 When the deflexion is only as many lOOths of an inch as 
 the beam is feet in length, a deflexion which should not be 
 greatly exceeded in shafts, on account of the irregular wear 
 on their gudgeons and bearings when the flexure is greater, 
 then 
 
 426 6 da 
 
 -L*- (xii.) 
 
 If the load be uniformly distributed over the length of a 
 uniform rectangular beam; then from art. 126 and 139, the 
 dimensions being all in inches, we have 
 
 5 c? 3 
 24 d 
 
 And since 
 
162 OF LATERAL STIFFNESS. [SECT. vm. 
 
 Comparing this equation with Equation vm., it appears 
 that a weight uniformly distributed will produce the same 
 depression in the middle as f ths of that weight applied in 
 the middle, as has been otherwise shown by Dr. Young.* and 
 Messrs. Barlow, f Duping and Duleau. 
 
 When w is the weight of the beam itself; then p being 
 the weight of a bar of the same matter 12 inches long, and 
 1 square, 
 
 and the deflexion of a beam by its own weight is 
 
 12 x 32 d 2 / ~~ 384 M d 2 (xiv.) 
 
 Where M is the height of the modulus of elasticity || in feet 
 (Equation v., art. 105). 
 
 258. In a uniform solid cylinder, the strength is to that 
 of a square beam as 1 : 1/7 nearly (art. 113); therefore, by 
 Equation x., art. 256, we have 
 
 25000 a (xv.) 
 
 Where L is the length between the supports in feet, d the 
 diameter in inches, and a the deflexion in inches produced by 
 the weight iv in fts. 
 
 If the load be uniformly diffused over the length, and s be 
 the load on 1 foot in length in Ibs. ; then w = L$, and the 
 effect will be the same as if f ths of this load were applied in 
 the middle (art. 257); consequently 
 
 \40000 a) (xvi.) 
 
 Therefore, if the load on a foot in length be the same, the 
 
 * Nat. Phil. vol. ii. art. 325 and 329. 
 
 t Treatise on the Strength of Timber, &c., art. 55. Idem, p. 97. 
 
 Essai The'orique et Experimental, art. 2 et 5. 
 
 || In theory this seems to furnish the most simple mode of obtaining the modulus, 
 but it is not so accurate in practice, because it is difficult to ascertain the exact 
 degree of flexure due to the weight. 
 
SECT. viii. ] OF LATERAL STIFFNESS. 163 
 
 diameter should be increased in direct proportion to the 
 length, so that the flexure may be the same. 
 
 If in Equation xv. we make the flexure proportional to the 
 length, and so that it may be yrro^h f an i ncn f r eacfl 
 in length. 
 
 Then, ' 
 
 250 
 
 
 ____ I - = M,. 
 
 250 ; V_^ (xvn.) 
 
 This equation will apply to uniform solid cylindrical shafts. 
 
 259. In a hollow shaft or cylinder, it will be only neces- 
 sary to fix on what aliquot part of the diameter the thickness 
 of metal should be, if its diameter were 1. Then, the 
 difference between twice the thickness of metal and 1, will be 
 the aliquot parts to be left hollow ; and calling these parts n, 
 
 it will be 
 
 d ^ 
 
 (l-n^ 
 
 the diameter of a hollow shaft of the same stiffness as the 
 solid one of the diameter d. (See Equation xviii., art. 115.) 
 And the weight a solid shaft will sustain multiplied by 
 (1 ft 4 ) will be the weight a hollow one of the same diameter 
 will sustain. 
 
 Examples. If the thickness of metal be fixed at ^th of 
 the diameter, then 
 
 ~ F ~ 5" = 
 
 (1 -6 4 )^ = '966 =t - - . 
 1-0352 
 
 And if the diameter of a solid cylinder be found by Equation 
 xv., xvi., or xvii., as the nature of the subject may require, 
 and the diameter so found be multiplied by 1*0352, it will 
 give the diameter of a hollow tube that will be of the same 
 stiffness, the hollow part being f ths of the whole diameter. 
 
164 OF LATERAL STIFFNESS. [SECT. viu. 
 
 In the same manner, if the thickness of metal be ^-th of 
 the diameter, multiply by 1*056. 
 
 And if the thickness of metal be <ft)ths of the diameter, 
 multiply by T07. 
 
 The weight a hollow cylinder will sustain when the thick- 
 ness of metal is exactly yth of the diameter, is 0'87 the 
 weight a solid cylinder of the same external diameter would 
 sustain with the same pressure ; for (1 n*} ='87. And its 
 stiffness is to that of a square prism of the same depth as 1 
 is to 2, nearly. 
 
 260. Example. Required the diameter of a solid cylin- 
 drical shaft, 21 feet in length, that would not be deflected 
 more than half an inch by a weight of 31 cwt., or 3472H)s., 
 applied in the middle. 
 
 By Equation xv., art. 258, 
 
 w L 3 3472 x 21 3 
 
 25000 a 25000 x -5 
 
 = cZ 4 = 2572, or d = 712 inches. 
 
 the diameter required. 
 
 261. Example. Required the diameter of a hollow shaft, 
 21 feet in length, the interior diameter j^ths of the exterior 
 one, that would not be deflected more than half an inch by 
 a load of 3472fts. applied in the middle of the length? 
 
 Find the diameter of the solid cylinder, as in the preced- 
 ing example, and multiply it by 1*07 (see art. 259). That 
 is, 
 
 7'1 2 x 1-07 = 7'62 inches, 
 
 the diameter required ; the thickness of metal will be 2 ^>tlis 
 of the diameter. 
 
SECTION IX. 
 
 RESISTANCE TO TORSION. 
 
 262. Definition. The resistance which a shaft or axis 
 offers to a force applied to twist it round is called the 
 resistance to Torsion. 
 
 263. If a rectangular plate be supported at the corners A 
 and B, fig. 29, Plate IV., and a weight be suspended from 
 each of the other corners C D, then the strains produced by 
 loading it in this manner will be similar to the twisting strain 
 which occurs in shafts, and the like. In a cast iron plate the 
 fractures would take place in the directions A B and C D at 
 the same time ; but, before the fracture, the one of the strains 
 would serve as a fulcrum for the other ; and the resistance to 
 the forces at C and D would be sensibly the same as if the 
 plate were supported upon a continued fulcrum in the direc- 
 tion A B. 
 
 Hence the strain may be considered a cross strain of the 
 same nature as has been investigated in art. 108, and dD or 
 c C the leverage the force at D or C acts with, the breadth of 
 the strained section being A B. 
 
 To find the breadth of the section of fracture, and the 
 leverage in terms of the length and breadth of the plate, we 
 have A B 3 the breadth, and by similar triangles, 
 
 A D x B D . 
 
 T-5 =D d the leverage. 
 
 A Jo 
 
166 RESISTANCE TO TORSION. [SECT. ix. 
 
 These values of the leverage and breadth being substituted in 
 the Equation, art. 110, it becomes 
 
 : 
 
 ~ 6xADxBD 
 
 or because 
 
 we have 
 
 = 
 
 ADxBD" 
 
 264. But when a force acts upon a shaft, it is usually at 
 the circumference of a wheel upon that shaft, and if R be the 
 radius of the wheel, then 
 
 2RW_ 
 ~BlT~ 
 
 the force collected at the surface of the shaft ; and therefore, 
 substituting this in the place of W, in the Equation above, 
 we have 
 
 2RW c 
 
 * 
 
 BD 6 A D x B D ' 
 _ fffi BD2 + AD2 
 
 If the length A D be / feet, and the leverage R be in feet ; 
 then for cast iron/ = 15,300 fbs., and we have 
 
 8-85 d(P+ 144 P) 
 Rt 
 
 But this equation has a minimum value when / = ^ ; there- 
 fore the resistance will be the same whatever the length may 
 be, provided the length be not less than the breadth. Con- 
 sequently, whenever the length exceeds the breadth, we have 
 
 212-4 & & 
 -^ = W.* (n.) 
 
 * In malleable iron the equation will be 
 238 
 
 W; for 212-4 x 1-12 = 238. 
 K 
 
SECT, ix.] KESISTANCE TO TOKS1UN. 167 
 
 But when b is to d in a less ratio than \/2 : 1 the shaft 
 will not bear so great a strain, and it will bear least when its 
 section is exactly square. 
 
 265. When a shaft is square, and its length / in feet, its 
 side d in inches, and the leverage R in feet, then, from Equa- 
 tion, art. 112, we obtain 
 
 And when/ = 1 5,300 fts., 
 
 (iii.) 
 
 In a square shaft also the resistance has a minimum value ; 
 that is, when V72 / = d ; hence, whenever the length is 
 greater than the diagonal of the section, the strength will be 
 
 Where R is the radius of the wheel in feet to which the 
 power W in pounds is to be applied, and d is the side of the 
 shaft or axis in inches. 
 
 266. In a cylindrical shaft the section of fracture is an 
 ellipse, and when / and R are in feet, and/= 15,300, d 
 being the diameter of the shaft in inches, we have by art. 114, 
 
 W=5|y x(eP + M4P). (v.) 
 
 267. Here again it may be shown, by the principles of 
 maxima and minima, that there is a particular line of fracture 
 where the resistance to torsion is a minimum ; in a cylindrical 
 body this happens when 1% I =* d ; that is, when the length 
 is equal to the diameter. 
 
 Consequently, in all cases where the length exceeds the 
 
 * In malleable iron shafts the equation will be 
 
168 RESISTANCE TO TORSION. [SECT. ix. 
 
 diameter, the Equation in art. 266 should be applied in the 
 form 
 
 As the equation reduces to this form by substituting - 2 for /. 
 
 268. In the same manner it may be shown that in a tube 
 or hollow cylinder of which the length is greater than the 
 diameter, the resistance to torsion is expressed by the 
 equation 
 
 124-8 *l-n') .. 
 
 W T here d is the exterior diameter in inches, and n d the 
 interior diameter. 
 
 It will be a good proportion in practice to make n = 0*6 ; 
 then the rule becomes 
 
 108d 3 
 
 j- = W. (vm.) 
 
 Where d is the exterior diameter in inches, and the thickness 
 of metal is exactly |th of the diameter ; R, as before, being 
 the radius of the wheel in feet, to the circumference of which 
 the power W in fts. is applied. 
 
 %,%.' Example. Let it be required to find the diameter of 
 a shaft for a water-wheel, the radius of the water-wheel 9 
 feet, and the greatest force that it will be exposed to at the 
 circumference, 2000 ft s. 
 
 If the shaft is to be a solid cylinder, then the diameter will 
 be found by Equation vi. art. 267 ; that is, 
 
 liff^^m^'^ . 
 
 And the cube root of 144*2 is 5^ inches, the diameter 
 required. 
 
 If the shaft is to be a hollow cylinder, Equation viii. will 
 apply, where 
 
 * For malleable iron make 140 d' A _ 
 
SECT, ix.] RESISTANCE TO TORSION. 169 
 
 W R 2000 x 9 
 
 To^-i08- 
 
 And the cube root of 166 '7 is 5| inches the diameter, when 
 the thickness of metal is ^th of this diameter. 
 
 270. But the lateral stress on a shaft will always be greater 
 than the twisting force, when the length of the shaft exceeds 
 ^th of the radius of the wheel ; yet the preceding equations 
 will often be of use in calculating the strength of journals,* 
 and these calculations should be made by Equation vi. in the 
 same manner as in the example in the preceding article ; only 
 as an allowance for wear the diameter should be ^th greater 
 than is given by the rules. 
 
 271. The preceding investigation has been confined to the 
 strength to resist twisting, but in shafts of great length in 
 respect to their diameters, the effect of flexure is considerable. 
 
 Let e be the extension the material will bear without 
 injury when the length is unity. This extension must 
 obviously limit the movement of torsion, or the angle of 
 torsion. But, since the line of greatest strain, in a bar of 
 greater length than its diameter, is always in the direction of 
 the diagonal of a square ; if a square were drawn on the 
 surface of the bar in its natural state, it would become a 
 rhombus by the action of the straining force, and the quantity 
 of angular motion would be nearly \/2 times the extension of 
 the diagonal ; or twice the extension of the length of the bar. 
 For if a line were wound round the bar at an angle of 45 
 with the axis, its length would be / \/2 ; / being the length 
 of the bar in feet. Therefore, / e \/2 = the extension, and 
 2 I * the arc described in feet, or 24 I = the arc in inches. 
 
 But if a be the number of degrees in an arc, and ^ its radius ; 
 
 0174533 being the length of an arc of one degree when its 
 radius is unity ; we have 
 
 * A journal is different from a gudgeon only in being exposed to a considerable 
 twisting strain. 
 
170 KESISTANCE TO TOKSION. [SECT, ix. 
 
 24 Je = ^y * -0174533; or 
 
 2750 I e 
 
 d " = ( 1X -) 
 
 That is, the angle of torsion a is as the length and exten- 
 sibility of the body directly, and inversely as the diameter. 
 
 If the value of e be taken for east iron, that is,"f^j, we 
 have 
 
 Here / is the length of the shaft or other body in feet ; d its 
 diameter in inches, and a the angle of torsion in degrees of a 
 circle. 
 
 Example. Thus, let the vertical shaft of a mill be 30 feet 
 in length, and the diameter 10 inches; then, when it is 
 strained to the extent of its elastic force, 
 
 - f degrees nearly. 
 
 In certain cases this degree of twisting may be of considerable 
 advantage in preventing the shocks incidental to machines 
 moved by wind, horses, or other irregular powers; but in 
 other cases it will be a disadvantage, because the motion will 
 neither be so accurate, nor so certain to produce the desired 
 effect. 
 
 272. Since the angle of torsion is as the extension, it will 
 be as the straining force ; and to estimate the stiffness of a 
 body to resist torsion, we have this analogy when the body 
 is a hollow cylinder; from Equation vii. and ix, of this 
 section, 
 
 2750 1 . ; . 124-8 d 3 (ln*) . = 124-8 d 4 o (l^ 4 ) 
 
 d ~~B~~ 2750 B I e 
 
 Or more generally, 
 
 1 1-965J 
 
 * In malleable iron, e = ; therefore. - = 
 
 1400 Or 
 
SECT, ix.] RESISTANCE TO TORSION. 171 
 
 336600 RZe 
 
 And if m be the weight of the modulus of elasticity (art. 105,) 
 
 336600 J R 
 
 When n=o the equation applies to a solid cylinder. 
 
 When a shaft is rectangular, the analogy from Equation ii. 
 and ix. becomes 
 
 2750 I . . . 212-4 <ff ft. w ^ 212-4 tftfa. 
 T~ R 2750 R I e ' 
 
 198900 RJ 
 
 We have now to show the application of these equations, and 
 to form practical rules from them. The value of m for cast 
 iron is, 18,400,000 fts. ; consequently Equation xi. applied 
 to cast iron is 
 
 IB 
 
 And equation xii. gives 
 
 9*5 gp*^ (xiv<) 
 
 I R 
 
 PRACTICAL RULES AND EXAMPLES FOR THE STIFFNESS OF 
 CYLINDRICAL SHAFTS TO RESIST TORSION. 
 
 273. In practical cases there will be known the length of 
 the shaft, the power, and the leverage the power acts with ; 
 and there must be fixed, by the person who applies the rule, 
 the number of degrees of torsion that will not affect the action 
 of the machine ; this being settled, the diameter of the shaft 
 will be determined by the rule. 
 
 * In malleable iron, 
 
 R 
 
 t In malleable iron, 
 
 lUd*b*a _ 
 
 in ~ w ' 
 
172 RESISTANCE TO TORSION. [SECT. ix. 
 
 Rule 1. To determine the diameter of a solid cylinder to 
 resist torsion, with a given flexure. 
 
 Multiply the power in pounds by the length of the shaft in 
 feet, and by the leverage in feet. Divide this product by 
 55 times the number of degrees in the angle of torsion, which 
 is considered best for the action of the machine; and the 
 fourth root of the quotient will be the diameter of the shaft. 
 
 Example. Let it be required to find the diameter for a series 
 of lying shafts 30 feet in length to transmit a power equal to 
 4000 fts. acting at the circumference of a wheel of 2 feet 
 radius, so that the twist of the shafts on the application of the 
 power may not exceed one degree ? 
 
 Here the whole length must be taken as if it were one 
 shaft, and by the rule, 
 
 4000 x 30 x 2 , 3G 4 
 55x1 ~ 
 
 and by a Table of powers,* the fourth root is found to be 
 8*13 inches, the diameter required. 
 
 If the machinery be required to act with much precision, 
 this will be as much flexure as can be allowed; but in 
 ordinary cases two degrees might be admitted, and then a 
 little less than 7 inches would be the diameter. 
 
 Where there is much wheel-work, the flexures should be 
 less ; indeed it does not appear to be desirable to exceed a 
 quarter of a degree for the shafts or axes. 
 
 274. Rule 2. To determine the diameter of a hollow 
 cylinder to resist torsion, when the thickness of metal is yth 
 of the diameter, and the flexure given. 
 
 Multiply the power in pounds by the length of the shaft in 
 feet, and by the leverage the power acts with in feet. Divide 
 the product by 48 times the angle of flexure in degrees ; the 
 fourth root of the quotient will be the diameter required in 
 inches. 
 
 * See Barlow's Mathematical Tables, Table III. 
 
SECT, ix.] RESISTANCE TO TORSION. 173 
 
 Example. Let the diameter of a hollow shaft be deter- 
 mined, so that it may be sufficient to withstand a force of 
 800 fts. acting at the circumference of a wheel of 4 feet radius 
 with a flexure of one degree ; the thickness of metal to be y 
 of the diameter, and the length 10 feet. 
 
 In this case 
 
 800x10x4 
 
 48x1 
 
 = 666-6; 
 
 and the fourth root of 666 '6 is 5*1 inches nearly; which is 
 the diameter required. 
 
 PRACTICAL RULE AND EXAMPLE FOR THE STIFFNESS OF 
 SQUARE SHAFTS TO RESIST TORSION. 
 
 275. Rules for square shafts are applications of Equation 
 xiv. ; and the same things are known as in the case of cylin- 
 drical shafts. 
 
 Rule. To determine the side of a square shaft to resist 
 torsion with a given flexure. 
 
 Multiply the power in pounds by the leverage it acts with 
 in feet, and also by the length of the shaft in feet. Divide 
 this product by 92*5 times the angle of flexure in degrees, 
 and the square root of the quotient will be the area of the 
 shaft in inches. 
 
 Example. Suppose the length of a shaft is to be 12 feet, 
 and it is to be driven by a power of 700 ft s. acting on a 
 pinion, on the shaft, of 1 foot radius to the pitch line, and 
 that a flexure of 1 degree will not affect the machinery. 
 
 By the rule, 
 
 The square root of 90'8 is 9'53, the area of the section in 
 inches; and the square root of 9' 53 is 3'1 inches nearly, for 
 the side of the shaft. 
 
 The reader may find further information on the torsion of 
 
174 RESISTANCE TO TORSION. [SECT. ix. 
 
 wires, and the laws of the oscillation of the torsion balance, in 
 Dr. Young's Lectures on Nat. Philos. vol. i. pp. 140, 141 ; 
 Dr. Brewster's Edinburgh Encyclopaedia, art. Mechanics, 
 p. 544 to 549 ; Dr. Brewster's edition of Ferguson's Lectures, 
 vol. ii. p. 234 ; or Professor Leslie's Elements of Nat. Philos. 
 vol. i. p. 243. 
 
SECTION X. 
 
 OF THE STRENGTH OF COLUMNS, PILLARS, OR OTHER SUPPORTS 
 COMPRESSED, OR EXTENDED, IN THE DIRECTION OF THEIR 
 LENGTH. 
 
 276. If the length of a column be considerable with respect 
 to its diameter, under a certain force it will bend ; but when 
 it becomes too short to bend, its strength is only limited by 
 the force which would crush it. Considering, however, that 
 it is imprudent to load even a short column beyond its elastic 
 force, an inquiry respecting the phenomena of crushing would 
 lead to nothing useful. 
 
 Let A A' be a column, fig. 30, supported at A', and sup- 
 porting a load at A ; and let this load have produced its full 
 effect in straining the column. Let E be the neutral axis, 
 B and D the centres of resistance, and A F the direction of 
 the straining force. Draw d D parallel to A F, then, by the 
 principles of statics, we have 
 
 2D : D A : : W (the weight) : rfp - 
 
 the compressive force in the direction A D. Also, 
 
 w. D A w. A F 
 
 the vertical pressure at D. 
 But, by similar triangles, 
 
 BF. dD 
 
 B D : B F : : tZD : A F 
 
176 RESISTANCE TO COMPRESSION AND TENSION. [SECT. x. 
 
 therefore 
 
 w. AF __ w. BF 
 
 dD BD"' (i.) 
 
 277. In a similar manner it may be proved that the strain 
 at B is expressed by 
 
 W. B P B D 
 
 BD~ () 
 
 Where it is obvious that when BD=BF this strain is 
 nothing ; that is, when the direction of the straining force 
 passes through the point D, or the neutral axis coincides with 
 the surface of the block. It also may be observed, that when 
 B F exceeds B D, this strain is expressed by a positive 
 quantity, indicating extension ; but when B F is less than 
 B D it is negative, indicating that it is a resistance to com- 
 pression. If B F = | B D, then both points are equally 
 compressed. 
 
 The force has been supposed to be perpendicular to the 
 plane of section for which the strains have been calculated, 
 but this is not essential to the investigation ; it is only the 
 most usual strain on columns and ties. For instead of the 
 force acting in the direction A F, let the force act in the 
 direction A G; and let the angle FAG be denoted by C. 
 Then, the investigation being resumed, we shall find Equa- 
 tion i. will become 
 
 (B F + A F. sin. C) W. cos. C , _. 
 
 BD~~ (iii.) 
 
 And, 
 
 (B F + A F. sin. C - B D) W. cos. C 
 
 BD = stress at B. (iy) 
 
 But when the force acts in an oblique direction, a further stay 
 is required to prevent the pillar overturning ; arid wherever 
 this stay is placed there the greatest strain would be in a 
 straight pillar. If it be stayed at D F, then the stay placed 
 
SECT, x.] RESISTANCE TO COMPRESSION AND TENSION. 177 
 
 there becomes a point of support; and the action of the 
 forces on the beam are similar to those already considered in 
 fig. 14, Plate II. But it was remarked in a note to art. 108, 
 that the mode of calculation there given is not correct when 
 the beam is not nearly horizontal ; the difference is owing to 
 a change of the position of the neutral axis caused by the 
 oblique direction of the force. The position of that axis, and 
 the strength of the section, we will now proceed to calculate, 
 and to develope the changes produced by altering the direc- 
 tion of the straining force. 
 
 278. It may be shown that the resistance of the section, 
 on either side of the neutral axis, is equal to the force of a 
 square inch multiplied by the area of that section and by the 
 distance of the centre of gravity from the neutral axis, and 
 divided by the distance of the compressed surface from the 
 neutral axis, when B or D is the centre of percussion of the 
 section.* 
 
 279. Let x be the distance of the neutral axis from the 
 middle of the depth ; y = E G the distance of the direction 
 A G of the straining force from the middle of the depth ; 
 d = the depth, b = the breadth, and / the resistance of a 
 square inch; then the area of the compressed part of the 
 section will be (-J- d + x] b, and the extended part of the section 
 (\ d x) b. Therefore, if n tyd + as) and n (^d x) be the 
 distances of the centres of percussion from the neutral axis, 
 and m (^d + #) and m (^d x) the distances of the centres 
 of gravity, we shall have 
 
 W. B G cos. C m f b (& d - x)- ^ W. (B G - B D) co.^C ^ 
 B L> <l~+ '' BD 
 
 
 B G x (i d-x)- = (B G - BD) x 
 
 * Emerson's Mechanics, 4to. edit, Prop. LXXVIL 
 
178 RESISTANCE TO COMPRESSION AND TENSION. [SECT. x. 
 
 and when the proper substitutions are made, this equation 
 reduces to 
 
 280. In a rectangular section n = f, and consequently we 
 find by the preceding equation 
 
 Also, since in this case m = |-, we have an equilibrium 
 between the compressing force and the resistance to com- 
 pression, when 
 
 W. BG.cos. C /&., 
 
 and substituting for B G, B D, and .2? their proper values, 
 this equation becomes 
 
 (d + Gy) cos. C * (v.) 
 
 But if B F be denoted by a ; and -|- -= A F, whence E G 
 will be = 
 
 A_F. sin. C _ I sin. C m 
 cos. C. 2 cos. G ' 
 
 and therefore, 
 
 I sin. C 
 
 * It is shown that 
 
 I sin. C 
 
 hence the distance of the neutral axis from the axis of the column is 
 
 and these axes must Coincide when C is an angle of 90, that is, when the direction 
 of the force is perpendicular to the axis of the column; but not in any other case. 
 
 For when C = 90, the tan. C is unlimited ; and consequently the fraction which 
 represents x is incomparably ssmall, or the axes coincide. 
 
SECT, x.] RESISTANCE TO COMPRESSION AND TENSION. 179 
 
 Consequently, 
 
 (d + tf y) cos. C "" d cos. C + 6 a. cos. C. + 3 1 sin. C ~~ ( y i-) 
 
 This equation will enable us to trace the particular con- 
 ditions of this important problem. 
 
 In the first place, if the points E and A be in a line 
 perpendicular to B G, then a =0, and the equation is 
 
 d. cos. G + 3 1 siu. C ~~ ' ' (vii.) 
 
 Secondly, if the force act in a direction parallel to B G, 
 then C = 90 degrees; and sin. C = 1, and cos. C = 0, and 
 Equation vi. becomes 
 
 3 = W ' (viii.) 
 
 We have in this case the same equation as in art. 110, for in 
 this instance / is double the length taken in that equation. 
 
 Thirdly, if the force act in a direction perpendicular to B G, 
 then cos. C=l, and sin. C = o, and consequently Equation vi. 
 becomes 
 
 ft>v 
 
 d + 6 ft v^v 
 
 Fourthly > when 0=0, or the direction of the force coincides 
 with the axis E, then 
 
 And, fifthly, if a = half the depth of the block, then 
 
 fbd -w 
 4 ~ W ' <*i.) 
 
 The Equations ix., x., and xi., apply to short columns, or 
 blocks, of which the length is not more than ten or twelve 
 times the least dimension of the section ; and from them are 
 derived the following practical rules ; 
 
180 RESISTANCE TO COMPRESSION AND TENSION. [SECT. x. 
 
 TO FIND THE AREA OF A SHORT RECTANGULAR COLUMN OR 
 BLOCK TO RESIST A GIVEN PRESSURE. 
 
 281. Rule. When the force is to be applied exactly in 
 the axis or centre of the section of the block, divide the 
 pressure or the weight in pounds by 15,300, and the quotient 
 will be the area of the section of the block in inches. But 
 since this requires a degree of ^precision in adjusting the 
 direction of the force which it is altogether impossible to 
 arrive at in practice, and when a force presses a block of 
 which a a is the axis, fig. 31, Plate IV., it is always probable 
 that the direction A A' of the force may act upon one edge 
 only of the end of the block, and therefore be at a distance 
 of half the least thickness from the axis ; which will reduce 
 the resistance of the block to ^th, and consequently the area 
 should always be made four times as great as is determined 
 by this rule. 
 
 When the distance of the direction of the force from the 
 axis is determined by the nature of the construction, the 
 following is a general rule. 
 
 282. Rule. To the thickness (or least dimension of the 
 section) in inches, add six times the distance of the direction 
 of the force from the axis in inches, and let this sum be 
 multiplied by the weight or pressure in pounds ; divide the 
 quotient by 15,300 times the square of the least thickness 
 in inches, and the quotient will be the breadth of the block 
 in inches. 
 
 This rule is the Equation ix., art. 280, in words at length, 
 and it applies to resistance to tension as well as to resistance 
 to compression. 
 
 283. The writer of the article < Bridge/ in the Supple- 
 ment to the Encycl. Brit., has shown that when the force 
 acts in the direction of the diagonal of the block, as is 
 shown in fig. 32, the strain will be twice as great as when 
 
SECT, x.] BESISTANCE TO COMPRESSION AND TENSION. 181 
 
 the same force acts in the direction of the axis.* Now the 
 reader will be satisfied, that, in consequence of settlements, 
 or other causes, a column is always liable to be strained in 
 this manner ; and therefore will carefully avoid enlarging the 
 ends of his columns, under the notion of gaining stability, 
 for the effect of the straining force will be still more in- 
 creased by such enlargement in the event of a change of 
 direction from settlement, as in fig. 33. In my ' Treatise on 
 Carpentry/ I have recommended circular abutting joints to 
 lessen the effect of a partial change in the position of the 
 strained pieces, f an idea which appears to have occurred, in 
 the first instance, to Serlio.J 
 
 284. A general solution of the equation expressing the 
 stress and strain, when the column is cylindrical, is compli- 
 cated, but in one particular case the result is extremely 
 simple; that is, when the neutral axis is in one of the 
 surfaces of the column. If d be the diameter of the column, 
 then *7854 d 2 = the area, and \d = the distance of the 
 centre of gravity, and therefore 
 
 W. B G. cos. C -7854 d*f 
 
 BD 2 
 
 But when the neutral axis is in the surface of the cylinder, 
 
 Iii this case the distance of the direction of the force from 
 the axis of the column will be -Jth of the diameter, the centre 
 of percussion being -| d distant from the neutral axis. 
 
 285. Hence it appears, that when the distance of the 
 direction of the force from the axis is -J- d, the strength of a 
 cylinder is to that of a circumscribed square prism, as seven 
 
 * Napier's Supp. to Encycl. Brit., art. ' Bridge/ Prop. I. p. 499. 
 t Tredgold's Elementary Prin. Carpentry, Sect. IX. p. 164. 
 J Serlio's Architecture, Lib. I. p. 13. Paris, 1545. 
 
182 RESISTANCE TO COMPRESSION AND TENSION. [SECT, x, 
 
 times the area of the cylinder, to eight times the area of the 
 prism; or nearly as 5*5 : 8, or as 1 : 1*46 nearly. 
 
 When the neutral axes are at or near the axes of the 
 pieces, the ratio of the strength of the cylinder to that of the 
 prism becomes 
 
 3 x -7854 
 
 : 1, or as 1 : 1*7, 
 
 as has been shown by Dr. T. Young ; * consequently in a 
 column, when both the resistances to compression and exten- 
 sion are brought into action, the ratio varies between 1 : 1*46 
 and 1 : 1'7 ; the mean being nearly 1 : 1/6. 
 
 O'P THE STRENGTH OF LONG PILLARS AND COLUMNS. 
 
 286. If a support be compressed in the direction of its 
 length, and the deflexion be sufficient to sensibly increase 
 the distance of the direction of the force from the axis, in 
 the middle of the length of the support, it is evident that 
 the strain will be increased; and since the curvature in 
 practical cases will be very small, we may suppose it to be an 
 arc of a circle. In a circle the square of the length of the 
 chord, in a small segment, is sensibly equal to the radius x 8 
 
 7 
 
 times the versed sine ; or -^ = radius. The deflexion will 
 
 o o 
 
 be greatest when the neutral axis coincides with the axis, and 
 taking this extreme case, we shall have this analogy ; as the 
 alteration of the length of the concave side is to the original 
 length, so is the J depth to the radius of curvature ; or, 
 
 Therefore 
 
 d ,. d 
 
 :: y : radius =-, 
 
 S-T = ; and 8 = = the deflexion in the middle, 
 
 85 2e 4 d. 
 
 * Dr. Young's Lectures on Nat. Philos. vol. ii. art. 339, B. 
 
SECT, x.] RESISTANCE TO COMPRESSION AND TENSION. 183 
 
 287. Let the distance of the direction of the force from 
 the axis, when first applied, be denoted by a, as in a pre- 
 ceding article (art. 280) ; then, in consequence of the 
 flexure, it will be equal to 
 
 consequently by Equation ix., we have 
 
 /5<* 2 _ W 
 
 ~ 
 
 In cast iron /== 15,300fts. and e = jm (art. 143 and 
 
 212) ; therefore, if / be the length in feet, b, d, and a in 
 inches, we obtain the following practical formula, for the 
 strength of a rectangular prism, viz. 
 
 15S005cZ 2 15300 I d* _ 
 
 ~ 
 
 d -r 6a + d* + 6da 
 
 288. If a = o } or the direction of the force coincides 
 with the axis, then the rule becomes 
 
 15300 5 d 3 
 
 It would, however, be improper in practice to calculate 
 upon the nice adjustment of the direction of the pressure in 
 the direction of the axis, which is supposed in the preceding 
 equation ; indeed, there are very few instances where its 
 direction may not in all probability be at the distance of half 
 the depth from the axis, and in that case a = \ d, and 
 
 For malleable iron, 
 
 17800 I d* 
 
 rf 2 + Qda + 
 For oak, 
 
 = W. 
 
 3960 I d* _ 
 
 7* ~ "' 
 
 6 d a + -,5 I* 
 
ISi RESISTANCE TO COMPRESSION AND TENSION. [SECT. x. 
 
 15300 
 
 4 d 2 + *18 I 2 
 
 = W. 
 
 289. As an approximate rule for the strength of a cylinder 
 to resist compression in the direction of its length, we have 
 
 15300 d 4 - _ . 
 
 /J'2 , .1 ij J0\ J0~ .10 7 > ~ " ' \ XV1 V 
 
 l't(d* + 18 i) d*+ -18 - " 
 
 290. And if the direction of the force be a inches distant 
 from the axis, the rule is 
 
 9562 d 4 
 
 d 3 + 3d a + -18 T 
 
 (xvii.) 
 
 If the force act in the direction of one of the surfaces of 
 the column, then a = \ d, and 
 
 By this rule the Table of columns (Table III. p. 26,) was 
 calculated, only the weight is there given in cwts. 
 
 In all the rules from Equation xiii. to xviii. / is the length, 
 A A', fig. 31, Plate IV., in feet, d either the diameter or the 
 least side in inches, b the greater side in inches, and W the 
 weight to be supported in Ibs. 
 
 291. Example 1. Required the weight that could be sup- 
 ported, with safety, by a cylindrical column, the length being 
 
 In malleable iron, 
 
 In oak, 
 
 17800 6_d 3 _ 
 
 3960 bd* = 
 
 d 2 + "5 Z ' 
 + In malleable iron, 
 
 11125 rf 4 _ 
 
 In oak, 
 
 2470 **_ 
 
SECT, x.] RESISTANCE TO COMPRESSION AND TENSION. 185 
 
 1 1 feet, and the diameter 5 inches, and supposing it probable 
 that the force may act in the direction A A', fig. 31, at the 
 distance of half the diameter from the axis ? 
 
 In this example Equation xviii., art. 290, should be used ; 
 and therefore 
 
 or a little above 22 tons. 
 
 In this manner may be calculated the strength of story- 
 posts for supporting buildings. When they are for houses, 
 ample allowance should be made for the weight of crowded 
 rooms, and when for warehouses the greatest possible weight 
 of goods should be estimated. 
 
 292. Example 2. It is proposed to determine the com- 
 pression a curved rib will sustain in the direction of its chord; 
 the greatest distance of the axis of the rib from the chord 
 line being 6 inches, the size of the rib 3 inches square, and 
 the length of the chord line 5 feet. 
 
 By Equation xiii., art. 287, 
 
 15300 b d* 15300 x 3< 
 
 Example 3. The piston-rod of a double-acting steam 
 engine is another interesting case to which these equations 
 will apply ; and the reader will excuse my having recourse 
 to algebraic notation in order to make the rule general. 
 
 Let D be the diameter of the steam cylinder in inches, 
 and p the greatest pressure of the steam on a circular inch 
 of the piston in fts. Then W = D 2 p. 
 
 But it has been shown in a note to art. 290, that in mal- 
 leable iron 
 
 _11125d_ 
 
 4^ 2 + -16 J 2 " 
 
 Therefore, 
 
RESISTANCE TO COMPRESSION AND TENSION". [SECT. x. 
 11125 d* 
 
 D = 53 d" V 
 
 p. (d* + -04 I 2 ) 
 
 Now in an extreme case we can never have the length in 
 feet greater than about three times the diameter in inches ; 
 substitute this value of /, and we have 
 
 If the pressure be 8fts. on the circular inch, that is, a 
 little more than lOfts. on the square inch, it gives -~- =- d. 
 That is, the piston-rod should never be less than j^th of the 
 diameter of the cylinder in a double-acting steam engine. 
 In practice it is usual to make them n>th, which does not 
 appear to be too great an excess of strength to allow for 
 wear. 
 
 OF THE STRENGTH OF BARS AND RODS TO RESIST TENSION. 
 
 293. When the effect of flexure is considered in bars to 
 resist tension, it makes an important difference, Instead of 
 the strength being diminished by flexure, it either has no 
 effect, or has a directly contrary effect. Hence in all works 
 executed in metals the tensile force of the materials should be 
 employed in preference to any other, except the bulk be con- 
 siderable in respect to the length. In wood we cannot employ 
 a tensile force to much advantage, because it is difficult to 
 form connexions at the extremities of sufficient firmness, but 
 in metals this creates no difficulty. 
 
 If a bar or rod be short, its resistance may be computed 
 by the -rules, art. 281 and 282. 
 
 But when it is long, and the bar is either curved, or the 
 force is not in the direction of the axis, then the effect of 
 flexure may be considered. 
 
SECT, x.] RESISTANCE TO COMPRESSION AND TENSION. 18? 
 
 The Equation ix., art. 280, will be applicable to all cases 
 where the direction of the force is parallel to the extremities 
 of the bar, that is, 
 
 (xviii.) 
 
 Qa 
 
 
 The flexure is found to be = 7-^ = 'r^ y when / is the 
 
 4 d d 
 
 length in feet, and e = ^^ But this flexure is to be 
 deducted from the distance from the axis. Hence 
 
 15300 6 d 3 
 
 W.* (xix.) 
 
 When the direction of the force is at the distance of half 
 the least side from the axis, then a = \ d, and 
 
 15800 6rf3 
 
 id*-'l$t* 
 
 And when the -direction of the force coincides with the axis, 
 
 15300 I d = W. (XXL) 
 
 When the bar is a cylinder, its strength is to that of a 
 square bar as 1 : 1'6 nearly (art. 285) ; hence, 
 
 9562 d* 
 
 Or, when the force is in one of the surfaces of the rod, 
 
 9562 d 4 
 
 = >W. (xxiii.) 
 
 It was desirable to show what constituted the advantage of 
 a tensile strain, but I do not intend to adopt these equations 
 
 * In malleable iron, 
 
 17800 & d 3 
 
 =rr \Y 9 
 
 In oak, 
 
 3960 &fP =w 
 
 2 + e a d -5 I 9 
 
188 RESISTANCE TO COMPRESSION AND TENSION. [SECT. x. 
 
 in practical rules, because they are not so simple and easily 
 applied as the rules already given in art. 281 and 282, which 
 will only err a small quantity in excess, when proper care has 
 been taken to take the greatest possible deviation of the 
 straining force from the axis of the piece. 
 
 Example 1. Required the weight that may be suspended 
 by a bar of cast iron of 4 inches by 8 inches ; under the sup- 
 position that the direction of the strain will be in one of the 
 wide surfaces of the bar ? Equation xviii. of this art. 
 applies to this case, wherein a is equal 2 inches, or half the 
 least dimension of the bar, that being the distance the 
 direction of the force is supposed to be from the axis ; and 
 therefore 
 
 fi*!~i!_4. = 1224 oon,, 
 
 d + 6 a 4 + 12 
 
 the weight required. See the rule in words at art. 282. 
 
 When it is considered that a very small degree of inac- 
 curacy in fitting the connexion may throw the strain all on 
 one side of the bar, the prudence of following this mode of 
 calculation will be apparent. 
 
 Example 2. It is proposed to determine the area that 
 should be given to the bars of a suspension bridge, if made 
 of cast iron, for a span of 370 feet ; the points of suspension 
 being 30 feet above the lowest point of the curve ; and the 
 greatest load, including the weight of the bridge itself, 
 500 tons. 
 
 The load being nearly uniformly distributed, the curve 
 assumed by the chains will not sensibly differ from a para- 
 bola;* and half the weight will be to the tension at the 
 lower point of the curve, as the rise is to ^th of the span ; 
 that is, 
 
 . 370 500 500 x 370 
 
 See Elementary Principles of Carpentry, art. 57. 
 
SECT, x.] RESISTANCE TO COMPRESSION AND TENSION. 189 
 
 This is equal to 1,727,040 Bbs., and by the rule art. 281, 
 we have 
 
 1727040 
 
 - =llo square inches 
 
 for the area of the bars, supposing the stress to be directly 
 in the axis of each ; and if we double this area it will provide 
 for a deviation equal to Jth of the diameter of each bar. 
 This will be a sufficient excess of force, considering the great 
 chance of its ever being covered with people, which is the 
 load I have estimated. Hence the sum of the areas of the 
 chains at the lowest point should be 230 square inches. 
 The area at any other point of the curve should be to the 
 area at the lowest point, as the secant of the angle a 
 tangent to the curve makes with a horizontal line, is to 
 the radius. In the present case, the sum of the areas at 
 the point of suspension should be 242 square inches. Cast 
 iron would be greatly superior to wrought iron for chain 
 bridges ; it would be more durable, less expensive to obtain 
 the same strength, and when made sufficiently strong, its 
 weight would prevent excessive vibration by small forces. 
 Most of the wrought iron bridges appear to be very slight 
 and temporary structures when examined by the rules I 
 have given, which appear to be founded on unquestionable 
 principles. 
 
 Example 3. To determine the area of a piston-rod for a 
 single-acting engine, the force on the piston being equivalent 
 to 11 fts. on a square inch, and allowing for the possibility 
 of the direction of the force being at half the diameter of the 
 rod from its axis. In this case 11 times the square of the 
 diameter of the piston in inches is equal to the stress, and if 
 D be the diameter of the steam piston, and d that of the 
 piston-rod, we have for wrought iron, 
 
190 RESISTANCE TO COMPRESSION AND TENSION. [SECT. x. 
 
 or very nearly, 
 
 D _ 
 
 20 
 
 That is, the diameter of the piston-rod should be 2 - th of the 
 diameter of the steam cylinder, when nothing is allowed for 
 wear ; or making the allowance which appears to be requisite, 
 the diameter should be ^th of the diameter of the steam 
 cylinder. 
 
SECTION XL 
 
 OF THE STRENGTH OF CAST IRON TO RESIST AN IMPULSIVE 
 
 FORCE.* 
 
 294. The moving force of a body, or of a part of a 
 machine, ought to be balanced by the elastic force of the 
 parts which propagate the motion ; for if the effect of the 
 moving force be greater than the elastic force of the parts, 
 some of them will ultimately break ; besides, a part of the 
 power of the machine will be lost at each stroke. 
 
 And since increasing the mass of matter to be moved 
 increases the friction in a machine, it is an advantage to 
 employ no more material in its moving parts than is abso- 
 lutely necessary for strength ; but, in other parts exposed to 
 pressive forces only, it is desirable that the materials should 
 always be capable of resisting the strains, with as small a 
 degree of flexure as is convenient, because steadiness is, in the 
 fixed parts of machines, a most desirable property. 
 
 A beam resists a moving force, as a spring, by yielding 
 and opposing the force as it yields, till it finally overbalances 
 it ; f and hence it is, that a brittle or very stiff body breaks, 
 because it does not yield sufficiently for destroying the force. 
 
 As the resistance of a beam under different degrees of 
 
 * Dr. Young has given the term resilience to this species of resistance ; and the 
 reader will find some interesting remarks on the importance of studying it, in his 
 Lectures on Nat. Phil. vol. i. p. 143. 
 
 t For a machine to produce the greatest effect, the time of bending the beam 
 should be as small as possible. 
 
192 RESISTANCE TO IMPULSION. [SECT. xi. 
 
 flexure can be calculated, the effect of that resistance in the 
 destruction of motion may be estimated by the principles of 
 dynamics : such inquiries are usually managed by the 
 method of fluxions ; but not being satisfied with the manner 
 of establishing the principles of that method, though I have 
 no doubt of the correctness of results obtained by it, I shall 
 briefly deduce the rules of this section by another mode of 
 calculation. 
 
 295. If the intensity of a force be variable, so that the 
 action upon the body moved at any point be directly as some 
 power, n, of the distance from a point B, fig. 23, towards 
 which it moves. Then, if the intensity of the force at A be 
 
 equal P, the intensity at any point C will, be ^L"'/. For, 
 by the definition, 
 
 Put S to denote the space A B ; and conceive this space S 
 to be divided into m equal parts, denoting any one of these 
 parts by x ; and in consequence of the smallness of these 
 parts, if we take the mean between the intensity at the 
 beginning, and that at the end of each part, and consider 
 each of these means a uniform intensity for the space it was 
 calculated for, then these uniform intensities may be repre- 
 sented by the following progression : 
 
 - x f + x n + x n + 2 n x n + 2* x n + 3 n x n + ...m 1" x n + m n x n ) 
 \ J 
 
 or, 
 
 l n + 2* + 3" +...OT 1 + 
 
 296. It is shown by writers on dynamics, that when the 
 intensity of a force is uniform, the square of the quantity of 
 force accumulated or destroyed is directly as the intensity 
 
SECT. XL] RESISTANCE TO IMPULSION. 193 
 
 multiplied by the quantity of matter moved, and by the space 
 moved through.* Therefore making W = the quantity of 
 matter, and g a constant quantity to reduce the proportion to 
 an equation, we find the square of the forces accumulated 
 or destroyed, in the space S, may be exhibited by the 
 progression 
 
 q P W x n + 1 
 
 
 And from the principles of the method of progressions,! 
 the accurate value of the square of the force accumulated or 
 destroyed in the space S is 
 
 p w s 
 
 297. When n = 0, or the intensity is uniform, the square 
 of the accumulated force is = g P W S. 
 
 298. The force of gravity near the earth's surface is nearly 
 uniform, and in this case we know from experiments on fall- 
 ing bodies that g = 64^, and P = W the weight of the body ; 
 therefore, 64 J W 2 S = the square of the accumulated force, 
 and 64^ may be substituted for g. 
 
 Hence the moving force of a falling body is W V64^ S. 
 
 299. If =J, we have 
 
 and as, in the resistance of beams, the intensity at any 
 deflexion is directly as the deflexion, the quantity 32^ P W S 
 represents the square of the force destroyed in producing a 
 deflexion equal to S. That is, when a beam is supported at 
 both ends, and S = the deflexion in the middle, in decimal 
 parts of a foot, then V 32^ P W S = the force that would be 
 
 * Dr. C. Button's Course of Math. vol. ii. p. 186, 5th edition. 
 f See Philosophical Magazine, vol. Ivii. p. 20 L 
 
194 KESISTANCE TO IMPULSION. [SECT. xi. 
 
 destroyed in producing the flexure S ; where P is the weight 
 that would produce the deflexion S.* 
 
 Having considered the effect of the resisting force of the 
 material in destroying an impulsive force, we must now con- 
 sider the circumstances which take place in the different cases 
 occurring in practice. 
 
 300. If the blow be made by a falling body in the direc- 
 tion of gravity, and the weight of the falling body be w, and 
 its velocity at the time of impact be v 9 then by the laws of 
 collision, in the case of equilibrium, 
 
 vw= V 32 P S (W + w). (i.) 
 
 In which equation the small acceleration that would be 
 produced by the action of gravity on the mass W + w, during 
 the flexure of the beam, is neglected. 
 
 301. If the blow were made horizontally by a body of the 
 weight w, moving with a velocity v, then the equation is 
 correct ; and even in the first case it is accurate enough for 
 practical purposes. 
 
 302. If the blow were made by a weight w falling from a 
 given height 7i, we have, by the laws of gravity (art. 298), 
 
 w v w 
 
 therefore, 
 
 = V 32 P S (W + w), or 
 2 w 2 h = P S (W + w). (ii.) 
 
 303. When the strain is occasioned by a force of an 
 intensity F, and velocity v, such for example as would be 
 occasioned by the sudden derangement of a machine in 
 motion with the velocity v, and force P, then 
 
 * The effect of elastic gases in producing or destroying motion is expressed by the 
 same equation, when the change of bulk is not so rapid as to cause cold in the one 
 case, or to develope heat in the other. The developement of heat by the sudden 
 compression of air materially affects the velocity of sound, and was first applied by 
 Laplace to correct the discrepancy between theory and experiments ; a subject which 
 has been further illustrated by the researches of Poisson, in an article "Sur la Vitesse 
 du Son," Annales de Chimie, tome xxiii. p. 5. 
 
SECT. XL] RESISTANCE TO IMPULSION. 195. 
 
 F v = V 32 P S W ; or, 
 F 2 z; 2 = 32 P S W. (iii.) 
 
 The last equation is applicable to the beams of steam 
 engines, and in general to reciprocating movements in 
 machines, such as the connecting rods, cranks, &c. 
 
 If a body be previously in motion in the direction of the 
 impulsive force, then the force F v should be the difference 
 between the forces of the impelling and impelled bodies. 
 
 304. A general number of comparison to exhibit the power 
 of a body to resist impulse, and which might be termed the 
 modulus of resilience, would be extremely convenient in calcu- 
 lations of this kind; and when we omit the effect of a 
 difference of density, which it is usual to do, we have an easy 
 method of forming such a number.* For in any case, if /be 
 the force which produces permanent alteration, and * the 
 corresponding extension, 
 
 PS:/. 
 
 And, since in bars of different materials placed in the same 
 circumstances the resistance to impulse may be considered pro- 
 portional to the height a body must fall to produce a permanent 
 change in the structure of the matter ; and as that height is 
 proportional to P S, and consequently to /e, when the effect 
 of density is neglected ; we may take f e the measure of the 
 power of a body to resist impulsion, that is, the modulus of 
 resilience ; and representing this modulus by R, 
 
 /=R. (iv.) 
 
 * The number might include the effect of density, if we were to measure the re- 
 sistance to impulse by the height a body should fall to produce permanent change by 
 its own weight ; for we easily derive from Equation ii. art. 302, 
 
 when s is the specific gravity. This expression, might be termed the specific resilience 
 of a body, and if it were denoted by 2, we should have 
 
 In cast iron, 
 
 5 = 1-762. 
 
196 RESISTANCE TO IMPULSION. [STSCT. xi. 
 
 In cast iron, 
 
 /=15300;ande:=--L; 
 therefore B = 127. 
 
 305. These equations flow from the principle that while 
 the elasticity is perfect, the deflexion or extension is as the 
 force producing it, but it also varies according to the manner 
 in which the material is strained. In some cases, of frequent 
 occurrence, the application is shown in the examples. 
 
 But it will be useful, before we proceed any further, to 
 inquire what velocity cast iron will bear, without permanent 
 alteration, in order that we may be aware whether such 
 velocity will ever take place in the parts of machines ; for if 
 any part of a machine be connected with others that will yield 
 to the force, and the material be capable of transmitting the 
 motion with greater velocity than the machine moves with, 
 it need be formed only for resistance to power or pressure. 
 
 306. It has been shown that V 32 P W S is equal to 
 the greatest force an elastic body can generate or destroy 
 (art. 299) ; if it were exposed to a greater force, its arrange- 
 ment would be permanently altered. Now, if V be the 
 greatest velocity the body is capable of transmitting, if com- 
 municated to its mass, we have 
 
 S = V W, or 
 
 307. It has also been shown, that in cast iron the 
 cohesive force/ == 15,300 ibs. (art. 143), and the extension, 
 
 and since S =- / (by art. 104), and P =- Idf (art. 103), 
 and Ib dp = W, where p = 3 ! 2 fts., the weight of a bar of 
 
SECT. XL] EESISTANCE TO IMPULSION. 197 
 
 iron 12 inches long and 1 inch square; therefore, when a bar 
 is strained in the direction of its length, 
 
 V \V I \f " ~ Ibdp V 3-2x1204 ~ 
 
 1T3 feet per second. 
 
 308. If a uniform bar be supported at the ends, we have 
 
 P = ! (art. 143) and S = 7' ft- (art. 212) 
 
 also, 
 
 for the mass of the beam would acquire only the same 
 momentum as half of it collected in the middle. Consequently, 
 
 / 32$ P S _ / 32^ x 850~x "02 x 2 _ _ 
 V W~~ = V 12 x 32 
 
 5 '3366 feet per second, nearly. 
 
 I have shown by a comparison of many experiments in art. 
 70, that about 3*3 times the force that produces permanent 
 alteration will break a beam; therefore, assuming the de- 
 flexion to continue proportional to the force till fracture takes 
 place, we have 
 
 32i x 3-3 P x 3-3 S 
 
 ~ ~~~ ==V;oi< 
 
 V ~w~ 
 Therefore, a velocity of 
 
 3'3 x 5'3366 = 17'6 feet per second, 
 
 would break a beam ; or a beam would break by falling a 
 height of about 5 feet. 
 
 309. Hence it is clear, that cast iron is capable of sus- 
 taining only a very small degree of velocity ; and a correct 
 
198 RESISTANCE TO IMPULSION. [SECT. xi. 
 
 knowledge of this limit is certainly of the first importance in 
 the application of this material in machinery. When a cast 
 iron bar is exposed to an impulsive force in the direction of 
 its length, the utmost velocity its mass should acquire must 
 never exceed 1 1 feet per second ; and when the force acts 
 in a direction perpendicular to the length, it should never be 
 capable of communicating to the mass of the bar a greater 
 velocity than about 5 feet per second ; and if it exceed 18 
 feet per second, the bar will break. 
 
 If the connecting rod of a steam engine were to move with 
 a greater velocity than 5 feet per second, the swag of its own 
 weight would produce permanent flexure. 
 
 If a ship with hollow cast iron masts should strike a rock 
 when it moved with a velocity of 12 miles per hour, the 
 masts would break ; and even with less velocity, for here we 
 neglect the effect of the wind on the masts. 
 
 310. To illustrate the use of the above investigation, or 
 rather, to prevent any one from disappointment, in applying 
 these rules for the resistance to impulsion, it may be useful 
 to consider how they should be applied to the parts of 
 machines. In a machine the motion is communicated from 
 the impelled to the working point by a certain number of 
 parts, and among these parts one at least should be capable 
 of resisting the whole energy of the moving power. If there 
 be many parts to transmit the power, then two or more of 
 them should be capable of resisting the energy of the moving 
 power, and they should be distributed so as to divide the 
 line of communication into nearly equal parts. If the inter- 
 mediate parts be made sufficient to resist the dead power of 
 the machine, that is, the power without velocity, they will 
 always be strong enough to convey the velocity, if it be less 
 than is stated in the preceding article, to other parts, that 
 will either forward it to the working point, or resist it entirely 
 during a momentary derangement of the action of the machine. 
 To make all the parts strong enough for this purpose would 
 
SECT. XL] RESISTANCE TO IMPULSION. 199 
 
 often cause a machine to be clumsy, and unfit for any 
 practical use. 
 
 311. Let the constant numbers for the strength and 
 deflexion in feet be/8. Then, 
 
 Also, let the weight of the beam itself be n times the weight 
 of the falling body. These values being substituted in 
 Equation i. art. 300, we have 
 
 n+l); or, . 
 
 = 
 
 312. If the like substitutions be made in Equation iii. art. 
 303, we obtain 
 
 F-' V 2 = (32* P S W) = 32* Z 6 cZ/5 W; 
 
 and if lit dp be the weight of the mass of the beam the force 
 acts upon, then 
 
 (vii.) 
 
 PRACTICAL RULES AND EXAMPLES. 
 
 313. Prop. i. To determine a rule for finding the dimen- 
 sions of a beam to resist the force of a body in motion. 
 
 It is evident by Equation vi. art. 311, that the error which 
 would arise from neglecting to allow for the effect of the 
 weight of the beam itself, would always be on the safe side 
 in calculating the dimensions of a beam to resist an impulsive 
 force ; and since, by such neglect, the rule is reduced to a 
 very simple form, instead of a very complicated one, I shall 
 apply the equation under the form 
 
200 RESISTANCE TO IMPULSION. [SECT. XL 
 
 314. Case 1. When the beam is uniform and supported 
 at the ends. In this case/ = 850 (see art. 143), and 6 in 
 
 
 feet = jz (by art. 212) hence, 
 
 315. Rule. Multiply the weight of the falling body in 
 pounds by the square of its velocity in feet per second; 
 divide this product by 45 '5 times the length in feet, and the 
 quotient will be the area in inches. 
 
 The depth should be at least sufficient to render the beam 
 capable of supporting its own weight, added to the weight of 
 the falling body, which may be readily found by Table II. 
 art. 6. 
 
 316. If the height of the fall be given instead of the 
 velocity of the falling body, then instead of multiplying by 
 the square of the velocity, multiply by sixty-four times the 
 height of the fall. 
 
 317. Example 1. To determine the area of a cast iron 
 beam that would sustain, without injury, the shock of a 
 weight of 170 fts. falling upon its middle with a velocity of 
 8 feet per second, the distance between the supports being 
 26 feet. By the rule 
 
 170 x 8 2 
 
 - = 9'2 inches, the area required. 
 
 45'5 x 26 
 
 Hence, if we make the depth 6 inches, the breadth will be 
 1*53 inches, and the beam would sustain a pressure of 
 1800 fcs. (see Table II.) to produce the same effect as the 
 fall of 170fts. It may also be observed, that half the 
 weight of the beam is 400 fts., making 570fi)s. for the 
 pressure the beam would have to sustain after the velocity 
 was destroyed, which is not quite ^rd of the weight the 
 beam would bear. 
 
 318. Example 2. If a bridge of 30 feet span were formed 
 
SECT. XL] RESISTANCE TO IMPULSION. 201 
 
 on beams of cast iron, of what area should the section of 
 these beams be, so that any one of them might be sufficient 
 to resist the impulsive force of a waggon wheel falling over a 
 stone 3 inches high, the load upon that wheel being 3360fts. ? 
 The height of the fall being *25 foot, the square of the 
 velocity acquired by the fall will be 64 x '25 = 16; therefore, 
 
 the area required. 
 
 This area is nearly 40 inches ; suppose it 40, then 40 x 1 5 x 
 3'2 == 1920 fts. = half the weight of the beam (that is, the 
 area in inches multiplied by half the length in feet, multiplied 
 by 3*2 fts., the weight of a piece of cast iron, 1 foot in 
 length, and 1 inch square) ; consequently 1920 -f 3360 = 
 5280 Sbs., the whole effective pressure on the beam, after the 
 velocity is destroyed. If we were to make the beam 20 inches 
 deep, and 2 inches in thickness, it may be found by Table II. 
 that the deflexion would be '9 of an inch, and it would 
 require a pressure of 45,328 fbs. to produce the same effect 
 as the fall of the wheel, above eight times the pressure of the 
 load and weight. 
 
 319. Case 2. When a beam is supported at the ends, the 
 breadth uniform, and the outline of the depths an ellipse. 
 
 This case applies to bridges or beams to withstand an 
 impulsive force at any point of the length. By art. 144, /= 
 850, and by art. 139, 
 
 therefore the equation 
 
 32|/8 I 
 
 in art. 313, becomes 
 
202 RESISTANCE TO IMPULSION. [SECT. xi. 
 
 320. Rule. Calculate by the rule, art. 315, with 58'5 as 
 a divisor instead of 45*5. 
 
 321. Case 3. When the breadth and depth of a beam are 
 uniform, and the section is as fig. 9, Plate I., and the beam 
 supported at the ends. 
 
 In this case/= 850 (I qp*) by art. 186, and 
 
 by art. 212; hence the equation (art. 313), 
 
 V 2 W _ V 2 W _ ^ 
 
 32^/5 I ~ 45-5 (1-p 3 q) I ~ 
 
 consequently the power of a beam to resist an impulsive force, 
 when the quantity of material is the same, is considerably 
 increased by giving this form to the section. 
 
 322. Case 4. If a beam of the form of section shown in 
 fig. 9, be the elliptical form of equal strength (see fig. 24, 
 Plate III.), then 
 
 when the beam is supported at both ends, and the impulsive 
 force acts at any point of the length. 
 
 323. Case 5. In an open beam, as fig. 11, Plate II., we 
 may consider the beam as bounded by a semi-ellipse, when 
 the breadth is uniform, and in this case 
 
 = & d. 
 
 58-5 I (1 - 
 
 324. Example. To determine the area of the section of an 
 open girder that would sustain the shock of 300 fts. falling 
 from a height of 1 foot, the length between the supports 
 being 26 feet, and the depth of the open part ^ of the whole 
 depth. 
 
 In this example 
 
 v 2 w 64 x 300 
 
 jp 3 ) = 58-5 x 26 (1 - -348) = 
 
SECT. XL] RESISTANCE TO IMPULSION. 203 
 
 20 inches nearly. This is perhaps as great an impulsive force 
 as it is probable a girder for a room will be likely to be 
 exposed to; and since this area of section would not be 
 sufficient for the greatest pressure, it appears unnecessary to 
 calculate the effect of moving force in the construction of 
 girders. 
 
 325. Prop. ii. To determine a rule for finding the dimen- 
 sions of a uniform beam to resist a moving force. 
 
 This proposition applies to the parts of machines ; and as 
 there are few people engaged in the construction of powerful 
 machines that are not competent to apply an equation, I shall 
 in this part give the rules in the form* of equations only. 
 
 326. Case 1. When a uniform beam is supported at the 
 ends, and the moving force acts at the middle of the length. 
 
 By art. 143,/ = 850, and by art. 212, 
 
 02 
 5= ^| = -00166 foot; 
 
 and since 3'2Sbs. =the weight of 1 foot in length, and 
 1 inch square, we shall have 
 
 *- T = 
 
 therefore, 
 
 F V F V F 
 
 (art 312) * 
 
 ^32^ x 850 x -0016 x 1-6 8>t> l 
 
 327. Rule. When F is the force in pounds, V its velocity 
 in feet per second, / the whole length in feet between the 
 supports, b the breadth, and d the depth in inches, then 
 
 328. Case 2. When a uniform beam rests upon a centre 
 of motion, and the moving force acts at one end, and is 
 opposed by a greater resistance at the other end. 
 By art. 153, 
 /= 212, and by art. 220, 8 =-Q8 (1 + r\ andp = ~; 
 
204 RESISTANCE TO IMPULSION, INSECT. XL 
 
 hence, 
 
 FV F v 
 
 8-6 I J&Tr 
 
 329. Huh. Make F = the force in pounds, V it s velocity in 
 feet per second, I = the length in feet between the centre of 
 motion and the point where the force acts, and /' = the length 
 in feet between the centre of motion and the point of resist- 
 ance ; It and d being the breadth and depth in inches ; then 
 
 v FV 
 
 7 = r, and ~~ = o a . 
 
 
 
 330. If / = /' we have 
 
 FV FV 
 
 8-6 I V2 12-2 I 
 
 331. Example. To determine the area of the section of the 
 beam for a steam engine, when it is to be of uniform depth ; 
 the length 24 feet, the centre of motion in the middle of the 
 length; the pressure upon the piston 5000 fts., and its 
 greatest velocity 4 feet per second. 
 
 By art. 330, 
 
 F V 5000 x 4 
 
 6 d = I9.0 V 10 = l0 ' mCheS nCarl F' 
 
 If this beam were made 30 inches deep, the deflexion by 
 such a strain would be about n>ths of an inch, and the 
 breadth would be 
 
 ^ = 4-57 inches, 
 
 and such a beam would bear a weight of about 12 times the 
 pressure on the piston, without destroying its elastic force. 
 
 332. Prop. in. To determine a rule for finding the area 
 of the middle section of a parabolic beam to resist a moving 
 force when the breadth is uniform. 
 
 The motion communicated to the arm of a lever is the 
 
SECT. XL] RESISTANCE TO IMPULSION. 205 
 
 same as if its whole weight were collected at its centre of gravity; 
 and as the length of the arm is to the distance of its centre of 
 gravity, so is the mass to the effect of that mass collected at 
 the extremity. Therefore, when the distance of the centre of 
 gravity is some part of the length, the effect of the mass of 
 the arm will be the same part of the whole of its weight 
 when acting at the extremity. 
 
 333. Case 1. When a parabolic beam is supported at both 
 ends, and the moving force acts at the middle of the length. 
 
 By art. 143,/ = 850, and by art. 224, 
 
 o = ^ = -0033 foot. 
 
 Also, because the area of a parabolic beam is f of one uni- 
 formly deep,* and the distance of the centre of gravity from 
 the centre of motion is f of the length ; f we have 
 
 Consequently, the equation (art. 312), 
 
 FV FV FV 
 
 850 
 
 "* l d ' 
 
 334. In beams supported at both ends, and of the. same 
 breadth, the power of a parabolic beam to resist a moving force, 
 is to that of a uniform beam, as 10 is to 8 nearly; and the 
 parabolic beam requires very little more than f rds of the 
 quantity of material. 
 
 335. Rule. When F is the force in pounds, V its velocity 
 in feet per second, I the whole length between the supports, 
 and b and d the breadth and depth in inches ; then 
 
 FV 
 
 336. Example. Let the force of a steam engine be applied 
 to the middle of its beam, so as to cause it to move an axis 
 
 * Dr. Button's Course, vol. ii. p. 126. t Idem, vol. ii. p. 327. 
 
206 RESISTANCE TO IMPULSION. [SECT. XL 
 
 by means of two cranks, placed so as to be impelled by the 
 ends of the beam. Let the greatest pressure on the piston 
 be 3000 Sbs., its greatest velocity 3 feet per second, and the 
 whole length 12 feet. 
 By the rule (art. 335), 
 
 F V 3000 x 3 
 
 = l d = 70 mches ' 
 
 337. Case 2. When a parabolic beam rests upon a centre 
 of motion, and a moving force acts at one end, and is opposed 
 by a greater resistance at the other. 
 
 By art. 153,/= 212, and by art. 227, 
 
 ^ = -16 (1 + r) . 
 
 also, 
 hence, 
 
 F V F V F V 
 
 = I d. 
 
 I V 32 x 212 x -853 x -16 (1 + r) 8-82ZV1 
 12 
 
 338. Rule. Make P the force in pounds, and V its 
 velocity in feet per second, / == the length in feet, from the 
 centre of motion to the point where the force acts, and /' the 
 length from the centre of motion to the resisted point ; also, 
 make b and d the breadth and depth in inches ; then 
 
 ' FV 
 
 339. If / ===== /', that is, when the centre of motion is in the 
 middle of the beam, 
 
 FV 
 
 340. In a steam engine the weight of the connecting 
 pparatus, the power applied to the air-pump, and the weight 
 
SECT. XL] RESISTANCE TO IMPULSION. 207 
 
 of the catch-pins, should be allowed for ; and when the engine 
 moves machinery, the beam should not be less than is deter- 
 mined by this rule. The depth of the beam is usually the 
 same as the diameter of the steam piston. 
 
 341. Example. If the pressure on the piston of a steam 
 engine be 15,000 fts. the whole length of the beam 24 feet, 
 and its velocity 3 feet per second, required the area of the 
 beam ? 
 
 In this case, 
 
 F V 15000 x 3 
 
 If the beam be made 48 inches deep, it should be 6^ inches 
 in breadth ; and the best method of forming such a beam is 
 to make it in two parts, each 3^ in breadth, placed at 12 or 
 14 inches apart, and well connected together. This arrange- 
 ment causes an engine to work with more steadiness, and the 
 parts are less troublesome to move and fix in their places than 
 a single mass would be. 
 
 342. Prop. iv. To determine a rule for finding the area 
 of the middle section of a beam of uniform breadth, the depth 
 at the end being half the depth in the middle, and the middle 
 of the depth open, to resist a moving force. 
 
 Let the parts be so arranged fhat the centre of gravity 
 may be considered to be at the middle of the length of the 
 arm of the beam, which will be very nearly true in practice, 
 and will render the computation somewhat easier. 
 
 343. Case 1. When an open beam is supported at the 
 ends, and the force is applied in the middle of the length. 
 
 By art. 200,/= 850 (1 / 3 ), and by art, 231, 
 
 W-wmtt 
 
 12 
 
 also, we have 
 
208 EES1STANCE TO IMPULSION. [SECT. xr. 
 
 and the Equation (art. 312), 
 
 FV FV 
 
 ~~ I ->/32| x 850 (1 p' 3 ) x -002725 x 1-6(1 p 7 ) ~ 
 FV 
 
 10-92 I V(l-2>' 3 ) x (1 JP') 
 
 344. Rule. Make F the force in pounds, V its velocity in 
 feet per second, / the whole length between the supports in 
 feet, p' that number which would be produced by dividing 
 the depth of the part left out in the middle, by the whole 
 depth ; (if this ratio were not fixed, the solution could not be 
 effected ;) and b and d the breadth and depth in the middle 
 in inches ; then 
 
 FV 
 
 t =~bd. 
 
 345. If p be made = *7, which is a convenient propor- 
 tion, then 
 
 346. Case 2. When an open beam is supported on a 
 centre of motion, and the moving force acts at one end, and 
 the resistance at the other. 
 
 By the same method as above we find 
 
 FV 
 
 10'92 I V (1 P 3 ) x (1 p) x (1 + r) ~ 
 
 347. Rule. Make F = the force in pounds, V its velocity 
 in feet per second, / the length from the point where the force 
 acts to the centre of motion in feet and I' the length from the 
 centre of motion to the point of resistance, b and d the 
 breadth and depth in inches in the middle of the beam, and 
 p the number arising from the division of the depth of the 
 part left out in the middle by the whole depth ; then, j= r, 
 and 
 
SECT. xi. J BESISTANCE TO IMPULSION. 209 
 
 FV 
 
 10-92 I V (1 -p 3 ) x (1 2>) x (1 + r) ' 
 
 348. If p = '7, the equation reduces to 
 
 F V 
 
 4-85 J. VTT7- 
 
 349. Also, when the centre of motion is in the middle of 
 the beam, and/? = *7, we have 
 
 F v 
 
 350. Example. As an example to the equation in the last 
 article, let us suppose the pressure on the piston of a steam 
 engine to be 15,000fts., its velocity 3 feet per second, and 
 the whole length of the beam 24 feet, which is the same as 
 the example (art. 341). In this case 
 
 And, if the depth be made 48 inches, then 
 
 inches the breadth, which is to be the same throughout the 
 length. The bulk of the metal in the upper and lower part 
 of the beam will be found by multiplying the depth by *7 ; 
 that is, '7 x 48 = 33'6 ; which, deducted from 48, leaves 
 14*4 inches, or 7 '2 inches for each side. 
 
 Fig. 34, Plate IV., shows a sketch for a beam of this kind, 
 drawn according to these proportions. 
 
 351. Any of the rules of this, or of the preceding Section, 
 may be applied to other materials by substituting the proper 
 values of the cohesive force, extensibility, and density ; these 
 are given for several kinds in the following Table. 
 
TABLE OF DATA, &c. 
 
 USEFUL IN VARIOUS CALCULATIONS; 
 
 ARRANGED ALPHABETICALLY. 
 
 The data correspond to the mean temperature and pressure of the atmosp/ficre, dry 
 materials ; and the temperature is measured by FahrenJieifs scale. 
 
 Am. Specific gravity 0-0012 ; weight of a cubic foot 0-0753 ft., or 527 
 grains (SHUCKBURGH) ; 13 '3 cubic feet or 17 cylindric feet of air weigh 
 
 1 Ib. ; it expands ~ Q or -00208 of its bulk at 32 by the addition of 
 one degree of heat (DuLONG and PETIT). 
 
 ASH. Specific gravity 0'76 , weight of a cubic foot 47'5Ibs. ; weight of 
 a bar 1 foot long and 1 inch square 0'33Bx ; will bear without 
 permanent alteration, a strain of 3540 Ibs. upon a square inch, and 
 
 an extension of ^ of its length ; weight of modulus of elasticity 
 for a base of an inch square 1,640,000 Ibs. ; height of modulus of 
 elasticity 4,970,000 feet ; modulus of resilience 7'6 ; specific resi- 
 lience 10. (Calculated from BARLOW'S Experiments.) 
 
 Compared with cast iron as unity, its strength is 0'23 ; its exten- 
 sibility 2-6 ; and its stiffness 0-089. 
 
 ATMOSPHERE. Mean pressure of, at London, 28-89 inches of mercury = 
 14-181)8. upon a square inch. (ROYAL SOCIETY.) The pressure of 
 the atmosphere is usually estimated at 30 inches of mercury, which 
 is very nearly 14| Ibs. upon a square inch, and equivalent to a 
 column of water 34 feet high. 
 
 BEECH. Specific gravity 0'696 ; weight of a cubic foot 45 -3 Ibs. ; weight 
 of a bar 1 foot long and 1 inch square 0-315 Ib. will bear without 
 permanent alteration on a square inch 2360 Ibs., and an extension of 
 g^j of its length ; weight of modulus of elasticity^ for a base of 
 an inch square 1,345,000 Ibs. ; height of modulus of elasticity 
 4,600,000 feet ; modulus of resilience 4*14 ; specific resilience 6. 
 (Calculated from BARLOW'S Experiments.) 
 
 Compared with cast iron as unity, its strength is 0-15; its 
 extensibility 2-1 ; and its stiffness 0-073. 
 
PROPERTIES OF MATERIALS. 211 
 
 BRASS, cast. Specific gravity 8*37 ; weight of a cubic foot 523 ft>s. ; 
 weight of a bar 1 foot long and 1 inch square 3 -63 Ibs. ; expands 
 
 93800 f its l en g tn by one degree of heat (TROUGHTON) ; melts at 
 1869 (DANIELL); cohesive force of a square inch 18,000 Ibs. 
 (RENNIE) ; will bear on a square inch without permanent alteration 
 6700 Ibs., and an extension in length of ^ ; weight of modulus of 
 elasticity for a base of an inch square 8,930,000 Ibs. ; height of 
 modulus of elasticity 2,460,000 feet; modulus of resilience 5; 
 specific resilience 0-6 (TREDGOLD). 
 
 Compared with cast iron as unity, its strength is 0435; its 
 extensibility 0'9 ; and its stiffness 0'49. 
 
 BRICK. Specific gravity 1-841 ; weight of a cubic foot 115 Ibs. ; absorbs 
 j of its weight of water ; cohesive force of a square inch 275 t>s. 
 (TREDGOLD); is crushed by a force of 562 Ibs. on a square inch 
 (RENNIE). 
 
 BRICK- WORK. Weight of a cubic foot of newly built, 1175>s. ; weight 
 of a rod of new brick- work 16 tons. 
 
 BRIDGES. When a bridge is covered with people, it is about equivalent 
 to a load of 120 Ss. on a superficial foot ; and this may be esteemed 
 the greatest possible extraneous load that can be collected on a 
 bridge; while one incapable of supporting this load cannot be 
 deemed safe. 
 
 BRONZE. See Gun-metal. 
 
 CAST IRON. Specific gravity 7'207 ; weight of a cubic foot 450 Rs. ; a 
 bar 1 foot long and 1 inch square weighs 3-2 Ibs. nearly ; it expands 
 
 102000 of its l en gth by one degree of heat (Roy) ; greatest change 
 of length in the shade in this climate ~ ; greatest change of 
 length exposed to the sun's rays ^ ; melts at 3479 (DANIELL), 
 and shrinks in cooling from ^ to ^ of its length (MUSCHET) ; is 
 crushed by a force of 93,000 ft>s. upon a square inch (RENNIE) ; will 
 bear without permanent alteration 15,300 fts.* upon a square inch, 
 and an extension of .-^ of its length; weight of modulus of 
 elasticity for a base of an inch square 18,400,000 Ibs. ; height of 
 modulus of elasticity 5,750,000 feet ; modulus of resilience 127 : 
 specific resilience 1-76 (TREDGOLD). 
 
 CHALK. Specific gravity 2-315; weight of a cubic foot 144 -7 Ibs. ; is 
 crushed by a force of 500 t>s. on a square inch. (RENNIE.) 
 
 CLAY. Specific gravity 2-0 ; weight of a cubic foot 125 Ibs. 
 
 * See note to art. 143. EDITOR. 
 
 p2 
 
212 PROPERTIES OF MATERIALS. 
 
 COAL. Newcastle. Specific gravity 1-269 ; weight of a cubic foot 
 79-31 Ebs. A London chaldron of 36 bushels weighs about 28 cwt., 
 whence a bushel is 87 Ibs. (but is usually rated at 84 Ibs.) A New- 
 castle chaldron, 53 cwt. (SMEATON.) 
 
 COPPER. Specific gravity 8 *7 5 (HATCHETT) ; weight of a cubic foot 
 549 K>s. ; weight of a bar 1 foot long and 1 inch square 3 '81 Ibs. ; 
 expands in length by one degree of heat ^^ (SMEATON) ; melts at 
 2548 (DANIELL) ; cohesive force of a square inch, when hammered, 
 33,000 Ibs. (RENNIE). 
 
 EARTH, common. Specific gravity 1'52 to 2 -00 ; weight of a cubic foot 
 from 95 to 125 Ibs. 
 
 ELM. Specific gravity 0*544 ; weight of a cubic foot 34Ibs. ; weight of 
 a bar 1 foot long and 1 inch square 0*236 Ibs. ; will bear on a square 
 inch without permanent alteration 3240 Ebs., and an extension in 
 length of ^ ; weight of modulus of elasticity for a base of an 
 inch square 1,340,000 Ibs. ; height of modulus of elasticity 5,680,000 
 feet ; modulus of resilience 7 '87 ; specific resilience 14 -4. (Calcu- 
 lated from BARLOW'S Experiments.) 
 
 Compared with cast iron as unity, its strength is 0*21 ; its exten- 
 sibility 2-9 ; and its stiffness 0*073. 
 
 FIR, red or yellow. Specific gravity 0'557 ; weight of a cubic foot 
 34*85)8. ; weight of a bar 1 foot long and 1 inch square 0*242 lb. ; 
 will bear on a square inch without permanent alteration 4290 Ibs., 
 
 = 2 tons nearly, and an extension in length of ^ ; weight of 
 modulus of elasticity for a base of an inch square 2,016,000 Ibs. ; 
 height of modulus of elasticity 8,330,000 feet ; modulus of resi- 
 lience 9*13 ; specific resilience 16*4. (TREDGOLD.) 
 
 Compared with cast iron as unity, its strength is 0*3 ; its exten- 
 sibility 2*6 ; and its stiffness 0*1154, = ~ & . 
 
 FIR, white. Specific gravity 0*47 j weight of a cubic foot 29*3 Jbs. ; 
 weight of a bar 1 foot long and 1 inch square 0*204 lb. ; will bear 
 on a square inch without permanent alteration 3630 Ibs., and an 
 extension in length of ^ ; weight of modulus of elasticity for a 
 base of an inch square 1, 830.000 Ibs. ; height of modulus of elas- 
 ticity 8,970,000 feet ; modulus of resilience 7*2 ; specific resilience 
 15-3. (TREDGOLD.) 
 
 Compared with cast iron as unity, its strength is 0*23 ; its exten- 
 sibility 2*4 : and its stiffness O'l. 
 
 FLOORS. The weight of a superficial foot of a floor is about 40 Ibs. 
 when there is a ceiling, counter-floor, and iron girders. When a 
 floor is covered with people, the load upon a superficial foot may be 
 calculated at 120 Ibs. Therefore 120 + 40 = 160 Ibs. on a super- 
 
PKOPERTIES OF MATERIALS. 213 
 
 ficial foot is the least stress that ought to be taken in estimating 
 
 the strength for the parts of a floor of a room. 
 FORCE. See Gravity, Horses, &c. 
 GKANITE, Aberdeen. Specific gravity 2-625 ; weight of a cubic foot 
 
 164 Ibs. ; is crushed by a force of 10,910ft>s. upon a square inch. 
 
 (RENNIE.) 
 
 GRAVEL. Weight of a cubic foot about 120Ibs . 
 GRAVITY, generates a velocity 32-J- feet in a second, in a body falling from 
 
 rest ; space described in the first second 1 6^ feet. 
 GUN-METAL, cast (copper 8 parts, tin 1). Specific gravity 8-153 ; weight 
 
 of a cubic foot 509 J Ibs. ; weight of a bar 1 foot long and 1 inch 
 
 square 3 '54 Ibs. (TREDGOLD) ; expands in length by 1 of heat ^^ 
 (SMEATON) : will bear on a square inch without permanent altera- 
 tion 10,000 Ibs., and an extension in length of ggo ; weight of 
 modulus of elasticity for a base of an inch square 9,873,000 Ibs. : 
 height of modulus of elasticity 2,790,000 feet ; modulus of resi- 
 lience, and specific resilience, not determined. (TREDGOLD.) 
 
 Compared with cast iron as unity, its strength is 0'65 ; its exten- 
 sibility 1-25 ; and its stiffness 0-535. 
 
 HORSE, of average power, produces the greatest effect in drawing a load 
 when exerting a force of 187J Ibs. with a velocity of 2J feet per 
 second, working 8 hours in a day.* (TREDGOLD.) A good horse 
 can exert a force of 480 Ibs. for a short time. (DESAGULIERS.) In 
 calculating the strength for horse machinery, the horse's power 
 should be considered 400 Ibs. 
 
 IRON, cast. See Cast Iron. 
 
 IRON, malleable. Specific gravity 7 '6 (MUSCHENBROEK) ; weight of a 
 cubic foot 475 Ibs. ; weight of a bar 1 foot long and 1 inch square 
 3-3 Ibs. ; ditto, when hammered, 3 -4 Ibs. ; expands in length by 1 of 
 heat 143000 (SMEATON) ; good English iron will bear on a square inch 
 without permanent alteration 17,800 Ibs.,t = 8 tons nearly, and an 
 extension in length of ; cohesive force diminished g^ by an 
 elevation 1 of temperature ; weight of modulus of elasticity for a 
 base of an inch square 24,920,000 K>s. j height of modulus of 
 elasticity 7,550,000 feet ; modulus of resilience, and specific resi- 
 lience, not determined (TREDGOLD). 
 
 Compared with cast iron as unity, its strength is T12 ; its exten- 
 sibility 0-86 : and its stiffness 1'3. 
 
 LARCH. Specific gravity -560 j weight of a cubic foot 35 Ibs. ; weight of 
 
 * This is equivalent to raising 3 cubic feet of water 2 feet per second, or 7 cubic feet 
 1 foot per second. See Buchanan's Essays, 3rd edition, by Mr, Eennie, page 88. 
 t Equivalent to a height of 5000 feet of the same matter. 
 
214 PROPERTIES OF MATERIALS. 
 
 a bar 1 foot long and 1 inch square 0*24 3 Ib. ; will bear on a square 
 inch without permanent alteration 2065 Ibs., and an extension in 
 
 length of 20 ') weight of modulus of elasticity for a base of an inch 
 square 10,074,000 Ibs. ; height of modulus of elasticity 4,415,000 
 feet; modulus of resilience 4 ; specific resilience 7 -1. (Calculated 
 from BARLOW'S experiments.) 
 
 Compared with cast iron as unity, its strength is 0*136 ; its 
 extensibility 2*3 ; and its stiffness 0*058.* 
 
 LEAD, cast. Specific gravity 11*353 (BRISSON) ; weight of a cubic foot 
 709 '5 Ibs. ; weight of a bar 1 foot long and 1 inch square 4*94 Ibs. ; 
 
 expands in length by 1 of heat -^^ (SMEATON) ; melts at 612 
 (CRTCHTON) ; will bear on a square inch without permanent altera- 
 tion 1500 Ifes., and an extension in length of -^ ; weight of modulus 
 of elasticity for a base of an inch square 720,000 Ibs. ; height of 
 modulus of elasticity 146,000 feet ; modulus of resilience 3*12 ; 
 specific resilience 0*27 (TREDGOLD). 
 
 Compared with cast iron as unity, its strength is 0*096 ; its 
 extensibility 2*5 ; and its stiffness 0*0385. 
 
 MAHOGANY, Honduras. Specific gravity 0*56 ; weight of a cubic foot 
 35 Ibs. ; weight of a bar 1 foot long and 1 inch square 0*243 Ib. ; 
 will bear on a square inch without permanent alteration 3800 Ibs., 
 
 and an extension in length of ^ ; weight of modulus of elasticity 
 for a base of an inch square 1,596,000 Ibs. ; height of modulus of 
 elasticity 6,570,000 feet; modulus of resilience 9*047; specific 
 resilience 16*1. (TREDGOLD.) 
 
 Compared with cast iron as unity, its strength is 0*24 ; its exten- 
 sibility 2*9 ; and its stiffness 0*487. 
 
 MAN. A man of average power produces the greatest effect when 
 exerting a force of 31^ l>s. with a velocity of 2 feet per second, for 
 10 hours in a day.t (TREDGOLD.) A strong man will raise and 
 carry from 250 to 300 Ibs. (DESAGULIERS.) 
 
 MARBLE, white. Specific gravity 2*706 ; weight of a cubic foot 169 Ibs. ; 
 weight of a bar 1 foot long and 1 inch square 1*1? Ibs. ; cohesive 
 
 force of a square inch 1811 Ibs. ; extensibility^ of its length; 
 weight of modulus of elasticity for a base of an inch square 
 2,520,000 Ibs. ; height of modulus of elasticity 2,150,000 feet; 
 
 * The mean of my trials on specimens of very different qualities places the strength 
 and stiffness of Larch much higher on the scale of comparison ; but I had not observed the 
 point where it loses elastic power. 
 
 t This is equivalent to half a cubic foot of water raised 2 feet per second ; or 1 cubic 
 foot of water 1 foot per second. See Buchanan's Essays, p. 92, edited by Mr. Rennie. ' 
 
PROPERTIES OF MATERIALS. 215 
 
 modulus of resilience at the point of fracture 1 -3 ; specific resilience 
 at the point of fracture '48 (TREDGOLD) ; is crushed by a force of 
 6060 5>s. upon a square inch (RENNIE). 
 MERCURY. Specific gravity 13*568 (BRISSON) ; weight of a cubic inch 
 
 0-4948 S>. ; expands in bulk by 1 of heat ^ (DULONG and PETIT) ; 
 weight of modulus of elasticity for a base of an inch square 
 4,417,000 Bis. ; height of modulus of elasticity 750,000 feet (Dr. 
 YOUNG from CANTON'S Experiments). 
 
 OAK, good English. Specific gravity 0'83 ; weight of a cubic foot 52 Ibs. ; 
 weight of a bar 1 foot long and 1 inch square 0"36 ft>. ; will bear 
 upon a square inch without permanent alteration 3960 Ibs. ; and an 
 
 extension in length of ^ ; weight of modulus of elasticity for a 
 base of an inch square 1,7 00,000 Ibs. ; height of modulus of elas- 
 ticity 4,730,000 feet ; modulus of resilience 9 -2; specific resilience 
 11. (TREDGOLD.) 
 
 Compared with cast iron as unity, its strength is 0*25 ; its exten- 
 sibility 2-8 ; and its stiffness 0-093. 
 
 PENDULUM. Length of pendulum to vibrate seconds in the latitude of 
 London 39 '1372 inches (KATER) ; ditto to vibrate half ^seconds 
 9-7843 inches. 
 
 PINE, American, yellow. Specific gravity 0'46 ; weight of a cubic foot 
 26| Ibs. weight of a bar 1 foot long and 1 inch square 0'186Ib. ; 
 will bear on a square inch without permanent alteration 3900 Ibs., 
 
 and an extension in length of j^ ; weight of modulus of elasticity 
 for a base of an inch square 1,600,000 Ibs. \ height of modulus of 
 elasticity 8,700,000 feet; modulus of resilience 9 -4 : specific resi- 
 lience 20. (TREDGOLD.) 
 
 Compared with cast iron as unity, its strength is 0-25 ; its exten- 
 sibility 2-9 and its stiffness 0-087. 
 
 PORPHYRY, red. Specific gravity 2-871 : weight of a cubic foot 179 K>s.; 
 is crushed by a force of 35,568 Ibs. upon a square inch. 
 (GAUTHEY.) 
 
 ROPE, hempen. Weight of a common rope 1 foot long and 1 inch in 
 circumference from 0-04 to O f 046Ib. ; and a rope of this size should 
 not be exposed to a strain greater than 200 Ibs. ; but in compounded 
 ropes, such as cables, the greatest strain should not exceed 120 Ibs.;* 
 
 * The square of the circumference in inches multiplied by 200 will give the number of 
 pounds a rope may be loaded with, and multiply by 120 instead of 200 for cables. 
 Common ropes will bear a greater load with safety after they have been some time in use, 
 in consequence of the tension of the fibres becoming equalized by repeated stretchings and 
 partial untwisting. It has been imagined that the improved strength was gained by their 
 being laid up in store ; but if they can there be preserved from deterioration, it is as 
 much as can be expected. 
 
216 PROPERTIES OF MATERIALS. 
 
 and the weight of a cable 1 foot in length and 1 inch in circumfe- 
 rence does not exceed 0*027 Ib. 
 
 ROOFS. Weight of a square foot of Welsh rag slating 11^ Ibs. ; weight 
 of a square foot of plain tiling 16|-K>s. ; greatest force of the wind 
 upon a superficial foot of roofing may be estimated at 40 Ibs. 
 
 SLATE, Welsh. Specific gravity 2752 (KIRWAN) ; weight of a cubic foot 
 172 K>s. ; weight of a bar 1 foot long and 1 inch square 1*1 9 Ibs. ; 
 cohesive force of a square inch 11, 5 00 Ibs. ; extension before frac- 
 ture 1370 ; weight of modulus of elasticity for a base of an inch 
 square 15,800,000 Ibs. ; height of modulus of elasticity 13,240,000 
 feet ; modulus of resilience 8'4 ; specific resilience 2. (TREDGOLD.) 
 
 SLATE, Westmoreland. Cohesive force of a square inch 7870 Ibs. ; exten- 
 sion in length before fracture ^ ; weight of modulus of elasticity 
 for a base of an inch square 12,900,000 Ibs. (TREDGOLD.) 
 
 SLATE, Scotch. Cohesive force of a square inch 9600 Ibs. ; extension in 
 
 length before fracture j ; weight of modulus of elasticity for a 
 base of 'an inch square 15,790,000 ft>s. (TREDGOLD.) 
 STEAM. Specific gravity at 212 is to that of air at the mean tempera- 
 ture as 0-472 is to 1 (THOMSON) ; weight of a cubic foot 249 grains; 
 modulus of elasticity for a base of an inch square 1 4| Ibs. ; when 
 
 not in contact with water, expands of its bulk by 1 of heat. 
 (GAY LUSSAC.) 
 
 STEEL. Specific gravity 7 '84 ; weight of a cubic foot 490 Ebs. ; a bar 
 1 foot long and 1 inch square weighs 3*4 Ibs. ; it expands in length 
 
 by 1 of heat ' l[)7m (RoY) j tempered steel will bear without per- 
 manent alteration 45,000 Ibs. ; cohesive force of a square inch 
 130,000 Ibs. (RENNIE); cohesive force diminished g^ by elevating 
 the temperature 1 ; modulus of elasticity for a base of an inch 
 square 29,000,000 t>s. ; height of modulus of elasticity 8,530,000 
 feet (Dr. YOUNG). 
 
 STONE, Portland. Specific gravity 2-113 ; weight of a cubic foot 132 Ibs. ; 
 weight of a prism 1 inch square and 1 foot long 0'92 Ib. ; absorbs 
 
 ig of its weight of water (R. TREDGOLD) ; is crushed by a force of 
 3729 Bbs. upon a square inch (RENNIE) ; cohesive force of a square 
 inch 857 Ibs. ; extends before fracture -^ of its length ; modulus 
 of elasticity for a base of an inch square 1,533,000 Ibs. ; height of 
 modulus of elasticity 1,672,000 feet ; * modulus of resilience at the 
 
 * In the stones, the modulus here given is calculated from the flexure at the time of 
 fracture ; when it is taken for the first degrees of flexure, it is a little greater. The 
 experiments are described in the Philosophical Magazine, vol. Ivi. p. 290. 
 
PROPERTIES OF MATERIALS. 217 
 
 point of fracture 0*5 ; specific resilience at the point of fracture 
 0-23 (TREDGOLD). 
 STONE, Bath. Specific gravity 1-975 ; weight of a cubic foot 123-4 Ibs. ; 
 
 absorbs ^ of its weight of water (R TREDGOLD) ; cohesive force of 
 a square inch 478 Ibs. (TREDGOLD). 
 STONE, Craigleith. Specific gravity 2-362 ; weight of a cubic foot 
 
 147 '6 Ibs. ; absorbs 6 - 3 of its weight of water; cohesive force of a 
 square inch 772 Ibs. (TREDGOLD) ; is crushed by a force of 5490 5>s. 
 upon a square inch (RENNIE). 
 STONE, Dundee. Specific gravity 2-621 ; weight of a cubic foot 163-8 Ibs. ; 
 
 absorbs ^ part of its weight of water ; cohesive force of a square 
 
 inch 2661 Ibs. (TREDGOLD); is crushed by a force of 6630 Ibs. upon 
 
 a square inch (RENNIE). 
 STONE-WORK. Weight of a cubic foot of rubble-work about 140 Ibs. ; 
 
 of hewn stone 160 Ibs. 
 TIN, cast. Specific gravity 7 -2 91 (BRISSON) ; weight of a cubic foot 
 
 455 '7 Ibs. ; weight of a bar 1 foot long and 1 inch square 3 -165 Ibs.; 
 
 expands in length by 1 of heat -^ (SMEATON) ; melts at 442 
 (CRICHTON) ; will bear upon a square inch without permanent altera- 
 tion 2880 Ibs., and an extension in length of ^ ; modulus of 
 elasticity for a base of an inch square 4,608,000 Ibs.; height of 
 modulus of elasticity 1,453,000 feet; modulus of resilience 1*8; 
 specific resilience 0-247 (TREDGOLD). 
 
 Compared with cast iron as unity, its strength is 0*182 ; its exten- 
 sibility 0-75 ; and its stiffness 0*25. 
 
 WATER, river. Specific gravity 1*000 ; weight of a cubic foot 62 -5 Ibs. ; 
 weight of a cubic inch 252*525 grains; weight of a prism 1 foot 
 long and 1 inch square 0*434 Ib. ; weight of an ale gallon of water 
 
 10 -2 Ibs. ; expands in bulk by 1 of heat (DALTON) ;* expands 
 in freezing ^ of its bulk (WILLIAMS) ; and the expanding force of 
 freezing water is about 35,000 Ibs. upon a square inch, according to 
 Muschenbroek's valuation ; modulus of elasticity for a base of an 
 inch square 326,000 5>s. ; height of modulus of elasticity 750,000 
 feet, or 22,100 atmospheres of 30 inches of mercury (Dr. YOUNG, 
 from CANTON'S Experiments). 
 WATER, sea. Specific gravity 1-0271 ; weight of a cubic foot 64*2 Ibs. 
 
 * Water has a state of maximum density at or near 40, which is considered an excep- 
 tion to the general law of expansion by heat : it is extremely improbable that there is 
 anything more than an apparent exception, most likely arising from water at low tem- 
 peratures absorbing a considerable quantity of air, which has the effect of expanding it ; 
 and consequently of causing the apparent anomaly. 
 
218 
 
 PROPERTIES OF MATERIALS. 
 
 WATER is 828 times the density of air of the temperature 60, and 
 
 barometer 30. 
 WHALE-BONE. Specific gravity 1-3 ; weight of a cubic foot 81 Ibs. ; will 
 
 bear a strain of 5600 Ibs. upon a square inch without permanent 
 
 alteration, and an extension in length of ^j 6 ; modulus of elasticity 
 for a base of an inch square 820,000 Ibs. ; height of modulus of 
 elasticity 1,458,000 feet ; modulus of resilience 38*3 ; specific 
 resilience 29. (TREDGOLD.) 
 
 WIND. Greatest observed velocity 159 feet per second (ROCHON) ; force 
 of wind with that velocity about 57 Jibs, on the square foot.* 
 
 ZINC, cast. Specific gravity 7*028 (WATSON); weight of a cubic foot 
 439| 5>s. ; weight of a bar 1 inch square and 1 foot long 3*05 5>s. ; 
 
 expands in length by 1 of heat ^oo (SMEATON) ; melts at 648 
 (DANIELL) ; will bear on a square inch without permanent alteration 
 5700 Ibs. = 0*365 cast iron, and an extension in length of ^ = J 
 that of cast iron (TREDGOLD) ; t modulus of elasticity for a base of 
 an inch square 13,680,000 Ibs. ; height of modulus of elasticity 
 4,480,000 feet ; modulus of resilience 2*4 ; specific resilience 0*34. 
 (TREDGOLD.) 
 
 Compared with cast iron as unity, its strength is 0*365 ; its 
 extensibility 0-5 ; and its stiffness 0'76. 
 
 * Table of the force of winds, formed from the Tables of Mr. Rouse and Dr, Lind, and 
 compared with the observations of Colonel Beaufoy. 
 
 Velocity 
 in miles 
 per hour. 
 
 A wind may be denominated when it does 
 not exceed the velocity opposite to it. 
 
 Velocity 
 per 
 second. 
 
 Force on a 
 square foot. 
 
 
 
 feet. 
 
 fts. 
 
 6-8 
 
 A gentle pleasant wind 
 
 10 
 
 0-229 
 
 13-6 
 
 A brisk gale . ; ' . 
 
 20 
 
 0-915 
 
 19-5 
 
 A very brisk gale 
 
 30 
 
 2-059 
 
 34-1 
 
 A high wind . , 
 
 50 
 
 5-718 
 
 47-7 
 
 A very high wind . ' . - '. 
 
 70 
 
 11-207 
 
 54-5 
 
 A storm or tempest -.- ^ . 
 
 80 
 
 14-638 
 
 68-2 
 
 A great storm . 
 
 100 
 
 22-872 
 
 81-8 
 
 A hurricane . ' ' . . 
 
 120 
 
 32-926 
 
 102-3 
 
 f A violent hurricane, that tears up "1 
 \ trees, overturns buildings, &c. . J 
 
 150 
 
 51-426 
 
 * Accurate observations on the variation and mean intensity of the force of winds would 
 be very desirable both to the mechanician and meteorologist. 
 
 t The fracture of zinc is very beautiful ; it is radiated, and preserves its lustre a long 
 time. 
 
NOTE ON THE ACTION OF CERTAIN SUBSTANCES ON 
 
 CAST IRON. 
 
 IN some circumstances cast iron will decompose, and be converted into a soft 
 substance resembling plumbago. A few instances of this kind I add here, as they 
 will be interesting to persons who employ iron for various purposes. 
 
 Dr. Henry observed that when cast iron was left in contact with muriate of lime, 
 or muriate of "magnesia, most of the iron was removed, the specific gravity of the 
 mass was reduced to 2*155, and what remained consisted chiefly of plumbago, and the 
 impurities usually found in cast iron.* 
 
 A similar change was produced in some cast iron cylinders used to apply the 
 weaver's dressing to cloth : this dressing is a kind of paste, made of wheat or barley- 
 meal. The corrosion of the cylinders took place repeatedly, and was so rapid that it 
 was found necessary to use wooden ones. Dr. Thomson ascribes the change to the 
 acid formed by the paste turning sour, f 
 
 Another instance of greater importance has been recorded by Mr. Brande. A portion 
 of a cast iron gun had undergone a like change from being long immersed in sea- water. 
 To the depth of an inch it was converted into a substance having all the external 
 characters of plumbago ; it was easily cut, greasy to the feel, and made a black streak 
 upon paper. J The component parts of this substance were in the ratio of 
 
 Oxide of iron , . .81 
 Plumbago . : . . .16 
 
 97 
 
 Mr. Brande qould not detect any silica in it ; and remarks, that anchors and other 
 articles of wrought iron, when similarly exposed, are only superficially oxidized, and 
 exhibit no other peculiar appearance. 
 
 Near the town of Newhaven, in America, a cannon ball was discovered, which it was 
 ascertained had lain undisturbed about forty-two years in ground kept constantly 
 moist by sea-water : the diameter of the ball was 3 '87 inches, and with a common 
 saw a section was easily made through a coat of plumbaginous matter, which at the 
 place of incision was half an inch thick ; but its thickness varied in different places. 
 The plumbago cut with the same ease, gave the same streak to paper, and had in 
 every respect the properties of common black lead. 
 
 A cannon ball had undergone a similar change, which was taken from the wreck of 
 a vessel that appeared to have been many years under water ; the ball was covered 
 by oysters firmly adhering to it, and its external part was converted into plumbago. 
 
 * Dr. Thomson's Annals of Philosophy, vol. v, p, 66. 
 
 f Idem, vol. x. p. 302. 
 
 Quarterly Journal of Science, vol. xii. p, 407. 
 
220 NOTE. 
 
 But an old cannon found covered with oysters did not, on the removal of its coating, 
 show any signs of such conversion.* 
 
 The reader who wishes to pursue this interesting subject may consult an article 
 " On the Mechanical Structure of Iron developed by Solution," &c., by Mr. Daniell,t 
 who has made several experiments with a view to determine the nature of the 
 substance resembling plumbago, which is found on the surface of iron after it has 
 been exposed to the action of an acid. 
 
 Mr. Daniell found that the structure of iron, as developed by solution, was very 
 different in different kinds ; and that it required three times as long to saturate a 
 given portion of acid when it acted on white cast iron, as when it acted on the gray 
 kind. 
 
 * Phillips's Annals of Philosophy, vol. iv. p. 77. 1822. 
 
 f Quarterly Journal of Science, vol. ii. p. 278. Much additional information on the 
 effect of water upon iron will be obtained from the Report of Mr. Mallet "upon the Action 
 of Sea and River Water, whether clear or foul, and at various temperatures, upon Cast 
 and Wrought Iron." British Association, vols. vii. and viii. EDITOR. 
 
EXPLANATION OF THE PLATES. 
 
 PLATE I. 
 
 FIG. 1. A bar supported at the ends, and loaded in the middle of the length. See 
 art. 8. 
 
 FIG. 2. A beam with the load uniformly distributed over the length, as the experi- 
 ment, art. 61, was tried. See art. 20 and 61. 
 
 FIG. 3. The form for a beam of uniform strength to resist the action of a load at C. 
 ACD and BCD are semi-parabolas, A and B being the vertices. The dotted 
 lines show the additions to this form to render it of practical use. See art. 27, 
 123, and 223-229. 
 
 FIG. 4. A form for a beam which is as nearly of uniform resistance as practical con- 
 ditions will admit of : it is bounded by straight lines, and the depths at the 
 ends are each half the depth in the middle. See art. 17 (Ex. 7), 28, 65, 127, 
 and 230-234. 
 
 FIG. 5. A variation of the last form for the case where the force sometimes acts 
 upwards and sometimes downwards. See art. 29, 127, and 230-234. 
 
 FIG. 6. A figure of uniform strength for a beam, when the depth is uniform. See 
 art. 30, 122, and 242-246. 
 
 FIG. 7. A modification of fig. 6, which is the most economical form of equal strength 
 for resistance to pressure. B' is the form of the end. See art. 30. 
 
 FIG. 8. The form of equal resistance for a load rolling along the upper side, as in a 
 railway ; or for a load uniformly distributed over the length. A C B is half an 
 ellipse. The dotted lines show the addition required in practice. See art. 32, 
 125, 240, and 241. 
 
 FIG. 9. The strongest form of section for a beam to resist a cross strain. AM is the 
 line called the neutral axis. See art. 40, 54, 116, 185-197, and 321. 
 
 FIG. 10 shows an application of the section, fig. 9, to form a fire-proof floor, the 
 projection serving the double purpose of giving additional strength, and forming 
 a support for the arches. See art. 40 and 194. 
 
 PLATE II. 
 
 FIG. 11. This is the figure of a very economical beam for supporting a load diffused 
 over its length : it is adapted for girders, beams to support walls, and the like. 
 An easy rule for proportioning girders is given in art. 50. "When this form is 
 used for a girder, the openings answer for inserting cross joists. A B and C D 
 show the sections at these places. See art. 21 (Ex. 12), 41, 43, 117, 198-210, 
 and 323. 
 
222 EXPLANATION OF THE PLATES. 
 
 FIG. 12. This figure shows a beam on the same principle as the preceding figure, 
 except that the load is supposed to act only at one point A. See art. 43, 44, 117, 
 and 198-210. 
 
 FIG. 13. The section of a shaft, commonly called a feathered shaft. See art. 46. 
 
 FIG. 14. A figure to illustrate the action of forces upon a beam, and to explain the 
 mode of calculation. See art. 106, 108, 131, and 154. 
 
 FIG. 15. A section of the beam in fig. 14, at BD. This section is supposed to be 
 divided into thin laminse. See art. 106. 
 
 FIG. 16. A figure to illustrate the method of calculating the deflexion of beams. In 
 these figures (figs. 14 and 16) I have regarded distinctness of the parts referred 
 to more than the true relation of the parts to one another. See art. 120. 
 
 PLATE III. 
 
 FIG. 17 is to illustrate the circumstances which take place in the deflexion of beams 
 fixed at one end. See art. 133 and 154. 
 
 FIG. 18. To explain the mode of calculating the strength of cranks. See art. 135. 
 
 FIG. 19. A figure to explain the manner of estimating the strength and deflexion of 
 a beam supported at the ends. See art. 136, 143, 146, 149, and 165. 
 
 FIG. 20. A figure to show how to calculate the strain upon a beam when a load is 
 distributed in any regular manner over it. The load being uniform, a d is the 
 line which represents its upper surface ; when the load increases as the distance 
 from the end A, c d is the line bounding it ; and when the load Increases as the 
 square of the distance from A, b d is the line bounding it. The second case is 
 the same as the pressure of a fluid against a vertical sheet fixed at the ends. See 
 art. 138-141, and 160. 
 
 FIG. 21. ACB is the form of equal strength for a uniform load : it is in this figure 
 applied to the cantilever of a balcony, and whatever ornamental form may be 
 given to the under side, it should not be cut within the line B C. See art. 34, 
 130, and 157. 
 
 FIG. 22. When the section is of the. form C' D', and the breadth uniform, the figure 
 of equal strength for a load in the middle is formed by two semi-parabolas (as in 
 fig. 3), shown by the dotted lines ; and it may be formed for practical application, 
 as shown in the figure. See art. 187, 223-227. 
 
 FIG. 23 is a figure to illustrate the nature of variable forces. See art. 295. 
 
 FIG. 24. If the section of a beam be C' D', and the breadth uniform, the form of 
 equal strength for a uniform load, as in girders for floors, is a semi-ellipse, 
 shown by the dotted lines ; and also when the load rolls or slides over it ; and it 
 may be formed for practical application, as the figure ; and an easy rule for 
 girders of this kind is given in page 42. See art. 21 (Ex. 13), 188, 193, 240, 
 241, and 322. 
 
 PLATE IV. 
 
 FIG. 25 represents a beam fixed at one end ; a' b' is its section ; the load acting at 
 the end C, the figure of equal resistance is a semi-parabola. See art. 196 and 
 223-229. 
 
 FIG. 26. Form suitable for the beam of a steam-engine to the form of section a' b'. 
 See art. 40, 196, 222, and 223-229. 
 
EXPLANATION OF THE PLATES. 223 
 
 FIG. 27. A sketch for a beam to bear a considerable load distributed uniformly over 
 its length, when the span is so much as to render it necessary to cast it in two 
 pieces. The connexion may be made by a plate of wrought iron on each side at 
 C, with indents to fit the corresponding parts of the beam. Wrought iron should 
 be preferred for the connecting plates, because, being more ductile, it is more 
 safe. See the next figure and art. 198-210. 
 
 FIG. 28 shows the under side of the beam in the preceding figure. The plates are 
 held together by bolts ; but it is intended that the strength should depend on 
 the incidents, the bolts being only to hold them together. No connexion is 
 wanted at the upper side of the beam, except a bolt c d, or like contrivance, to 
 steady it. See art. 199. 
 
 FIG. 29. A figure to explain the nature of the resistance to twisting or torsion. See 
 art. 263. 
 
 FIG. 30. A figure to illustrate the action of the straining force on columns, posts, 
 and the like. See art. 276. 
 
 FIG. 31. To explain the effect of settlement or other derangement of the straining 
 force. See art. 10 and 281. 
 
 FIG. 32. Another case of settlement or derangement of the straining force on a 
 column considered. See art. 283. 
 
 FIG. 33. To show why columns should not be enlarged at the top or bottom. See 
 art. 283. 
 
 FIG. 34. A sketch for an open beam recommended for an engine beam. See art. 
 350. In small beams the middle part may be left wholly open, except at the 
 centre. Capt. Kater has used this form for the beam of a delicate balance. 
 
A LIST OF AUTHORS 
 
 QUOTED IN THE ALPHABETICAL TABLE, WITH THE TITLES OF THE WORKS 
 FROM WHICH THE DATA HAVE BEEN QUOTED. 
 
 BARLOW. Essay on the Strength and Stress of Timber, London, 1817 ; reprinted 
 with many additions in 1837. 
 
 BEAUFOY. Thomson's Annals of Philosophy. 
 
 BRISSON. Dr. Thomson's System of Chemistry, fifth edition, London, 1817. 
 
 CRICHTON. Philosophical Magazine, vol. xv. 
 
 DALTON. Dr. Thompson's System of Chemistry, fifth edition. 
 
 DANIELL. Quarterly Journal of Science, vol. xi. p. 318. 
 
 DESAGULIERS. Course of Experimental Philosophy, London, 1734. 
 
 DULONG and PETIT. Annals of Philosophy, vol. xiii. 
 
 GAUTHEY. Rozier's Journal de Physique, tome iv. p. 406. 
 
 GAY LTJSSAC. Dr. Thomson's System of Chemistry, fifth edition. 
 
 HATCHETT. Dr. Thomson's System of Chemistry, fifth edition. 
 
 KATER. Philosophical Magazine, vol. liii., 1818. 
 
 KIRWAN. Elements of Mineralogy. 
 
 LIND. Dr. Thomas Young's Lectures on Natural Philosophy. 
 
 MUSCHET. Philosophical Magazine, vol. xviii. 
 
 MUSCHENBROEK. Dr. Thomson's System of Chemistry, fifth edition. 
 
 RENNIE. Philosophical Transactions for 1818, Part I. 
 
 EICE. Annals of Philosophy for 1819. 
 
 ROCHON." Dr. Thomas Young's Natural Philosophy, vol. ii. 
 
 ROUSE. Smeaton's Experimental Inquiry of the Power of Wind and Water. 
 
 ROY. Account of the Trigonometrical Survey of England and Wales, vol. i. 
 
 ROYAL SOCIETY. Dr. Thomas Young's Natural Philosophy, vol. ii. 
 
 SHUCKBURGH. Dr. Thomson's System of Chemistry. 
 
 SMEATON. Reports, vol. iii. and Miscellaneous Papers. 
 
 THOMSON. Annals of Philosophy, vol. iii., New Series, 1822. 
 
 R. TREDGOLD. Elementary Principles of Carpentry, London, 1820, of which an 
 enlarged edition has been given by Mr. Barlow in 1840. 
 
 TREDGOLD. Philosophical Magazine, vol. Ivi., 1820 ; Elementary Principles of 
 Carpentry ; Buchanan's Essays on Mill-work, second edition, or third edition 
 by Mr. Rennie in 1841 ; and Experiments of which the details have not been 
 published. ^-- 
 
 TROTJGHTO'N. Dr. Thomas Young's Natural Philosophy, vol. ii. 
 
 WATSON. Chemical Essays. 
 
 WILLIAMS. Dr. Thomas Young's Natural Philosophy, vol. ii. 
 
 YOUNG. Natural Philosophy, vol. ii. 
 
 
Fiq.l 
 
 F iff. 14. Fig. IS. Fig. 16 
 
 Fig. I 
 
 Fio.20. 
 
 Iuf.21. 
 
 J ii 
 
 If 
 
 ;! /r ./. ll'.-',-,fp,iit //,. Architectural. L 
 
Plated 
 
1' I II -It A 
 
 OF 
 
PlaU.Jff. 
 
 F i. 42. 
 
 d 
 
 
 
 f LIE 
 
 UN IV h 
 
Plate IV. 
 
 HC/t.Seei 
 
EXPERIMENTAL RESEARCHES 
 
 ON THE 
 
 STRENGTH AND OTHER PROPERTIES 
 
 OF 
 
 CAST IKON. 
 
 FORMING A SECOND VOLUME TO TREDGOLD'S 
 
 PRACTICAL ESSAY ON THE STRENGTH OF CAST IRON 
 AND OTHER METALS. 
 
 BY EATON HODGKINSON, F.RS. 
 
EXPERIMENTAL RESEARCHES 
 
 ON THE 
 
 STRENGTH AND OTHER PEOPEETIES 
 
 OP 
 
 CAST IRON: 
 
 WITH THE DEVELOPMENT OF NEW PRINCIPLES; 
 CALCULATIONS DEDUCED FROM THEM; 
 
 AND 
 
 INQUIRIES APPLICABLE TO RIGID AND TENACIOUS BODIES 
 GENERALLY. 
 
 BY EATON HODGKINSON, F.B.S. 
 
 LONDON: 
 
 JOHN WEALE, 59, HIGH HOLBORN. 
 1860-1. 
 
 1 4 4 
 
LONDON : 
 BRADBURY AND EVANS, PRINTERS, WHITEFRtARS. 
 
CONTENTS OF PART II. 
 
 PAGE 
 
 INTRODUCTION. Art. 1 , . - . ! U- ' . -'. / . 227 
 
 Tensile Strength of Cast Iron. Art. 2 13 . ; . . '. . . 228 
 Strength of Cast Iron and other Materials to resist Compression. 
 
 Art. 14, 15 . -...,;. -.': ....; * - .; 232 
 
 Resistance of short Masses to a crushing Force. Art. 16 34 . 232 
 
 Strength of long Pillars. Art. 35 44 . ; .: ., . ' . . 240 
 
 Strength of short flexible Pillars. Art. 45 GO . . < <. 248 
 
 Comparative Strength of long similar Pillars. Art. 61 68 . 254 
 On the Strength of Pillars of various Forms, and different Modes 
 
 of fixing. Art. 6973 . ' * > / . ; -. V. ' . . 257 
 Comparative Strengths of long Pillars of Cast Iron, Wrought Iron, 
 
 Steel, and Timber. Art. 74 . -. .-' . . . .; 259 
 
 Power of Pillars to sustain long-continued Pressure. Art. 75 * 259 
 
 Euler's Theory of the Strength of Pillars. Art. 7678 \ > - . 260 
 Results of Experiments on the Resistance of solid uniform Cylin- 
 ders of Cast Iron to a Force of Compression : 
 
 Table I. Solid Columns with rounded Ends V. . 262 
 
 Table II. Solid Columns with flat Ends . . . . 266 
 Table III. Hollow cylindrical Pillars, rounded at the 
 
 Ends 270 
 
 Table IV. Hollow Pillars with flat Ends . ' .^ . . 274 
 
 Transverse Strength. Art. 79 . . & . "... " - \ - ^ - . . 277 
 
 Long-continued Pressure Tipon Bars or Beams. Art. 80 v?> . . 277 
 Table of Experiments by W. Fairbairn, Esq., on the Strength of 
 
 Bars to resist long-continued Pressure. Art. 81 . . .279 
 
 Observations on these Experiments. Art. 82, 83 . . . 280 
 
 Effects of Temperature on the Strength of Cast Iron. Art. 84, 85 280 
 
vi CONTENTS OF PART II. 
 
 PAGE 
 
 On the Strength of Cast Iron Bars or Beams under ordinary cir- 
 cumstances, the Time when the Elasticity becomes impaired, 
 and the erroneous Conclusions that have been derived from a 
 Mistake as to that Time. Art. 8691 , . , . , 282 
 Experiments to determine the Transverse Strength of uniform Bars 
 
 of Cast Iron. Art. 92 - , 288 
 
 Table of Experiments by W. Fairbairn, Esq., on the Strength of 
 uniform rectangular Bars of Cast Iron. Art. 93 : 
 
 English Irons . '.',.-.'.. . . . 290 
 Scotch Irons . . . . ; . . . . 294 
 Welsh Irons '}.'<:; .= ; ,. - ; .-j , *: T T > -, ; > . . 295 
 Welsh Anthracite Irons . . '".' < - ^ ^ ^ 29g 
 Table of Mean Results of Experiments by the Author on the 
 Transverse Strength and Elasticity of uniform Bars of Cast 
 Iron, of different Forms of Section. Art. 94 : 
 
 Rectangular Bars of English Iron . ,*,,... . , 299 
 
 Scotch Iron . .. . f . . . 300 
 
 Bars of Scotch Iron (Carron) . . .*., . . .301 
 Remarks on the preceding Experiments. Art. 95 . . . 302 
 
 Table of Abstract of Results obtained from the whole of the Expe- 
 riments, both of Mr. Fairbairn and the Author . . .304 
 Defect of Elasticity. Art. 99107 , . ; .., . . . 307 
 Of the Section of greatest Strength in Cast Iron Beams. Art. 108 
 
 126 .... ,.* . .. 'fjjjfo.. . Y .., 311 
 
 Experiments to ascertain the best Forms of Cast Iron Beams, and 
 
 the Strength of such Beams. Art. 127 134 . . . . 317 
 Tabulated Results of the preceding Experiments : 
 
 Table I. Art. 135 . . >. . , A . . 320 
 Table II. Art. 136 .'..< ;, ;,' . -' ;.'.' . ^ . . 328 
 Table III. Results of Experiments upon Beams of the 
 
 usual Form. Art. 137 . .* , . 332 
 
 Remarks upon the Experiments in Table I. Art. 138141 . . 334 
 Remarks on Table II. Art. 142, 143 . -* . . . ,. 338 
 Simple Rule for the Strength of Beams. Art. 144146 . -;; . 338 
 Another approximate Rule. Art. 147 152 .* , > .* , ~= 340 
 Experiments on large Beams. Art. 153 . * >. , ,-.,, . . 346 
 Experiments on Beams of different Forms. Art. 154 159. t ; 347 
 Mr. F. Bramah's Experiments on Beams. Art. 160, 161 . . . 350 
 Table of Mr. Bramah's Experiments upon solid Cast Iron Beams. 
 
 Art. 162 352 
 
CONTENTS OF PART II. vii 
 
 PAGE 
 
 Table of Experiments on Beams differing in Section from the 
 
 former only in having a Portion taken away from the Middle. * 
 
 Art. 163 353 
 
 Remarks on Mr. Bramah's Experiments. Art. 164 . . 354 
 
 Mr. Cubitt's Experiments on Beams. Art. 165 167 . . . 354 
 
 Table to Mr. Cubitt's Experiments . . . 356 
 
 Remarks on Mr. Cubitt's Experiments. Art. 168 179 . . . 360 
 
 Comparative Strength of Hot and Cold Blast Iron. Art. 180 183 363 
 Tables of Experiments on Scotch Iron . U ; -. ::> . K . .364 
 General Summary of Transverse Strengths, and computed Powers 
 
 to resist Impact. Art. 184188 . . . . . . 368 
 
 Theoretical Inquiries with regard to the Strength of Beams. Art. 
 
 189194 . . . . t . . . .' i . . .370 
 
 On Resistance to Torsion. Art. 195 199 . . ... 378 
 
 Experiments by Geo. Rennie, Esq., on the Strength of Bars of Cast 
 
 Iron to resist Fracture by Torsion. Art. 200, 201 . . 380 
 Remarks on the Experiments of Messrs. Rennie, Bramah, Dunlop, 
 
 Bevan, Cauchy, and Savant. Art. 202215 . . . 380 
 
PART II. 
 
 EXPERIMENTAL EESEARCHES 
 
 ON THE 
 
 STRENGTH AND OTHER PROPERTIES OF CAST IRON; 
 
 WITH 
 
 THE DEVELOPMENT OF NEW PRINCIPLES ; CALCULATIONS DEDUCED FROM THEM J AND 
 INQUIRIES APPLICABLE TO RIGID AND TENACIOUS BODIES GENERALLY. 
 
 BY EATON HODGKINSON, F.K.S. 
 
LI Bit ART 
 
 { VEHSITY OF 
 
 EXPERIMENTAL RESEARCHES, ETC. 
 
 1. IN the preceding Work the very ingenious Aoithor has 
 confined his reasonings chiefly to the effects produced upon 
 bodies by forces, which were small comparatively to those 
 necessary to produce fracture. In this Second, or Additional 
 Part, I shall generally give the ultimate strength of the bodies 
 experimented upon, and endeavour to show the laws, or 
 illustrate the phenomena, attendant upon fracture. The 
 conclusions in this Part will be drawn from experiments 
 mostly made since Mr. Tredgold wrote his Work upon Cast 
 Iron, at which time there was confessedly a want of experi- 
 mental information upon the subject in this country ; and we 
 were but slightly acquainted with what had been done upon 
 the Continent. Having for many years devoted a portion of 
 my time to experimental research on the strength of materials, 
 in which the expense was borne by my liberal friend William 
 Fairbairn, Esq., whose extensive mechanical establishment, at 
 Manchester, enabled him to offer me every facility for the 
 purpose, I have obtained a large mass of facts on most of the 
 subjects connected with the strength of materials. Mr. 
 Fairbairn has likewise published the results of a great number 
 of well-conducted experiments upon the transverse strength of 
 bars of cast iron. An abstract, therefore, of the leading 
 experiments made at Mr. Fairbairn's Works, and of those 
 given by Navier, Rennie, Bramah, and others, with theoretical 
 considerations, is all that can be attempted in this Additional 
 
 Q 2 
 
228 TENSILE STRENGTH. 
 
 Part; pointing out, as I proceed, whatever has a bearing upon 
 the conclusions of Tredgold in the body of the Work. 
 
 TENSILE STRENGTH OF CAST IRON. 
 
 2. To determine the direct tensile strength of cast iron, I 
 had models made of the form in Plate I. fig. 1. 
 
 The castings from these models were very strong at the 
 ends, in order that they might be perfectly rigid there, and 
 had their transverse section, for about a foot in the middle, of 
 the form in fig. 2. This part, which was weaker than the 
 ends, was intended to be torn asunder by a force acting per- 
 pendicularly through its centre. The ends of the castings 
 had eyes made though them, with a part more prominent 
 than the rest in the middle of the casting, where the eye 
 passed through ; fig. 3 represents a section of the eye. The 
 intention of this was that bolts passing through the eyes, and 
 having shackles attached to them, by which to tear the 
 casting asunder, would rest upon this prominent part in the 
 middle, and therefore upon a point passing in a direct line 
 through the axis of the casting. Several of the castings were 
 torn asunder by the machine for testing iron cables, belonging 
 to the Corporation of Liverpool ; the results from these are 
 marked with an asterisk. Others were made in the same 
 manner, but of smaller transverse area ; these were broken by 
 means of Mr. Pairbairn's lever (Plate II. fig. 40), which was 
 adapted so as to be well suited for the purpose. 
 
 The form of casting here used was chosen to obviate the 
 objections made by Mr. Tredgold (art 79 and 80) and others 
 against the conclusions of former experimenters. The results 
 are as follow : 
 
TENSILE STRENGTH. 
 
 229 
 
 3. Results of Experiments on the Tensile Strength of Cast Iron. 
 
 Description of iron. 
 
 Area of 
 section in 
 inches. 
 
 Breaking 
 weight in 
 
 0)3. 
 
 Strength 
 per sq. in. 
 of section. 
 
 Mean in Ibs. per 
 square inch. 
 
 Carron iron, No. 2, hot blast . 
 
 4-031 
 
 56,000 
 
 13,892*) 
 
 tons. cwt. 
 
 Do. do. do. 
 
 17236 
 
 22,395 
 
 12,993 } 
 
 13,505 = 6 O-i- 
 
 Do. do. do. 
 
 17037 
 
 23,219 
 
 13,629 ) 
 
 
 Carron iron, No. 2, cold blast . 
 Do. do. do. 
 
 17091 
 1-6331 
 
 28,667 
 27,099 
 
 16,772 ) 
 16,594 i 
 
 16,683 = 7 9 
 
 Carron iron, No. 3, hot blast . 
 Do. do. do. 
 
 1-7023 
 1-6613 
 
 28,667 
 31,019 
 
 16,840 
 18,671 
 
 17,755 = 7 18i 
 
 Carron iron, No. 3, cold blast . 
 Do. do. do. 
 
 1-6232 
 1-6677 
 
 22,699 
 24,043 
 
 13,984 
 14,417 
 
 14,200=6 7 
 
 Devon (Scotland) iron, No. 3, 
 hot blast . . . . 
 
 
 4-269 
 
 93,520 
 
 21,907* 
 
 21,907 = 9 15* 
 
 Buffery iron, No. 1, hot blast 
 
 
 3-835 
 
 51,520 
 
 13,434* 
 
 13,434 = 6 
 
 Buffery iron, No. 1, cold blast . 
 
 4-104 
 
 71,680 
 
 17,466* 
 
 17,466 = 7 16 
 
 Coed-Talon (North Wales) ) 
 iron, No. 2, hot blast . . \ 
 Do. do. do. 
 
 1-586 
 1-645 
 
 25,818 
 28,086 
 
 16,279 i 
 17,074 S 
 
 16,676=7 9 
 
 Coed-Talon (North Wales) 
 iron, No. 2, cold blast . . 
 
 
 1-535 
 
 30,102 
 
 19,610 i 
 
 
 Do. do. do. 
 
 
 1-568 
 
 28,380 
 
 18,100 1 
 
 18,855 = 8 8 
 
 Low Moor iron (Yorkshire), 
 No. 3, from 5 experiments 
 
 
 1-540 
 
 22,385 
 
 14,535 
 
 14,535 = 6 10 
 
 Mixture of iron, 4 experi- 
 ments further on (art. 7.) . 
 
 
 ... 
 
 
 
 16,542 = 7 7| 
 
 Mean from the whole . . . . . 
 
 16,505 = 7 7J 
 
 4. The preceding Table, excepting the two last lines, is 
 extracted from my Report on the strength and other pro- 
 perties of cast iron obtained by hot and cold blast, in vol. vi. 
 of the British Association. 
 
 5. In the second volume published by the Association, 
 there are given the results of a few experiments, which I 
 made to determine the tensile strength of cast iron of the 
 following mixture : Blaina No. 2 (Welsh), Blaina No. 3, and 
 W. S. S., No. 3 (Shropshire), each in equal portions.* 
 
 6. In these experiments the intention was to determine, 
 first, the direct tensile strength of a rectangular mass, when 
 drawn through its axis, and next the strength of such a mass, 
 when the force was in the direction of its side. The castings 
 for the experiments on the central strain were of the form 
 previously described ; and in the others the force was exactly 
 
 * This mixture of iron is the same as I had employed in some experiments on the 
 strength and best forms of cast iron beams (Memoirs of the Literary and Philosophical 
 Society of Manchester, vol. v., Second Series), of which an account will be given 
 further on. 
 
230 
 
 TENSILE STRENGTH. 
 
 along the side. The experiments were made by means of the 
 Liverpool testing machine. 
 
 7. Force up the middle. 
 
 Experi- 
 ments. 
 
 Area of section 
 in parts of an 
 inch. 
 
 Breaking 
 weight in 
 tons. 
 
 Strength per square inch. 
 
 1 
 2 
 3 
 4 
 
 3-012 
 2-97 
 3-031 
 2-95 
 
 22-5 
 21-0 
 25-5 
 19-5 
 
 7-47 ) tons. 
 7-07 } mean 7'65 -17136 its. 
 8-41) 
 6 "5 9 different quality of iron. 
 
 8. Force along the side. 
 
 Experi- 
 ments. 
 
 Area of section 
 in parts of an 
 inch. 
 
 Breaking 
 weight in 
 tons. 
 
 Strength per square inch. 
 
 5 
 6 
 
 4-83 
 4-815 
 
 11-5 
 13-75 
 
 |gj? 5 j mean 2'62 tons. 
 
 9. Whence we see that the strength of a rectangular piece 
 of cast iron, drawn along the side, is about one -third, or a 
 little more, of its strength to resist a central strain, since 
 3 x 2-62=7-86 is somewhat greater than 7'65. Mr. Tredgold 
 computed that, if the elasticity remained perfect, the strength 
 in these two cases would be as 4 to 1 (art. 281). 
 
 10. The following Table, calculated from one given by 
 Navier (Application de la Mecanique), contains the results of 
 several experiments made in 1815 by Minard an'd Desormes 
 upon the direct tensile strength of cylindrical pieces of cast 
 iron, of which the specific gravity was 7*074. The results 
 from the experiments which they have given on defective 
 specimens are here rejected. 
 
 N"n nf 
 
 
 Area of 
 
 Weight producing rupture. 
 
 1> O. OI 
 
 Experi- 
 ments. 
 
 Temperature. 
 
 transverse 
 section. 
 
 Total. 
 
 Per square 
 millimetre. 
 
 Per English 
 inch square. 
 
 
 Degrees of 
 
 
 
 
 
 
 
 centigrade 
 thermometer. 
 
 Degrees of 
 Fahrenheit. 
 
 Square 
 millimetres. 
 
 Kilo- 
 grammes. 
 
 Kilogrammes. 
 
 Tons. 
 English. 
 
 1 
 
 6 
 
 21-2 
 
 330 
 
 3392 
 
 10-2788 
 
 6*5277 
 
 2 
 
 5 
 
 23 
 
 346 
 
 3542 
 
 10-2370 
 
 6-5011 
 
 3 
 
 5 
 
 23 
 
 363 
 
 3092 
 
 8-5179 
 
 5-4094 
 
 4 
 
 15 
 
 5 
 
 363 
 
 3720 
 
 10-2479 
 
 6-5080 
 
 5 
 
 + 60 
 
 140 
 
 353 
 
 4020 
 
 11-3881 
 
 7-2321 
 
 6 
 
 + 3 
 
 37-4 
 
 147 
 
 1920 
 
 13-0612 
 
 8-2946 
 
 7 
 
 + 5 
 
 41 
 
 165 
 
 1920 
 
 11-6364 
 
 7-3898 
 
 8 
 
 + 5 
 
 41 
 
 165 
 
 2140 
 
 12-9691 
 
 8-2362 
 
 9 
 
 + 5 
 
 41 
 
 165 
 
 2360 
 
 14-3030 
 
 9-0833 
 
 13' 
 
 + 5 
 
 41 
 
 346 
 
 3670 
 
 10-6069 
 
 6-7360 
 
TENSILE STRENGTH. 231 
 
 11. Since a square millimetre is "001550059 of an English 
 square inch, and a kilogramme = 2*205 of a pound avoir- 
 dupois, multiplying the kilogramme per square millimetre, in 
 the last column of the Table above, by 2'205, and dividing 
 the product by "001550059, and by 2240, the number of 
 pounds in a ton, gives the number of tons per square inch 
 which the iron required to tear it asunder. Or if we 
 multiply the numbers in the last column but one by '63506, 
 we obtain the same result; and thus the last column was 
 formed. 
 
 12. We find from these experiments that the strength of the 
 weakest specimen was 5'09 tons per square inch, that of the 
 strongest 9 '08 tons, and the mean strength from all the 
 specimens was 7*19 tons. 
 
 13. In my own experiments given above, in which every 
 care was taken both to form the castings in such a manner as 
 to obviate theoretical objections, and to obtain accurate 
 results, the strengths varied from 6 tons to 9f tons per square 
 inch, the mean from twenty-five experiments being 16,505fts. 
 or 7 '37 tons nearly. These experiments were upon iron 
 obtained from various parts of England, Scotland, and Wales ; 
 and in no case, except one, was it found to bear more than 
 8^ tons per square inch. With these facts before the reader 
 he will, I conceive, be unable to see how the mean tensile 
 strength of cast iron can properly be assumed at more than 
 7 or 1\ tons per square inch ; but some of our best writers 
 have, by calculating the tensile strength from experiments on 
 the transverse, arrived at the conclusion that the strength of 
 cast iron is 10, or even 20 tons, or more. Mr. Barlow conceives 
 it to be upwards of 10 tons (Treatise on Strength of Timber, 
 Cast Iron, &c., p. 222), and Mr. Tredgold makes it at least 
 20 (art, 72 to 76). The reasoning of Mr. Tredgold, by 
 which he arrives at this erroneous conclusion, with others 
 resulting from it, will be examined at length under the head 
 " Transverse Strength." Navier, too, (Application de la 
 
232 STRENGTH OF CAST IRON. 
 
 Mecanique, article 4,) calculates the tensile strength of cast 
 iron from principles somewhat similar to those assumed by 
 Tredgold, and finds it much too high. 
 
 STRENGTH OF CAST IRON AND OTHER MATERIALS TO 
 RESIST COMPRESSION. 
 
 14. On this subject there was acknowledged to exist a 
 greater want of experimental research than on any other 
 connected with the strength of materials. Feeling this to be 
 the case, I have done all that I could, without too lavish a 
 use of the means intrusted to me, to supply the deficiency. 
 
 15. The matter will be classed under two heads. 1st. The 
 resistance of bodies which are so short, compared with their 
 lateral dimensions, as to be crushed with little or no flexure. 
 2nd. The resistance of pillars long enough to break by 
 flexure. 
 
 RESISTANCE OF SHORT MASSES TO A CRUSHING FORCE. 
 
 16. On this subject I shall give, as before, an abstract 
 from my Report on the strength and other properties of hot 
 and cold blast iron, in the sixth volume of the British 
 Association, referring for more information to the Work itself. 
 Great diversity exists in the conclusions of former experi- 
 menters upon the matter. Eondelet found (Traite de TArt 
 de Mtir) that cubes of malleable iron, and prisms of various 
 kinds of stone, were crushed with forces which were directly 
 as the area, whilst from Mr. Rennie's experiments, both upon 
 cast iron and wood, it would appear that the resistance 
 increases, particularly in the latter, in a much higher ratio 
 than the area (Mr. Barlow's Treatise on the Strength of 
 Materials, art. 112). With respect to M. Rondelet's con- 
 clusions, that cubes of malleable iron resisted crushing with 
 forces proportional to their areas, and that to such a degree, 
 
RESISTANCE TO CRUSHING. 233 
 
 that when in his experiments the area was increased four 
 times, this ratio did not differ from the result so much as a 
 fiftieth part, I am strongly persuaded that wrought iron does 
 not admit of such precision of judging when crushing 
 commences, as to enable any conclusion to be easily drawn 
 with respect to its proportionate resistance to crushing. A 
 prism of that metal becomes slightly flattened and enlarged 
 in diameter with about 9 or 10 tons per square inch, and this 
 effect increases as the weight is increased; but there is no 
 abrupt change in the metal by disunion of the parts, as in 
 cast iron, wood, &c. 
 
 17. With respect to the experiments of Mr. Rennie, the 
 lever used in performing them* would not during its descent 
 act uniformly upon all parts of the specimen ; and therefore 
 the results would be liable to objection. I endeavoured there- 
 fore, by repeating, with considerable variations, in the Report 
 above named, the ingenious experiments of Mr. Rennie, to 
 arrive at some definite conclusions upon the subject. 
 
 18. In order to effect this, it was thought best to crush 
 the object between two flat surfaces, taking care that these 
 were kept perfectly parallel, and that the ends of the prism 
 to be crushed were turned parallel, and at right angles to 
 their axes ; so that when the specimen was placed between 
 the crushing surfaces its ends might be completely bedded 
 upon them. For this purpose a hole, 1^ inch diameter, was 
 drilled through a block of cast iron, about 5 or 6 inches 
 square, and two steel bolts were made which just fitted this 
 hole, but passed easily through it ; the shortest of these bolts 
 was about 1 \ inch long, and the other about 5 inches ; the 
 ends of these bolts were hardened, having previously been 
 turned quite flat and perpendicular to their axes, except one 
 end of the larger bolt, which was rounded. The specimen 
 was crushed between the flat ends of these bolts, which were 
 kept parallel by the block of iron in which they were inserted. 
 See fig. 4, where A and B represent the bolts, with the prism 
 
234 
 
 RESISTANCE TO CRUSHING. 
 
 C between them, and D E the block of iron. During the 
 experiment the block and bolt B rested upon a flat surface of 
 iron, and the rounded end of the bolt A was pressed upon by 
 the lever. There was another hole drilled through the block 
 at right angles to that previously described ; this was done in 
 order that the specimen might be examined during the experi- 
 ment, and previous to it, to see that it was properly bedded. 
 
 19. The specimens were crushed by means of the lever 
 represented in Plate II. fig. 40, the bolt A (Plate I. fig. 4) 
 being placed under it in the manner the pillar is there 
 described to be. In the experiments the lever was kept as 
 nearly horizontal as possible. 
 
 The results of the first experiments I made are given in the 
 following Table : 
 
 20. Tabulated results of experiments made to ascertain the weights necessary to crush 
 given cylinders, &c., of cast iron, of the quality No. 2, from the Carron Iron Works. 
 The specimens in the first three columns of results were from cylinders cast for the 
 purpose, and turned to the right size ; the ^ inch from those of about f inch, &c. 
 
 4 
 
 r 
 
 J" . 
 
 r 
 
 
 Right prisms, 
 
 
 
 
 I 
 
 1 
 
 .^ _j 
 
 lit 
 
 ' '5 
 
 ll| 
 -g'-s 3 
 .ss^ 
 
 11 1 
 
 ;s|i 
 
 base an equila- 
 teral triangle, 
 circumscribing 
 an ^ iuch cylin- 
 
 Right prisms, 
 bases squares, 
 inch the side, 
 cut out of an 
 
 Right prisms, 
 base a rectangle 
 l-OOX'261 inch, 
 cut out of a bai 
 
 C 
 
 4fe 
 
 If 
 
 ill 
 
 '*' j| ^C 
 
 ^3 
 
 i|ii 
 
 der, its sides 
 being '866 in., 
 
 inch square bar, 
 area "250. 
 
 li inch square, 
 area '251 inch. 
 
 1 
 
 
 lal 
 
 "a ^ 3 
 
 ^ I* 
 
 area '3247. 
 
 Crushing 
 
 Crushing 
 
 
 W 
 
 fli 
 
 i" 
 
 |i5 
 
 ^ 6 
 
 Crushing 
 weight. 
 
 weight. 
 
 weight. 
 
 
 fca. 
 
 fcs. 
 
 fibs. 
 
 fcs. 
 
 
 
 
 
 ( 87^7 ) 
 
 
 
 
 
 
 
 i inch. 
 
 \ O/Ol / 
 
 18145J 
 
 18,882 
 
 30,461 
 
 
 
 
 
 i do. 
 
 6818 
 
 16,474 
 
 26,983 
 
 
 
 
 
 I do. 
 
 6563 
 
 13,736 
 
 26,412 
 
 
 
 
 
 4 do. 
 
 6301 
 
 13,638 
 
 24,210 
 
 
 35,548 
 
 25,721 
 
 27,187 
 
 1 do. 
 
 6309 
 
 14,156 
 
 
 38,671 
 
 
 
 
 ^ do. 
 
 5980 
 
 15,059 
 
 23,465 
 
 35,888 
 
 33,448 
 
 24,191 
 
 
 1 do. 
 
 
 
 
 35,888 
 
 
 
 
 1 do. 
 
 5798 
 
 14,877 
 
 22,867 
 
 
 31,348 
 
 23,950 
 
 25,151 
 
 li do. 
 
 
 14,190 
 
 24,177 
 
 
 
 
 
 1J do. 
 
 
 14,143 
 
 23,453 
 
 
 
 
 
 2 do. 
 
 
 13,800 
 
 21,828 
 
 
 
 
 
 21. By comparing the results in each vertical column, we 
 see that the shorter specimens generally bear more than the 
 larger ones of the same diameter, or dimensions of base. In 
 the shortest specimens fracture takes place by the middle 
 becoming flattened and increased in breadth, so as to burst 
 
RESISTANCE TO CRUSHING. 235 
 
 the surrounding parts, and cause them to be crumbled and 
 broken in pieces. This is usually the case when the lateral 
 dimensions of the prism are large compared with the height. 
 When they are equal to, or less than the height, fracture is 
 caused by the body becoming divided diagonally in one or 
 more directions. In this case the prism, in cast iron at least, 
 either does not bend before fracture, or bends very slightly ; 
 and therefore the fracture takes place by the two ends of the 
 specimen forming cones or pyramids, which split the sides 
 and throw them out ; or, as is more generally the case in 
 cylindrical specimens, by a wedge sliding off, starting at one 
 of the ends, and having the whole end for its base; this 
 wedge being at an angle which is constant in the same 
 material, though different in different materials. In cast iron 
 the angle is such that the height of the wedge is somewhat 
 less than f of the diameter. The forms of fracture in these 
 cases may be seen from Plate L, in which fig. 5 represents a 
 cylinder before fracture, and fig. 6 the same cylinder after- 
 wards ; a part in the form of a wedge having separated from 
 one side of it, and the remainder being shortened and bulged 
 out in the middle, which is very obvious in experiments on 
 soft cast iron. Tigs. 7 and 8 represent another cylinder 
 before and after fracture : in fig. 8 the sides are separated by 
 the action of two cones, having the ends of the cylinder for 
 the bases, and the vertices with sharp edges or points formed 
 near to the centre of the cylinder, but inclined a little from 
 the axis, so that they may slip past each other, and divide the 
 mass without injury to the cones. Figs. 9 and 10 show the 
 same thing as the two preceding figures; and fig. 11 is a 
 representation of one of the cones, the vertex being sharp, 
 as above mentioned. Fig. 12 is another cylinder, of soft cast 
 iron, showing the directions of the fractures, but not sepa- 
 rated. Fig. 13 was a cylinder too short to be crushed in the 
 ordinary way ; but the centre shows the rudiments of a cone, 
 throwing out the surrounding parts. Fig. 14 represents a 
 
236 RESISTANCE TO CRUSHING. 
 
 rectangular prism; and figs. 15 and 16 the modes of its 
 fracture. Figs. 17 and 18 represent a whole and fractured 
 prism ; and the same is the case with respect to figs. 19 and 
 20. In fig. 20 the sharp-pointed pyramid, with the lower 
 side of the specimen for its base, is very clearly shown ; it has 
 cut up the prism, separating the sides, and left a number of 
 sharp-edged parts, all of which have slid off in the angle of 
 least resistance. Figs. 21 and 22 exhibit the appearance of 
 a triangular prism, before and after fracture ; two pyramids, 
 formed as usual, with their bases at the ends, and the vertices 
 towards the centre, have thrown out the angular parts. The 
 parts, so separated, have in prisms of every form a general 
 resemblance, and the form of the pyramidal wedge has con- 
 siderable interest, as it is that of least resistance in cast iron, 
 and furnishes hints as to the best forms of cutters for that 
 metal. Further investigations upon the subject of this article, 
 of a theoretical nature, will be given on a future occasion. 
 
 22. The mode of fracture is the same, and the strength of 
 the specimen very little diminished, by any increase of its 
 height, whilst its lateral dimensions are the same, provided 
 the height be greater than the diameter, when the body is a 
 cylinder, but not greater, in cast iron, than four or five times 
 the diameter, or least lateral dimension in specimens not 
 cylindrical. If the length be greater than this, the body 
 bends with the pressure, and though it may break by sliding 
 off as before, the strength is much decreased. In cases 
 where the length is much greater than as above, the body 
 breaks across, as if bent by a transverse pressure. 
 
 23. The preceding Table was formed by taking the means 
 from results on the resistance to crushing of specimens of 
 equal size in Carron Iron, No. 2, one Table being on hot 
 blast iron, and the other cold; the mean from the results 
 being given here in one Table, in preference to the two 
 Tables at large, as in the volume of the Association ; since 
 most of the results thus obtained are means from several ex- 
 
RESISTANCE TO CRUSHING. 237 
 
 periments ; and there was very little difference in the strength 
 of the hot and cold blast iron of this description to resist 
 crushing. 
 
 24. Comparing the results in the different vertical columns 
 of the Table, it appears that the strength of the specimens 
 was nearly as the area of their transverse section. Thus the 
 cylinders -| inch in diameter bore nearly four times as much 
 as those of J inch. The falling off from this proportion in 
 the strength of some of the larger specimens, I attribute to 
 those having been cut out of larger (and consequently softer) 
 masses of the iron than the small specimens. 
 
 25. To obtain further evidence on the matter I crushed in 
 the same manner twelve right cylinders of Teak-wood, \ inch, 
 1 inch, and 2 inches diameter, four of each, the latter eight 
 out of the same piece of wood ; the height in each case was 
 double the diameter. The pressure was in the direction of 
 the fibres. The strengths were as below. 
 
 Cylinders 
 \ inch diameter. 
 2335 ) 
 2543 f mean 
 2543 ( 2439 tts. 
 2335) 
 
 Cyli 
 1 inch ( 
 10507 
 9499 
 10507 
 10171 
 
 aders 
 iiameter. 
 
 mean 
 10171 fts. 
 
 Cyl 
 2 inches 
 38909 
 39721 
 41294 
 41294 
 
 mean 
 40304 ffis. 
 
 26. The mean crushing weights above are nearly as 25, 
 100, and 400, which is the ratio of the areas of the sections 
 of the cylinders. The strengths are therefore as the areas, 
 though these vary as 4 and 16 to 1. 
 
 27. In this and every other kind of timber, like as in iron 
 and crystalline bodies generally, crushing takes place by 
 wedges sliding off in angles with their base, which may be 
 considered constant in the same material : hence the strength 
 to resist crushing will be as the area of fracture, and con- 
 sequently as the direct transverse area ; since the area of 
 fracture would, in the same material, always be equal to the 
 direct transverse area, multiplied by a constant quantity. 
 
 28. In estimating the resistance, per square inch, of the 
 iron above to a crushing force, I shall .mostly confine myself 
 
233 
 
 RESISTANCE TO CRUSHING. 
 
 to such specimens as vary in height from about the length of 
 the wedge, which would slide off to twice its length, say, 
 such as have their height from 1^ to 3 times the diameter, 
 thus avoiding the results from prisms which, through their 
 shortness, could not break by a wedge sliding off at its 
 proper angle, and therefore must offer an increased resis- 
 tance ; and those whose extra length would enable them to 
 bend before sliding off, and thus have their strength reduced. 
 I shall here take the results from my experiments upon the 
 hot and cold blast iron separately, that the difference may be 
 seen ; the iron being of the same materials in the two cases, 
 and affected only by the heat of the blast used in its 
 preparation. 
 
 The results are in the following Tables. 
 
 29. Resistance of Hot Mast Iron (Carron, No. 2) to a crushing force. 
 
 Dimensions of base of 
 specimen. 
 
 Number of 
 experiments 
 derived from. 
 
 Mean 
 crushing 
 weight. 
 
 Mean crushing 
 weight per 
 square inch. 
 
 General mean per 
 square inch. 
 
 
 
 iba. 
 
 Ibs. 
 
 
 Right cylinder, ) 
 diameter \ inch . . \ 
 
 3 
 
 6,426 
 
 130,909 
 
 From the cylinders 
 
 >> 8 
 
 4 
 
 14,542 
 
 131,665 
 
 121,685 Ibs. =54 
 
 ,, rV - 
 
 5 
 
 22,110 
 
 112,605 
 
 tons 6| cwts. 
 
 s*=-* ' 
 
 1 
 
 35,888 
 
 111,560 
 
 
 Right prism, 
 area '50 x '50 in. . , j 
 
 3 
 
 25,104 
 
 100,416 
 
 ) From the prisms 
 
 
 
 
 
 100,739 Ibs. = 44 
 
 Ditto area 1-00 x -26 
 
 2 
 
 26,276 
 
 101,062 
 
 ) tons 194 cwts. 
 
 Mean per sq. in. from the whole of the experiments, 114,703 Ibs. = 51 tons 4 cwts. 
 
 30. Resistance of Cold Blast Iron (Carron, No. 2) to a crushing force. 
 
 Dimensions of base of 
 specimen. 
 
 Number of 
 experiments 
 derived from. 
 
 Mean 
 crushing 
 weight. 
 
 Mean crushing 
 weight per 
 square inch. 
 
 General mean per 
 square inch. 
 
 Right cylinder, dia- 
 meter 5 inch . . 
 
 I ',', - ' 
 
 
 2 
 
 4 
 
 7 
 
 Ibs. 
 6,088 
 
 14,190 
 24,290 
 
 Iba. 
 124,023 
 
 128,478 
 123,708 
 
 | From the cylinders 
 } 118,211 Ibs. = 52 
 tons 15A cwts. 
 
 45 and 
 75 high . . . 
 
 
 2 
 
 15,369 
 
 96,634 
 
 1 
 
 Equilateral triangle, 
 side -866 inch 
 
 
 2 
 
 32,398 
 
 99,769 
 
 ^ 
 
 Square, area -50 x 
 50 inch . 
 
 
 2 
 
 24,538 
 
 98,152 
 
 From the prisms 
 101,964 Ibs. = 45 
 
 Rectangle, area 
 l-OOx-243 . . 
 
 
 3 
 
 26,237 
 
 107,971 
 
 tons 10 J cwts. 
 
 Mean per sq. in. from the whole of the experiments, 111,248 ibs. =49 tons 13 cwts. 
 
RESISTANCE TO CRUSHING. 
 
 239 
 
 31. Comparing the results in these two Tables, it will be 
 seen, as has been mentioned before, that the Carron Iron, 
 No. 2, offers but little difference of resistance to a crushing 
 force, whether the iron be prepared with a hot or a cold 
 blast. The falling off in the resistance per square inch in the 
 latter class of experiments, in each Table, compared with the 
 former, has been attempted to be accounted for by the iron 
 out of which the larger specimens were cut being softer and 
 weaker than the thin cylinders out of which the smaller 
 specimens were obtained. 
 
 32. The resistances of other species of cast iron to a 
 crushing force, obtained in the same manner as above, are as 
 in the following Table. 
 
 Description of Iron. 
 
 Form of 
 specimen. 
 
 Numbei' of 
 experi- 
 ments de- 
 rived from. 
 
 Mean strength per square inch. 
 
 
 
 
 Ibs. tons. cwts. 
 
 Devon (Scotch) iron, 
 No. 3, hot blast . . 
 
 Cylinder. 
 
 4 
 
 145,435 = 64 18 
 
 Buffery (near Birming- 
 
 
 
 
 ham) iron, No. 1, hot 
 blast 
 
 Do. 
 
 4 
 
 86,397=38 11J 
 
 Do., cold blast 
 
 Do. 
 
 4 
 
 93,385 = 41 134 
 
 Coed-Talon (Welsh) ) 
 iron, No. 2, hot blast \ 
 
 Do. 
 
 4 
 
 82,734 = 36 18 
 
 Do., cold blast ."*'; 
 
 Do. 
 
 4 
 
 81,770 = 36 10 
 
 Carron (Scotch) iron, 
 No. 2, hot blast . . 
 
 Cylinders and 
 prisms. 
 
 u 
 
 114,703 = 51 4 
 
 Do., cold blast . 
 
 Do. 
 
 22 
 
 111,248 = 49 13i 
 
 Carron iron, No. 3, hot ) 
 blast . . . I 
 
 Prisms. 
 
 3 
 
 133,440 = 59 11J 
 
 Do., cold blast . 
 
 Do. 
 
 4 
 
 115,442 = 51 lOf 
 
 Low Moor (Yorkshire) } 
 iron, No. 3, cold > 
 blast . , .) 
 
 Cylinder. 
 Rectangle. 
 
 3 
 
 2 
 
 1 1 f> Q1 1 ) mean. tons. cwts. 
 lot692 109,801=49 OJ 
 
 
 Cylinders '508 
 
 } 
 
 
 Mixture of iron, same as 
 
 and -6 inch. 
 
 ( * 
 
 
 in my experiments on 
 
 diameter, 3 of 
 
 I 
 
 100,049 
 
 the strength of beams 
 (see note to art. 5). 
 
 each. 
 Rectangles cut 
 out of a beam. 
 
 \ 3 
 
 110,908 = 49 10J 
 121,767 
 
 33. The ratios of the forces necessary to crush and tear 
 equal masses of cast iron may now be obtained ; the experi- 
 ments of which I have given the results, in this and the 
 
240 
 
 STRENGTH OF LONG PILLARS. 
 
 preceding article, will supply those ratios which are in the 
 following Table. 
 
 
 Crushing 
 
 Tensile 
 
 
 
 Description of Metal. 
 
 force per 
 square inch 
 
 force per 
 square inch 
 
 Ratio. 
 
 
 
 in fi)s. 
 
 in fi>s. 
 
 
 
 Devon iron, No. 3, hot blast . ' > 
 
 145,435 
 
 21,907 
 
 6-638:1-, 
 
 
 Buffery iron, No. 1, hot blast . . 
 
 86,397 
 
 13,434 
 
 6-431 : 1 
 
 
 Do. No. 1, cold blast ^>'>?1 
 
 93,385 
 
 17,466 
 
 5-346 : 1 
 
 
 Coed-Talon iron, No. 2, hot blast . 
 
 82,734 
 
 16,676 
 
 4-961:1 
 
 7j 
 
 Do. No. 2, cold blast . 
 
 81,770 
 
 18,855 
 
 4-337 : 1 
 
 *c 
 
 Carron iron, No. 2, hot blast . , . ... 
 
 114,703 
 
 13,505 
 
 8-493 : 1 
 
 Oi 
 
 Y^ 
 
 Do. No. 2, cold blast. ' '^ * ? 
 
 111,248 
 
 16,683 
 
 6-668:1 
 
 <b 
 
 Do. No. 3, hot blast 
 
 133,440 
 
 17,755 
 
 7-515:1 
 
 a 
 
 rt 
 
 Do. No. 3, cold blast . 
 
 115,442 
 
 14,200 
 
 8-129:1 
 
 6 
 g 
 
 Low Moor iron, No. 3, cold blast \ .- 
 
 109,801 
 
 14,535 
 
 7-554 : 1 
 
 ^ 
 
 Mixture of iron used in my experiments ) 
 on beams (art. 5-7) . . ) 
 
 110,908 
 
 17,136 
 
 6-472 : 1 
 
 
 34. 'From this Table it appears that cast iron requires from 
 4*337 to 8*493 times as much force to crush it as to tear it 
 asunder, the mean being 6*595. I conceive, however, this 
 mean to be too low ; it would have been between 7 and 8 if 
 the prisms which were crushed had been cut out of the same 
 masses as those which were torn asunder ; but they were in 
 many instances out of larger castings, which, being softer, 
 offered less resistance. 
 
 STRENGTH OF LONG PILLARS. 
 
 35. I shall here consider such pillars as are too long to 
 break by sliding off, as in the last article. This class will 
 include all such as are usually made of iron or timber, the 
 lengths of the shortest of which are generally many times 
 their lateral dimensions ; and it has been remarked before, 
 that in some cases right cylinders of cast iron, whose length 
 did not exceed 5 times the diameter, became bent under a 
 direct force of compression, so as to break nearly straight 
 across in the manner of longer columns. 
 
 The acknowledged want of practical information upon this 
 
STRENGTH OF LONG PILLARS. 241 
 
 subject,* and its great importance, made me anxious to 
 undertake an extensive series of experiments upon it, such as 
 would confirm or show the error of existing theories, and give 
 such information as~would be of real service to the engineer 
 and architect, whilst they tended to unfold the laws that 
 regulate the strength of pillars. This wish was, as on other 
 occasions, cheerfully responded to by my friend William 
 Eairbairn, Esq., at whose expense the extensive series of 
 experiments, of which the following is an abstract! was 
 made. 
 
 36. In the original Paper the experiments are contained in 
 thirteen Tables, J as below. 
 
 CAST IKON. 
 
 No. of 
 
 Experiments. 
 
 Table I. Solid uniform cylindrical pillars, with rounded ends (fig. 23) . 55 
 II. Do., with flat ends (tig. 26) 65 
 
 III. Solid uniform square pillars, with rounded ends (fig. 28.) . . 7 
 
 IV. Solid uniform cylindrical pillars, with discs (fig. 27) ... 12 
 V. Do., with ends rounded (fig. 23), rounded and flat (fig. 24), and 
 
 both ends flat (fig. 26) 23 
 
 VI. Solid cylindrical pillars, enlarged middle, rounded ends (fig. 31) . 7 
 
 VII. Do., do., discs on ends (fig. 32) 4 
 
 VIII. Hollow uniform cylinders, rounded ends (fig. 36) ... 19 
 
 IX. Do., with flat ends (fig. 37) 11 
 
 X. Short hollow uniform cylinders, with flat ends (fig. 37) . . 13 
 
 XI. Pillars, hollow and solid, of various forms, and different modes of 
 
 fixing (figs. 33, 34, 35, 38, 39) 10 
 
 WROUGHT IRON AND STEEL. 
 
 XII. Solid uniform cylindrical pillars of these metals (figs. 23, 24, 25, 
 
 26,27) 30 
 
 WOOD. 
 XIII. Pillars of oak and red deal, square and other rectangular forms 
 
 (figs. 28, 29, 30, &c.) . 21 
 
 Number of experiments 277 
 
 * Dr. Robison (Mechanical Philosophy, vol. i.) and Mr. Barlow (Report to the 
 British Association) have strongly expressed their opinion of our want of information 
 upon it ; and the British Association have mentioned such experiments as among the 
 desiderata of Practical Science. 
 
 t See Experimental Researches on the Strength of Pillars of Cast Iron and other 
 Materials. Phil. Trans, of the Royal Society, Part II. 1840. 
 
 Abstracts of Tables I, II., VIII., IX., X., are given in the present work. 
 
 R 
 
242 STRENGTH OF LONG PILLARS. 
 
 37. The drawing, Plate II. fig. 40, will show how the 
 experiments were made. The pillars were placed vertically, 
 resting upon a flat smooth plate of hardened steel, laid upon 
 a cast iron shelf, E, made very strong, and lying horizontally. 
 The pressure was communicated to the upper end of the 
 pillar by means of the lever, A B, acting upon a bolt, C, of 
 hardened steel, 2-| inches diameter, and about a foot long, 
 kept vertical by being made to pass through a hole bored in 
 a deep mass, D, of cast iron, the hole being so turned as just 
 to let the bolt slide easily through without lateral play. The 
 top of the bolt was hemispherical, that the pressure from the 
 lever might act through the axis of it ; and the bottom was 
 turned flat to rest upon the pillar, I K. The bottom of this 
 bolt, and the shelf on which the pillar stood, were necessarily 
 kept parallel to each other ; for the mass through which the 
 bolt passed, and that on which the shelf rested, were parts of 
 the same large case, D E G, of iron, cast in one piece, and so 
 formed as to admit shelves at various heights for breaking 
 pillars of different lengths. The case had three of its four 
 sides closed ; circular apertures were, however, made through 
 them, that the experimenter might observe the pillar without 
 danger. 
 
 38. With a view to develope the laws connecting the 
 strength of cast iron pillars with their dimensions, they were 
 broken of various lengths, from 5 feet to 1 inch, and their 
 diameters varied from ^ an inch to 2 inches in solid pillars ; 
 and in hollow ones the length was increased to 7 feet 6 inches, 
 and the diameter to 3^ inches. My first object was to supply 
 the .deficiencies of Euler's theory of the strength of pillars 
 (Academic de Berlin, 1757,) if it should appear capable of 
 being rendered practically useful ; and if not, to endeavour to 
 adapt the experiments so as to lead to useful conclusions. 
 As the results of the experiments were intended to be com- 
 pared together, it was desirable that all the pillars of cast iron 
 should be from one species of metal; and the description 
 
STRENGTH OF LONG PILLARS. 243 
 
 chosen was a Yorkshire iron, the Low Moor, No. 3, which is 
 a good iron, not very hard, and differs not widely from that 
 called No. 2. The pillars were mostly made cylindrical, as 
 that seemed a more convenient form in experiments of this 
 kind than the square ; for square pillars do not bend or break 
 in a direction parallel to their sides, but to their diagonals, 
 nearly. The experiments in the first Table were made on 
 solid uniform pillars rounded at the ends to render them 
 capable of turning easily there, and that the force might be 
 through the axis ; and in the second Table the pillars were 
 uniform and cylindrical, as before, but had their ends flat and 
 at right angles to the axis ; so that the whole end of the 
 pillar might be pressed upon, instead of the axis only, as in 
 the last case ; thus rendering the pillar incapable of moving at 
 the ends. In the fifth Table (art. 36) uniform pillars, with 
 one end rounded and one flat, were used ; and to prove the 
 constant connexion (in any particular material) between the 
 results of these three kinds of pillars, all of equal dimensions, 
 enters largely into the other Tables, which are occupied like- 
 wise in inquiries into the strength of hollow cylindrical 
 pillars, and others of different forms. 
 
 39. RESULTS. 
 
 1st. In all long pillars of the same dimensions, the 
 resistance to fracture by flexure is about three times greater 
 when the ends of the pillar are flat and firmly bedded, than 
 when they are rounded and capable of turning.* 
 
 2nd. The strength of a pillar, with one end round and the 
 other flat, is the arithmetical mean between that of a pillar of 
 the same dimensions with both ends rounded, and with both 
 ends flat. Thus, of three cylindrical pillars, all of the same 
 length and diameter, the first having its ends rounded, the 
 
 * This will be seen by comparing the results from the longer pillars of equal size in 
 Tables I. and II, page 211, of which abstracts are given further on. 
 
 B 2 
 
244 STRENGTH OF LONG PILLARS. 
 
 second with one end rounded and one flat, and the third with 
 both ends flat, the strengths are as 1, 2, 3, nearly.* 
 
 3rd. A long uniform pillar, with its ends firmly fixed, 
 whether by discs or otherwise, has the same power to resist 
 breaking as a pillar of the same diameter, and half the length, 
 with the ends rounded or turned so that the force would pass 
 through the axis. 
 
 The preceding properties were found to exist in long pillars 
 of steel, Avrought iron, and wood. 
 
 4th. The experiments in Tables VI. and VII. (art. 36) 
 show that some additional strength is given to a pillar by 
 enlarging its diameter in the middle part ; this increase does 
 not, however, appear to be more than yth or -|th of the 
 breaking weight. 
 
 5th. The index of the power of the diameter, to which the 
 strength of long pillars of cast iron, with rounded ends, is 
 proportional, is 3*76, nearly; and 3*55 in those with flat 
 ends ; as appeared from means between the results of a great 
 number of experiments ; f or the strength of both may be 
 taken as following the 3 '6 power of the diameter, nearly. 
 
 6th. In cast iron pillars of the same thickness the strength 
 is inversely proportional to the 1*7 power of the length, 
 nearly. J 
 
 40. Thus the strength of a solid pillar of that material, 
 with rounded ends, the diameter of the pillar being d and the 
 length /, is, as 
 
 * This was proved by Table V., page 241, and the next conclusion was obtained by 
 a like comparison of results. 9 
 
 t The pillars with rounded ends, compared together, varied in diameter (or side of 
 square) from ^ inch to 2 inches. There were seventeen of them, and the largest 
 exponent of the diameter, to which the strength was in any case proportional, is 
 3'928, and the smallest 3*425. The pillars with flat ends, and those with discs, com- 
 pared as above, were eleven, and their exponents varied from 3*922 to 3*412. 
 
 J The exponent 1*7 for the strength, according to the inverse power of the length, 
 is from experiments upon nineteen pillars, varying in length from 60*5 to 3'78 inches; 
 the highest and lowest exponents being 1*914 and 1*424. 
 
STKENGTH OF LONG PILLARS. 245 
 
 41. The breaking weights of solid cylindrical cast iron 
 pillars, as appeared from the experiments, are nearly as below. 
 
 In solid pillars with their ends rounded and moveable as 
 above, fig. 23, 
 
 ^3-6 
 
 Strength in tons = 14'9 x -yy. 
 
 In solid pillars with their ends flat, and incapable of 
 motion, fig. 26, 
 
 d 
 Strength in tons 44-16 x y ; 
 
 where / is in feet, and d in inches. 
 
 In hollow pillars nearly the same laws were found to 
 obtain ; thus, if D and d be the external and internal 
 diameters of a cast iron pillar, whose length is /, the strength 
 of a hollow pillar of which the ends were moveable, fig. 36, 
 (as in the connecting rod of a steam engine,) would be 
 expressed by the formula below. 
 
 J)3-6 _ 0T3.8 
 
 Strength in tons = 13 x 
 
 
 In hollow pillars, whose ends are flat, and firmly fixed 
 (fig. 37, &c.) by discs or otherwise, I found from the results 
 of numerous experiments as before, 
 
 D 3 - 6 - 
 Strength in tons = 44*3 x -- - 
 
 42. The co-efficients given above in the formula? for the 
 strength of solid pillars were obtained as below : The 14*9 
 is the mean result from 18 pillars in Table I., art. 36, 
 varying in length from 121 times the diameter down to 15 
 times. The 44*16, for pillars with flat ends, is similarly 
 obtained from 11 pillars in Table II., varying in length from 
 78 to 25 times the diameter. Plat-ended pillars, shorter 
 than 25 or 30 times the diameter, require a modification of 
 the above rule for their strength ; as I found them to be 
 
246 STRENGTH OF LONG PILLARS. 
 
 crushed as well as bent by the pressure, and therefore to have 
 their strength decreased: the mode I used in this case will be 
 seen in arts. 45 to 51, further on. The co-efficients for the 
 strength of hollow pillars were obtained in the same manner 
 as those for solid ones ; the 13 is the mean obtained from 19 
 experiments in Table VIII. upon hollow pillars, with rounded 
 caps upon the ends, to make them moveable there like solid 
 ones with rounded ends ; this number is too low, as some of 
 the first hollow pillars were bad castings. The 44 '3, for 
 those with flat ends, was obtained from 11 pillars in Table 
 IX. ; taking pillars only whose length was more than 25 times 
 the external diameter, as in solid ones. 
 
 The Tables from which these results were derived are 
 given, slightly abridged, further on. The formulae for hollow 
 pillars were obtained by adapting the results of Euler's theory 
 to those of experiment, and were found to answer well when 
 so altered. According to that theory the strength varies as 
 
 p (Poisson, Mecanique, vol. i., 2nd edition, art. 315). 
 
 43. As the above expressions for the strength of pillars 
 contain fractional powers of the diameters and lengths, these 
 may be taken from the Tables below ; the first Table com- 
 prising the involved values in inches of most of the diameters 
 in common use ; and the second those of the lengths in feet. 
 
STRENGTH OF LONG PILLARS. 
 
 247 
 
 TABLE I. Powers of Diameters. 
 
 1-0 3 - 6 = 1 
 
 4-25 3 - 6 = 182-S9 
 
 6-8 3 - B - 993-19 
 
 l-25 3 - 6 = 2-2329 
 
 4-3 3 - 6 = 190-76 
 
 6-9 3 - 6 = 1046-8 
 
 1-5 3>6 = 4-3045 
 
 4-4 3 - 6 = 207-22 
 
 7-0 3 -e= 1102-4 
 
 l-75 3 - 6 = 7-4978 
 
 4-5 3 - 6 = 224-68 
 
 7-1 3.e = H60-2 
 
 2-0 3 - 6 = 12125 
 
 4-6 3 - 6 = 243-18 
 
 7-2 3 - 6 = 1220-1 
 
 2-1 3.e = 14.454 
 
 4-7 3>6 = 262-76 
 
 7'25 3 - 6 = 1250-9 
 
 2-2 3 - 6 = 17-089 
 
 4-75 3 - fi = 272-96 
 
 7-3 3 - G = 1282-2 
 
 2-25 3 - G = 18-529 
 
 4-8 3 - 6 = 283-44 
 
 7-4 3 - 6 = 1346-6 
 
 2-3 3 - 6 = 20-055 
 
 4-9 3>6 = 305-28 
 
 7'5 3 - 5 = 1413-3 
 
 2-4 3>6 = 23-3755 
 
 5-0 36 = 328-32 
 
 7-6 36 = 1482-3 
 
 2-5 3 - 6 = 27-076 
 
 5-1 3 - G = 352-58 
 
 77 3 ' 6 = 1553-7 
 
 2-6 3 ' 6 = 31-182 
 
 5-2 36 = 378-10 
 
 7-75 3 - 6 = 1590-3 
 
 2-7 3 ' 6 = 35-720 
 
 5-25 3 - 6 = 391-36 
 
 7-8 3 - 6 = 1627-6 
 
 2-75 3 - 6 = 38-159 
 
 5-3 36 = 404-94 
 
 7-9 3 -s= 1704-0 
 
 2-8 3 ' 6 = 40-716 
 
 5-4 3 - 6 =433-13 
 
 8-0 3 - 6 = 1782-9 
 
 2-9 3 ' 6 = 46-199 
 
 5-5 3 - 6 = 462-71 
 
 8-25 3 ' 6 = 1991-7 
 
 3-0 36 = 52-196 
 
 5-6 3 - 6 =493-72 
 
 8-5 3 - 6 = 2217-7 
 
 3-1 3 - 6 = 58-736 
 
 5-7 3>6 = 526-20 
 
 8-75 3 - 6 =2461-7 
 
 3-2 3 - 6 = 65-848 
 
 5.753.6 = 543-01 
 
 9-0 3>6 = 2724-4 
 
 3-25 3 ' 6 = 69-628 
 
 5-8 3 - 6 = 560-20 
 
 9-25 3 - 6 = 3006-85 
 
 3-3 3 - 6 = 73-561 
 
 5-9 3 - 6 = 595-75 
 
 9-5 3 - 6 = 3309-8 
 
 3-4 3 - 6 = 81-908 
 
 6-0 3 - 6 = 632-91 
 
 9-75 3 - 6 = 3634-3 
 
 3-5 3 ' 6 = 90-917 
 
 6-1 3 - 6 =671'72 
 
 10-0 3 -= 3981-07 
 
 3-6 3 - 6 = 100-62 
 
 6-2 36 = 7l2-22 
 
 10-25 3 ' 6 = 4351-2 
 
 3-7 3 - 6 = 111-05 
 
 6-25 3 - 6 - 733-11 
 
 10-5 3fi = 4745-5 
 
 3-75 3>6 = 116-55 
 
 6-3 3 - c = 754-44 
 
 10-75 3 - 6 = 5165-0 
 
 3-8 3 - 6 = 122-24 
 
 6-4 3 - 6 = 798-45 
 
 11-0 3 - 6 = 5610-7 
 
 3-9 3 - 6 = 134-23 
 
 6-5 3 6 = 844-28 
 
 ll-2f s - 6 = 6083-4 
 
 4-0 3 - 6 = 147'03 
 
 6-6 3 - 6 = 891-99 
 
 11-5 3 - 6 = 6584-3 
 
 4-1 3-6 = l'60-70 
 
 6-7 3 ' 6 = 941-61 
 
 ll-75 3 - 6 =7114-4 
 
 4-2 3 - 6 = 175-26 
 
 675 3 ' 6 = 967-15 
 
 12-0 3 - 6 = 7674'5 
 
 TABLE II. Powers of Lengths. 
 
 1 1 '7= 1 
 
 9 1 -?= 41-900 
 
 17 1>7 = 123-53 
 
 2 ll7 = 3-2490 
 
 10 J -7= 50119 
 
 18 1 -" = 136-13 
 
 31-7= 6-4730 
 
 ll 1 -"^ 58-934 
 
 19 J -"= 149-24 
 
 4 1 -5' = 10-556 
 
 12 1 -7= 68-329 
 
 20 1 '"= 162-84 
 
 5 l -' = 15-426 
 
 13 1 -7= 78-289 
 
 21 1 -7= 176-92 
 
 6 J -7 = 21-031 
 
 14 1 -7= 88-801 
 
 22 1 * = 191-48 
 
 7 1> 7 = 27'332 
 
 15 1 -"= 99-851 
 
 23 1 -?= 206-51 
 
 S 1 -^ 34-297 
 
 lG l -< = 111-43 
 
 24 1 -7 = 222'00 
 
 44. As an example, suppose it were required to find the 
 strength of a hollow cylindrical cast iron pillar, 14 feet long, 
 6*2 inches external diameter, and 4*1 inches internal; the 
 pillar being flat, and well supported, at the ends. 
 
 Prom the Tables we obtain U^-SS'SOl, 6-2 3 - 6 =712'22, 
 and 4-l 3 - 6 = 160-70. Whence strength = 44-3 x" 
 
 44-3 x 
 
 712-22160-70 
 88-801 
 
 ==275-1 tons. 
 
248 STRENGTH OF SHORT FLEXIBLE PILLARS. 
 
 STRENGTH OF SHORT FLEXIBLE PILLARS. 
 
 45. The formulae above apply to all pillars whose length is 
 not less than about 30 times the external diameter; for 
 pillars shorter than this, it will be necessary to modify the 
 formulae by other considerations, since in these shorter pillars 
 the breaking weight is a considerable proportion of that 
 necessary to crush the pillar. 
 
 46. Thus, considering the pillar as having two functions, 
 one to support the weight, and the other to resist flexure, 
 it follows that when the material is incompressible (supposing 
 such to exist), or when the pressure necessary to break the 
 pillar is very small, on account of the greatness of its length 
 compared with its lateral dimensions, then the strength of the 
 whole transverse section of the pillar will be employed in 
 resisting flexure ; when the breaking pressure is half of what 
 would be required to crush the material, one half only of the 
 strength may be considered as available for resistance to 
 flexure, whilst the other half is employed to resist crushing ; 
 and when, through the shortness of the pillar, the breaking 
 weight is so great as to be nearly equal to the crushing force, 
 we may consider that no part of the strength of the pillar is 
 applied to resist flexure. 
 
 47. This reasoning is supported by the results from a 
 number of short pillars of various lengths, from 26 times the 
 diameter down to twice the diameter. These pillars were 
 reduced to about half the strength, as calculated by the 
 preceding formulae, when the length was so small that the 
 breaking weight was half of that which would crush the 
 pillar ; and the results from short pillars of other lengths were 
 in accordance with the preceding reasoning. (See Researches 
 on the strength of pillars, &c., art. 39, Phil. Trans. 1840.) 
 
 48. We may therefore separate these effects by taking, in 
 imagination, from the pillar (by reducing its breadth, since 
 the streng h is as the breadth,) as much as would support the 
 
STRENGTH OF SHORT FLEXIBLE PILLARS. 249 
 
 pressure, and considering the remainder as resisting flexure to 
 the degrees indicated by the previous rules. 
 
 49. Suppose, then, c to be the force which would crush the 
 pillar without flexure ; d the utmost pressure the pillar, as 
 flexible, would bear to break it without being weakened by 
 crushing (as was shown to take place with a certain pressure 
 dependent on the material) ; b the breaking weight, as 
 calculated by the preceding formulae for long pillars ; y the 
 real breaking weight. 
 
 Then supposing a portion of the pillar, equal to what 
 would just be crushed by the pressure d, to be taken away, 
 we have c ^=the crushing strength of the remaining part, 
 
 and yd the weight actually laid upon it. Whence ~r d 
 the part of this remaining portion of the pillar which has to 
 resist crushing, 
 
 c d c d 
 
 the part to sustain flexure. 
 
 50. But the strength of the pillar, if rectangular, may be 
 supposed to be reduced, by reducing either its breadth, or the 
 computed strength of the whole, to the degree indicated by 
 the fraction last obtained. In circular pillars this mode is 
 not strictly applicable ; but we obtain a near approximation 
 to the breaking weight y, by reducing the calculated value of 
 b in that proportion. 
 
 Whence ^ x ^^ = y, the strength of a short flexible 
 pillar, b being that of a long one, /. b cby=cydy t 
 
 and?/ 
 
 b + c 
 
 51. It was shown (Experimental Researches, art. 5-7) that 
 cast iron pillars with flat ends uniformly bore about three 
 times as much as those of the same dimensions with rounded 
 ends ; and this was found by experiment to apply to all 
 pillars from 121 times the diameter down to 30 times. 
 
250 STRENGTH OF SHORT FLEXIBLE PILLARS. 
 
 52. In flat-ended cast iron pillars, shorter than this, there 
 was observed to be a falling off in the strength; and the same 
 was found to be the case in pillars of other materials, on 
 which many experiments were made, to ascertain whether the 
 results previously mentioned, as obtained from the cast iron 
 pillars, were general. The cause of the shorter pillars falling 
 off in strength, as mentioned above, was rendered very 
 probable by the experiments upon wrought iron ; for in that 
 metal a pressure of from 10 to 12 tons per square inch pro- 
 duced a permanent change in, and reduced the length of 
 short cylinders, subjected to it, (art. 60 of Paper above ;) 
 and about the same pressure per square inch of section, when 
 required to break by flexure a wrought iron pillar with flat 
 ends, produced a similar falling off* in strength to that which 
 was experienced when a weight per square inch, not widely 
 different from this, was required to break a cast iron pillar 
 with flat ends. The fact of cast iron pillars sustaining a 
 marked diminution of their breaking strength by a weight 
 nearly the same as that which produced incipient crushing in 
 wrought iron, and a falling off in the strength of wrought iron 
 pillars, rendered it extremely probable that the same cause 
 (incipient crushing or derangement of the parts) produced the 
 same change in both these species of iron. 
 
 53. The pressure which produced the change mentioned 
 above in the breaking of cast iron pillars was about ^th of that 
 which crushed the material, as given from the experiments 
 upon the metal there used. I shall therefore assume here, as 
 I did there, that one-fourth of the crushing-weight is as great 
 a pressure as these cast iron pillars could be loaded with, 
 without their ultimate strength being decreased by incipient 
 crushing; and it was there shown that the length of such a 
 pillar, if solid and with flat ends, would be about 30 times its 
 diameter. 
 
 54. We shall have, therefore, d=\, in the preceding 
 formula 
 
STRENGTH OF SHOUT FLEXIBLE PILLARS. 
 
 251 
 
 be 
 
 whence in cast iron of the kind used, (Low Moor, No. 3,) 
 
 be 
 
 y = - 
 b 
 
 3_ 
 
 4 
 
 55. To find the force necessary to crush a square inch of 
 the iron mentioned above, in order that the value of c, which 
 is that which would crush the whole pillar if inflexible, might 
 be computed, I made (art. 55 of the Paper before referred to) 
 experiments upon it, both upon cylinders and rectangles; arid 
 the mean strength from five of the results gave, per square 
 inch, 109,801 fts.=49-018 tons. 
 
 56. The value of y above is compounded of two quantities, d 
 the strength as obtained from one of the formulae for long 
 flexible pillars (art. 41 of the present Work), and c the 
 crushing force. 
 
 57. The following Table, which gives the dimensions and 
 breaking weights of eleven short solid pillars with flat ends, 
 together with the calculated values of b, c, and y, will show 
 what degree of approximation the calculated strength bears to 
 the real. 
 
 Here 6 the strength in fts. (arts 39 and 41), 
 
 /Z3-55 
 
 = 98922 x jfj-. 
 58. Short solid pillars flat at the ends, fig. 26. 
 
 
 
 
 
 
 Calculated break- 
 
 
 
 
 
 
 ing weight from 
 
 
 
 
 
 
 the formula 
 
 Diameter 
 
 
 Value 
 
 Value 
 
 Breaking 
 
 be 
 
 of pillar. 
 
 Length of pillar. 
 
 of 6. 
 
 of c. 
 
 weight. 
 
 11 I 3C 
 
 inches. 
 50 
 
 ft. inches. 
 1-008 = 12-1 
 
 fts. 
 
 8327 
 
 21559 
 
 ibs. 
 7195 
 
 tbs. 
 7328 
 
 50 
 
 840 = 10083 
 
 11353 
 
 21559 
 
 8931 
 
 8872 
 
 50 
 
 630= 7-5625 
 
 18515 
 
 21559 
 
 11255 
 
 11508 
 
 50 
 
 315= 3-7812 
 
 60155 
 
 21559 
 
 17468 
 
 16992 
 
 777 
 
 1-681 = 20-166 
 
 16713 
 
 52064 
 
 15581 
 
 15604 
 
 775 
 
 1-260 = 15-125 
 
 27005 
 
 51797 
 
 21059 
 
 21241 
 
 785 
 
 1-008 = 121 
 
 41300 
 
 53142 
 
 24287 
 
 27043 
 
 768 
 
 840 = 10-083 
 
 52096 
 
 50865 
 
 25923 
 
 29363 
 
 777 
 
 630= 7-5625 
 
 88547 
 
 52064 
 
 32007 
 
 36130 
 
 1-022 
 
 1-681 = 20-1666 
 
 44218 
 
 90074 
 
 31804 
 
 35631 
 
 1-000 
 
 1-260 = 15125 
 
 66746 
 
 86238 
 
 40250 
 
 43797 
 
252 STRENGTH OF SHORT FLEXIBLE PILLARS. 
 
 59. The next Table (abridged from Table X., art. 36) 
 contains the dimensions of thirteen short hollow cylinders of 
 the same iron, together with their real and calculated breaking 
 weights, for comparison as before. Here, as before, the ends 
 are flat, and the lengths less than 30 times the external 
 diameter. 
 
STRENGTH OF SHORT FLEXIBLE PILLARS. 
 
 253 
 
 
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254 STRENGTH OF SIMILAR PILLARS. 
 
 Some of these pillars broke into many pieces ; several of 
 them were bored inside, and turned on the outside. They 
 were, with the two exceptions named, very good castings. 
 By the ratio T : C is to be understood the depth of the part 
 extended to that compressed in the section of fracture. v By 
 the value of b is to be understood the breaking weight, as 
 
 O o ' 
 
 calculated from the formula 
 
 D 3.55 $ 3-55 
 
 6= 99338 j-^j , 
 
 for the strength of long hollow pillars in fts., which is given, 
 somewhat abridged, in art. 41. 
 
 It will be observed that the calculated strengths agree 
 moderately well with the real ones in both of the preceding 
 Tables, showing that the resistance to crushing is an element 
 of the strength of sJtort flexible pillars at least. 
 
 COMPARATIVE STRENGTH OF LONG SIMILAR PILLARS. 
 
 61. It has been stated (art. 39, results 5 and 6) that the 
 
 strength of solid pillars with rounded ends varied as^ 6 , and 
 
 73.55 
 that of those with flat ends as /177 . This was when the former 
 
 pillars were not shorter than about 15, nor the latter than 
 about 30 times the diameter. 
 
 62. In the research for the above numbers, I was led to 
 conclude that, if the material had been incompressible, the 
 3*76 and 3*55 would each have become 4, and the T7 have 
 been 2 (see arts. 24 and 33 of the Paper above referred to). 
 
 In that case the strength would have varied as ^ 4 , which is the 
 
 ratio of the strength of pillars according to the theory of 
 Euler ; which theory was intended to apply to the power of 
 pillars to resist incipient flexure, whilst my inquiry was as to 
 the breaking strength. In similar pillars the length is to the 
 diameter in a constant ratio : calling then the length n d, 
 
SIMILAR PILLARS. 255 
 
 where n is a constant quantity, we have, in these different 
 cases, the strength as 
 
 dW d 3 ' 53 d* 
 
 Dividing, these become 
 
 d 2 - B 
 
 63. In the first of these cases the strength varies as a 
 power of the diameter somewhat higher than the square ; in 
 the second somewhat lower ; and in the third, as the square. 
 We may therefore conclude, that in similar pillars the strength 
 is nearly as the square of the diameter, or of any other linear 
 dimension ; and as the area of the section is as the square of 
 the diameter, the strength is nearly as the area of the 
 transverse section. 
 
 64. In deducing the conclusion in the last article, Euler 
 remarks that if of two similar pillars of the same material, one 
 be double the linear dimensions of the other, the larger will 
 but bear four times as much as the smaller, though its weight 
 is eight times as great. Berlin Memoirs, 1757. 
 
 65. The following Table, containing the results from such 
 of my experiments on solid uniform cylindrical pillars as were 
 from models similar in form, will show how far the above 
 conclusions agree with the results of experiments. 
 
256 
 
 SIMILAR PILLARS. 
 
 Diameters of 
 pillars compared. 
 
 Length of 
 pillars com- 
 pared. 
 
 Breaking 
 weight of 
 pillars. 
 
 Powers of the dimensions 
 to which the breaking 
 weights are proportional. 
 
 1 
 
 inch. 
 
 C '497 
 99 
 
 inches. 
 7-5625 
 15-125 
 
 Ibs. 
 5262 
 19752 
 
 1-908 i 
 
 . 
 
 3 - 
 
 76 
 1-52 
 
 15-125 
 30-25 
 
 9223 
 32531 
 
 1-819 
 
 
 1 
 
 % 
 
 99 
 
 a-97 
 
 30-25 
 60-5 
 
 6105 
 25403 
 
 2-057 
 
 to" 
 
 
 f -51 
 1-56 
 
 20-166 
 60-5 
 
 3830 
 28962 
 
 1-841 
 
 -1 
 
 I 
 
 50 
 997 
 
 30-25 
 60-5 
 
 1662 
 6238 
 
 1-9081 
 
 1 
 
 fl 
 c5 
 o 
 
 I s 
 
 S 
 
 51 
 1-02 
 
 15-125 
 30-25 
 
 6764 
 21844 
 
 1-6913 
 
 
 - 
 
 50 
 
 a -022 
 
 10-083 
 20-166 
 
 8931 
 31804 
 
 1-8323 
 
 
 66. In the preceding Table, the pillars being from similar 
 models were assumed to be similar, notwithstanding slight 
 deviations in the measures. It appears that the power of the 
 lineal dimensions, according to which their strengths vary, is 
 somewhat lower than the second. 
 
 67. If long pillars be so formed as to resist being crushed 
 by the breaking weight, as has been mentioned before, they 
 will be similar. 
 
 We have seen (art. 52-3) that when pillars require a force 
 to break them by flexure, which exceeds a certain portion of 
 the force which would crush them, if they were not flexible, 
 the pillar sustains a considerable diminution in its power of 
 resistance to flexure in consequence of a partial crushing, or 
 crippling of the material. Suppose c d* the crushing force 
 of the pillar (d being the diameter), or that pressure which 
 would cause rupture in it, if it were too short to break by 
 flexure ; and n c d* 1 that part of this pressure which is the 
 
PILLAES OF VAEIOUS FORMS. 257 
 
 utmost it would, as flexible, sustain without apparent 
 crippling or crushing. Then, since the strength in fts. to 
 resist fracture by flexure in pillars, with both ends rounded 
 
 and both flat, was 33379 ~ and 98922 f^, respectively, as 
 appeared from my experiments, / being in feet and d in inches, 
 we have these quantities each equal to n c d 1 *, in the cases 
 where short pillars, which break by flexure, are bearing, at 
 the time of fracture, the greatest weights they can sustain 
 without any apparent crushing. Whence, in pillars with 
 rounded ends, 
 
 in pillars with flat ends, 
 
 68. In the former of these cases, / varies somewhat faster 
 than as the first power of the diameter, and in the second 
 somewhat slower ; the two showing that, in the case of pillars 
 equally loaded to resist crushing by the weight, the length to 
 the diameter will be nearly in a constant ratio, or the pillars 
 must be similar. 
 
 ON THE STRENGTH OF PILLARS OF VARIOUS FORMS, AND 
 DIFFERENT MODES OF FIXING. 
 
 69. In hollow pillars of greater diameter at one end than 
 the other, or in the middle than at the ends, as in Table XI. 
 (art. 36), it was not found that any additional strength was 
 obtained over that of uniform cylindrical pillars; on the other 
 hand, the strength of these seemed to be the greater ; with 
 respect to this, however, the conclusions were not very 
 decisive. The result from the comparison is in agreement 
 with what may be deduced from Euler's theoretical values of 
 
258 PILLAES OF VARIOUS FORMS. 
 
 the strengths of uniform cylindrical solid pillars, and of those 
 
 o ^ 
 
 in the shape of a truncated cone (Berlin Memoirs, 1757) ; his 
 formulae for these being 
 
 These values are to one another, cateris parilus, as D 4 to 
 D' 2 E 2 ; where D is the diameter of the uniform pillar and 
 D' E the diameters of the two ends of that in the form of a 
 truncated cone. But if we compute the diameter of a 
 uniform cylindrical pillar of the same length and solid content 
 as one with unequal diameters, we shall find the uniform 
 pillar stronger than the other, and the more so according as 
 the inequality of the diameters of the latter is greater. 
 
 70. The strength of a pillar in the form of a connecting 
 rod of a steam engine was found to be very small ; indeed, 
 less than half the strength that the same metal would have 
 given if cast in the form of a uniform hollow cylinder. The 
 ratio of the strength, according to my experiments, was 
 17578 to 39645. 
 
 71. A pillar irregularly fixed, so that the pressure would 
 be in the direction of the diagonal, is reduced to one-third of 
 its strength, the case being nearly similar to that of a pillar 
 with rounded ends, the strength of which has been shown to 
 be only ^rd or that of a pillar with fiat ends.* 
 
 72. Uniform pillars fixed at one end and moveable at the 
 other, as in those flat at one end and rounded at the other, 
 break at -grd of the length (nearly) from the moveable end ; 
 therefore, to economise the metal, they should be rendered 
 stronger there than in other parts. 
 
 73. Of rectangular pillars of timber it was proved experi- 
 
 * Tredgold, art. 283 of his Work on Cast Iron, and in his Treatise on Carpentry, 
 following the idea of Serlio ia his Architecture, recommends circular abutting joints, 
 to lessen the effect of irregularity in the strains upon columns, from settlements and 
 other causes ; but this, we see, is voluntarily throwing away two-thirds of the full 
 strength of the material to prevent what may often be avoided. 
 
CONTINUED PRESSURE UPON PILLARS. 259 
 
 mentally that the pillar of greatest strength, where the length 
 and quantity of material is the same, is a square. 
 
 COMPARATIVE STRENGTHS OF LONG PILLARS OF CAST IRON, 
 WROUGHT IRON, STEEL, AND TIMBER. 
 
 74. It results from the experiments upon pillars of the 
 same dimensions, but different materials, that if we call the 
 strength of cast iron 1000, we shall have for wrought iron 
 1745, cast steel 2518, Dantzic oak 108'S, red deal 78'5. 
 The numbers, all but the last, were obtained from the pillars 
 with rounded ends, and the computations made by the rules 
 used for cast iron. 
 
 POWER OF PILLARS TO SUSTAIN LONG CONTINUED 
 PRESSURE. 
 
 75. In all the experiments of which an account has been 
 given, the pillars were broken without any regard to time, 
 and an experiment seldom lasted longer than from one to 
 three hours. To determine, therefore, the effect of a load 
 lying constantly upon a pillar, Mr. Fairbairn had at my 
 suggestion four pillars cast of the same iron as before, and all 
 of the same length and diameter; the length of each was 
 
 6 feet, and the diameter 1 inch, and they were rounded at the 
 ends. The first was loaded with 4 cwt., the second with 
 
 7 cwt., the third with 10 cwt., and the fourth with 13 cwt. ; 
 this last was loaded with f^- Q of what had previously broken 
 a pillar of the same dimensions, when the weight was 
 carefully laid on without loss of time. The pillar loaded with 
 13 cwt. bore the weight between five and six months and 
 then broke ; that loaded with 10 cwt. is increasing slightly 
 in flexure; the others, though a little bent, do not alter. 
 They have now borne the loads three years. The deflexion 
 of the first pillar is *01 inch, that of the second *025, and of 
 
 's2 
 
260 EULER'S THEORY. 
 
 the third '409. The deflexion of this last pillar, when first 
 taken, was '230 ; and after each succeeding year it was '380, 
 380, and "409, as at present. 
 
 EULER'S THEORY OF THE STRENGTH F PILLARS. 
 
 76. It appeared from the researches of this great analyst, 
 that a pillar of any given dimensions and description of 
 material required a certain weight to bend it, even in the 
 slightest degree ; and with less than this weight it would not 
 be bent at all (Acad. de Berlin, 1757). Lagrange, in an 
 elaborate essay in the same work, arrives at the same 
 conclusion. The theory as deduced from this conclusion is 
 very beautiful, and Poisson's exposition of it, in his 'Me- 
 canique/ 2nd edition, vol. L, will well repay the labour of a 
 perusal. 
 
 77. I have many times sought, experimentally, with great 
 care for the weight producing incipient flexure, according to 
 the theory of Euler, but have hitherto been unsuccessful. 
 So far as I can see, flexure commences with weights far below 
 those with which pillars are usually loaded in practice. It 
 seems to be produced by weights much smaller than are 
 sufficient to render it capable of being measured. I am there- 
 fore doubtful whether such a fixed point will ever be obtained, 
 if indeed it exist. With respect to the conclusions of some 
 writers, that flexure does not take place with less than about 
 half the breaking weight ; this, I conceive, could only mean 
 large and palpable flexure ; and it is not improbable that the 
 writers were in some degree deceived from their having 
 generally used specimens thicker, compared with their 
 length, than have been usually employed in the present 
 effort. 
 
 Some results of the theory of Euler, as given by Poisson 
 (Mecanique, vol. i. 2nd edit.), have been of great service in 
 the- course of the inquiry. 
 
INCIPIENT FLEXUEE. 261 
 
 78. I will now give the leading results, abridged from four 
 of the Tables of experiments on cast iron pillars, enumerated 
 in art. 36, p. 241 ; and the reader who wishes for further 
 information upon them, or upon those of wrought iron or 
 timber, is respectfully referred to the original paper in the 
 Philosophical Transactions of the Royal Society, Part II. 
 1840. 
 
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 !> <*! -^tl CO CO "* i 1 
 Tf( CO CO CO CM 10 CO 
 
 10 10 10 
 
 I ' rH i 1 rH 
 
 r-l rH r-l rH CM CM CM 
 
 
 
 
 CO CM ^ CM 
 
 CM 
 
 GO CO CM O i 1 C^l CO *O O 1O *O .- CM rH *O CO Tfi GO 
 
 -^ c- o i>. : 
 
 1O 
 
 *^t* O CM * CO CO t>- CN CO CM CO " CO OS rH CO CO CM OS 1^ 
 
 C^ -* CO "H 
 
 rH 
 
 rH T 1 ' r-t r t i IrH' i i CM CM CN rH i i CM 
 
 
 
 -^CM CM- -*J^. Ot 
 
 - 
 
 rH 
 
 O O CO '. & O rH G^l 9 O ^ I & O CM *O ^O ^O O i ( ^ 
 
 P *( O O 4 
 
 
 - '-" 
 
 rO 
 
 1O 
 
 
 
 CM CO 
 
 CO CO CO 
 
 O 
 
 rH rH 
 
 f i- ' J|* o 
 
 rH : rH t rH 
 
 
 CO rH CO T* 
 
 CO t- CO O ^ Jt CO 
 
 , , 
 
 
 i> i> 1^- CO OS C?J O* 
 
 1O 1O 1O 
 
 rH rH rH rH 
 
 rH rH rH rH rH rH rH 
 
 
 10 >010 V* 
 
 1O O 1O 1O 1O 1O *O 
 
 CM CM CM 
 
 O O O O 
 
 O O O O 
 
 o o o 
 
 CO CO CO CO 
 
 CO CO CO 50 CO O CO 
 
 CO CO CO 
 
264 
 
 SOLID COLUMNS 
 
 
 
 8 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 "S 
 
 
 
 
 
 
 
 
 s 
 
 
 
 | 
 
 R 
 
 
 M 
 
 
 rri 
 
 
 
 
 r 1 N 
 
 H 
 
 
 E 
 
 
 ( - 1 
 
 4" 
 
 
 p 
 
 
 
 s 
 
 
 JJ 
 
 I 
 
 
 VH 
 
 
 
 
 i 
 
 
 I 
 
 ^ 
 
 
 
 I, 
 
 
 
 
 R 
 
 $3 
 
 
 bD ^ 
 
 o 
 
 
 
 
 I 
 
 J 
 
 
 8 
 
 * 
 
 
 
 
 Qt 
 1 
 
 fl 
 
 
 II 
 
 II 
 
 
 
 
 -+o 
 B 
 
 s 
 
 
 o s 
 
 o - 
 
 
 
 
 
 ep) 
 
 
 CO IO 
 
 CO CO 
 
 
 
 
 2 
 jg 
 
 1 
 
 
 1 a 
 
 s * 
 
 
 
 
 ^ 
 
 U) 
 
 
 ^^ 
 
 CN CN 
 
 
 o 
 
 
 
 <~. 
 
 
 
 
 
 
 
 CO r-l 
 CN CN 
 
 CO CN 
 
 ^H CN CO 
 IO "5 ^ 
 
 CO CO 
 
 
 r^ TH 
 
 1 
 
 
 CO CO 
 
 OS 
 
 J 1 
 
 OS rH O 
 OS rH 
 I-H 71 
 
 CO 10 
 
 CO CO 
 
 
 OS OS 
 CO CO 
 
 III 
 
 
 
 
 
 
 
 
 **- ' E^j r^ 
 
 BJ 01 
 
 O 
 
 CO 
 
 CO 
 
 co 
 
 o 
 
 o 
 
 J- " tlfl 
 
 *- 
 
 
 
 IO 
 
 
 CO 
 
 os 
 
 E 
 
 r*q 
 
 CN 
 
 
 r-l 
 
 CO 
 
 IO 
 
 71 
 
 CO 
 
 rH 
 
 ^ 
 
 
 
 
 ^^ 
 
 OS CO CO 
 
 
 
 o o 
 
 
 r^ fcO 
 
 , ^ 
 
 
 
 i- 10 
 
 r-l 00 CO 
 
 CO CO 
 
 IO O 
 
 OS 10 10 
 
 
 
 
 10 os 
 
 
 
 CN OS 
 
 os co co 
 
 I* 
 
 " 
 
 CO CO 
 
 I 1 I 1 
 
 CM ^ O 
 CO CO CO 
 
 10 10 
 
 
 rH rH rH 
 
 i bC .^ 
 
 
 
 
 
 
 
 
 glf 
 
 o g 
 
 "Jr^OS 
 
 ^ti co ^ co 
 
 CO r-l CO i-H 
 
 O OS .OS 
 r-H CO ; i 1 
 
 ; ; 
 
 
 OS !> rH i> CO rH 
 
 CO OS O IO IO O 
 
 co os co co co co 
 
 
 
 
 71 I I-I 1 71 
 
 <?} CO 'CM 
 
 
 ' 
 
 
 III 
 
 
 ^ CO CM 1^ 
 O r-i O 
 
 i 1 t>- CO > 1 
 
 ooocq 
 
 ^HCN :*~ 
 
 : : 
 
 : : 
 
 >O rH O "* CO O 
 O CO CM CO O CN 
 
 .|s 
 
 
 
 
 
 
 
 
 fil 
 
 . . 
 
 OS 
 
 OS 
 
 CN 
 
 
 CO 
 
 . 
 
 f J 
 
 1 i'~ 
 
 OS 
 
 CN 
 
 T 1 
 
 
 
 
 
 : : :^ 
 
 fi 
 
 
 
 
 
 
 
 
 J 
 
 i 
 
 Is 
 
 OS OS 
 Ol OS 
 
 OS OS 
 CN CN 
 
 rH 7 1 
 
 <M CO TH 
 >O IO O 
 i 1 i 1 r 1 
 
 O <M 
 
 O 
 rH rH 
 
 IO 
 CO IO 
 
 O O 
 
 ip IP IP 
 
 5 
 
 M 
 
 
 >o o 
 
 IO IO 
 
 IO IO IO 
 
 CO CO 
 CO CO 
 
 CO CO 
 
 co co 
 co co 
 
 10 IO 10 
 CM CN CN 
 
 a 
 
 
 
 CO CO 
 
 O 
 CO CO 
 
 O 
 CO CO CO 
 
 O 
 CN CN 
 
 O 
 CN CN 
 
 IO 10 IO 
 i i rH rH 
 
WITH ROUNDED ENDS. 
 
 265 
 
 
 
 ^1 
 
 i* 
 s & 
 
 2 ^ 
 
 -1 
 >> 
 
 ll 
 
 38 
 
 rd C3 
 
 *fl 
 
 lH 
 M 1s 
 
 -a fccni 
 g a o 
 
 ^'a 
 
 % 
 H 
 
 
 
 O2 
 1 
 
 
 
 O 
 
 ^J 
 
 C5 
 -jj 
 
 "a, 
 w 
 
 2 
 
 O 
 
 
 
 'Sn 
 
 CO 
 02 
 
 2 
 
 F 
 
 In these two experiments the area compressed 
 seemed greater than the extended area. 
 The ends were split by the pressure. 
 
 05 CO CO 
 CM CM CN 
 
 CO CO O5 
 
 ^h o "* 
 
 CM C5 O 
 ^ CO CO 
 
 .b- O 
 
 o co co 
 
 
 O t~*O 
 <N rH rH 
 
 T-I CM -rt< 
 CO CO CO 
 
 CM CO 
 ^ CO 
 
 XO rH 
 CO "* 
 
 
 CO 
 CM 
 CM 
 
 at 
 
 M 
 
 >o 
 Jt^ 
 
 Ci 
 
 I 
 
 o 
 
 
 
 10 
 i^ 
 
 i i 
 
 <M 
 CO 
 CM 
 K) 
 
 OO 
 
 O5 
 CM 
 CM 
 
 i>- 
 
 
 
 o 
 
 GO CO CO 
 
 co -* oo 
 
 r-H 1>- 1- 
 
 O Ol 1 
 
 CO C3i VO 
 CO IO 
 r-( C1 CO 
 O O5 OS 
 
 CO CO <M 
 
 oo co ^o 
 
 CO CD i i 
 CD ?0 05 
 
 OO 00 05 
 
 OO I T i 
 
 r-- 10 
 CO ^ ^O 
 
 CO CO 
 O5 O 
 CO O 
 CO CM 
 
 CO I-H 
 CO CO 
 CN O5 
 
 o -^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 : : : 
 
 : : : 
 
 
 
 ; ; 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 : : co 
 : t- 
 
 : : es 
 : : cj 
 
 <o 
 t~ 
 
 . lr^ 
 
 C5 
 
 ' '. "* 
 
 1^ 
 Jt-- 
 
 " o 
 
 : 10 
 
 I>- to 1O 
 
 i-l-i- 
 
 0> 05 C5> 
 Od CV Ci 
 
 co co Jt^ 
 t^. i>- J>- 
 
 r-l O O5 
 >O -^ -^1 
 
 I^Jt^ 
 i-- J>- 
 
 O 
 
 >o o 
 
 
 
 
 
 
 
 8$s 
 
 kTJ IO O 
 CM C<I C^J 
 
 CO CO CO 
 
 
 
 U5 >O O 
 CM <N CN 
 
 CO CO CO 
 O O VC5 
 
 O 
 CN CM 
 
 CO CO 
 O W5 
 
 CM CM 
 r-i T 1 
 CO CO 
 t-.t- 
 
 u-i b *b 
 
 b o c 
 
 O O O 
 
 t- t~ t 
 
 t-t- 
 
 CO CO 
 
 
 
 
 
 
 
266 
 
 SOLID COLUMNS 
 
 It 
 
 !! 
 
 1! 
 
 si 
 
 
 iiameter upon 
 ie ends turned 
 
 
 
 
 Remarks. 
 
 ro jaj 
 
 a jj 
 
 ""* "OT "5 
 CM 0) O 
 
 
 
 
 
 I f. * 
 
 
 
 
 
 111 
 
 
 
 
 
 |ll 
 
 
 
 
 
 r 
 
 
 
 
 i 
 
 
 
 
 
 tjO . 
 i- 25 
 
 S 
 
 I 
 
 00 
 CO 
 
 CO 
 
 CO 
 rH 
 
 if 
 
 CO r-l 
 .OO OS 
 
 CO O 
 CM (M 
 
 rH Ihi CO 
 
 oo os os 
 
 CO 
 
 Jt~ CO i 
 
 CM CO CO 
 5 CO CO 
 CO 10 CO 
 I 1 I 1 t-H 
 
 ill 
 
 , : 
 
 CM CO O OO CO -<J< 
 CO CO CO 00 CO OO 
 
 co ^5 co t- co t^ 
 
 CM t^ <M CO (M CO 
 
 XO CO rH !>.!>. 
 
 CO CO rH CO 
 
 3 |S 
 
 | 
 
 rH <M CM i-H CN CM 
 
 ^COrh^^O 
 
 rH -*! CO i 1 VO 
 rH i 1 rH rH i 1 
 
 ill 
 
 1 i 
 
 t-<M O O O t~ 
 
 O CM CO O rH rH 
 
 O O O ^ -rH O 
 
 ^ *- >0 VO 
 
 |l| 
 
 
 
 
 
 
 ^ 
 
 E5 
 
 00 
 
 rH O 
 
 1 
 
 g. 
 
 ! 
 
 i I 
 T-H 
 
 OS O 
 
 -> CM 
 
 | 
 
 
 ^ 
 
 os ; 
 
 OS 
 
 *l 
 
 g 
 
 
 
 rH ' 
 
 1 
 
 |S? 
 
 r? : :. ' 
 
 CS r-l OS 
 OS OS 
 r-l 
 
 O OS 00 
 CO CM CM 
 rH rH rH 
 
 I 
 
 "So o 
 c co co 
 
 
 
 o ^o *o 
 
 O O 
 CO O O 
 
 00 
 
 CO CO CO 
 
WITH FLAT ENDS. 
 
 267 
 
 1 
 
 "S 
 
 
 
 
 g 
 
 
 
 
 'o 
 
 
 <u 
 
 
 1 
 
 
 i 
 
 
 1 
 
 
 1 
 
 a 
 
 1 
 
 a 
 
 o 
 
 
 j 
 
 1 1 
 
 1 
 
 
 g 
 
 i 
 
 s 
 
 
 1 
 
 a 
 
 
 
 
 
 ^ 
 
 QJ 
 fcO 
 
 
 I 
 
 u 
 
 
 
 1 
 
 1 
 
 J 
 
 
 j 
 
 o 
 
 CO 
 
 
 m 
 
 OS XT5 
 CO CO 
 
 O 
 
 
 
 
 .. 
 
 CM 
 
 rH 
 
 
 co 
 
 CO CO 
 
 CM 
 
 CM 
 
 
 o 
 
 
 co 
 
 
 
 05 
 
 <0 
 
 00 
 
 CO 
 
 00 
 
 rH 
 
 CO 
 
 
 CM 
 
 
 
 f] 
 
 O OO CO 
 CO OS 10 
 
 CM 
 
 to 
 
 cs a o 
 
 CO CM 7-1 
 
 CM OS "* J^ 
 CO O >*l OS 
 
 rH CO CO 
 
 CO 
 
 
 rH CO OO CO 
 
 t- i- 00 
 
 
 oo as oo 
 
 OS 00 rH r-l 
 
 CM CO CM 
 
 
 
 rH rH CM CM 
 
 *-' 
 
 O O 
 os o 
 
 t~ l>. t- rH Jt- 
 
 kQ rH O O> 00 !> 
 
 CO O O *O rH UT5 OS 
 
 ^ CO O OS i i TH CO OO 
 
 CO rH -^ CO I 05 SO 
 rH O t rH VO OS rH 
 CM CM CM CM CM CM CO 
 
 os co 
 
 CO t- OS O CM rH 
 
 "* t^ CM ^ CO OS 
 
 ; OO OO X>- t^ OO CO OO 
 ^ CO t^* J>* "^ t 1 * O 
 rH p- r-l CM i-H rH CM 
 
 CO rH CM O -* 'O 00 
 rH CM O rH rH CM ^1 
 
 un co 
 
 O rH 
 
 00 CM ^H O O OS 
 O CM O O rH i 
 
 * IO OS O <O VO t^ CO 
 lOrHprHOprH 
 
 
 
 
 
 
 HJH 
 
 'I 
 
 
 i- 
 
 O 
 
 rH 
 
 j 
 
 
 CO 05 CO 
 
 H 
 
 
 
 CN CM CM 
 
 
 
 
 
 
 
 
 rH 
 
 * 
 
 lp 
 
 * 
 
 O 
 rH 
 
 i ":' o 10 
 
 rH rH rH 
 
 O 
 
 00 00 
 
 rH CM O 
 O p p O 
 r-i iH rH rH 
 
 IO O * 
 
 a 
 
 o o us 
 
 CM CM <M 
 
 CM CM CM <M 
 
 
 
 o 
 
 CO 
 
 O O 
 CO CO CO 
 
 O O O 
 CO CO CO CO 
 
26S 
 
 SOLID COLUMNS 
 
 
 
 rH 0> 
 
 1 
 
 
 
 
 
 
 
 -+j c5 5 * 1 "" 1 
 
 o 
 
 
 
 
 
 
 
 |SS| 
 
 a 
 
 
 
 
 
 
 
 O d (D 
 
 co . 
 
 
 
 
 
 Remarks. 
 
 
 & s"^ 
 
 places near tl 
 at neutral lin 
 
 1 
 
 
 
 
 
 
 JJPl 
 
 -M O 
 
 
 
 
 
 
 
 ' fe -2 rt 
 
 'o 
 
 
 
 
 
 
 
 s 
 
 pq 
 
 
 
 
 
 O 
 
 H 
 
 
 CO 
 CO 
 
 CO OS 
 
 OS 
 rH 
 
 
 
 CM CM CM 
 
 1 
 
 
 3 
 
 SS 
 
 CO 
 
 
 
 co co oo 
 
 IJf 
 
 O 
 
 1 1 
 
 T 1 
 
 00 
 
 IO 
 rH 
 
 o 
 
 co 
 
 CO 
 
 CO 
 
 OS 
 
 o 
 
 O 
 10 
 <M 
 
 
 os 
 
 fco . 
 
 
 
 ^-^ > 
 
 ^-^-^. 
 
 s-~~x~-~. 
 
 _-+_>,_*-_ 
 
 
 
 ^^__ 
 
 11 
 I s 
 
 . 
 
 ro cO CO 
 jg CO 00 
 
 co co 
 
 rH J>> ITS 
 
 >O OO 
 t^ CO CO 
 CO ^K 
 rH rH rH 
 
 O CO 
 
 CO CO 
 
 >o o t^ 
 
 CO 1^ CO 
 
 OS CO CO rH 
 Jt^ CO C> OS 
 rH CO CO 
 T-H CM OS rH 
 CM CM rH CM 
 
 <M 00 
 
 rH CO 
 rH CO 
 
 O5 T 1 
 CO * 
 
 t ^+t CO 
 CM Oi CO 
 
 .P 
 
 . 
 
 o 55 
 
 ISO 
 CO 
 
 to 
 
 212 
 
 CO 
 rH 
 
 os 
 
 CM 
 
 i 
 
 rH 
 
 *f 
 
 2 - 
 
 * 
 
 r 
 
 
 
 CO 
 
 
 | 
 
 N .^ 
 
 G> <X> l~- 
 
 CO 
 
 -rH CO CO 
 rH rH rH 
 
 OS OS OO OS 
 CM CM C^ CM 
 
 
 
 O O 
 i 1 i 1 rH 
 
 
 IrHrH 
 
 (M (N CM 
 
 ^^ 
 
 
 
 CO CO 
 
 
 i| 
 
 J 
 
 J>- 
 
 CM 
 cp 
 
 * 
 
 g 
 
 o 
 
 O 
 
 'O 
 
 
 
 
 
 
 
 
 1 
 
 |rHrH 
 
 OO OO Jt^ 
 
 CO rH 
 
 
 rH r-H rH 
 
 OS t-CO OO 
 
 
 
 p p 
 
 O O O 
 IO >0 IO 
 
 s 
 
 
 
 rH T 1 
 
 
 
 
 
 d 
 
 bo 
 
 .CO CO 
 01 CO CO 
 
 CO CO CO 
 
 O co co 
 co co co 
 
 CO CO 
 
 co co 
 co co 
 
 8S8 
 
 CM <M CM <M 
 
 CM CM 
 
 
 1 
 3 
 
 o T 1 T 1 
 
 fl O O 
 
 o o o 
 
 O 
 
 us ib ib 
 
 IO ^O *O IO 
 
 xb>b 
 
 CM CM CM 
 
 
 
 
 
 i 
 
 
 
 
WITH FLAT ENDS. 
 
 269 
 
 
 
 "5 % 2 
 
 
 Is 
 
 i|| 
 
 J^fcg 
 
 F^\- * 1 
 
 
 
 j Sir 
 
 
 ubt respectin 
 at more than 
 
 ay bending, a 
 owed a short i 
 mentioned al 
 
 *S : I 
 
 ree-quarters 
 in the direc- 
 
 
 
 
 
 g-|>| 
 
 
 5-q 
 
 T3 
 
 '-M 
 
 5^s 
 
 ^ >,- 
 
 i^ 
 
 ^5g 
 
 
 
 j^ ^i 
 
 
 a^- 
 'a ^ 
 
 ^^_2 
 1 1 g 
 
 
 d J 
 
 ri 2 
 
 |l 
 
 
 
 bese generally bro 
 always in the midc 
 however, usually 
 centre, which tenc 
 
 
 here was a good de 
 neutral line; but 
 half was compress 
 
 ||o 
 
 ' M rS 3 
 -*3 1 
 
 Ujfl 3 
 
 lie first bent, and e 
 other bent and CK 
 the middle. 
 
 ^5 
 
 rM f \ < ^* 
 
 
 
 H 
 
 
 B 
 
 6j 
 
 H 
 
 PQ 
 
 Ci 
 
 CM CM CO 
 
 
 CO 00 CO 
 
 
 O 
 
 
 
 CO rH rH 
 
 rH rH rH 
 
 rH 
 
 rH rH 7 1 
 
 10 
 
 CO 
 
 
 
 rH rH 
 
 CO CO CO 
 
 rH 
 
 CM CM CM 
 
 tj 
 
 o 
 
 
 
 00 
 
 rH rH rH 
 
 
 rH rH rH 
 
 
 CM 
 
 
 
 I 
 
 
 CO 
 
 
 
 oo 
 
 
 
 OO 
 
 CO 
 
 
 VO 
 
 o 
 
 
 
 CO 
 
 
 
 Ci 
 
 Oi 
 
 CM 
 
 
 
 
 co 
 
 co 
 
 ^gj 
 
 co 
 
 VO 
 
 
 CM 
 
 J;^. 
 
 
 
 
 
 CM 
 
 
 CO 
 
 I-H 
 
 CN 
 
 CM 
 
 O CO VO vo 
 
 CO O CO CO 
 VO rH 10 CO 
 CM CM CM <M 
 
 t^ CO CO 
 
 
 
 00 OO Ci 
 
 rH rH rH Oi 
 Ci CO CO 00 
 Tfl VO O rH 
 I VO VO VO 
 CM CM CM CM 
 
 Ci CO CO 
 rH rH rH 
 
 VO rH VO 
 
 CM O Ci 
 
 CM CO rH 
 
 OO OO CO 
 
 Ci CO .t- 
 
 t- OS CO 
 
 i>- 1^ co 
 
 CO 
 CO CM 
 
 CM CM 
 
 CO t~ Ci 
 CO -^1 CO 
 Ci t-rH 
 CO -*< VO 
 CM CM CM 
 
 MM 
 
 
 
 HN 
 
 -, 
 
 
 
 
 rH 
 
 
 
 CO 
 
 i 
 
 
 
 
 i^ oo oo I 
 
 rH rH rH r-t 
 
 co oo co 
 
 
 O CO CO 
 
 H? 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 IO 
 
 
 OO 
 
 
 ^ 
 
 
 
 
 CO 
 
 o 
 
 co 
 
 o 
 
 i^- 
 
 
 
 CM 
 
 CM 
 
 
 
 b- 
 
 
 ' 
 
 
 VO 
 
 vp 
 
 OO Ci Ci 00 
 
 J^ 1- 1- i- 
 
 O O 
 VO VO VO 
 
 OO^COCO 
 
 O O 
 vo KS vo 
 
 CO CO t^ 
 
 O O 
 
 CM CM 
 
 CM CM CM 
 VO O ^ 
 
 
 
 
 
 
 
 
 
 
 CO CO CO 
 
 CO CO CO CO 
 CO CO CO CO 
 
 O IO IO 
 CM <M CM 
 
 10 IO 
 
 CM CM fM 
 
 CM CM CM 
 
 
 
 
 oo co co 
 
 CO CO CO CO 
 
 CO CO CO 
 
 CO CO CO 
 
 co oo co 
 
 
 
 rH rH rH rH 
 
 O O 
 
 O w O O 
 
 10 10 10 
 
 vo vo vo 
 
 !> .1-- i'^ 
 
 
 
 <M CM CM (M 
 
 O O 
 
 o o o o 
 
 t-t^t- 
 
 t- 1- 1- 
 
 CO CO CO 
 
 <M <M 
 
 rH rH rH 
 
 
 
 
 
 
 
 
 
270 
 
 HOLLOW CYLINDRICAL PILLARS 
 
 388 
 
 
 % %*,% 
 
 S 
 
 -E 
 
 
 
 
 s>. -2 
 
 
 ^ ^ pj "& re 
 
 -4J ^2 
 
 J 
 
 ^ ^ "^ 
 
 
 O* "tS * 
 
 
 -* f>^ O ^ ^ 
 
 d o 
 
 oS 
 
 O _, ^4-1 
 
 
 oo ^2 s S 
 
 
 3 -w pn ? O 
 
 O g 
 
 
 rf rd O 
 
 
 
 
 r^ ~V d -S r^3 
 
 
 f 
 
 ,?H ^O ^ 
 
 
 i " J *! 
 
 
 ~'_3 0! -^ 
 
 05 ^ 
 
 D 
 3 
 
 <-. ^ O 
 
 
 .S ^ -2 "* 
 
 
 C CO T3 t*_( 
 
 ^ O 
 
 J3 
 
 s 
 
 
 
 
 p , , Q fr ~* 
 
 ^ ^ 
 
 o 
 
 
 
 ^V.5 
 
 
 * g'o'S 
 
 "11 
 
 J 
 
 i J J 
 
 
 |-| 11, 
 
 
 i^ni 
 
 
 O 
 
 5^ i 
 
 
 "S ^s ^ 
 
 
 S-S'S 4 ^!. 
 
 I 9 8 I* fl 
 
 t 5^ 
 
 J 
 
 ^ > 
 
 ro ^ 
 
 
 "^ ^ ?*^ "S^ 
 
 
 ^ ^4 
 
 "5 m 
 
 -> -^h 
 
 ^^^ 
 
 
 on the strength of hollou 
 force might act through 
 irs were in most cases, e 
 wn, are for the whole len 
 
 Remarks. 
 
 fjlll 
 
 ^ ffl.15 ^ 
 
 iSili 
 
 SlJrN 
 
 iif-sl 
 
 10^ J=-r: ' 
 
 jj-i p j 
 
 ^^ - s b 8 
 1 1.2 S 
 
 ^ 2 i : 
 
 r sunk with the weight 
 than necessary, and wa: 
 lent, as before. 
 
 :ness of the metal at the 
 opposite sides as 19 to 
 
 ! of metal on opposite s 
 4 nearly. The thin si 
 ssed, and the same was 
 illars. 
 
 
 
 
 
 TO LJ O 
 
 M 
 
 
 
 s I s 5s ^ 
 
 
 rt ^ ^ a ^ S 
 
 a oi-g 
 
 ^2% 
 
 cS ^ ^"* 
 
 
 'I *~ 
 
 
 "** ^ t^-*^ fe 
 
 
 3 
 
 a & 1 o 
 
 
 SCQ S? 
 03 ^) CO 
 
 
 *^ O O C3 r (-1 
 
 .-S a > S S 2 
 
 ~l s 
 
 -4J 
 
 OJ O -4J 
 
 /^ a o o 
 
 
 <^ 5 -N 
 
 
 ^ 
 
 fl 
 
 E 
 
 5 
 
 
 K w ^ ^ 
 
 .Jig 
 
 
 
 
 
 
 ft J "^ 
 
 
 
 
 
 
 
 a 3 v 
 
 ijs "* it *) S S 
 
 ^^ 
 
 CO 
 
 o 
 
 CO 
 
 TH 
 
 < ^ o "w 
 
 ** S ^ 1 ^ 'S 5 .3 
 
 a <5? 
 
 I 
 
 
 
 
 ft? - -g - 
 
 "o S ? ^ ^ $2 
 
 jg M< 
 
 1 
 
 S 
 
 CO 
 
 CO 
 
 o 
 
 S JP **"% 
 
 3 4n Q ^* C ^ d 
 
 OO 
 
 OS 
 
 00 
 
 OS 
 
 CO 
 
 x-^>^| _ 03 
 
 i5 || rC^* 3 
 
 
 
 
 
 
 <! ^j 
 
 > H IjgJ 
 
 
 
 
 
 
 s^- i ! 
 
 
 Ui 
 
 I 1 
 
 ^ 
 
 OS 
 
 OS 
 
 HH ^ ^ E^ 
 
 t u ** ^'6^ c 
 
 TO CO 
 
 jfi 
 
 J^ 
 
 CO 
 
 oo 
 
 
 
 i 
 
 E * i 
 
 ^S'^- 16 
 
 10 
 
 5 
 
 CO 
 
 
 (N 
 
 tT's s * 
 
 w ^-^ ^ 
 
 
 
 
 
 
 ^ gn S 
 ^ "^ 3 O 
 
 *J ** 
 
 
 
 
 
 
 SI <>!> 
 
 Iff I 
 
 tin 
 
 t^ OS CO ' 
 to CO <M CO 
 ig (N 00 CO 
 
 ^ CN CO 
 
 rH CO O 
 
 CO -* O 
 <N CO 1-1 
 (M CO CO 
 
 CO OS 
 
 rH CM -^ 
 r-H .-1 r-( 
 
 CO CO 
 CM CO r 1 
 CN OO CN 
 
 |flt 
 
 uotxayao: 
 
 ^CO (N OS 
 C O CO "etl 
 
 (N CO CO 
 
 O CO X~ 
 
 <* t^ (M 
 |>1 CO t- 
 
 rH CM O5 
 
 O CM CO 
 
 o S "1 *8 
 
 
 .r- 
 
 
 
 
 
 jljj 
 
 
 d - " : ' 
 
 - 1 
 
 - 
 
 .a S, 
 
 d jo 
 
 ^ "^ ^ 05 
 
 c3 
 
 ^ CO ^^ 
 
 -rt< OO -"W 
 
 r-\ i ( |(N 
 
 CO O -W 
 
 CO -rf< r ~' 
 
 tg 
 
 S 
 
 i^^So 
 
 1-^H 
 ^^^ 
 
 O -* CO 
 
 CO t- CO 
 
 
 ft 
 
 
 ^ 
 
 
 . b 
 
 ll 
 
 fc r * fe 
 
 - '< 
 
 o 
 
 3 J ,g 
 
 Q) r% 
 
 s ^ 
 
 1 -s 
 
 
 [II. Hollow cyli 
 i (the Low Moor, . 
 ?nds move freely. 
 ' in dry sand, boti 
 
 p 
 
 If 
 
 External diame 
 Internal do. 
 Weight of cylin 
 
 External diame 
 Internal do. 
 Weight of cylin 
 
 External diame 
 Internal do. 
 Weight of cylin 
 
 External diame' 
 Internal do. 
 Weight of cylin 
 
 g 1 3 1 
 
 fuocnuodxa 
 
 i i 
 
 (M 
 
 CO 
 
 
 
 id 
 
 M 
 
 jo aaqtun^j 
 
 
 
 
 
 
ROUNDED AT THE ENDS. 
 
 271 
 
 
 ^r|is 
 
 f 
 
 *" 
 
 i 
 
 
 o 
 
 .2 
 
 2 
 
 
 "^ 'O "S 
 
 CQ 
 
 ci 
 
 
 
 <M 
 
 -g 
 
 5 
 
 
 **! PW 
 
 O) 
 
 O 
 
 *0 
 
 
 2 
 
 3 
 
 1 
 
 
 i j's 
 
 
 
 'i 
 
 o 
 
 
 -g 
 
 p 
 
 IM 
 
 
 1 "^ ^ 
 
 c3 
 
 > 
 
 2 
 
 
 c5 
 
 O 
 
 o 
 
 
 g^ 
 
 Q 
 
 g 
 
 'Ei 
 
 
 45 
 
 3 
 
 
 
 
 III 
 
 
 1 
 
 1 
 
 
 ?S 
 
 9 
 
 1 
 
 1, 
 
 
 ^-^ d 
 
 _ 
 0) 
 
 c 
 
 3 
 
 
 % 
 
 o 
 
 J 
 
 
 * ci ^ 
 
 I 
 
 IH 
 
 o 
 
 w 
 
 
 A 
 
 <D 
 
 ^ 
 
 
 111 
 
 S 2> 
 
 2 
 
 1 
 
 d 
 I 
 
 
 
 
 1 
 
 
 
 3 
 
 i 
 
 
 5 ii 
 
 S g 
 
 8 
 
 .2 
 
 
 P 
 
 -g 
 
 IM 
 
 o 
 
 
 11* . 
 
 o 2 
 
 1 
 
 o 
 
 
 02 
 DQ 
 
 13 
 
 
 
 .J QJ -U 0) 
 
 rr^ <D S ^ 
 
 1 | 
 
 a 
 
 3 
 
 
 1 
 
 1 . 
 
 I 
 
 
 f- 1 :2 - * 
 
 f- erf 
 
 o 
 
 ^3 "* 
 
 
 .2 
 
 'o ^ 
 
 o 
 
 
 H O fl ^ 
 
 > 
 
 
 _^j -_ 
 
 
 J 
 
 ""^ 
 
 P^H 
 
 
 p5 "^ >> 
 
 1^ 
 
 DQ 
 
 rj + 
 
 
 +H 
 
 w _O 
 
 
 
 |||| 
 
 |! 
 
 q 
 1 
 
 ^ CO 
 
 la 
 
 
 it 
 
 Jos 
 
 ci 
 
 ll 
 
 
 "E'rS't ( ^ 
 
 05 d 
 
 53 os 
 
 2 | 
 
 
 ii 
 
 I'l 
 
 
 
 |if^ 
 
 g s 
 
 1" 
 
 rd *" 
 
 H 
 
 
 r 
 
 H 
 
 |2 
 
 
 CM 
 
 
 
 
 CO 
 
 
 
 
 
 
 cp 
 
 CM 
 
 ip 
 
 os 
 
 
 >o 
 
 ^n 
 
 1 
 
 
 Jb- 
 
 cb 
 
 CO 
 
 6s 
 
 6s 
 
 6s 
 
 6s 
 
 
 
 
 os 
 
 
 co 
 
 
 
 
 Oi 
 
 
 rH 
 
 OS 
 
 
 rH 
 
 OS 
 
 OS 
 
 o 
 
 rH 
 
 CO 
 
 CO 
 
 rH 
 
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 CO 
 rH 
 
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 7-H 
 OS 
 CO 
 I-H 
 
 K5 
 
 co 
 
 I-H 
 
 
 OS 
 
 T 1 
 
 rH 
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 rH 
 
 CO 
 
 i 
 
 3 
 
 CO 
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 co 
 
 CM 
 
 | 
 
 co 
 
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 OS 
 
 o 
 
 r-l IO OS 
 T* Ml CO 
 
 CM CM 
 
 CO CO OS 
 CM CM CO 
 
 1-H I 1 t 
 
 I-H 05 CO 
 
 rH CO CO O 
 -T* O OO rH 
 
 CO IO t 
 O O CO 
 
 gOSrH 
 
 1O OS CO 
 rH rH CM 
 
 rH CM CO 
 
 i i * co 
 
 CM CM CM 
 rH rH 
 
 CO 
 * CO 
 rH 
 
 rH CO CM 
 ^ CO OS 
 
 CM CO CO 
 
 cc i-- co 
 
 rH rH CO 
 CM OS CO CO 
 
 1 1 I-H 
 
 co i r 
 
 CO <N CM 
 rH CM 
 
 CO CM * 
 
 CO rH 1>- 
 
 CM CM 
 
 "I CM OS 
 
 CO rH 
 rHCNCM 
 
 I-H CO CO 
 CO O 
 rH CO ^ 
 
 ^O 
 
 
 0.. 
 
 C-. 
 
 
 J 
 
 
 
 
 rH -* CO 
 O CO CO 
 
 OCM 
 
 r-H O CM 
 
 ' 
 
 rH CO *M 
 
 OCM CO 
 
 . . . 
 
 ri ^ OS 1 O 
 
 S o TJI co 
 
 Q CO O 
 
 CM ^ O 
 O rH rH 
 
 Jt^ CO IO 
 CM t^ 
 
 OS CM O 
 CO rH 
 
 
 
 
 
 
 
 - 
 
 rH 
 
 
 rH 
 
 * 
 
 .2 
 
 .2 1 
 
 .2 g 
 
 2 J 
 
 
 5 
 
 
 
 * CO W 
 |i 
 
 s* 
 
 !?3T 
 
 rt ,> 
 
 i^ co " 
 
 SS5! 
 
 CO ' 
 <*! CO O 
 
 CO t^ 1 " 1 
 t-rH CO 
 CM CM "** 
 
 '-> 
 
 Ss- 
 
 $4 
 
 rH CO -W 
 O -* 
 COCN 10 
 _, fn 
 
 CO oo <5i 
 
 In 
 
 rS 
 
 J4 " 
 
 " . d 
 
 a o "3 
 
 l'^ 
 
 <_, 
 
 14 
 
 d " Js 
 
 v JS 
 
 -*- *"O 
 
 2 6.2 
 
 Jl* 
 
 w 
 
 -^ 
 
 III 
 
 External 
 Internal 
 Weight o 
 
 11 
 II 
 
 ill 
 
 External 
 Internal 
 Weight o 
 
 iii 
 
 External 
 Internal 
 Weight o 
 
 External 
 Internal 
 Weight o; 
 
 ill 
 
 External 
 Internal 
 Weight o 
 
 CO 
 
 <- 
 
 CO 
 
 cs 
 
 
 
 1 1 
 
 rH 
 
 rH 
 
 CM 
 
 CO 
 rH 
 
 r^ 
 
272 
 
 HOLLOW PILLARS (ROUNDED ENDS). 
 
 
 o 
 
 
 
 
 1 11 
 
 
 
 
 
 
 
 c3 &r ^ 
 
 
 
 
 
 
 "fH 2 
 
 
 1 
 
 
 
 . 
 
 w 5 
 
 
 B 
 
 
 
 01 
 
 ."3 ^^ 
 
 
 *-i 
 
 
 
 o 
 
 '3 TJ. 
 
 
 o 
 
 
 
 
 
 
 
 
 
 o 
 
 ^s CO C3 
 
 
 "S 
 
 
 
 r-{ 
 
 | 3 1 
 
 
 1 
 
 
 
 (D 
 
 
 TO 
 
 ~rf 
 
 
 
 03 
 
 r^4 ?? 3 
 
 
 tn 
 1 
 
 
 
 f 
 
 :! 
 
 I 
 
 1 
 
 
 
 
 
 "** "^ S 
 
 O 
 
 
 
 
 
 d ^ QJ 
 
 H 
 
 o 
 
 
 
 Qj 
 
 
 
 a 
 
 
 
 O 
 
 ^ M -2* 
 
 
 4-* 
 
 
 
 d 
 
 ^-* ^ r^ 
 
 
 .9 
 
 
 
 o 
 
 rf qj "^ 
 
 
 i 
 
 a 
 
 
 
 1 
 
 .2 *> S ^ 
 
 ^11% 
 
 
 **^ 
 
 
 
 o 
 
 r-1 ^H ^ .^ ) 
 
 j l 
 
 a 
 
 
 
 w 
 
 B 
 
 a*lll 
 
 
 o 
 
 1 
 
 
 
 d 
 
 <u ,_< o^ t> 
 
 S ^ tjoo o 
 
 
 ff 
 
 
 
 H 
 
 s d s 
 
 P 11411 
 
 
 
 
 
 
 *** ^c3 L j || ,0 ^ g 
 
 j ! 
 
 
 CO 
 
 
 
 ^ 3 **" 1 ' . * ^ c3 
 
 
 CO 
 
 CO 
 
 CO 
 
 o 
 
 *O 3 i 1 ^ ^ JT3 
 
 
 
 CO 
 
 
 *o 
 
 5 
 
 
 "^ co 
 
 01 
 
 Ol 
 
 o 
 
 co 
 
 2 <S Ifi Qi C 2 r^ 
 
 CO 
 
 o 
 
 OS 
 
 
 
 
 1 "fjsf 
 
 
 rH 
 
 
 rH 
 
 
 fail's 
 
 ri 
 
 1 
 
 CO 
 Oi 
 CO 
 
 CO 
 CO 
 
 CO 
 CO 
 
 O .r^T~* .r^ *P( 5j 
 
 s9 ^ 
 
 
 CO 
 
 
 co 
 
 w Msji 
 
 
 01 
 
 1 1 
 
 CO 
 
 CO 
 
 bo ( . 
 
 
 
 
 
 
 *^ .2 o s 
 
 IO *O "^ft rH Tj* 
 
 rH rH CO 
 
 
 
 ^, 
 
 ^ -2 
 
 CD 1O rH O1 rH Oi 
 CO rH CO IO -^ 
 
 rH in Ol 
 
 
 
 co 
 
 10 
 
 |*J 
 
 ^ CO CO CO CO O5 
 rH CO 'sji -5H 
 
 CM Jt~ 
 
 
 
 Ol 
 
 CO 
 
 
 *' ^ Cft ^ 
 
 "Q Ol ^O 
 
 
 
 
 uotxayoa 
 
 ^ ^ C> CO Oi O 
 
 g 004 
 
 
 
 CO 
 
 
 
 ^-^>_^ 
 
 Jj 
 
 d 
 
 a S 
 
 Description of Pillar. 
 
 .2 '^ 
 
 r^ **"* f 1 
 
 ^ CO cp t^ 
 "SO CN1 *" 
 
 II 1 
 
 || ^ .S 
 
 fill 
 
 d uniform pillar rounded 
 ; the ends . 
 neter 2 "24 in. 
 ght 93 Ibs. 
 
 f External diameterl'781 
 Internal do. 1'21 
 Length 4 feet 9 inches. 
 
 External diameter 2 - 31 i 
 Internal do. T67 
 Length 4 feet 9 inches. 
 
 External diameter T85 i 
 Internal do. T36 
 Length 2 feet 7 inches. 
 Weight of 2 feet 5 inch 
 1 =8fos. 15^ ox. 
 
 
 
 "S s ' s 
 
 ^ c^ c5 O 
 
 'OAOQTJ OSOTT1 
 
 
 Kr5^ 
 
 10 S^ 
 
 ireift sq'jSnoi .T9;.toqs jo s.tunTd 
 
 srsss 
 
 10 
 
 I 1 
 
 CO 
 rH 
 
 rH 
 
 CO 
 
 II 
 
 Ci 
 
VALUE OF CO-EFFICIENT. 273 
 
 The value of #, obtained in the sixth column of the 
 preceding Table, is the strength of a solid pillar, 1 inch 
 diameter, and of the same length as those above. For, when 
 the length is constant, the strength W varies as D 3 ' 76 d 3 ' 76 , 
 (arts. 39 & 41) ; and therefore, to find x, the strength of a 
 solid pillar, 1 inch diameter and 7 feet 6f inches long, we 
 
 have D 376 ^ 376 : I 3 ' 76 : : W : ^pj^,,. The mean from 
 the values of so found in the Table above is 932*76Ibs., and if 
 this be multiplied by l l '\ where /=7'5625 feet (=7 feet 
 6f inches), we obtain the strength in pounds of a solid pillar, 
 1 foot long and 1 inch diameter, rounded at the ends ; and 
 this, divided by 2240, to reduce it into tons, is the co-efficient 
 12' 9 7 9, called 13, in the formula for the strength, 13 x 
 
 D L ~ , (art. 41), where the index 3*6 is put as a mean 
 
 between the 3 '76 of pillars with rounded ends, and the 3' 5 5 
 of those with flat ends. The remarks here made will apply to 
 the values of w in the next Table. 
 
274 
 
 HOLLOW PILLARS 
 
 
 s ^ 
 
 Ci e*i, 
 
 1 fill 
 
 fl H;) ft 01 co C) _ | 
 
 Imm 
 
 tSMSS*- - 
 
 O ni r O 
 
 be 
 ced 
 tle 
 
 d 
 h 
 
 y 
 u 
 litt 
 be 
 
 is cylin 
 ner as t 
 fracture 
 half-wa 
 the red 
 was a li 
 whole b 
 
 .CO TH 
 
 jco oo 
 
 t-t^-l CO 
 
 rH CO <M >H 
 
 CN TH OS CO 
 
 T i OS J>- CO 
 
 73 rH 00 rH ^ .g 
 
 do. 
 ind 
 
 h 7 
 
 "i-ss- 
 
 2 x too.3 o 
 gH.g^ ^J 
 ^ 'V^r* ^ 
 ^.S 8 35 
 
 isMsr* 
 s H^^^ 
 
 I |"S 
 
 " 
 
 
 Exte 
 Inter 
 
WITH FLAT ENDS. 
 
 275 
 
 T3 
 
 a -2 
 
 CD 
 
 Ij 
 
 * fl 2 r jf 
 
 O t fl -S 
 
 || 
 
 III!] 
 
 
 O Q} 
 
 s 
 
 r-l C 00 "1 d 
 
 r-] ,,, 
 
 
 '% 
 
 II 
 
 II 
 
 fl!iiS 
 
 rt rt 
 
 _fl a^S 2 
 
 ^ "" <D 
 
 'o.S 
 
 gi 
 
 O fl -Jj 
 
 1 
 
 4 
 ij 
 
 OQ ""2 
 g 
 l> o 
 
 9 <D 
 
 O 
 
 4) fe 
 O ^ 
 
 4Jj 
 
 j! ? 
 
 ader was the same as th 
 mt 3 of the last Table. Ii 
 quite straight, and its ( 
 bedded : it was reduced, 
 lit 2, and it broke in th 
 three of the reduced plac 
 
 II 
 
 ls| 
 
 <l) o ^ 
 
 a <u g 
 
 nder was the same as thai 
 ible ; it was now reduced 
 . the same manner, and to 
 , as in the preceding ones, 
 middle, and at one of th 
 near to the middle. 
 
 3 fl a 
 
 "~O ~^ 
 
 rQ "O 
 
 . i q,; v ^_j 
 
 ^a^ 
 
 ^ C-H .3 $ 00 
 
 III 
 
 .9 a 
 
 ^3 o3 
 
 go 
 fl 
 
 111 111 
 
 |u 
 
 II ll*t 
 
 H 
 
 ^ 
 
 02 
 
 p 
 
 
 JH 
 
 
 
 o 
 
 rH 
 
 
 
 CM 
 
 rH 
 
 rH 
 
 rH 
 
 feV 
 
 OS 
 
 rH 
 
 rH 
 
 * 
 
 IQ 
 
 CO 
 
 * 
 
 OS 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 
 CO 
 
 CO 
 
 o 
 
 4^ 
 
 OO 
 
 6s 
 
 
 
 os 
 
 rH 
 
 co 
 
 
 10 
 
 *o 
 
 
 
 cs 
 
 <o 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 CM 
 
 co 
 
 1 
 
 CO 
 
 rH 
 
 OS 
 OCi 
 
 o 
 
 CO 
 10 
 CO 
 CO 
 
 1 
 
 co 
 o> 
 
 8 
 
 CO 
 
 CO 
 
 
 
 
 1^1^ OS 
 
 co t^ o 
 
 <M -rH 1O 
 
 OS CO l>-rH 
 
 GO o .r o 
 
 >o t- os co 
 
 CO ^H OS rH 
 rH CM CO 
 
 OS .t r^;, CO t 
 
 oo co ce >o i^ 
 
 CO CO OS 
 CO CO i i CO OS 
 rH a CM CN 
 
 3 
 
 rH t- >T3 i-- rH 
 
 iO O eg i 1 -* 
 C-l CO O OS >O 
 
 "* CM'O CM CM 
 
 CO O rH 
 *O CO 
 ^ CO CM J^ 
 rH U5 -* CO 
 rH CM CM 
 
 
 
 
 3 
 
 3 
 
 
 
 r-c OS IO 
 
 -*rl GO lr~ CM 
 
 O CO 1O 
 
 CO CO CO CO 
 O O O CO *O 
 
 "fl O i ^ VO 
 S rH O rH CM 
 
 "S O O O 
 5 CM kp OS 
 
 - 
 
 
 
 
 
 1 
 
 4 
 
 
 
 g 
 
 
 OS 
 
 !> 
 
 
 
 
 
 CO 
 
 
 
 
 ww 
 
 
 
 "^ 
 
 4 
 
 
 00 
 
 lo 
 
 
 
 a 
 
 1O T ~* 
 
 ^l CO 
 
 rH CO ^3 
 
 rH rH 
 
 OS rH O 
 
 CO Trl 
 
 !- rH CM 
 
 O ^f 
 
 O CO *-i 
 
 -^ 
 
 OS CO '^ 
 
 CN 10 J3 
 
 rH rH ^ 
 
 <N T-l 
 
 CM i i ^~ 
 
 CM rH 
 
 rH T( 
 
 (M tH 2 
 
 * | 
 
 $ d 
 
 1 -1 
 
 - s 
 
 J .si 
 
 1 fll 
 
 Ff- 
 
 r rt 
 
 |4Tj 
 
 2 ^JH 
 ^3 C! 10 
 
 ^rH 1 ^ 00 
 
 |-*sS 
 
 l*fl 
 
 lHi*t? 
 
 6 ? o 
 
 S rr; *o S 
 
 1 dl 
 ^-^' Q 
 
 0> . "- ** 
 rrt ** **"* 
 
 III 
 
 Sa5^ 
 
 Uri 
 
 Hr3^^ 
 
 lilt 
 
 W fl C3 K-J^. 
 
 1^5 
 
 o - to to 
 
 IIJI 
 
 T rt * 
 
 S fl 5 rd 
 
 S js tc t>o 
 tj -S fl 'a> 
 
 ^J^ 
 
 ||f-f 
 
 * 
 
 
 
 CO 
 
 - 
 
 CO 
 
 cs 
 
 T2 
 
276 
 
 HOLLOW PILLARS. 
 
 CO O r-l 00 
 
 rH 00 CM Jt^ 
 
 ** O T-I CM 
 
 r-l <N C* 
 
 1O O >O O 
 
 I I 1O Oi r-H 
 r-l CO O J^ 
 000 O rH 
 ri rH GS GN 
 
 IJ 
 
 
 
TRANSVERSE STRENGTH. 277 
 
 TRANSVERSE STRENGTH. 
 
 79. The transverse strain is that to which cast iron and 
 some other materials are most frequently subjected, and 
 therefore experiments have been oftener made that way than 
 any other. Still, as regards cast iron, whose uses are 
 multiplying every day, the knowledge of the practical man 
 has hitherto been far from equalling his wants ; and 
 accordingly various efforts have lately been made, and 
 doubtless will continue to be made, to obtain extra informa- 
 tion upon so important a subject. 
 
 I will give an account of some of these, the objects of 
 which are as below. 
 
 1st. To ascertain what alteration takes place in bars of cast 
 iron subjected to long-continued strains. 
 
 2nd. To determine the effects of changes in the temperature 
 of bars upon their strength. 
 
 3rd. To inquire into the elasticity and strength of cast iron 
 bars, under ordinary circumstances, the time when the 
 former becomes impaired, and the erroneous conclusions that 
 have been deduced from it. 
 
 4th. To find the best forms of beams, and the strength of 
 beams of particular forms. 
 
 80. LONG-CONTINUED PRESSURE UPON BARS OR BEAMS. 
 
 To ascertain how far cast iron beams might be trusted 
 with loads permanently laid upon them, Mr. Fairbairn made 
 the following experiments. (Export on the Strength of Cast 
 Iron, obtained from the Hot and Cold Blasts, vol. vi. of the 
 British Association). 
 
 He took bars both of cold and hot blast iron (Coed Talon, 
 No. 2), each 5 feet long, and cast from a model 1 inch square; 
 and having loaded them in the middle with different weights, 
 
278 LONG-CONTINUED PRESSURE. 
 
 with their ends supported on props 4 feet 6 inches asunder, 
 they were left in this position to determine how long they 
 would sustain the loads without breaking. They bore the 
 weights, with one exception, upwards of five years, with 
 small increase of deflexion, though some of them were loaded 
 nearly to the breaking point. Since that time, however, less 
 care has been taken to protect them from accident, and three 
 others have been found broken. They are carefully examined 
 and have their, deflexions taken occasionally, which are set 
 down in the following Table, which contains the exact 
 dimensions of the bars, with the load upon each. These 
 experiments were undertaken by Mr. Fairbairn at my 
 suggestion, as I was led to conceive, from experiments I had 
 made in a different way upon malleable iron, that time would 
 have little eifect in destroying the power of beams to bear a 
 dead weight. 
 
LONG-CONTINUED PRESSURE. 
 
 279 
 
 '0^0- 1 
 
 noji jjwqq ^OH 
 
 .8* 
 
 o -p r o" g ^JJ 
 
 l 
 
 CO iO VO t~l>- O5 
 
 g ^uorauodxg; 
 
 '000- T 
 
 UO.TT ^SB^q 
 
 '020- T -"*q jo q^daa 
 
 "080. T 
 
 nen 
 each 
 
 h a pe 
 laid u 
 
 ion 
 392 
 
 fi^ 
 
 >o co o i>-ia co co 
 
 i 10 <O CO i^ CO 1 
 
 JC-CMOO rH CM 
 
 0> O OS O O 
 l-^ co t- co co 
 
 CO O t~ ~* CM i-H ^ 
 1-1 CM CO CM CM O CM 
 t- i- Jt~ t- Jt- CO CO 
 
 o -* "* co -^ooio^oco 
 
 i-HCMCMCM CM i I CM (M CM 
 COCOGOGO QOCOCOCOOO 
 
 h 
 la 
 
 OSO-T Jq JO q^pTOjg 
 
 CM 1O CO 
 O O> CO 
 )O lO O 
 
 -*CO7-HrH rHOsOCOCO 
 CMCOCOCO COi-HCMCMrH 
 10 10 IO U7 ^O 
 
 rH ^ft CO CO 00 CO CO 
 t~ t~ CO <X> CO OS i I 
 CM CM CM CM CM CM CO 
 
 COOCOCO IOC5CO1OIO 
 CMCOCOCO COCOCOCOCO 
 
 K 
 
 'OTO-T -req jo qqp'co.ig 
 
 050- 1 
 
 UOJT ^stsiq 
 'I ^uorau 
 
 D 
 lo 
 
 OO CM CO CO CO rH t^- 
 
 1^ co co oo co as o 
 
 O O O ^ O> O rH 
 
 O O 
 
 i-H CO 
 OS C5 
 
 O <M b- CN rH XO CO 
 
 CO CO CO -^ TH ^* CO 
 
 O5 OS OS OS C5S OS OS 
 
 OO51OCO O^rHCOt^ 
 O5O5O3OS OSOSO5O5O 
 
 JO 9UIT^ ^t? JTB 
 
 oq^ jo 9jn^t?J8dca9X 
 
280 EFFECTS OF TEMPERATURE. 
 
 82. Looking at the results of these experiments, and the 
 note upon the first and fourth, it appears that the deflexion in 
 each of the beams increased considerably for the first twelve 
 or fifteen months ; after which time there has been, usually, 
 a smaller increase in their deflexions, though from four to five 
 years have elapsed. The beam in Experiment 8, which was 
 loaded nearest to its breaking weight, and which would have 
 been broken by a few additional pounds laid on at first, had 
 not, perhaps, up to the time of its fracture, a greater deflexion 
 than it had three or four years before ; and the change in 
 deflexion in Experiment 1, where the load is less than frds of 
 the breaking weight, seems to have been almost as great as in 
 any other; rendering it not improbable that the deflexion will, 
 in each beam, go on increasing till it become a certain 
 quantity, beyond which, as in that of experiment 8, it will 
 increase no longer, but remain stationary. The unfortunate 
 fracture of this last beam, probably through accident, has left 
 this conclusion in doubt. 
 
 83. These important experiments show that cast iron may 
 be trusted with permanent loads far greater than has 
 previously been expected ; it having been generally admitted 
 that about ^rd of the breaking weight was as far as it was 
 safe to load a beam with in practice. It was conceived that a 
 load greater than this would break the beam or other body in 
 time, since the elasticity was thought to be injured with about 
 this weight; and it was accounted unsafe to load a body 
 beyond its elastic force (see arts. 70, 71, Part I. of this 
 volume). 
 
 EFFECTS OF TEMPERATURE UPON THE STRENGTH OF 
 CAST IRON. 
 
 84. Mr. Fairbairn gave, in the report previously referred 
 to, the results of several experiments to determine how far the 
 strength of cast iron bearers is influenced by such changes of 
 
EFFECTS OF TEMPERATURE. 
 
 281 
 
 temperature as they are occasionally subjected to. He had a 
 number of bars cast, of Coed Talon Iron, Nos. 2 and 3, part 
 of them being of iron made with a heated blast, and part 
 with cold. These bars were from models 1 inch square and 
 2 feet 6 inches long. They were laid on supports 2 feet 3 
 inches asunder, and broken by weights hung at the middle. 
 The experiments were made in winter. Some bars were 
 broken in the open air ; some when immersed in frozen water 
 or covered with snow ; some in melted lead ; and others when 
 heated red hot. The results are in the following Table, the 
 first column of which shows the temperature under which the 
 experiments were made ; the second and third columns give 
 the breaking weights of the bars in pounds, when reduced by 
 calculation to exactly 1 inch square ; and the fourth column 
 gives the ratio of the strengths of the cold and hot blast irons 
 in the two preceding columns. 
 
 Temperature, 
 Fahrenheit. 
 
 Coed Talon, 
 cold blast. 
 
 Coed Talon, 
 hot blast. 
 
 Ratio of the strengths 
 of the two irons. 
 
 
 No. 2 Iron. 
 
 No. 2 Iron. 
 
 
 
 Ibs. 
 
 Ibs. 
 
 
 16 
 
 
 800-3 
 
 
 26 
 
 851-0 
 
 823-1 
 
 1000 : 967-2 
 
 
 mean 
 
 mean 
 
 
 32 
 
 940-7) 9 , 9 . 6 
 958-5 \ y4Jb 
 
 933-4) - 
 906-0 \ yiy7 
 
 1000 : 977-6 
 
 190 
 
 743-1 
 
 823-6 
 
 1000 : 1108-3 
 
 Red in the dark 
 
 723-1 
 
 829-7 
 
 
 Perceptibly red ) 
 in daylight \ 
 
 663-3 
 
 
 
 
 No. 3 Iron. 
 
 No. 3 Iron. 
 
 
 
 mean 
 
 
 
 212 
 
 905-0) Q9 ,.o 
 943-6 i 924 ' 
 
 818-4 
 
 1000 : 885-4 
 
 
 
 mean 
 
 
 600 
 
 909-3) 1fm .i 
 1157-0 \ 1( 
 
 834-1) 9 ~. fi 
 917-5 \ 8 ' 5 ' 
 
 1000 : 847-7 
 
 85. It would appear from these experiments, though the 
 results are somewhat anomalous, that the strength of cast iron 
 is not reduced when its temperature is raised to 600, which 
 is nearly that of melting lead ; and it does not differ very 
 widely whatever the temperature may be, provided the bar be 
 not heated so as to be red hot. 
 
282 STRENGTH OF BARS. 
 
 ON THE STRENGTH OF CAST IRON BARS OR BEAMS UNDER 
 ORDINARY CIRCUMSTANCES, THE TIME WHEN THE ELAS- 
 TICITY BECOMES IMPAIRED, AND THE ERRONEOUS CONCLU- 
 SIONS THAT HAVE BEEN DERIVED FROM A MISTAKE AS TO 
 THAT TIME. 
 
 86. It is, as has been before observed, to ascertain the re- 
 sistance of materials to a transverse strain, that the efforts of 
 experimenters have chiefly been directed : one reason for this 
 seems to be the great facility* with which bodies can be 
 broken this way comparatively with others, which require 
 large weights, or complex machinery, and often considerable 
 attention to theoretical requirements. 
 
 The inquiry into the strengths of hot and cold blast cast 
 iron, which I undertook, in conjunction with Mr. Fairbairn, 
 for the British Association, and whose results were inserted 
 in the sixth volume of their Reports, induced me to make a 
 number of experiments on the transverse strength of bars. 
 In these experiments, most of which were made before 
 Mr. Fairbairn's avocations enabled him to attend to the matter, 
 I had a number of bars cast from models 5 feet long and 1 
 inch square; and the castings were supported on props 4 
 feet 6 inches asunder, and broken by weights suspended from 
 the middle. These bars were made long and slender, in 
 order that small variations in the elasticity might be rendered 
 obvious. My earliest experiments upon the Carron Iron, and 
 particularly those upon the BufFery, convinced me that the 
 elasticity of bars is injured much earlier than is generally sup- 
 posed ; and that instead of it remaining perfect till one-third 
 or upwards of the breaking weight was laid on, as is generally 
 admitted by writers (Tredgold, art. 70, &c.), it was evident 
 that ^th or less produced in some cases a considerable set, or 
 defect of elasticity ; and judging from its slow increase 
 afterwards, I was persuaded that it had not come on by any 
 
DEFECT OF ELASTICITY. 283 
 
 sudden change, but had existed, though in a less degree, from 
 a very early period. I mentioned the fact and my convic- 
 tions some time afterwards to Mr. Fairbairn, and, after 
 examining the matter with more attention than before, 
 expressed a desire to have bars cast of a greater length than 
 the preceding ones, to render the defect more obvious. I had, 
 therefore, two bars of Carron Iron, No. 2, hot blast, cast from 
 the same model, each 7 feet long ; they were uniform through- 
 out, and the form of the section of A c 
 
 each was as in the figure. They were ~1 p~ * 
 
 laid on supports, 6 feet 6 inches JH n 
 
 asunder, and broken by weights suspended from the middle ; 
 the former with the rib downward, in which experiment the 
 flexure would be, almost wholly, owing to the extension of 
 that rib ; and the latter with the rib upward, in which the 
 flexure would be owing to the compression of the rib. In 
 both of these bars, the dimensions of the parallelogram A B 
 was the same, = 5 x '30 = 1*50 square inches, the thickness 
 being -30 inch. In the former of these cases, CD = 1-55, 
 and D E (which may represent the uniform thickness of the 
 rib DP) = -36, inch. In the latter casting CD = 1'56, and 
 D E = '365 inch. The results are as follow. 
 
284 
 
 EARLY DEFECT IN ELASTICITY. 
 
 FIRST BAR. 
 Broken with the vertical rib downwards. 
 
 T 
 
 SECOND BAR. 
 Broken with the vertical rib upwards. 
 
 1 
 
 "Weight in 
 Ibs. 
 
 Deflexion in 
 inches. 
 
 Deflexion 
 (load removed). 
 
 Weight in 
 Ibs. 
 
 Deflexion in 
 inches. 
 
 Deflexion 
 (load removed). 
 
 7 
 
 015 
 
 visible 
 
 7 
 
 
 not visible 
 
 14 
 
 032 
 
 001? 
 
 14 
 
 "'02o' 
 
 visible 
 
 21 
 
 046 
 
 002 
 
 21 
 
 045 
 
 002 
 
 28 
 
 064 
 
 004 
 
 28 
 
 065 
 
 OU3 
 
 56 
 
 130 
 
 005 
 
 56 
 
 134 
 
 005 
 
 112 
 
 273 
 
 020 
 
 112 
 
 270 
 
 015 
 
 168 
 
 444 
 
 035 
 
 224 
 
 580 
 
 058 
 
 224 
 
 618 
 
 058 
 
 336 
 
 895 
 
 101 
 
 280 
 
 813 
 
 093 
 
 448 
 
 1-224 
 
 155 
 
 336 
 
 1-030 
 
 130 
 
 560 
 
 1-585 
 
 235 
 
 364 
 
 broke 
 
 
 
 672 
 
 1-985 
 
 330 
 
 
 784 
 
 2-410 
 
 490 
 
 .. "Ultimate deflexion = 1-138 
 
 896 
 
 
 
 722 
 
 
 1008 
 
 4 : 140 ' 
 
 1-040 
 
 
 1064 
 
 
 
 
 
 
 1120 
 
 broke 
 
 
 
 
 During the fracture a wedge 2'92 inches 
 
 
 1 nnrl I'fiK 
 
 
 deep, broke out, \_ J 
 
 
 of the form ~^\^ 
 
 i 
 
 . . Ultimate deflexion = 5-00. 
 
 In the first of these experiments, it will be seen that the 
 elasticity was sensibly injured with 7 fts., and in the latter 
 with 14fbs. ; the breaking weights being 364fts. and 1120fts. 
 In the former of these cases a set was visible with 5 - 2 nd, and 
 in the other with ^th of the breaking weight, showing that 
 there is no weight, however small, that will not injure the 
 elasticity. The ratio of the breaking weights in these two 
 experiments was as 1 : 3 '07 ; showing that a bar of this form 
 was more than three times as strong one way up as the 
 opposite way. 
 
 87. The mode I used to observe when the elastic force 
 became injured was as follows. When a bar was laid upon 
 the supports for experiment, a " straight edge " was placed 
 over it, the ends of which rested upon the bar directly over 
 the points of support. These ends were slides which enabled 
 the straight edge to be raised or lowered at pleasure. In this 
 manner it was easy to bring it down to touch in the slightest 
 
MR. TREDGOLD'S CONCLUSIONS, 285 
 
 degree a piece of wood tied upon the middle of the bar. A 
 candle was then placed at the side of the bar, opposite to 
 where the observer stood, by the light of which, distances 
 extremely minute could be observed. 
 
 88. The results from these experiments will enable us to 
 see the mode by which Tredgold deduced the erroneous con- 
 clusion, as to the high tensile strength of cast iron, adverted 
 to in the note to art. 143, page 113 ; and which has produced 
 an effect on many of the numerical conclusions throughout his 
 work. 
 
 The experiment (No. 2, art. 56), from which, and others, 
 he infers that a cast iron bar, 1 inch square and 34 inches 
 between the supports, will bear 300 Ibs. on the middle, with- 
 out injury to the elasticity, has not, I conceive, been examined 
 in its progress with adequate care to form the basis of 
 important conclusions. It is evident, from my experiments 
 given above, that the elastic force was injured with much less 
 weight than that which formed the set he first noticed in the 
 beam. 
 
 Tredgold calculated the direct tensile strength of the cast 
 iron in these experiments, from the results of the transverse 
 strength. He assumed in his calculations, that the position 
 of the neutral line remains fixed, and in the middle of a 
 rectangular or circular section, during the whole experiment ; 
 and the resistance of a particle, at equal distances on each 
 side of the neutral line, to be the same. 
 
 From these principles he calculated that the most extended 
 surface of the bar was supporting a tension of 15,300 fts. per 
 square inch, without the elasticity being injured. This 
 number which is greater than was required to tear asunder 
 the specimen in many of the preceding experiments (art. 3, 
 Part II.), Tredgold adopts, throughout the work, "as the 
 direct tensile strength of cast iron, when not strained beyond 
 the elastic force. 
 
286 POSITION OF THE NEUTKAL LINE. 
 
 From these principles he calculates (art. 212 of his work) 
 that the greatest extension which cast iron will bear without 
 injury to its elastic force is ju^th part of its length. In art. 
 70, he calculates the weights which would be required to 
 destroy the elasticity of a number of cast iron bars, the 
 breaking weights of which are severally given. He compares 
 these weights, and concludes that a body requires about 
 three times as much to break it as to destroy its elastic force. 
 Hence he would conclude that the absolute strength, per 
 square inch, of this iron, is 15,300 x 3 = 45,900 fts. nearly, 
 or more than 20 tons. Tredgold computes, in several cases, 
 (arts. 72 to 76), the ultimate tensile strength of cast iron bars 
 from the weights which broke them transversely. He found 
 the strength to vary from 4 0,000 to 48,200 fts. per square 
 inch, the mean being 44,620 fts., or nearly three times 
 15,300, as mentioned above. 
 
 My own experiments, which were made by tearing asunder 
 25 castings, prepared with great care, from cast iron obtained 
 from various parts of England, Scotland, and Wales, gave as 
 a mean 16,505 fts.=: 7*37 tons per square inch; and in no 
 case, except one, was the strength found to be more than 8-J 
 tons per square inch (art. 3, page 229). 
 
 89. According to the principles assumed by Tredgold (art. 
 37 and 100), the position of the neutral line must remain 
 unchanged during flexure by different weights ; but experi- 
 ment shows that the neutral line shifts as the weights are 
 increased ; and at the time of fracture it is frequently near to 
 the concave side of the beam. This seldom can be dis- 
 covered in fractures of beams by simple transverse pressure, 
 but it may sometimes be discovered in those that have been 
 broken by a blow, and then it will perhaps seem to have been 
 at ^th or ^th of the depth of a rectangular beam, as I have 
 occasionally observed, the smaller part being compressed. 
 
 90. From the experiments on the strength of cast iron 
 
POSITION OF THE NEUTRAL LINE. 287 
 
 pillars (Tables I. and II., art. 78) we get additional evidence 
 upon this matter ; for in these the neutral line was frequently 
 well defined ; and in the longest pillars, those which required 
 the least weight to break them, the ratio T : C, which is that 
 of the depths of the parts submitted to tension and com- 
 pression, was, in different cases, 126 : 26, 145 : 36, 153 : 43, 
 and 120 : 37. In two rectangular pillars, each 60^ inches 
 long, and 1^ inches square, the area of the compressed part, 
 was less than yth of the whole section. In the pillars 
 mentioned above the weight necessary to break them was 
 very small, compared with that which would have been 
 required to crush them without flexure; and there was 
 scarcely a pillar broken in which the part compressed 
 exceeded the part extended, though some had sustained very 
 great pressures. 
 
 In the longest pillars mentioned above we may, I conceive, 
 consider the position of the neutral line as not widely different 
 from what it would have been if the pillar had been broken 
 as a beam, transversely ; the only difference in the two cases 
 being, that the compressed part, compared with the extended 
 part, would be greater in the pillar, through the weight laid 
 upon it, than in the beam. And we have seen that, in the 
 pillars alluded to, the depth of the compressed part varied 
 from -Jrd to less than ^th of that of the extended part ; and 
 in a beam the depth of the compressed part would be still 
 smaller. In the experiments on tension and compression 
 (arts. 33-34, Part II.), it has been shown that cast iron 
 resists crushing with about seven times* as much force as it 
 does tearing asunder; and some experiments not yet published 
 have inclined me to believe that the neutral line in a 
 rectangular beam, at the time of fracture, divides the section 
 in the proportion of six or seven to one, or one not widely 
 different from this ; but I draw the conclusion with consider- 
 able diffidence. * 
 
 91. The subject here treated of will be adverted to in a 
 
288 TRANSVERSE STRENGTH OF BARS. 
 
 future article, after the experiments which I am intending to 
 give upon the transverse strength of bars have, with those on 
 the tensile strength (art. 3, page 229), furnished data for 
 the purpose. 
 
 EXPERIMENTS TO DETERMINE THE TRANSVERSE STRENGTH 
 OF UNIFORM BARS OF CAST IRON. 
 
 92. I will now give the results of a very extensive series of 
 experiments upon rectangular bars, all cast from the same 
 model, and including irons from the principal Works in the 
 United Kingdom. 
 
 The great accumulation of specimens of iron, which were 
 obtained, but could not be used in our inquiry, before 
 mentioned, respecting the strength of hot and cold blast iron, 
 afforded a good opportunity to acquire the relative values of 
 many of the leading irons. Mr. Fairbairn, therefore, under- 
 took the matter, and had castings made from the whole ; and 
 having increased them by subsequent additions of iron, the 
 variety of results from the whole, both of Mr. Fairbairn's and 
 my own experiments, has become great, especially when those 
 are added to them which we derived from the hot blast 
 inquiry, and these will be found abridged in the following 
 pages. The experiments of Mr. Fairbairn were very carefully 
 made, some of them by myself, as those on the anthracite 
 iron, &c., and all with the same attention to accuracy. The 
 results are published at length in the Manchester Memoirs, 
 vol. vi., second series; but are here given in an abridged 
 form, only one series of results being set down for each kind 
 of iron : every result in each series being a mean between 
 values derived from the same weights in different experiments. 
 The bars were cast from a model, 5 feet long and 1 inch 
 square, and they were during the experiment, laid upon 
 supports 4 feet 6 inches asunder, and bent by weights 
 suspended from the middle. After each load had been laid 
 
TRANSVERSE STRENGTH OF BARS. 289 
 
 on, which was done with care and witli small additions of 
 weight, the deflexion was obtained by means of a long scale in 
 the form of a wedge, graduated along its side, so that very 
 minute distances could be measured. The beam was then 
 unloaded in order that the defect of elasticity, or set, might 
 be obtained. These experiments and those made by my 
 friend for the British Association, were conducted in the same 
 manner as others which I had made some time before on the 
 Carron, Buffery, and other irons, mentioned in art. 86 ; and 
 the remark which I had made of the very early defect of 
 elasticity of cast iron, as shown by my experiments, received 
 a further confirmation from these ; as the quantity of set was, 
 from that time, always carefully observed. The results of all 
 the bars were afterwards reduced by calculation in the same 
 manner as those in my Report (British Association, vol. vi.), 
 in order to preserve uniformity in the whole. 
 
290 
 
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302 REMARKS ON THE PRECEDING EXPERIMENTS. 
 
 REMARKS ON THE EXPERIMENTS IN THE LAST ARTICLE. 
 
 95. The objects sought for in every experiment were to obtain 
 the deflexion of the beam with given weights, generally in- 
 creasing by equal increments, and the set, or defect of elasti- 
 city, as exhibited by the deviation of the beam from its original 
 form, after the weight was taken off; for reasons before men- 
 tioned (art. 86). As the beams were generally broken, the 
 deflexion at the time of fracture could not often be obtained 
 by direct admeasurement ; it was therefore usually calculated, 
 for each separate experiment, from the breaking weight, and the 
 last weight, with its observed deflexion previous to fracture. 
 
 Many of the experiments were on the Carron Iron, No. 2, 
 both hot and cold blast ; and as these two denominations of 
 iron differ in transverse strength only as 99 to 100, from the 
 results of a great number of experiments, we may consider 
 them as of the same strength. Some of the experiments 
 were made to test admitted conclusions with respect to the 
 strength of materials, and their objects will now be described. 
 
 The experiments Nos. 5 and 6 were on bars somewhat 
 greater than 1 inch square; and the bars Nos. 10 and 11 
 were nearly of the same breadth as those, but had their 
 depths 3 and 5 inches respectively. Hence, supposing each 
 bar to be 1 inch broad, and the depths 1, 3, 5 inches respec- 
 tively, the strengths should be as 1, 9, 25, the square of the 
 depths. The mean strength from the bars Nos. 5 and 6 was 
 490 fts., but as these were larger than 1 inch square, I will 
 state the reduced results from a number of experiments. 
 The strength per square inch from five cold blast bars of this 
 iron was 467 Tbs., and from five hot blast bars it was 459 fts. ; 
 mean from the whole 463 fts. Multiplying this mean by 9 
 and 25, gives for the strength of the bars, 3 and 5 inches 
 deep, 4167 and 11,575 fts. respectively. The 3 and 5-inch 
 bars, in Nos. 10 and 11, are rather more than 1 inch broad, 
 and their mean strengths are 3890 and 1 0,298 fts. And, if 
 
REMARKS ON THE PRECEDING EXPERIMENTS. 303 
 
 the breadths were reduced to 1 inch, the strengths would be 
 3795, 10,067 fts. respectively. But, as these numbers are 
 less than 4167, 11,575, as above, it appears that the strength 
 of rectangular bars, of the same length and breadth, increases 
 in a ratio somewhat lower than as the square of the depth. 
 
 It having been often asserted that, if the external part or 
 crust of a cast iron bar be taken away, the strength of the 
 internal part will be much less than that of a bar of the same 
 dimensions, retaining its outer crust ; the result of the 
 experiments in No. 9,- compared with those of Nos. 5 and 6 
 will show that the falling off in strength, if any, is not great. 
 
 It was asserted by Emerson, in his f Mechanics,' 4to, page 
 114, that if a beam be made in the form of a triangular 
 prism, and -g-th of the height be taken from the vertex, parallel 
 to the base, the remaining part will be stronger than the 
 whole beam: this result was obtained on a supposition that 
 materials are incompressible. Tredgold, in art. 118 of this 
 volume, computes the same, on the supposition of equal 
 extensions and compressions from equal forces, and finds that 
 if j^th of the depth of such a beam be taken away from the 
 vertex, the strength will be about the greatest. The experi- 
 ments Nos. 12 and 13 were intended to show how far this 
 was true ; and it appears that the frustrum, with j^th of the 
 depth taken away, instead of being stronger than the whole 
 triangle, was weaker than it in the proportion of 702 to 766. 
 The object of the experiments Nos. 14 and 15 was principally 
 to .show that beams of cast iron, of the same dimensions, 
 might be made to bear, when turned one way upwards, 
 several times the weight which they would bear when turned 
 the opposite way up. In this case the strengths were as 
 1015 to 273, or as 4 to 1 nearly. This was first shown in 
 the Author's Paper on the ' Strength and best Form of Iron 
 Beams ' (Manchester Memoirs, vol. v.), and of which an 
 abstract will be given in this volume. The experiments No. 16 
 have in part the same object as those in art. 86, before described. 
 
304 
 
 TABULATED RESULTS. 
 
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 O CO ** i < O5 O >O O O O O O >O t CM O CO <O >O O rs >O IO "* 1> O CO CO OJ 
 r-( CO CO CO t~ m OS CO O t- O rH C5 OO r-l 'O i-" (M t- CO O CO O ri i- -H (M 1-- CO 
 
 O 1>- O 
 
 CO (M CO 
 
 O CO 00 
 r i OJ CO O O5 O O O> O O O T 1 O O CD O O> O O O O i * O ry-( C^ O O O O 
 
 & 
 
 qo9 uo s^ngui] JL 
 -odx9 jo J9qtun^j 
 
 Wjji : fliiii i 
 
 l^l-| I*l5Tl^ff%1 -g^ 
 
 JO 9^08 9-q^ Tit 
 
 uoax jo aaqcuii^j 
 
TABULATED RESULTS. 
 
 305 
 
 gjggssjsj 
 
 . 
 i 
 
 i 
 
 
 
 is 
 
 ,3ra 
 
 -I 
 
 E -2 
 
 .0 
 
 tc^j .a bo'-rfco .d . ,3 w-g ^^ 
 
 ii Ns^s-iNsi ri-2 Hit iril 
 
 
 Jd 
 
 I I 
 
 CJ O^ oq 
 
 69Z 192, 
 
 T-! 00 O OO 
 
 e>t-t^i-COCOO<Oi-IOOOS<M-<*O<Mr-IO 
 
 <MOOi-bos(NO CO <M(M 
 
 "* t~ 10 OO CO 
 
 ;O <N ^ OO -* , O Jt>- 
 <M CO <M I O 00 CO 
 ^44 <M I r^ ro 
 
 ^ CO CO CO 
 
 CO b- 
 
 co us w> rs cq co o b- (M 
 
 ^ -^ o "^ co ^ 
 
 O O <> O O O 
 
 O Ol>- ^O 
 
 O O<N CO 
 
 -tt< O CO O -* 
 
 i-l -*^1 IO ^f< 
 
 '^ CO O * 
 
 J> CO i I O O 
 
 CO O?O T-H -^H 
 
 JC^ CO CO O 
 
 OO 1OCO (M OOCOOOCOCN 
 OSO OS OS*QOCO*0 
 OrH OS OOCNr^HCO 
 
 i^iis^-. 
 
 '.^1 d d^ d^d c a^l-g o og 
 
 s ^^ i- K >-. *ry ~ h3r O o n-( H i m tr* 
 
 H br ^ 
 
 .*i#<fc 
 
 Lf^^^illl* 
 
 ^ ""S. &l a" b^ 3 ^^ S T bS ^ -"-a ^ S 43 P- J P. '< - - 
 
 lilltllfi^lillll-iiil-tli tS 
 
 1 1" -3 1" Tc| 111 I i i a v 
 
 no . 1 Q_J . 1 ^ r-rl r$l \^. r~\ r*\ ^ n% ri r> ^j . T ^ 
 
 g4f 
 
 ^^||44^4^ 
 
306 EXPLANATION OF PRECEDING TABLE. 
 
 96. In the preceding Table, the result from each bar is 
 reduced to exactly 1 inch square ; and the transverse strength 
 which may be taken as a criterion of the value of each iron, is 
 obtained from a mean between the reduced results of the 
 original experiments upon it ; first on bars 4 feet 6 inches 
 between the supports, and next on those of half the length, or 
 2 feet 3 inches between the supports. All the other results 
 are deduced from the 4 ft. 6 inch bars. In all cases the 
 weights were laid on the middle of the bar. 
 
 97. Since the experiments above were given to the public, 
 some others, upon bars of the same dimensions, and having 
 their results reduced in the same manner as these, have been 
 published by Mr. David Mushet. Other experiments on the 
 Ystalyfera iron have been given to the public by Mr. Evans : 
 those above are results obtained from experiments upon two 
 samples of each kind, sent to Mr. Eairbairn from the 
 proprietors. 
 
 Mr. Fairbairn has likewise recently sent to the Institution 
 of Civil Engineers the results of experiments made for him by 
 the Author, upon bars of the same size as those in the pre- 
 ceding pages, and on four other kinds of cast iron, viz. : iron 
 obtained from Turkish ores ; iron from the island of Elba ; 
 and two kinds of Ulverston (English) iron. 
 
 98. Explanation and Uses of the preceding Table. 
 
 1st. Explanation. The column representing the number 
 of experiments refers to those from which the strength of the 
 beams was obtained. 
 
 The specific gravity was obtained, generally, from a mean 
 of about half a dozen experiments, on small specimens, 
 weighed in and out of water. 
 
 The modulus of elasticity was usually obtained from the 
 deflexion caused by 112fts. on the 4 feet 6 inch bars, 
 calculated from the value of m in the formula, ^= jj^> (Part 
 I. art. 256). The numbers representing the power to resist 
 impact were obtained from the product of the breaking weight 
 
DEFECT OF ELASTICITY. 307 
 
 of the bars, by their ultimate deflexion ; as it appeared from 
 the experiments in the Author's Paper 'On Impact upon 
 Beams/ (British Association of Science, fifth Report,) that the 
 conclusions of Tredgold (art. 304), with respect to a modulus 
 of resilience, applicable so long as the elasticity was un- 
 injured, might be extended to the breaking point in cast 
 iron. 
 
 2nd. Uses of the Table. These are numerous, but two 
 only of the most common will be mentioned. If b and d be 
 the breadth and depth of a rectangular beam in inches, / the 
 distance between the supports in feet, w the breaking weight 
 in fts., w any other weight, d' its deflexion, and m the 
 modulus of elasticity in lbs.,1 for a square inch : putting 
 4*5 for the distance 4 feet 6 inches, above, we have 
 
 w = . The value of s being taken from the Table above. 
 
 , _m 6 d 3 d' ) (Part I. art. 256) the value of the modulus m being obtained from 
 432 P f the Table. 
 
 DEFECT OF ELASTICITY. 
 
 99. In all the preceding experiments on rectangular bars, 
 the defect of elasticity, measured by the deflexion remaining 
 in the bar after the load had been removed, was observed, for 
 reasons previously given (arts. 86, 92) ; and to show the law 
 which regulates this defect, its value, with equal additions of 
 weight, will be collected from the mean results upon each 
 iron, and placed under the corresponding weights in the 
 following Table. 
 
 x 2 
 
303 
 
 DEFECT OF ELASTICITY, 
 
 100. Defect of elasticity, or set, as obtained from the mean deflexion of lars cant from 
 models 1 inch square, laid on supports '5 ft. asunder ; using only those irons upon 
 which experiments had been made, as to the set, upon all the weights set down. 
 
 1 
 
 
 
 
 
 
 
 
 
 FIRST SERIES. 
 
 56 
 
 112 
 
 168 
 
 224 
 
 230 
 
 336 
 
 392 
 
 448 
 
 No. 1 Irons. 
 
 
 
 
 
 
 
 
 
 Elsicar, cold blast . .,.,:. * I 
 
 
 020 
 
 038 
 
 054 
 
 075 
 
 102 
 
 135 
 
 176 
 
 Muirkirk, cold blast . ; ' .""'. \ 
 
 
 Oil 
 
 028 
 
 051 
 
 081 
 
 121 
 
 172 
 
 255 
 
 Bute, cold blast . . ,., , s / 
 
 
 010 
 
 022 
 
 042 
 
 067 
 
 098 
 
 133 
 
 185 
 
 No. 2 Irons. 
 
 
 
 
 
 
 
 
 
 Corbyn'sHall 
 
 
 .015 
 
 036 
 
 062 
 
 088 
 
 122 
 
 171 
 
 234 
 
 Beaufort, hot blast .... 
 
 
 .008 
 
 017 
 
 030 
 
 049 
 
 073 
 
 103 
 
 136 
 
 Pentwyn . . .;,<.;, *$*'.; 
 
 
 .008 
 
 018 
 
 030 
 
 048 
 
 073 
 
 101 
 
 139 
 
 Frood, cold blast . . . , 
 
 
 .016 
 
 032 
 
 056 
 
 089 
 
 131 
 
 188 
 
 273 
 
 No. 3 Irons. 
 
 
 
 
 
 
 
 
 
 | Carron, hot blast . $ t . 
 
 
 006 
 
 Oil 
 
 021 
 
 035 
 
 052 
 
 075 
 
 103 
 
 Gartsherrie, hot blast 
 
 
 018 
 
 034 
 
 056 
 
 081 
 
 114 
 
 155 
 
 209 
 
 Dundy van, cold blast .f*. ... *^& 
 
 
 008 
 
 021 
 
 037 
 
 059 
 
 089 
 
 126 
 
 181 
 
 Coed-Talon, do. ;' . . < ? 
 
 
 on 
 
 020 
 
 033 
 
 048 
 
 066 
 
 089 
 
 116 
 
 Ponkey, do. . . . ^j . 
 
 
 010 
 
 020 
 
 031 
 
 048 
 
 068 
 
 094 
 
 126 
 
 Maesteg, number unknown . 
 
 
 013 
 
 030 
 
 056 
 
 090 
 
 140 
 
 191 
 
 274 
 
 No. 2 Irons. 
 
 
 
 
 
 
 
 
 
 Oldberry, cold blast . J . . . . 
 
 003 
 
 012 
 
 031 
 
 054 
 
 083 
 
 122 
 
 175 
 
 253 
 
 Pontypool . ;" ,9'/Oii/> ..- 
 
 005 
 
 020 
 
 038 
 
 065 
 
 097 
 
 142 
 
 204 
 
 296 
 
 Brimbo 
 
 002 
 
 014 
 
 033 
 
 058 
 
 088 
 
 124 
 
 172 
 
 237 
 
 Carron, cold blast . 
 
 003 
 
 006 
 
 012 
 
 022 
 
 034 
 
 053 
 
 075 
 
 105 
 
 No. 3 Irons. 
 
 
 
 
 
 
 
 
 
 Blaina, cold blast . sr.v-'v o&t*:4 
 
 006 
 
 018 
 
 035 
 
 059 
 
 093 
 
 141 
 
 197 
 
 290 
 
 Coed-Taloii, hot blast . 
 
 004 
 
 Oil 
 
 024 
 
 039 
 
 057 
 
 077 
 
 101 
 
 136 
 
 Means from sets from nineteen ) 
 kinds of iron . . . . . ) 
 
 
 0124 
 
 026 
 
 045 
 
 069 
 
 100 
 
 140 
 
 196 
 
 Sets computed from the formula 
 W 2 
 
 
 
 
 
 
 
 
 
 x = , where x is the set, 
 
 
 0117 
 
 026 
 
 047 
 
 073 
 
 105 
 
 143 
 
 187 
 
 and w the weight in cwts. . . 
 
 
 
 
 
 
 
 
 
 SECOND SERIES. 
 
 
 
 
 
 
 
 
 
 No. 2 Irons. 
 
 
 
 
 
 
 
 
 
 Adelphi, cold blast . 5 , 
 
 002 
 
 014 
 
 034 
 
 060 
 
 093 
 
 138 
 
 201 
 
 
 Eagle Foundry, hot blast . . . 
 
 003 
 
 013 
 
 030 
 
 051 
 
 078 
 
 113 
 
 159 
 
 
 Level, hot blast . , 
 
 002 
 
 Oil 
 
 022 
 
 038 
 
 061 
 
 088 
 
 121 
 
 
 Pant 
 
 005 
 
 014 
 
 025 
 
 042 
 
 059 
 
 076 
 
 100 
 
 
 No. 3 Irons. 
 
 
 
 
 
 
 
 
 
 Wallbrook 
 
 003 
 
 013 
 
 028 
 
 049 
 
 071 
 
 103 
 
 138 
 
 
 Carron, cold blast . it* itt '& 
 
 005 
 
 Oil 
 
 024 
 
 043 
 
 066 
 
 094 
 
 129 
 
 
 Means from the six kinds of iron . 
 
 0037 
 
 0127 
 
 027 
 
 047 
 
 071 
 
 102 
 
 141 
 
 
 Means from the last six in former ) 
 part of Table . . . . \ 
 
 0038 
 
 0128 
 
 029 
 
 051 
 
 080 
 
 115 
 
 157 
 
 
 Means from the twelve kinds of iron 
 
 0037 
 
 0127 
 
 028 
 
 049 
 
 076 
 
 109 
 
 149 
 
 
 Sets computed, as before, from 
 
 
 
 
 
 
 
 
 
 W 2 
 
 formula x = , x and w being 
 
 0030 
 
 0122 
 
 027 
 
 049 
 
 073 
 
 110 
 
 149 
 
 
 as above 
 
 
 
 
 
 
 
 
 
 101. Comparing the mean sets, or defects of elasticity, in 
 each series of the preceding Table, with the computed ones, it 
 
LAW OF DEFECTIVE ELASTICITY. 309 
 
 appears that the defects vary nearly as the square of the 
 weights; the set being the abscissa and the weight the 
 ordinate of a parabola. 
 
 Hence there is no force, however small, that will not injure 
 the elasticity of cast iron. 
 
 102. When bars of a J_ form of section are bent, so as to 
 make the flexure to depend upon the extension or compression 
 of the vertical rib (as in arts. 86, and 94, experiments 14 to 
 16), the set is nearly as the square of the extension or com- 
 pression ; these being measured by the deflexions. 
 
 103. In all the preceding experiments, the weight laid upon 
 the beam acted in a vertical direction ; and the weight of the 
 beam, independently of the other weight, had a small 
 tendency to deflect the beam; the deflexions given in the 
 Table being measured as commencing from that position 
 which the beam had taken in consequence of its own weight. 
 This, therefore, introduced an error which, though very small, 
 on account of the great strength of cast iron compared with 
 its weight, ought, if possible, to be avoided ; especially where 
 the object was, not only to prove that defects of elasticity 
 were produced by weights which were not hitherto supposed 
 capable of injuring the elasticity, but also to seek for the law 
 which regulated these defects. Other objections to these 
 results might be urged, as for instance : when a beam is laid 
 upon two supports, and bent by a weight in the middle or 
 elsewhere ; the friction between the ends of the beam and the 
 supports will have a slight influence upon the deflexion, a 
 matter which has been submitted to calculation by Professor 
 Moseley* in his able work on engineering. To meet the 
 objections above, I had an apparatus constructed with four 
 friction wheels, two to support each end of the beam; one 
 wheel acting horizontally and the other vertically. The 
 horizontal wheels were intended to destroy the friction arising 
 from the weight of the beam, and the vertical ones that from 
 
 * " Mechanical Principles of Engineering and Architecture," art. 389. 
 
310 DEFECT OF ELASTICITY. 
 
 the weight applied ; this weight, in its descent, being made to 
 act horizontally upon the beam, by means of a cord passing 
 over a pulley. The results obtained in this way confirm the 
 truth of the former ones ; and by being freed from small 
 errors, are much more consistent among themselves than they 
 would otherwise have been. 
 
 104. A bar of the L form of section, bent so as to 
 compress the vertical rib, with weights varying from 112 to 
 1344fts., gave, from a mean of two experiments very 
 carefully made, the set, as the T88 power of the deflexion, 
 measuring the compression of the rib. In these experiments, 
 each weight was allowed to remain on the beam five minutes, 
 and the set was taken twice, at intervals of one and five 
 minutes after unloading ; it having been found that a greater 
 length of time produced but little change in the quantity of 
 the set. 
 
 105. Experiments made to extend the vertical rib, the bar, 
 during flexure, being turned the opposite side upwards, gave 
 the set as a power of the deflexion, or extension, somewhat 
 higher than as above. 
 
 106. Supposing the set to arise wholly from the extension 
 or compression of the rib, which is very probable, it will 
 therefore be nearly as the square of the extension or com- 
 pression, as above observed. If, therefore, % represent the 
 quantity of extension or compression, which a body has 
 sustained, and a x the force producing that extension or 
 compression, on the supposition that the body was perfectly 
 elastic ; then, the real force /, necessary to produce the 
 extension or compression x, will be smaller, than on the 
 supposition of perfect elasticity, by a quantity b x* ; and we 
 shall have/= ax bo?. 
 
 107. The law of defective elasticity, as here given, and its 
 application to other materials, as stone, timber, &c., was 
 discovered by the author in July, 1843, and laid before the 
 British Association of Science, at its meeting in Cork. 
 
SECTION OF GEEATEST STRENGTH, 311 
 
 OF THE SECTION OF GREATEST STRENGTH IN CAST IRON BEAMS. 
 
 108. The very extensive and increasing use made of cast 
 iron beams renders it excedingly desirable that they should be 
 cast in the form best suited for insuring strength ; and that, if 
 possible, formulae should be obtained by which the strength 
 can be estimated. Without these the engineer and founder 
 must be in constant uncertainty ; and either endanger the 
 stability of erections, costing many thousands of pounds, and 
 perhaps supporting hundreds of human beings, or incur the 
 risk of employing an unnecessary quantity of metal, which, 
 besides its expense, does injury by its own weight. 
 
 109. The earliest use of this most valuable material for 
 beams has been but of recent date : so far as I can learn it was 
 first used by Boulton and Watt, who in 3800 employed beams, 
 whose section was of the form ^ , in building the cotton mill 
 for Messrs. Philips and Lee in Salford. These, the earliest 
 cast iron beams, differed from the J_ formed beams of the 
 present day, in having the lower portion of the vertical part 
 thicker than the modern ones. Both have had the same 
 object in their construction, that of supporting arches of 
 brick- work for the floors of fire-proof buildings ; and as they 
 were well suited for that purpose, and of a convenient form 
 for casting, besides being very strong, particularly the modern 
 ones, comparatively with rectangular beams of that metal, their 
 use has been very general, and they are still employed ; though 
 they have now been supplanted in most of the large erections 
 of Manchester and its neighbourhood, and many other parts 
 of the kingdom, by another form derived from experiments of 
 which I gave an account in the fifth volume of the ' Memoirs 
 of the Literary and Philosophical Society of Manchester ' 
 (second series) published in 1831. I propose giving here 
 extracts from the leading results and reasonings in that Paper. 
 
 110. In the application of a material like cast iron to 
 purposes to which it had not been before applied, it could not 
 
312 SECTION OF EARLIEST FOEMS OF CAST IRON BEAMS. 
 
 be expected that the form best suited for resistance to strain, 
 any more than the quantity necessary to support that strain, 
 could be at once attained. The J_ form of cast iron beam 
 mentioned above, was, however, by no means a bad one ; it 
 had undergone modifications and improvements by different 
 parties, and had had various experiments made upon it, some 
 of which, made by my friend Mr. Fairbairn, on a large scale, 
 I gave in the Paper mentioned above. This form, however, 
 Mr. Tredgold saw, was not the best, and gave, in his article 
 on the ' Strongest Form of Section ' (Part I. Section IV.), a 
 representation of what he considered the best (Plate I. fig. 9), 
 a beam with two equal ribs or flanges, one at the top and the 
 other at the bottom. Mr. Tredgold proposed this form, 
 assuming that, whilst the elasticity of a body is perfect, it 
 resists the same degree of extension or compression with equal 
 forces ; and therefore he concluded that a beam, to bear the 
 most, should have equal ribs at top and bottom, as it ought 
 not to be strained so as to injure its elasticity.* 
 
 111. Having myself given a Paper on the 'Transverse 
 Strength of Materials/ in the fourth volume of the ' Memoirs 
 of the Literary and Philosophical Society of Manchester/ 
 published in 1824, containing the mathematical development 
 of some principles to which I attached importance, besides 
 some experiments to ascertain the position of the neutral line 
 in bent pieces of timber, I felt persuaded that the form 
 proposed by Mr. Tredgold was not the best to resist fracture 
 in cast iron. It was evident that that metal resisted fracture 
 by compression with much greater force than it did by 
 tension, though the ratio was then unknown : and I was 
 convinced that the transverse strength of a bar depended in 
 some manner upon both of these forces ; the situation of 
 the neutral line being changed before fracture in consequence 
 of their inequality. 
 
 jlr-. Tredgold did not suspect that tko elasticity was injured by forces however 
 small, set- urt. i6, &e. 
 
PRELIMINARY EXPERIMENTS. 313 
 
 112. To obtain further information on this subject, I 
 adopted, about the year 1828, a mode of analysing, separately 
 the forces of extension and compression, in a bent body ; 
 results of which have been given in arts. 86 and 94, from 
 experiments, since made for other purposes. I had bars cast 
 from a model, 5 feet long, whose section was of the 
 
 f 2 It was in all parts of an inch 
 
 form, ' ~~j F~~ B thick, and uniform in breadth and 
 Jr-js depth, the thickness being as 
 
 small as the castings could be run to make them uniform and 
 sound ; the breadth A B of the flange was 4 inches, and the 
 depth E E of the rib running along its middle was 1*1 inch. 
 
 113. When the castings so formed had their ends placed 
 horizontally upon supports, and weights were suspended 
 from the middle, the flexure would depend almost entirely 
 upon the contraction or extension of the rib E E. When the 
 rib was upwards, the deflexion would arise from the contrac- 
 tion of that rib, and, when downwards, from its extension. 
 
 114. To ascertain the resistance to fracture in these two 
 cases, I took two castings, apparently precisely alike, and 
 placing the ends of each of them upon two props 4 feet 3 inches 
 asunder, broke them by weights in the middle, one with the 
 rib upwards, and the other with it downwards, as in the figure. 
 
 That with the rib downwards bore 2J cwt., and broke 
 with 2^ cwt. The other casting bore 8f cwt., and broke 
 with 9 cwt. Deflexion of the latter in middle with 4 cwt. = 
 *6 inch, with 8J cwt. = 1*8 inch. 
 
 115. The strength of the castings was, therefore, nearly as 
 2^- to 9, or as 10 to 36, accordingly as they were broken one 
 or the other way upwards. 
 
 116. When the second broke, a piece flew out, whole, 
 from the compressed side of the casting, of the following 
 form, A, D, B, where A B = 4 inches, and C D = *98 inch; 
 the point D at the bottom being in or near to the neutral line 
 of the bar, a side view of which is represented in the figure. 
 
314 SECTION OF GREATEST STRENGTH. 
 A C B 
 
 The side A B of the wedge-like piece broken out, was, as 
 will be seen, in the direction of the length of the casting, and 
 the weights were laid on at C. Hence, as the depth of the 
 
 casting was found to be T35 inch, C 1) = ^ = ^ of the 
 depth, nearly. In the experiment on the second bar (art. 
 86), made since that time, ^ D=i||=~ nearly. 
 
 117. These experiments are interesting ; they show the 
 effect of the position of the casting on the strength ; give the 
 situation of the neutral line ; and may, from the peculiar form 
 of the wedge, which, as represented here, is more perfect than 
 usual, throw some additional light on the nature of the strain. 
 
 118. Those who with Tredgold (Part I. art. 37, &c.) sup- 
 pose the strength to be bounded by the elasticity, and that 
 the same force would destroy the elastic power, whether it was 
 applied to extend or compress the body, must have conceived 
 these castings, and indeed those of every other form, to be 
 equally strong, whichsoever way upwards they were turned ; 
 a conclusion which we see would lead to very erroneous results, 
 if applied to measure the ultimate strength of cast iron. 
 
 Other experiments were made at that time upon bars of the 
 same form as the preceding, to ascertain the deflexions with 
 given forces when the rib C D was subjected alternately to 
 tension and compression ; and it was shown as might be 
 expected, that the extensions and compressions, measured by 
 the deflexions, were nearly equal from the same forces, though 
 the extension was usually somewhat greater than the com- 
 pression, the difference increasing with the weight, through 
 the whole range to fracture. This always took place by the 
 rib being torn asunder, the compression necessary to produce 
 
PRELIMINARY EXPERIMENTS. 315 
 
 fracture being several times as great as the extension re- 
 quired to do it, as may be inferred from art. 33. 
 
 119. The object of the preceding experiments being to 
 prepare, in some degree, the way to an inquiry into the best 
 forms of beams of cast iron ; we will now reconsider the strain 
 to which they are subjected, with a view to their adaptation 
 to bear a given load with the least quantity of metal. 
 
 120. Suppose a beam supported at its ends, and bent by a 
 weight laid at any intermediate point upon it : since all 
 materials are both extensible and compressible, it is evident 
 that the whole of the lower fibres are in some degree of 
 extension, less or more, and the whole of the upper fibres are 
 in a compressed state ; there being some point, intermediate 
 between both, where extension ends and compression begins. 
 If then we suppose all the forces of extension and com- 
 pression, in the section of the beam where the deflecting force 
 is applied, to be separately collected into two points, one over 
 the other, the beam will offer the greatest resistance, the 
 quantity of metal being the same, when these points are as far 
 asunder as possible, since the leverage is then the greatest. 
 
 121. When the depth of the beam is limited, this object 
 would, perhaps, be best attained by putting two strong ribs, 
 one at the top and the other at the bottom, the intermediate 
 part between the ribs being a thin sheet of metal, to keep the 
 ribs always at the same distance, as well as to serve another 
 purpose which will be mentioned further on. 
 
 122. As to the comparative strength of the ribs, in beams 
 of different materials, that depends on the nature of the body, 
 and can only be derived from experiment. Thus, suppose 
 the same force were required to injure the elasticity to a 
 certain extent, or to cause rupture, whether it acted by 
 extension or compression, then the strengths of the ribs 
 should be equal ; and this would be the case whatever the 
 thickness of the part between the ribs might be, providing it 
 was constant. But, supposing the thickness of the part 
 
316 SECTION OF GREATEST STRENGTH. 
 
 between the ribs was so small that its resistance might be 
 neglected, and the metal to be of such a nature that a force P 
 was needed to injure the elasticity to a certain degree by 
 stretching it, and another force G to do the same by com- 
 pressing it, it is evident that the size of the ribs should be as 
 G to F, or inversely as their resisting power, that they may be 
 equally affected by the strain. Or if, the resistance of the part 
 between the ribs being neglected, it took equal weights F and 
 G' to break the material by tension and compression, the beam 
 should have ribs as G' to Y to bear the most without fracture. 
 
 123. This last matter must be considered with some 
 modifications : it would not, perhaps, be proper to make the 
 size of the ribs just in the ratio of the ultimate tensile to the 
 crushing forces, as the top rib would be so slender that it 
 would be in danger of being broken by accidents ; and the 
 part between the ribs, though thin, has some influence on the 
 strength. 
 
 124. The thickness, too, of the middle part between the 
 ribs is not a matter of choice : independent of the difficulty of 
 casting, and the care necessary to prevent irregular cooling, 
 and contraction, in beams whose parts differ much in thick- 
 ness, the middle part cannot be rendered thin at pleasure, but 
 must: have a certain thickness, though in long beams the 
 breaking weight is small, and a very small strength in the 
 middle part is all that is necessary. 
 
 125. The neutral line being the boundary between two 
 opposing forces, those of tension and compression, it seems 
 probable that bending the beam would produce a tendency to 
 separation at that place. Moreover, the tensile and com- 
 pressive forces are, strictly speaking, not parallel; they are 
 deflected from their parallelism by the action of the weight, 
 which not only bends the beam, but tends to cut it across in 
 the direction of the section of fracture ; and this last tendency 
 is resisted by all the particles in the section. This com- 
 pounded force will then tend to separate the compressed part 
 
EXPERIMENTS TO OBTAIN THE BEST FORM OF BEAMS. 317 
 
 of the beam, in the form of a wedge, and this tendency must 
 be resisted by the strength of the part between the ribs or 
 flanges. We have had several instances of fracture this way 
 (arts. 86, 94), and there will occur several others in the course 
 of the following experiments, as in art. 135, Experiment 12, &c. 
 
 126. We see then that there are three probable ways in 
 which a beam may be broken : 1st, by tension, or tearing 
 asunder the extended part; 2nd, by the separation of a wedge, 
 as above ; and 3rd, by compression, or the crushing of the 
 compressed part. I have not, however, obtained a fracture, 
 by this last mode, in cast iron broken transversely. 
 
 * 
 
 EXPERIMENTS TO ASCERTAIN THE BEST FORM OF CAST IRON 
 BEAMS, AND THE STRENGTH OF SUCH BEAMS. 
 
 127. In the commencement of these experiments the form 
 I first adopted was one in which the arc, bounding the top of 
 the beam, was a semi-ellipse, with the bottom rib a straight 
 line; but the sizes of the ribs at top and bottom were in 
 various proportions. The ribs in the model were first made 
 equal, as in the beam of strongest form according to the 
 opinion of Mr. Tredgold (Section IV., art. 37) ; and when a 
 casting had been taken from it, a small portion was taken 
 from the top rib, and attached to the edge of the bottom one, 
 so as to make the ribs as one to two ; and when another 
 casting had been obtained, a portion more was taken from the 
 top, and attached to the bottom, as before, and a casting got 
 from it, the ribs being then as one to four. In these altera- 
 tions the only change was in the ratio of the ribs, the depth 
 and every other dimension in the model remaining the same. 
 
 128. Finding that all these beams had been broken by the 
 bottom rib being torn asunder, and that the strength by each 
 change was increased, I had the bottom rib successively 
 enlarged, the size of the top rib remaining the same. The 
 bottom rib still giving way first, I had the top rib increased, 
 
318 EXPERIMENTS TO OBTAIN THE BEST FORM OF BEAMS. 
 
 feeling that it might be too small for the thickness of the 
 middle part between the ribs. The bottom rib was again 
 increased, so that the ratio of the strengths of the bottom and 
 top ribs was greater than before ; still the beam broke by the 
 bottom rib failing first, as before. As the strength continued 
 to be increased more than the area of the section, though the 
 depth of the beam and the distance between the supports 
 remained the same, I pursued, in the future experiments, the 
 same course, increasing by small degrees the size of the ribs, 
 particularly that of the bottom one, till such time as that rib 
 became so large that its strength was as great as that of the 
 top one ; or a little greater, since the fracture took place by a 
 wedge separating from the top part of the beam. I here 
 discontinued the experiments of this class, conceiving that the 
 beams last arrived at, were in form of section nearly the 
 strongest for cast iron. 
 
 129. In most of the experiments the beams were intended 
 to have been broken by a weight at their middle; and, 
 therefore, the form of the arcs, bounding the top of the 
 beams, was, in this inquiry, of little importance : in making 
 them elliptical, they were too strong near to the ends for a 
 load uniformly laid over them ; the proper form is something 
 between the ellipse and the parabola. It is shown, by most 
 of the writers on the strength of materials, that if the beam 
 be of equal thickness throughout its depth, the curve should 
 be an ellipse to enable it to support, with equal strength in 
 every part, a uniform load ; and if there be nothing but the 
 rims, or the intermediate parts be taken away, the curve of 
 equilibrium, for a weight uniformly laid over it, is a parabola. 
 When, therefore, the middle part is not wholly taken away, the 
 curve is between the ellipse and the parabola, and approaches 
 more nearly to the latter, as the middle part is thinner. 
 
 130. The instrument used in the experiments was a lever 
 (Plate II. fig. 40) about 15 feet long, placed horizontally, one 
 end of which turned on a pivot in a wall, and the weights 
 
EXPERIMENTS TO OBTAIN THE BEST FORM OF BEAMS. 319 
 
 were hung near to the other ; the beams being placed 
 between them and the wall, at 2 or 3 feet distance from it. 
 
 131. All the beams in the first Table (Table I. following) 
 were exactly 5-| inches deep in the middle, and 5 feet long, 
 and were supported on props just 4 feet 6 inches asunder. 
 The lever was placed at the middle of the beam, and rested 
 on a saddle, which was supported equally by the top of the 
 beam and the bottom rib, and terminated in an arris at its 
 top, where the lever was applied. The deflexions were taken 
 in inches and decimal parts, at or near the middle of the 
 beam, as mentioned afterwards. The weights given are the 
 whole pressure, both from the lever and the weights laid on, 
 when reduced to the point of application on the beam. The 
 dimensions of section in each experiment were obtained from 
 a careful admeasurement of the beam itself, at the place of 
 fracture, which was always very near (usually within half an 
 inch of) the middle of the beam ; the depth of the section being 
 supposed to be that of the middle of the beam, or 5-g- inches. 
 
 132. As the experiments were made at different times, and 
 there might be some variation in the iron, though it was 
 intended always to be the same, a beam of the same length 
 and depth as the others, but of the usual JL form, always 
 from the same model, was cast with each set of castings for 
 the sake of comparison. The results of the experiments upon 
 these beams are given in the third Table. 
 
 133. The first six beams, in the first Table, were cast 
 horizontal, that is, each beam lay flat on its side in the sand ; 
 all the rest were cast erect, that is, each beam lay in the sand 
 in the same posture as when it was afterwards loaded, except 
 that the casting was turned upside down, when in the sand. 
 
 134. In all the experiments the area of the section was 
 obtained with the greatest care ; it includes, besides the parts 
 of which the dimensions are given, the area of the small 
 angular portions at the junction of the top and bottom ribs 
 with the vertical part between them. 
 
EXPEK1MENTS TO OBTAIN THE 
 
 TABLE 
 
 135. Tubulated Results of experiments to ascertain the best form of cast iron beams, all the 
 depth in the middle, where the load was applied, =5$ inches. All dimensions in the 
 
 Form of section of beam in 
 middle. 
 
 Area of top 
 rib in middle 
 of beam. 
 
 Area of but - 
 torn rib in 
 middle of 
 beam. 
 
 Thickness of 
 vertical part be- 
 tween the ribs. 
 
 Area of section of 
 beam at place of 
 fracture. 
 
 Weight of beam. 
 
 Deflexions in parts 
 of an inch. 
 
 Corresponding 
 weights in Ibs. 
 
 1st Exp< 
 
 Beam (Plate 
 with equal 
 and bottom. 
 H 
 
 M 
 
 ;riment. 
 
 III. fig. 41) 
 ribs at top 
 
 ^ 
 
 n 
 
 1-75 x -42 
 = 735 
 
 1-77 x -39 
 
 = 690 
 
 29 
 
 2-82 
 
 
 
 
 
 2nd Exp 
 
 Beam with 
 tion of top , 
 rib as 1 to 2 
 
 E^; 
 
 HI 
 
 eriment. 
 
 area of sec- 
 ind bottom 
 
 23 
 9 
 
 1-74 x -26 
 = 45 
 
 1-73 x -55 
 = 98 
 
 30 
 
 2-87 
 
 39 lfcs. 
 
 
 3rd Exp< 
 
 Beam with 
 tion of top 
 rib as 1 to 4 
 
 i 
 
 
 
 jriment. 
 
 area of sec- 
 and bottom 
 
 a 
 
 H 
 
 1 07x-30 
 
 = 32 
 
 2-1 x -57 
 
 32 
 
 3-02 
 
 
 
 
 
 4th Experiment. 
 
 Beam from the same mo- 
 del as the last, but cast 
 the opposite way up. 
 
 1-05 x -32 
 = 0-34 
 
 2 15 x -56 
 = 1-20 
 
 33 
 
 3-08 
 
 394K.3. 
 
 
BEST FORM OF BEAMS. 
 
 321 
 
 learns being made 5 feet long and laid on supports 4 feet 6 indies asunder, and having the 
 Table are in inches, and the weights in pounds, except otherivise mentioned. 
 
 
 N 
 
 *1I* 
 
 III 
 
 
 
 Breaking 
 weight 
 in fts. 
 
 rj'o 
 
 li-g" 
 
 **ffs 
 
 A S3 g:g 
 
 fi2|1 
 
 Jfl "c5 
 O rj 
 
 "rt^ 
 
 CJ 0-0 
 
 Form of fracture. 
 
 Remarks. 
 
 
 p 
 
 Oil! 
 
 ill 
 
 
 
 
 CO 
 
 ^"o o 8 
 
 O ^ 
 
 
 
 6678 fts. 
 
 6678 
 
 
 1 This is represented 
 
 
 = 59 cwt. 
 70 fts. 
 
 2-82 
 = 2368 
 
 2584 
 
 -4 
 
 by the line bnrt, 
 (Plate III. fig. 41,) 
 where tr='6, and 
 
 
 
 
 
 
 b n =2 - 5, the figure 
 
 
 
 
 
 
 being a side view 
 
 
 
 
 
 
 
 of the beam. The 
 
 
 
 
 
 
 distances t r, bn 
 
 
 
 
 
 
 are measured ver- 
 
 
 
 
 
 
 tically. 
 
 
 7368 fts. 
 = 65 cwt. 
 88 fts. 
 
 7368 
 2-87 
 = 2567 
 
 2584 
 
 __^ 
 
 Nearly same as in 
 Exp. 1 ; here t r 
 (Plate III. fig. 41,) 
 = '55 inch. Here, 
 
 
 
 
 
 
 and in all other 
 
 
 
 
 
 
 cases, t r is mea- 
 
 
 ' S, 
 
 
 
 
 sured vertically, 
 as before. 
 
 
 
 
 
 
 
 
 8270 fts. 
 
 8270 
 
 
 
 Nearly as in Expe- 
 
 
 = 73 cwt. 
 94 fts. 
 
 3-02 
 = 2737 
 
 2584 
 
 ^nearly. 
 
 riment 1 ; and t r 
 = 6 (fig. 41), 
 
 
 8263 fts. 
 = 73 cwt. 
 89 fts. 
 
 826:t 
 3-08 
 
 = 2683 
 
 2792 
 
 -A 
 
 nearly. 
 
 Nearly as figure to 
 Exp. 1, but here 
 b 7i = 2'5, and tr 
 55 (fig. 41). 
 
 
322 
 
 EXPERIMENTS TO OBTAIN THE 
 
 TABLE I. 
 
 Form of section of beam in 
 middle. 
 
 Area of 
 top rib in 
 middle of 
 beam. 
 
 Area of 
 bottom rib 
 in middle 
 of beam. 
 
 o&l 
 
 ||| 
 
 0*3 
 
 Jlo 
 
 a 
 
 05 T* 
 0*0 
 
 Corresponding 
 weights in Ibs. 
 
 5th Experiment. 
 
 Beam with area of section 
 of ribs as 1 to 4 nearly. 
 
 1-05 x -34 
 = 0-357 
 
 3-08 x -51 
 = 1-570 . 
 
 305 
 
 ;! 
 
 3-37 
 
 1 
 
 
 
 
 
 6th Exp< 
 
 Ratio of ri 
 nearly. 
 IB! 
 
 jriuient. 
 bs 1 to 4 
 
 WML 
 
 1-6 x -315 
 = 0-5 
 
 4-16 x -53 
 = 2-2 
 
 38 
 
 4-50 
 
 57R>s. 
 
 4 
 45 
 52 
 
 1118G 
 12698 
 13706 
 
 
 7th Experiment. 
 
 Beam differing from 
 last, having a broader 
 bottom flange. Ratio 
 of ribs 1 to 5 nearly. 
 
 l-56x-315 
 = 0-49 
 
 5-17 x -56 
 = 2-89 
 
 34 
 
 5 
 
 
 24 
 36 
 40 
 42 
 45 
 48 
 49 
 53 
 
 8288 
 12698 
 13706 
 14210 
 15218 
 15722 
 16226 
 16730 
 
 
 8th Exp 
 
 Ij 
 
 t^ 7 ^- 
 
 eriment. 
 HI 
 
 H^^^ 
 
 2-3 x -31 5' 
 = 72 
 
 4-06 x -57 
 = 2-314 
 
 33 
 
 4-628 
 
 
 
 
 1 
 
BEST FORM OF BEAMS. 
 
 323 
 
 (Continued.) 
 
 Breaking 
 weight 
 in fts. 
 
 P' 9 
 
 a "o " 
 m'c s8 
 
 1 
 
 Form of fracture. 
 
 Remarks. 
 
 10727 fts. 
 
 10727 
 
 
 
 Here tr (fig. 41) 
 
 Broke by tension, small flaw 
 
 = 95 cwt. 
 871fes. 
 
 3-37 
 = 3183 
 
 2792 
 
 f nearly 
 
 =* '6 inch. 
 
 in bottom rib, at place ofj 
 fracture. 
 
 
 14462 
 
 
 
 Here 6 n (fig. 41) 
 
 Broke by tension 1 inch from 
 
 14462 fts. 
 = 129 cwt. 
 14 Ibs. 
 
 4-5 
 = 3214 
 
 2693 
 
 i nearly 
 
 = 2-5 inches. 
 
 the middle, 
 
 
 16730 
 
 
 
 
 After having borae the last- 
 
 16730 fts. 
 = 149 cwt. 
 
 5 
 
 = 3346 
 
 2693 
 
 I nearly 
 
 
 named weight some mi- 
 nutes, it broke by tension 
 very near the middle. 
 
 1502 -I ft.s. 
 = 134 cwt. 
 16 fts. 
 
 15024 
 
 T628" 
 = 3246 
 
 
 
 
 Broke by tension very nearly 
 in the middle. 
 Tiiis beam and those in all 
 the experiments, except the 
 last, were of the form (PI. 
 III., figs. 42 and 43), being 
 uniform in height, and hav- 
 ing a large bottom rib ta- 
 pering towards the ends. 
 
 Y 2 
 
324 
 
 EXPERIMENTS TO OBTAIN THE 
 
 TABLE I. 
 
 
 
 H o 1 -< 
 
 
 
 
 
 
 \ 
 
 ' 
 
 a o 
 
 | 
 
 
 
 oj 
 
 :> . 
 Form of section of beam in 
 middle. 
 
 Area of top 
 rib in middle 
 of beam. 
 
 Area of bot- 
 tom rib in 
 middle of 
 beam. 
 
 & 
 
 141 
 
 * 
 
 3 
 
 Lions in j 
 ' an inch 
 
 l.s 
 
 8,5 
 
 n 
 
 
 
 
 p 
 
 |f 
 
 1 
 
 So 
 1 
 
 Q 
 
 I! 
 
 9th Experiment. 
 
 2-35 x -29 
 
 5-43 x -537 
 
 35 
 
 5-292 
 
 1 -12 
 
 6218 
 
 From the same model 
 
 = 68 
 
 = 2-916 
 
 
 
 15 
 
 7598 
 
 as that used in Experi- 
 
 
 
 
 
 18 
 
 8288 
 
 ment 8, except that 
 
 
 
 
 
 20 
 
 9309 
 
 the bottom rib is in- 
 
 
 
 
 
 
 22 
 
 10330 
 
 creased in breadth. 
 
 
 
 
 
 
 25 
 
 11338 
 
 
 
 
 
 
 
 26 
 
 12346 
 
 
 
 
 
 
 
 29 
 
 13354 
 
 
 
 
 
 
 31 
 
 14371 
 
 
 
 
 
 
 33 
 
 15393 
 
 L 
 
 
 
 
 
 
 53 
 
 16401 
 
 10th Experiment. 
 Beam from the same 
 
 
 6-8 x -502 
 = 3-413 
 
 
 
 644 Ibs. 
 
 16 
 
 18 
 
 6218 
 7598 
 
 model, but with fur- 
 
 
 
 
 
 
 19 
 
 8288 
 
 ther increase of bottom 
 
 
 
 
 
 
 21 
 
 9309 
 
 rib. 
 
 
 
 
 
 
 22 
 
 10331 
 
 
 
 
 
 
 
 24 
 
 11339 
 
 
 
 
 
 
 
 26 
 
 12341 
 
 
 
 
 
 
 
 28 
 
 13351 
 
 11 tli Experiment. 
 Beam from same model 
 as in last experiment. 
 
 2-3 x -28 
 = 64 
 
 6-61 x -54 
 = 3-57 
 
 34 
 
 5-86 
 
 68*ft>3. 
 
 26 
 29 
 30 
 33 
 35 
 
 12057 
 12777 
 repeated 
 14345 
 15913 
 
 
 
 
 
 
 
 36 
 
 16697 
 
 
 
 
 
 
 
 43 
 
 18265 
 
 12th Experiment. 
 
 2-33 x -31 
 
 6-67 x -66 
 
 266 
 
 6-4 
 
 71 Ibs. 
 
 j 
 
 T 
 
 = 72 
 
 = 4-4 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 ' 
 
 1 ^j%i%ji% %jaa%iik 
 
 
BEST FORM OF BEAMS. 
 
 325 
 
 (Continued.) 
 
 
 2 
 
 d o^o 
 
 >, . 
 
 
 
 
 H. 
 
 *l|j 
 
 ill 
 
 
 
 Breaking 
 weight 
 
 s.| 
 
 
 
 Form of fracture. 
 
 Remarks. 
 
 infos. 
 
 ^o.a 
 
 'S)- d 
 
 tlllf 
 
 IK 
 
 
 
 
 J.2 
 
 2 si -g a 
 
 i 8 1 
 
 
 
 
 05 
 
 & o 3 o 8 
 
 6 ,0 
 
 
 i 
 
 
 16905 
 
 
 
 
 Broke by tension. 
 
 
 5-292 
 
 
 
 
 
 
 = 3194 
 
 
 
 
 
 16905 fcs. 
 
 
 
 
 
 
 = 150 cwt. 
 
 
 
 
 
 
 105fts. 
 
 
 
 
 
 
 
 
 
 
 
 This broke by tension, and 
 
 
 
 
 
 
 ought to have borne con- 
 
 
 
 
 
 
 siderably more than the last 
 
 
 
 
 
 
 beam; but its iron must 
 
 
 
 
 
 
 have been of a less tenacious 
 
 
 
 
 
 
 kind than the others ; as is 
 
 14336 fts. 
 nearly 
 
 
 
 
 
 evident by comparing their 
 deflexions, this beam having 
 
 = 128 cwt. 
 
 
 
 
 
 bent little more than half 
 
 
 
 
 
 
 what the preceding one did 
 
 
 
 
 
 
 before it broke. 
 
 
 
 
 
 
 
 
 19441 
 
 
 
 
 This beam broke very nearly 
 
 
 5-86 
 
 
 
 
 in the middle, by tension, 
 
 
 = 3317 
 
 
 
 
 as before. 
 
 19441 fta. 
 
 
 
 
 
 
 = 1734 cwt. 
 
 
 
 
 
 
 26084 fts. 
 
 26084 
 
 
 
 A wedge separated 
 
 
 = 11 tons 
 
 6'4 
 
 
 
 from its upper 
 
 
 13 cwt. 
 
 =4075 
 
 2885 
 
 upwards 
 off 
 
 side, as shown in 
 the fig. below, 
 
 
 
 
 
 
 which is a side 
 
 
 
 
 
 
 view of the beam, 
 
 
 
 
 
 
 where a d c is the 
 
 
 
 
 
 
 wedge, a c = 5'1 
 
 
 
 
 
 
 inches, b d = 3'9 
 
 
 
 
 
 
 inches, angle ad c 
 
 
 I 
 
 
 
 
 at vertex = 82. 
 
 
 
 
 
 
 
 . 
 
326 
 
 EXPERIMENTS TO OBTAIN THE 
 
 TABLE L 
 
 
 
 
 h 
 
 3<S 
 
 I 
 
 1 
 
 bo,; 
 
 Form of section of beam in 
 middle. 
 
 Area of top 
 rib in middle 
 of beam. 
 
 Area of bot- 
 tom rib in 
 middle of 
 beam. 
 
 ness of ver 
 between 1 
 ribs. 
 
 i of sectioi 
 M at place 
 fracture. 
 
 ght of bea 
 
 xions in p 
 if an inch. 
 
 .2*9 
 
 11 
 
 
 
 
 P 
 
 2 
 4* 
 
 g 
 
 o o 
 
 I 
 
 o 3 
 6$ 
 
 13th Experiment. 
 From the same model as 
 
 2-3 x -28 
 = 64 
 
 6-63 x -65 
 = 4-31 
 
 335 
 
 6-5 
 
 74ffcs. 
 
 22 
 24 
 
 9328 
 11397 
 
 in the last Experiment. 
 
 
 
 
 
 
 25 
 
 12777 
 
 
 
 
 
 
 
 26 
 
 14345 
 
 
 
 
 
 
 
 30 
 
 15913 
 
 
 
 
 
 
 
 34 
 
 17481 
 
 
 
 
 
 
 
 36 
 
 18265 
 
 
 
 
 
 
 
 38 
 
 19049 
 
 
 
 
 
 
 
 43 
 
 20617 
 
 
 
 
 
 
 
 47 
 
 22185 
 
 
 
 
 
 
 48 
 
 22969 
 
 
 
 
 
 
 
 50 
 
 V 
 
 14th Experiment. 
 
 1-54 x -32 
 
 6-50 x -51 
 
 34 
 
 5-41 
 
 TOffite. 
 
 26 
 
 9327 
 
 Elliptical beam, differ- 
 
 = 493 
 
 = 3-315 
 
 
 
 
 27 
 
 10707 
 
 ing from that in Ex- 
 
 
 
 
 
 
 28 
 
 11397 
 
 periment 7, in having 
 a larger bottom rib; 
 
 
 
 
 
 
 30 
 31 
 
 12087 
 12777 
 
 ratio of ribs 6 to 1. 
 
 
 
 
 
 
 34 
 
 14345 
 
 
 
 
 
 
 
 35 
 
 15913 
 
 s^npn 
 
 
 
 
 
 
 42 
 
 16697 
 
 j 
 
 
 
 
 
 
 43 
 
 17481 
 
 1 
 
 
 
 
 
 
 46 
 
 19849 
 
 1 
 
 
 
 
 
 
 50 
 
 19833 
 
 1 
 
 
 
 
 
 
 54 
 
 20617 
 
 1 
 
 
 
 
 
 
 
 
 
BEST FORM OF BEAMS. 
 
 327 
 
 (Continued.) 
 
 
 2 
 
 H 
 
 Jlli 
 
 3 a 
 fill 
 
 
 
 Breaking 
 weight 
 
 
 liffi 
 
 !fl 
 
 Form of fracture. 
 
 Remarks. 
 
 
 II 
 
 ||||i 
 
 ||| 
 
 
 
 
 02 
 
 oh M 
 
 0^ 
 
 
 
 
 23249 
 
 2885 
 
 ^nearly. 
 
 
 Broke in the middle by 
 tension. 
 
 6-5 
 = 3576 
 
 23249 Ibs. 
 = 10 tons 8 
 cwt. nearly 
 
 
 
 
 
 
 
 21009 
 5'41 
 = 3883 
 
 2885 
 
 upwards 
 of* 
 
 Nearly as in figure 
 to first experi- 
 ment, 6 n = 1*8 
 inch. 
 
 
 21009 Ibs. 
 = 9 tons 8 
 cwt. nearly 
 
 ! 
 
 
 
 
 
328 
 
 EXPERIMENTS ON BEAMS OF BEST FORM. 
 
 g 
 
 12 
 
 (M^-ii^eob->. 
 
 COOiC^COi IOO(NT*<t O^OOCOi lOii 
 O>t~'*<OOCO 
 
 II 
 
 - 
 
 1 
 
 o8 
 
 P3 
 
 Hill I 
 
 ,3V, ftfe i ? 
 
 
 ! -9 
 
 *<* 
 
 CO 
 
 t 
 
 9, 
 
 i" 
 
 </3 -4^ Q- 
 
 Bit 
 
 .3 
 
 !i 
 
EXPERIMENTS ON BEAMS OF BEST FORM. 
 
 329 
 
 . 
 
 T-l 
 
 Sfi - 
 
 O> -^ Oi <M 
 
 '3H |S S r S a ' <a -S'0 ,: SJ 
 
 |l|||15i|l||ii 
 
 <M O5 Mi r-t OO "*< 
 CO CO CO Oi ^ 
 O r-l ^ O t~ OS 
 
 CO CO CO OS to OO 
 1 CO CO -jfl -rji >0 >p 
 
 OOO0OOOOO(NO 
 
 (N CO >p 
 
 X p 
 
 o "7 
 
 f 
 
 
 - . L 
 
330 
 
 EXPERIMENTS ON BEAMS OF BEST FORM. 
 
 !a 
 
 -2 
 
 
 inches from the 
 where there were 
 ll defects whose 
 
 th of an inch 
 ib. The ex- 
 herefore im- 
 
 m 
 as 
 
 e 9 
 dle, 
 
 ke 
 id 
 o 
 a w 
 he 
 me 
 ec 
 
 Bro 
 mi 
 tw 
 
 in 
 peri 
 per 
 
 *i 
 
 IJf 
 
 O5 OO t^ O 
 CM OS CO CO 
 00 CO OS O 
 
 S 
 
 dle, 
 s in 
 the 
 
 2- 
 
 25 
 
 mid 
 e a 
 a c, 
 
 in the 
 a wedge 
 Here a 
 wedge, 
 depth, 
 
 out 
 ent 6. 
 f the 
 Id, its 
 
 It broke 
 throwing 
 Experim 
 length of 
 inches; I 
 inches. 
 
 OS CO WW 
 
 "^ 
 
 to co co co co co co 
 
 01 00 t~ CO r-l O 
 CM O> CD ^* 00 >O 
 OO OO OS rH <M CO ^ 
 
 O CO O O <M * ^ 
 
 OS CS O CO U5 OO O 
 
 rH rH rH rH Cl 
 
 I 
 
 T 
 
 ii 
 
 W 
 
 
 CO 
 CO 
 
 CO 
 
 t>. 
 
 X CO 
 
 d 
 
 a s^ 
 
 'S -3 *" 
 
 III 
 
 H * 
 
 PQ 
 
EXPERIMENTS ON BEAMS OF BEST FORM. 
 
 331 
 
 3 * 
 
 a 
 
 CO 
 
 co b 
 
 O O> 2 (3 ^ f-Q 
 
 flips 
 
 i ffgl 
 
 
 T ICOCDOrHCOiO 
 ilr- (T-ii C<I<N<N 
 
 CO 1O 
 rH <n Cp 
 
 din 
 
 
 
 w 
 
332 
 
 EXPERIMENTS ON BEAMS OF USUAL FORM. 
 
 li 
 
 S: 
 
 $ 
 
 2S 1 
 
 11 
 
 s M 
 
 a-*l 
 
 PQ ^^ 
 
 5 si 
 
 g o 
 
 J8 
 
 
 ;<3 
 
 1 f 
 i^l 
 
 <ri 
 
 squ 
 n in 
 
 w 
 
 sa 
 
 
 t-o 
 
 50 
 
 00 . 01 
 
 cojco^ 
 
 
 co 
 1 
 
 i 
 
 i 
 
 
 OO CO 
 i-H CO 
 (M i-l 
 
 co co 
 
 CS CO 
 
 
EXPERIMENTS ON BEAMS OF USUAL FORM. 
 
 333 
 
 -^COOOCOOOCOOOCOOOCOOOCOOOO 
 
 ,,. 
 
 ^ co QO 
 * 
 
 
 oooocooocooocooocooocoeoooocq 
 
 CO" 
 
 H 
 
 
334 REMARKS ON TABLE I. 
 
 REMARKS UPON THE EXPERIMENTS, IN TABLE I., TO OBTAIN THE 
 BEST FORM OF CAST IRON BEAMS OF WHICH THE LENGTH 
 AND DEPTH WERE INVARIABLE. 
 
 138. It has been mentioned before, that these experiments 
 were begun with that form of section of beam which Tredgold 
 (Part I., art. 40) was induced to consider as the strongest, 
 the top and bottom flanges in it being equal. This form 
 was, however, found to be ^th weaker to resist fracture than 
 that in common use (the -L form) ; though it would, perhaps, 
 be nearly the strongest in wrought iron. The top flange, in 
 the model before used, was next reduced, and the part taken 
 off it was added to the bottom one. This alteration gave an 
 increase of strength, and the beam, in Experiment 3, was 
 somewhat stronger than that of the usual form cast with it 
 for comparison. It did not now seem advisable to decrease 
 further the top flange ; and as every beam had been found to 
 break by tension, or through the weakness of the bottom 
 part, I thought it best to keep increasing the bottom flange, 
 by small degrees, till such time as the beam broke by the 
 rupture of some other part. Proceeding thus, the beam, in 
 Experiment 5, had its bottom and top ribs, or flanges, as 4J 
 to 1 ; and the result from that form of section was a gain in 
 strength of about yth. Before increasing the bottom rib any 
 further, I added a little to the top one, as the vertical part of 
 the beam, or that between the ribs, would be perhaps strong 
 enough for much larger ribs. In Experiments 6, 7, and 14, 
 the top rib and vertical part of the model were the same, the 
 only difference being in the increasing breadth of the bottom 
 rib. From these, the gain in strength, above what was 
 borne by beams of the usual form, was respectively ^th, ^th, 
 and between ^th and ^rd. 
 
 In Experiments 8, 9, 10, 11, 12, 13, the top rib of the 
 model was the same; but it was somewhat larger than in 
 
REMARKS ON TABLE I. 335 
 
 Experiments 6, 7, and 14, and the bottom rib was the only 
 part intended to be varied. In the 12th Experiment of the 
 Table (being the 19th made) the section of the bottom rib at 
 the place of fracture, the middle, was more than double the 
 rest of the section there ; and the ratio of the top and bottom 
 ribs was as 1 to 6. In all the beams before this, fracture 
 had taken place through the weakness of the bottom rib. In 
 this it took place by the separation of a wedge from the top 
 rib and the vertical part of the beam, which part happened 
 to be thinner than usual : the gain in strength, arising from 
 the form of the section in this beam, was upwards of f ths of 
 what the beam of usual form bore ; and the saving from the 
 general form of the beam was nearly ^ths of the metal. This 
 experiment was repeated in Experiment 13, but the beam, 
 though cast from the same model, had its vertical part thicker 
 than in the former ; and its strength, per square inch of sec- 
 tion, was less. Fracture took place in it by the rupture of the 
 bottom rib, as in all the preceding experiments except the last. 
 
 The form of section in Experiment 12 is somewhat better 
 than that in Experiment 14 ; it is the best which was arrived 
 at for the beam to bear an ultimate strain ; and it is, doubt- 
 less, nearly the strongest which can be attained : for the 
 vertical part between the flanges is as thin as it can, pro- 
 bably, be cast ; and the remainder of the metal is disposed in 
 the flanges, and consequently as far asunder as possible, the 
 section of the flanges being in the ratio of 6 to 1, or nearly in 
 that of the mean crushing and tensile strength of cast iron 
 (Part II., art. 33). 
 
 The strengths, per square inch, borne by the beams in 
 Experiments 12, 13, and 14, were 4075, 3576, 3883 fbs. 
 respectively, the mean being 3845 flbs. And the strength of 
 the beam of the common form (Table III.), cast with them 
 for comparison, was 2885 Jbs. per square inch. The differ- 
 ence, or strength gained from a mean among the results, 
 was therefore 960 fts., or upwards of th of that mean. 
 
336 FORM OF BEAM ALTERED. 
 
 Some of the beams, it will be noticed, had their bottom rib 
 considerably thicker than the vertical part between the ribs ; 
 the line of junction being tapered from the thick to the 
 thinner part. This tapering was more gradual, and higher 
 up the beam, than is represented in the forms of section. 
 The castings obtained were very good, as might be inferred 
 from the strength of them ; but as additional care is requisite 
 to obtain good castings, when the parts differ much in 
 thickness, we should bear in mind that it is not absolutely 
 necessary, but convenient, to make one part thicker than 
 another. The same strength would have been obtained by 
 making the bottom rib broader and thinner than that in the 
 beams tried, leaving the quantity of metal the same. 
 
 FORM OF BEAM ALTERED. 
 
 139. After Experiment 7, it was evident that the bottom 
 flange in the future beams, if made of equal size throughout, 
 as heretofore, would become very heavy ; I had, therefore, the 
 form of the beams, near to the ends, changed, leaving the 
 section in the middle, as before, and making the height of the 
 beam equal throughout, instead of being elliptical, as in the 
 previous experiments. The bottom flanges were both made 
 to taper towards the ends in the form of a double parabola 
 whose vertex was in the middle of the beam ; and the ratio of 
 the sections of the flanges were, throughout the length of the 
 beam, the same as that in the middle. (See Plate III. figs. 
 42 and 43, the former being a plan, and the latter an 
 elevation of the beam.) 
 
 140. From the great quantity of matter in the flanges, 
 particularly the bottom one, in the subsequent experiments, 
 and the small thickness of the part between the flanges, I was 
 convinced that nearly all the tensile force would be exerted by 
 the bottom flange, whilst the rest of the beam would serve for 
 little- more than a fulcrum; the centre of resistance to 
 
FORM OF BEAM ALTERED. 337 
 
 compression, or of that fulcrum, being very near to the top ; 
 it being perhaps at the point r, in the experiments in the 
 
 Table (Table!.). 
 
 Suppose then D to be the vertical distance from the centre 
 of compression, at any part of the beam, to the centre of 
 tension in the bottom flange ; and if T be the direct tensile 
 strength of the bottom flange at that part, T multiplied by 
 some function of D, (or T D nearly,) will represent the 
 strength of the beam there. But D throughout the same 
 beam will be a constant quantity, or nearly so ; the strength 
 of the beam at any part, will, therefore, be nearly in pro- 
 portion to that of its bottom flange at that part ; and as the 
 strain will be less towards the ends than elsewhere, the bottom 
 flange will be reduced there likewise. Suppose the bottom 
 flange to be formed of two equal parabolas, the vertex of one 
 of them, A C B, being at C, in the figure annexed ; then, by 
 
 the nature of the curve, any ordinate ^eisasA^xB^; the 
 strength of the bottom flange, therefore, and consequently 
 that of the beam at that place, will be as this rectangle. It is 
 shown too, in Part I., and by writers on the strength of 
 materials generally, that the rectangle A e x B e is the 
 proportion of strength which a beam ought to have, at any 
 distance A e from A, to bear equally the same weight in every 
 part, or a weight laid uniformly over it. The conclusions 
 above were verified by several experiments, in which beams 
 were broken by weights applied at half the distance between 
 the middle and the ends. 
 
 141. From the experiments in Table I., in which the length 
 and depth of the beam were always the same, it would appear, 
 that when the size of the top flange and the thickness of the 
 
338 SIMPLE RULE FOR THE STRENGTH 
 
 vertical part remain unaltered, the latter being small, the 
 strength of the beam is nearly in proportion to the size of the 
 bottom flange ; double the size of that flange giving nearly, 
 but not quite, double strength. 
 
 REMARKS ON TABLE II. 
 
 142. The first four beams in this Table were all cast 7 feet 
 6 inches long, and they were supported by props 7 feet 
 asunder. They were all from the same model, which varied 
 only in the breadth of its vertical part between the flanges, 
 the depth of the beam being all that was intended to vary. 
 The depths were nearly 4, 5, 6, and 7 inches, but accurate 
 admeasurements are given with the sections. The vertical 
 part in these beams was rendered too strong comparatively 
 with the size of the bottom flange, in order that they might 
 all break by tension, or in the same manner, to furnish the 
 means of judging correctly of their relative strength. 
 
 143. Prom these experiments it appears that the ultimate 
 strength, in sections like the preceding, is, cceteris paribus, 
 nearly as the depth, but somewhat lower than in that 
 ratio. 
 
 SIMPLE RULE FOR THE STRENGTH OP BEAMS APPROACHING 
 TO THE BEST FORM IN THE PRECEDING EXPERIMENTS. 
 
 (TABLES i. AND n.) 
 
 ' 144. It appears, from art. 141, that when the length, 
 depth, and top flange, in different cast iron beams, with very 
 large flanges, are the same, and the thickness of the vertical 
 part between the flanges is small and invariable, the strength 
 is nearly in proportion to the size of the bottom flange. It 
 appears, too, from the last article, that in beams which vary 
 only in depth, every other dimension being the same, the 
 strength is nearly as the depth. 
 
OF BEAMS OF THE BEST FORM. 
 
 339 
 
 145. Hence in different beams whose length is the same, 
 the strength must be nearly as their depth multiplied by the 
 area of a middle section of their bottom flange ; and when the 
 length is different, the strength will be as this product divided 
 by the length. 
 
 where W=the breaking weight in the middle of the beam, 
 #=the area of a section of the bottom flange in the middle, 
 d=ihe> depth of the beam there, / =the distance between the 
 supports, and c=a quantity nearly constant in the best forms 
 of beams, and which will be supplied from the results of the 
 experiments in Tables I. and II. 
 
 146. We will seek, by means of this approximate formula, 
 for the value of c considered as constant, obtaining it from each 
 of the experiments ; and, for that purpose, confining ourselves 
 to those forms in which the section of the bottom flange in 
 its middle is more than half the whole section of the beam, 
 take the mean from among them all for c. 
 
 Since W - 
 
 cad 
 
 _ 
 
 ad 
 
 Taking the dimensions in inches, and the breaking weight in 
 cwts., and separating the results of the beams which were 
 cast erect from those cast on their side, we shall have 
 
 In beams cast erect. 
 Experiment Value of c. 
 
 7 
 9 
 
 11 
 
 12 
 
 13 
 
 14 
 
 1 
 
 2 
 
 4 
 
 Table I. 
 
 Table II. 
 
 545 
 545 
 512 
 558 
 507 
 596 
 558 
 472 
 529 
 
 Mean 536 cwt. 
 
 In beams cast on their side. 
 Experiment Value of c. 
 
 Table II. 
 
 10 
 11 
 
 494 
 484 
 489 
 571 
 531 
 
 Mean 514 cwt. 
 
 Since 536 or 514 is the mean value of c to give the breaking- 
 weight in cwts., according as the beam has been cast erect or 
 
 z 2 
 
340 
 
 ANOTHER APPROXIMATE RULE 
 
 on its sides, one-twentieth of these numbers, or 26 'S and 
 2 5 '7, will be the value of c to give the breaking weight in 
 tons. Neglecting the decimals, and taking 26 for the mean 
 value of c, we have 
 
 , co_d_26 ad 
 "~T = ~~l~ 
 
 for the strength in tons where the dimensions d and / are in 
 inches. 
 
 But if /, the distance between the supports, be taken in feet, 
 the value of c will be ff = 2'166, and the strength in tons 
 will be 
 
 w= 
 
 This rule is formed on the supposition, that the strength of 
 the flanges is so great that the resistance of the middle part 
 between them is small in comparison, and may be neglected. 
 
 Another approximate rule, for the strength of the beams in 
 Tables I. and II., and which includes the effect of the vertical 
 part between the flanges, may be deduced as below. 
 
 147. Since cast iron resists rupture by compression with 
 about 6^ times the power that it does by extension, (art. 33,) 
 we may consider it as comparatively incompressible, and 
 suppose that the operation of the top flange of the beam, 
 when bent, is only to form a fulcrum upon which to break the 
 bottom flange and the part between the flanges. 
 
 Let then A D E represent the section of the beam in the 
 middle, D E being its bottom flange and A the top one, round 
 which it turns. 
 
FOR THE STRENGTH OF BEAMS. 341 
 
 Let W = breaking weight. 
 
 / = distance between supports. 
 
 d = A C = the whole depth. 
 
 d'= A B = the depth to the bottom flange. 
 
 b = D E = the breadth of the bottom flange. 
 
 #'= F G = the thickness of the vertical part. 
 
 f= the tensile strength of the metal per unit of 
 
 section. 
 
 n = a constant quantity. 
 
 To seek for the strength of the beam we may estimate, 
 first, the resistance of a rectangular solid whose depth is A C 
 and breadth D E, as shown by the dotted lines, and subtract 
 from that the resistance which would be offered by that part 
 which the beam wanted to make it such a uniform solid as 
 the above. 
 
 = moment of resistance of the particles in the 
 
 rectangular solid A D E, 
 and - = moment of resistance of the part necessary 
 
 to make the beam a solid rectangle, where /' is the 
 strain of the particles at the distance AB. 
 When the beam is supposed to be incompressible, as in the 
 present case, n is equal to 3, and when it is equally extensible 
 and compressible, n is equal to 6 (Tredgold, art. 110). 
 
 Substituting this value for f in the latter moment of 
 resistance, and subtracting the result from the former moment, 
 gives the moment of resistance of the solid equal to 
 
 fid 2 _ f_ (b - b'} d 1 2 
 n d n 
 
 But, from the property of the lever, this moment is equal to 
 
 w i w 
 1 1 x - = , or to half the weight acting with a leverage of 
 
 half the length. 
 
342 
 
 APPROXIMATE RULE 
 
 Whence lw - f J _ 
 
 ' ~ 
 
 In the same iron, /and n are constants ; putting c = -', w< 
 shall have 
 
 -_ 
 a I 
 
 This formula gives 
 
 The numerical value of c, calculated by this formula, from 
 each of the experiments in Tables I. and II., taking the 
 breaking weight in Ibs., the length in feet, and the other 
 dimensions in inches, is as below. 
 
 Table I. 
 
 Tablel I. 
 
 Value of c. 
 ,' 1932-78 
 1566-36 
 1593-82 
 1607-46 
 1604-63 
 1623-98 
 1627-43 
 1547-49 
 
 Value of c. 
 1897-93 
 1735-94 
 1666-31 
 1638-92 
 1744-10 
 1725-90 
 1631-74 
 1785-96 
 1570-80 
 183514 
 1671-01 
 U797-87 
 Mean value of c from the whole twenty beams, 1690 '28. 
 
 This value of c is in fts., and dividing it by 2240, gives *7544 
 for its value in tons. 
 
 The iron in my experiments on beams was of a strong 
 kind, made with a cold blast ; and many of the beams were 
 cast erect in the sand, which gives them a little additional 
 strength. We may, therefore, expect that the value of c, just 
 obtained, will be somewhat too great for the generality of hot- 
 blast castings; and for large beams, the iron of which is 
 usually softer than that of small ones. We will, therefore, 
 collect here its values, to obtain the strength in tons, com- 
 
FOR THE STRENGTH OF BEAMS. 343 
 
 puted from the results of other experiments on a large scale 
 given further on. Taking them in the order in which they 
 are inserted, we have as follows : 
 
 From Messrs. Marshall's beams (art. 153), cast from the cupola, 
 
 we obtain . . . . . . . " . . . . c '625 
 
 ....... c = -710 
 
 Prom Mr. Gooch's beam (art. 154), we obtain c = '679 
 
 Messrs. Marshall's beam (art. 153), cast from the air furnace, gives c = '795 
 
 Mean 
 671 
 
 Mr. Cubitt's beams (art. 165), taking the results from the 
 sound ones only, and the value of l> from a mean where that 
 dimension varies, give as below. 
 
 From Experiment 1 
 i> 11 *> 
 
 4 
 6 
 9 
 
 10 
 
 
 
 c = -6467 
 c '7276 
 
 Mean 
 
 . . . ' 
 
 c = -7473 
 c = -6703 
 C--7746J 
 
 ^ -7086 
 
 Taking a mean value of c, as obtained from the whole of the 
 beams cast from the cupola, thirty in number, we should have 
 it considerably more than *7 ; the means from twenty experi- 
 ments being "7544 ; from three experiments "671 ; and from 
 seven experiments *7086. The lowest of these means differs 
 but little from f ; and adopting this as a safe approximate value 
 for c, from which to compute the strength of beams generally, 
 we have in the preceding formula for W its value as below. 
 
 where W is in tons, / in feet, and the rest are in inches. 
 
 148. The preceding formula for the strength of a beam 
 depends on the two following suppositions : 1st, that all the 
 particles, except those of the top part or flange of a bent 
 beam, are in a state of tension ; 2nd, that the resistance of 
 each particle is as its distance from the top of the beam. 
 Neither of these suppositions can be regarded otherwise than 
 as an approximation. We know that the former, which is 
 almost tantamount to the exploded assumption of Galileo, 
 that materials are incompressible, is not strictly true of any 
 
344 
 
 APPROXIMATE RULE 
 
 oodies whatever ; and the second supposition is subject to the 
 double inaccuracy of the leverage of the particles being 
 estimated as from the top of the beams, and therefore rather 
 too great ; and of the force of the fibres being as their 
 extension, whilst, in reality, it is in a less ratio than that, as 
 shown in preceding articles (99 to 107). 
 
 If, as is expected, the formula should be allowed to give 
 results agreeing moderately well with those of experiment at 
 the time of fracture, it will appear evident that the 2nd 
 assumption above is favourable to that of incompressibility, in 
 estimating the transverse strength of cast iron. 
 
 149. To obtain further evidence on this subject, we will 
 seek, by means of the experiments in this work, for the value 
 
 of n in the formula, w =-^j-> f r the strength of a rectangular 
 bar, fixed at one end and loaded at the other, b being the 
 breadth, d the depth, I the length of leverage, w the weight 
 at the end, and the rest as in art. 147. 
 
 150. Selecting from the experiments, in articles 3 and 96, 
 the mean tensile and transverse strengths of all the irons in 
 which both these properties were obtained, we have as in the 
 following Table. 
 
 
 11 
 
 Mil 
 
 fill"!* 
 
 
 
 Z* 
 
 J3 m "y R" 
 
 jj^ 1 fl ^) 03 
 
 Value of n 
 
 
 S!Q 
 
 co f3 2 *** 
 
 QQ ^jH ^ ^J ^ rC s-^. 
 
 from formula 
 
 Description of Iron. 
 
 
 2-S 1 * c 
 
 02 a ^M^ =3- 
 
 fbd* 
 
 
 ^ M 
 
 ^^9 
 
 i^s-si fe 
 
 n ~Tw~ ' 
 
 
 o o 
 
 r- Cj P ^ 
 
 f3 05*73^ O ^ 
 
 
 
 
 IS* 
 
 l^ll^? 
 
 
 
 Ibs. 
 
 fts. 
 
 Ibs. 
 
 
 Carron iron, No. 2, cold blast . . . 
 
 16,683 
 
 476 
 
 238 
 
 7i = 2'o9 
 
 Carron iron, No. 2, hot blast . . . 
 
 13,505 
 
 463 
 
 23U 
 
 7i = 2'lG 
 
 Carron iron, No. 3, cold blast ... 
 
 14,200 
 
 446 223 
 
 w=2'36 
 
 Carron iron, No. 3, hot blast ... 
 
 17,755 
 
 527 2634 
 
 w=2-50 
 
 Devon iron, No. 3, hot blast 
 
 21,907 
 
 537 2684 
 
 w = 3'02 
 
 Buffery iron, No. 1, cold blast ... 
 
 17,466 
 
 4ff3 2314 
 
 7i = 2'79 
 
 Buffery iron, No. 1, hot blast ... 
 
 13,434 
 
 436 218 
 
 n=2-28 
 
 Coed -Talon iron, No. 2, cold blast 
 
 18,855 
 
 413 2064 
 
 7i = 3'33 
 
 Coed-Talon iron, No. 2, hot blast 
 
 16,676 
 
 416 
 
 208 
 
 n = 2'96 
 
 Low Moor iron, No. 3, cold blast 
 
 14,535 
 
 467 
 
 2334 
 
 = 230 
 
 
 
 
 
 Mean "alue 
 
FOR THE STRENGTH OF BEAMS. 345 
 
 151. The transverse strength of a rectangular body being 
 directly as the product of the breadth, the square of the depth 
 and the strength of its fibres, and inversely as the length, the 
 value of w, in the formula w = -y, will depend upon the 
 
 value of n ; and this last quantity will, as we have seen, depend 
 on the comparative resistance of the fibres to extension and 
 compression. Thus if, according to the general assumption, 
 the extensions and compressions of the particles are equal 
 from equal forces, and as the forces, the neutral line will be in 
 the middle of the body, and the value of n equal to 6 (Part L, 
 art. 110). If, according to Galileo, the body were incom- 
 pressible, and the forces of the fibres were as their extension, 
 we should have n = 3 ; and if, on the supposition of incom- 
 pressibility, the forces of the fibres were the same for all degrees 
 of extension, we should have n = 2. (See my Paper on the 
 Strength of Materials, ' Memoirs of the Literary and Philo- 
 sophical Society of Manchester,' vol. iv. 2nd series, p. 243.) 
 
 The value of n, in the preceding Table, obtained from 
 numerous experiments upon ten kinds of cast iron, varies 
 from 2*16 to 3 "38, the mean being 2*63. This result shows 
 that the assumption of the incompressibility of cast iron may 
 be admitted so long as we assume that the forces are directly 
 as the extension of the fibres ; and it might be admitted still, 
 if we were to make the more improbable assumption, that the 
 forces are the same for all degrees of extension ; for the value 
 of n in the former case would be 3, and in the latter 2, and 
 the mean result is 2 '63, somewhat nearer to the former than 
 the latter. 
 
 The mean value of n obtained from the fracture of different 
 kinds of stone, in numerous experiments not yet published, is 
 not widely different from 3. The value of n, being assumed 
 by Tredgold as 6, has, when applied at the time of fracture, 
 caused the errors pointed out in notes to arts. 68, 143, &c., 
 of Part I. 
 
346 EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 
 
 152. To obviate the anomalies above, and to obtain results 
 consistent with experiment in the fracture of beams of cast 
 iron taking the neutral line in its proper position we 
 shall assume the forces of extension and compression to be of 
 the form/ = #-<(#), where/ is the force, x the extension 
 or compression, a a constant quantity, and <f> (x) a function 
 representing the diminution of the force / in consequence of 
 the defect of elasticity. If < (x) be assumed as equal to 
 6 cc n , as in art. 106, we shall have n = 2 nearly, if the 
 experiments are made on the transverse flexure of bars ; but 
 it is more desirable that the value of n should be obtained 
 from the direct longitudinal variations of the body experi- 
 mented upon. This subject will be resumed in a future 
 article. 
 
 EXPERIMENTS ON LARGE BEAMS. 
 
 153. I have been favoured, through Mr. J. O. March, of 
 Leeds, with the results of three experiments upon beams 
 cast for the mill of Messrs. Marshall and Co., of that town, in 
 1838. They were from drawings supplied by Mr. Eairbairn, 
 of Manchester; and were of a moderately good form of 
 section, according to my experiments, though the bottom 
 flange was rather too small. The beams were broken to 
 ascertain the ultimate strengths, as well as to test the 
 difference of strength between those cast from the cupola and 
 the air furnace. The experiments, Mr. March states, were 
 very carefully made, under the inspection of Messrs. Marshall 
 and Co. ; and the beams were cast from Bierley pig iron. 
 The form, dimensions, and results are as follow : 
 
 Dimensions of the beams. 
 
 12 inches deep at one end, and 
 104 deep at the other. 
 
 2^ = breadth of rib on the upper edge at the ends. 
 
 4 = breadth of flange at the ends. 
 
EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 
 
 347 
 
 Dimensions of section in middle, in inches, 
 
 Area of top rib 3'00 x -75 = 2-25. 
 
 Area of bottom rib 8'25 x 1'25 = 10'31, 
 or 11 square inches, the increase from the brackets, at the 
 junction of the bottom rib with the vertical part, being 
 included. 
 
 Thickness of vertical part, f = '625 inch. 
 
 Depth of beam, in middle, 17 inches. 
 
 Seams proved at Messrs. Marshall and Co.'s, Leeds ; the distance between the supports 
 18 feet. Deflexions in 5Qths of an inch. 
 
 Tons. 
 
 2 
 
 4 
 
 G 
 
 8 
 
 10 
 
 12 
 
 14 
 
 1618 
 
 20 
 
 22 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 1st cupola casting. 
 Broke with 22 tons. 
 
 Deflected. 
 
 5 
 
 9 
 
 13 
 
 20 
 
 25 
 
 30 
 
 36 
 
 40 
 
 47 
 
 54 
 
 58 
 
 
 
 
 
 Permanent 
 Deflexion. 
 
 
 
 
 4 
 
 5 
 
 
 
 r7 
 
 
 8 
 
 10 
 
 
 
 
 
 
 
 Deflected. 
 
 
 7 
 
 
 12 
 
 
 26 
 
 
 3G 
 
 
 47 
 
 54 
 
 61 
 
 
 
 2nd cupola casting. 
 Broke with 25 tons. 
 
 Permanent 
 Deflexion. 
 
 
 
 
 4 
 
 
 6 
 
 
 G 
 
 
 8 
 
 10 
 
 121 
 
 
 
 
 Deflected. 
 
 
 7 
 
 
 16 
 
 
 32 
 
 
 35 
 
 
 47 
 
 53 
 
 60 
 
 (55 
 
 67 
 
 72 
 
 
 Air furnace casting. 
 Broke with 28 tons. 
 
 Permanent 
 Deflexion. 
 
 
 
 
 1 
 
 
 2 
 
 
 4 
 
 
 7 
 
 8 
 
 24 
 
 12 
 
 15 
 
 171 
 
 
 154. In an experiment made upon a beam by Mr. Gooch, 
 whilst he was superintending the formation of the Leeds and 
 Manchester Railway, the particulars are as follow : 
 
 Dimensions of section in middle, in inches. 
 Top rib . . 6 x 11 = 9. 
 Bottom rib . 8 x 11 = 12. 
 Thickness of middle pai't 11. 
 Depth of beam .... 9. 
 
 Distance between supports, 11 feet 8 inches ; weight 
 of casting, 11| cwt. 
 
 Weights laid on, 
 in tons. 
 
 Deflexions, in parts 
 of an inch. 
 
 41 
 51 
 61 
 71 
 81 
 
 10 
 
 17 
 
 20 
 
 Broke 81 inches from centre. 
 
 10 
 15 
 175 
 22 
 25 
 30 
 33 
 1-10 
 
348 EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 
 
 After bearing 17 tons, the beam was unloaded, and the 
 elasticity seemed to be very little or not at all injured. 
 The mixture of metal was 
 
 1 ton Colebrook Vale, cold blast. 
 14 Staffordshire, hot blast. 
 14 Scotch, 
 
 Mr. Gooch observes, in his letter giving an account of the 
 experiment, "The cross section and some of the other 
 dimensions are not of the most favourable or economical 
 form, but circumstances required the adoption of them in the 
 case of this girder." 
 
 155. The two following beams were cast for a viaduct 
 forming the junction of the Liverpool and Manchester with 
 the Leeds Railway, passing through Salford. They are not 
 of forms best adapted for resisting fracture, but their great 
 size will give additional interest to experiments upon them. 
 
 156. As several beams were cast from the same models, I 
 was requested, by the Messrs. Ormerod, of Manchester, the 
 founders, to superintend an experiment upon a beam from 
 each ; but the strength was so great that the experiment 
 could not well be made in the usual way, that of applying a 
 weight in the middle. I had, therefore, the beam inverted 
 during the experiment, its bottom flange being turned 
 upwards. The middle was supported laterally by stays, and 
 it rested upon a cross bar, and other apparatus, on which it 
 turned as on an axis ; this cross bar being made to rest on two 
 other beams of nearly equal magnitude to that intended to be 
 tried. One end of these beams, and the corresponding end 
 of the beam to be bent, were connected by means of a strong 
 bolt ; and the other end of the latter beam was connected, as 
 described below, with the opposite ends of the two supporting 
 beams. The object was to break the top beam in the middle 
 by a force applied at one end, whilst the other end was fixed; 
 and to effect the required pressure, a powerful lever, forged 
 
EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 
 
 349 
 
 for the purpose, was applied to the moveable end ; it being 
 evident that the pressure at the end would only be half of the 
 effect produced in the middle. (See Plate V.) 
 
 157. To enable the deflexions to be observed, a straight 
 edge of the same length as the beam was used, and made to 
 rest upon it, touching it only at the ends. The quantities, 
 which the deflexions varied in consequence, of different 
 weights, were measured by inserting a long wedge-like 
 body, graduated on the side, between the straight edge and 
 the top of the beam. The observed distances, when the 
 beam was bearing a given load, were subtracted from the 
 observed distance when there was no load upon it, for the 
 deflexion. 
 
 The experiments were made with a very complete apparatus 
 and every attention to accuracy. 
 
 158. The first beam had small ribs, or 
 flanges, at top and bottom, and a strong vertical 
 plate between them, as in the annexed section. 
 The dimensions and results from the beam are 
 as follow : 
 
 Section of top rib . . 5*1 x 2 '33 inches. 
 
 bottom rib . 121 x 207 
 Thickness at a, 2 -06 ) 
 
 I, 2-12 V mean 2-08. 
 
 c, 2-07 ) 
 
 Depth of beam in middle 30'5 inches. 
 Distance between supports 27 feet 5 inches. 
 Whole length of beam 28 feet 9 inches. 
 
 Weight applied at the end, 
 in tons. - 
 
 13-4 
 17-6 
 21-8 
 26-0 
 30-2 
 34-4 
 38-3 
 
 Weight applied in the 
 middle, in tons. 
 
 Deflexion in middle, in 
 parts of inch. 
 
 18-4 
 
 31 
 
 26'8 
 
 47 
 
 35-2 
 
 61 
 
 43-6 
 
 73 
 
 52-0 
 
 87 
 
 60-4 
 
 1-02 
 
 68-8 
 
 1-16 
 
 76-6 
 
 ' 1-29 
 
 With this, 76*6 tons in middle, it broke apparently in 
 consequence of an accidental shake. 
 
350 
 
 EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 
 
 159. The second beam was much 
 heavier and stronger than the former ; 
 its section is as in the annexed figure, 
 and its dimensions and results are as 
 follow : 
 
 Mean thickness of bottom flange 312 in. 
 Breadth of 23'9 
 
 Whole depth of beam . . . 36'1 
 Thickness of vertical rib at a . 314 
 6 . 3-36 
 e - 3-38 
 
 Distance between supports 23 feet 1 inch. 
 Whole length of beam 24 6 
 Weight of beam, 6 tons 1 cwt. 1 qr. 
 
 Weight applied at end of 
 beam, in tons. 
 
 Weight applied at middle ot 
 beam, in tons. 
 
 Deflexion in middle of beam, 
 in inches. 
 
 13-4 
 
 26-8 
 
 10 
 
 17-6 
 
 35-2 
 
 14 
 
 21-8 
 
 43-6 
 
 17 
 
 26-0 
 
 52-0 
 
 20 
 
 30-2 
 
 60-4 
 
 23 
 
 34-4 
 
 68-8 
 
 27 
 
 38-6 
 
 77-2 
 
 31 
 
 42-8 
 
 85-6 
 
 35 
 
 47-0 
 
 94-0 
 
 38 
 
 51-2 
 
 102-4 
 
 42 
 
 55-4 
 
 110-8 
 
 46 
 
 59-6 
 
 119-2 
 
 51 
 
 63-8 
 
 127-6 
 
 55 
 
 68-0 
 
 136-0 
 
 59 
 
 72-2 
 
 144-4 
 
 64 
 
 76-4 
 
 152-8 
 
 68 
 
 With this load, 153 tons in the middle nearly, the experi- 
 ment was discontinued, as the apparatus was overstrained. 
 
 MR, FRANCIS BRAMAH S EXPERIMENTS ON BEAMS. 
 
 160. In the second volume of the Institution of Civil 
 Engineers, there is a paper containing experiments made in 
 the year 1834, by Mr. A. H. Renton, for Mr. Bramah. 
 They were upon beams of which the section is in the form JL, 
 some of them being solid throughout their length, and others 
 having apertures in them. The following is an abstract of 
 
EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 351 
 
 such of the experiments as were pursued to the time of 
 breaking the beam. 
 
 161. Beams, the section of which is of the JL form, are, as 
 we have seen, much weaker than those of another which has 
 been arrived at (art. 135) ; but they are not without interest, 
 as they are easily cast, and have considerable strength; except 
 those with open work in them, which form the subject of Mr. 
 Bramah's second Table, given hereafter ; his first Table con- 
 taining the results of experiments on solid beams. The 
 former beams, though recommended by Tredgold (Part I., 
 art. 41), are very weak ; as will be seen by comparing the 
 breaking weights of the beams, 3 inches deep, in the two 
 Tables, and taking into consideration the weights of the 
 beams. The great weakness of beams Vith apertures in them 
 was shown, too, in my experiments on beams, ' Memoirs of 
 the Literary and Philosophical Society of Manchester/ second 
 series, vol. v., 1831. 
 
352 
 
 EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 
 
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EXPERIMENTS ON BEAMS OF DIFFERENT FORMS. 
 
 353 
 
 
 g I ^ 5 
 
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 K 
 
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354 MR. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 164. Mr. Bramah's experiments were made to support 
 certain principles adopted by Tredgold, some of which have 
 been controverted in this work. Mr. Bramah states that it 
 was " a principal feature in these experiments, and essential 
 to the accuracy of the results, to note that point where the 
 elastic power becomes impaired, and the specimens take a 
 permanent set," &c. But I have shown (arts. 86 and 101) 
 that there is no elastic limit in cast iron ; and if there were, 
 the depth, 3 inches, of the beams on which Mr. Bramah's 
 experiments were made, was much too great, with their small 
 distance between the supports, 3 '08 3 feet, to enable him to 
 discover when the defect of elasticity first took place. In 
 neglecting his reasonings, as very disputable, and not suited 
 to this place where the strength considered is the ultimate, 
 and Bramah's (following Tredgold) the incipient, of the 
 material I shall merely observe, that the results with respect 
 to fracture, seem to be in accordance with those from my own 
 experiments (arts. 112-115). Mr. Bramah draws no con- 
 clusions from the breaking weights, but they show the great 
 difference in the ultimate strength of a beam, according as it 
 is turned one way up, or the opposite ; and will serve as a 
 beacon to warn the public of the danger of using beams with 
 apertures in them. 
 
 MR. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 165. Sir Henry De la Beche and Thomas Cubitt, Esq., 
 having in 1844 been appointed by Government to inquire 
 into the circumstances respecting the fall of a cotton mill at 
 Oldham, in Lancashire, and part of a prison at Northleach, in 
 Gloucestershire; these gentlemen accompanied their Report 
 with the results of the experiments made at Manchester, and 
 given more at length in this 2nd Part ; adding in particular 
 those with respect to the strength and best forms of iron 
 beams and pillars. 
 
MR. CUBITrS EXPERIMENTS ON BEAMS. 355 
 
 166. Mr. Cubitt likewise, feeling "impressed with the 
 importance of further researches on the forms of cast iron 
 beams, whether for the purpose of confirming or of extending 
 the views hitherto taken," (Report, page 7,) caused eight 
 experiments to be made on a moderately large scale. These 
 experiments were published with the Report, and an abstract 
 of them is in the following Table. 
 
 167. Tabular results of experiments on cast iron beams 
 each cast 16 feet long laid during the experiment, on 
 supports 15 feet asunder, and intended to be of the same 
 weight. The first six beams were uniform throughout, the 
 seventh and eighth tapered towards the ends. 
 
 A A 2 
 
356 
 
 ME. CTJBITT'S EXPERIMENTS ON BEAMS. 
 
 poo 
 
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 qstsa.i o^ aoMotl 
 
 
 
 
 
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 JO GOUSIQ 
 
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 (M OS 
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 - 
 
MR. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 357 
 
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 O 00 O <O O 00 O 
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 CC >O t^ O5 r-i 
 
358 
 
 MK. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 TABLE Continued. 
 
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 !)srsaj o^ aoAi 
 
 -jjps O 
 
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 999-Tpq 
 
 CM (M <N 
 
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 UOT^OSS jo 130.IY 
 
 qqdop IB^OI 
 
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 uioq^oq jo azig 
 
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MR. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 359 
 
 
 
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360 MR. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 168. Mr. Cubitt states that his object in making these 
 experiments " was to show the great difference of strength of 
 cast iron that can be got by taking certain forms." (Appendix 
 to Report, page 39.) This object he has realized in a certain 
 manner, as will be seen from the last two columns in the 
 preceding Table, in which the beams, though of equal weight 
 and section, are successively made to increase rapidly in 
 strength above the top ones, which are the weakest. 
 
 169. Now as these weakest beams are not very different 
 in form of section from, though somewhat weaker than, those 
 which I have arrived at from a long course of inductive 
 experiments, and considered as nearly the strongest in cast 
 iron (arts. 135 and 136) ; and as the circumstance may be 
 considered as bearing deeply upon the character of my pub- 
 lished results with respect to beams, it will be incumbent on 
 me to analyze the effort of Her Majesty's Commissioner with 
 more freedom than I would otherwise willingly have done. 
 
 170. Speaking plainly, then, it appears to me that Mr. 
 Cubitt, by increasing the strength through increasing the 
 depth, the area being the same, has shown nothing more 
 than would have been predicted from the slightest knowledge 
 of theory ; and that several of his beams, instead of showing 
 greater strength, exhibit only weakness and inferiority of 
 form. 
 
 171. To give a simple illustration of this statement, we 
 will suppose a number of rectangular beams to be formed, of 
 the same length and area of section, but of different breadths 
 and depths. Then the strength of each being as b d 2 , varies 
 as d, since b d, the area of the section, is constant. 
 
 172. We will select from Mr. Cubitt's Table the results 
 of the different experiments, and attach to them a column 
 containing the results which would have been derived from 
 rectangular beams, of the same length and area, and varying 
 in depth as Mr. Cubitt's did. It must, however, be under- 
 stood that the rectangular section, which is comparatively a 
 
MR. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 861 
 
 weak one, is introduced only for illustration, as its strength 
 for different depths is easily computed; and it may be 
 presumed that the strength of other sections, not so easily 
 calculated, would increase, by augmenting the depth, in 
 some such ratio as that does. 
 
 Number 
 of 
 Experi- 
 ment. 
 
 Total depth 
 of beam 
 iu middle. 
 
 Comparative 
 strength 
 reduced to 
 equal areas. 
 
 Comparative strength of 
 rectangular beam of 
 the same depth as Mr. 
 Cubitt's, and of constant 
 
 Remarks. 
 
 
 
 
 area and length. 
 
 
 1 
 
 7-15 
 
 100 
 
 100 
 
 The strength of the 
 
 
 
 
 
 first rectangular beam 
 
 o 
 
 M 
 
 7-17 
 
 98-9 
 
 10C-27 
 
 is assumed as 100, 
 
 
 
 
 
 same as the first of 
 
 
 
 
 
 Mr. Cubitt's. 
 
 3 
 
 1075 
 
 157-8 
 
 150-3 
 
 
 4 
 
 1075 
 
 152-4 
 
 150-3 
 
 
 5 
 
 12-75 
 
 153-9* 
 
 178-3 
 
 * Bottom flange un- 
 
 
 
 
 
 souud. 
 
 6 
 
 12-8 
 
 186-2 
 
 179-0 
 
 
 7 
 
 14-0 
 
 147* 
 
 195-8 
 
 * The bottom flange 
 
 
 
 
 
 of both beams seems 
 
 8 
 
 17-25 
 
 194-2* 
 
 241'2 
 
 to have been slightly 
 
 
 
 
 
 defective. 
 
 173. If we compare the results in the third and fourth 
 columns, showing the comparative strengths of Mr. Cubitt's 
 beams and of rectangular ones of the same depth, we shall 
 see that the increased strength of Mr. Cubitt's beams, in the 
 lower part of the Table, above the strength of that at the 
 top, with which they are compared, is derived wholly from 
 the depth ; and as his latter beams give generally much lower 
 results than are obtained from the rectangular section, we 
 may infer that, they are of inferior forms to that with which 
 they are compared. The defect in the bottom flange of three 
 out of four of them (an uncommon occurrence in properly 
 cast beams) renders it probable that, if sound, they would 
 have borne a little more than they did ; but affords no pro- 
 bability that their increase of strength would have been equal 
 to that of the rectangular section; which no doubt would 
 
362 ME. CUBITT'S EXPERIMENTS ON BEAMS. 
 
 have been the case, or nearly so, if the form in the first 
 experiment had been used. 
 
 174. A further confirmation of the conclusion here arrived 
 at is derived from the fact, that the strength of beams of the 
 best form was found from my experiments to be, cceteris 
 paribus, nearly as the depth ; and the material in the section 
 was but little increased with a large addition to the depth. 
 
 175. From the experiments and reasoning above, Mr. 
 Cubitt has drawn the conclusion of " our knowledge of the 
 best forms and arrangements of cast iron beams not being based 
 upon principles the correctness of which cannot be questioned, 
 (Report, page 10,) and they are offered in confirmation or 
 extension' 3 
 
 176. It would appear that Mr. Cubitt had mistaken the 
 object of my experiments " on the strength and best forms 
 of cast iron beams/' It was virtually to seek for the form 
 into which a given quantity of iron could be cast, so as to 
 bear the greatest weight, the length and the depth of the 
 beam being constant. Mr. Cubitt's experiments seem to 
 have been intended to show that a given quantity of iron 
 cast into beams all of the same length the section being 
 of various forms (as the J. section which I had represented 
 as comparatively weak), would be made to bear more than 
 others which I had represented as approaching to the 
 strongest; this being effected merely by increasing the 
 depth. 
 
 177. A little more attention to theoretical considerations 
 might equally well have shown that increasing the depth a 
 privilege I did not allow myself when seeking for the best 
 form of beam had a great influence on the strength ; and 
 this might perhaps have prevented Mr. Cubitt offering to the 
 public, under Her Majesty's sanction, additional examples, 
 on a large scale, of weak beams. 
 
 178. In the tabular extract of Mr. Cubitt's experiments 
 I ha've given the sets or defects of elasticity as obtained by 
 
STRENGTH OF HOT AND COLD BLAST IRON. 363 
 
 that gentleman; but the length of his beams was not a 
 sufficient number of times their depth for the results of the 
 early sets, however carefully taken, to be any thing but an 
 approximation. 
 
 179. As Mr. Cubitt's experimental results with respect to 
 the strength of beams seem to be in accordance with my 
 own, and might generally be computed from them, whatever 
 opinion he may have formed to the contrary, I see no reason 
 to doubt that the best form of beam is obtained from the 
 reasonings and experiments previously given (arts. 108 to 
 144) ; and according to which many thousands of tons have 
 been cast. I am preparing to repeat the leading experiments 
 in my former effort on beams, on a very large scale, and to 
 extend them considerably, through the liberality of an Iron 
 Company. 
 
 COMPARATIVE STRENGTH OF HOT AND COLD BLAST IRON. 
 
 180. Having, in conjunction with Mr. Fairbairn, been 
 requested, by the British Association for the Advancement of 
 Science, to ascertain by experiment the comparative strengths 
 of irons made by a heated and a cold blast, I will give here 
 the results from my ' Report on the Tensile, Crushing, and 
 Transverse Strengths of several kinds of Iron/ (Brit. Assoc. 
 vol. vi. 1838,) attaching to them, in conclusion, the results 
 from Mr. Fairbairn's experiments, which were on the latter 
 kind of strain. 
 
 181. As the modes in which the different kinds of experi- 
 ment were made, and many of the results obtained, are given 
 in the earlier pages of this work, it will not be necessary here 
 to enter into detail upon that subject. I shall, therefore, 
 content myself with stating, that the experiments were made 
 with great care ; and in devising the apparatus, the utmost 
 attention was paid to theoretical requirements. 
 
 182. Taking only the means from all the experiments, in 
 the report above mentioned, and attaching to each result a 
 
364 
 
 STRENGTH OF HOT AND COLD BLAST IRON. 
 
 number, in a parenthesis, indicative of the number of experi- 
 ments from which it has been derived, we have as follows : 
 
 Carron Iron, No. 2 (Scotch). 
 
 
 Cold blast. 
 
 Hot blast. 
 
 Ratio represent- 
 ing cold blast 
 by 1000. 
 
 Tensile strength in Ibs. per square 
 inch 
 
 
 16683 (2) 
 
 13505 (3) 
 
 1000 : 809 
 
 
 Compressive (crushing) strength 
 
 
 
 
 
 
 
 in Ibs. per square inch ; from spe- 
 cimens cut out of castings pre- 
 
 , 
 
 ! 
 
 106375 (3) 108540 (2) 
 
 1000:1020, 
 
 
 viously torn asunder . 
 
 
 
 
 
 
 
 OS 
 
 Crushing strength obtained from 
 prisms of various forms 
 
 
 100631 (9) 
 
 100738 (5) 
 
 1000:1001 f 1 
 
 Do. from cylinders 
 
 125403(13): 121685(13)11000: 970^ 
 
 Transverse strength from all the ) 
 experiments . . . , J 
 
 (11) (13) I 1000: 991 
 
 
 Computed power to resist impact . 
 
 (9) (9) 1000:1005 
 
 
 Transverse strength of bars, 1 inch 
 
 
 
 
 
 
 square, and 4 feet 6 inches be- 
 
 
 476 (8)| 463 (3) 1000: 973 
 
 
 tween the supports, in Ibs. . 
 
 
 
 I 
 
 
 Ultimate deflexion of do. in inches . 
 
 1-313 (3) 1-337 (3) 1000:1018 
 
 
 Modulus of elasticity in Ibs. per } 
 square inch (Part I. art. 256) . j 
 
 17270500 (2) 
 
 16085000 (2)' 1000 : 931 
 
 
 Specific gravity .... 
 
 7066 
 
 7046 (f) 1000 : 997 
 
 
 Devon Iron, No. 3 (Scotch). 
 
 
 Cold blast. 
 
 Hot blast. 
 
 Ratio repre- 
 senting cold 
 blast by 1000. 
 
 Tensile strength per square inch 
 
 
 21907 (1) 
 
 
 Compressive strength do. . 
 
 
 
 145435 (4) 
 
 
 
 Transverse do. from the experi- } 
 inents generally . . .'j 
 
 (5) 
 
 (5) 
 
 1000:1417 
 
 Power to resist impact 
 
 (4) 
 
 (4) 
 
 1000:2786 
 
 Transverse strength of bars, 1 in. } 
 
 
 
 
 square, and 4 feet 6 inches be- > 
 
 448 (2) 
 
 537 (2) 
 
 1000:1199 
 
 tween the supports . . . ) 
 
 
 
 
 Ultimate deflexion, do. . . - *79 (2) 
 
 1-09 (2) 
 
 1000:1380 
 
 Modulus of elasticity, do. . . 22907700 ('2) 
 
 22473650 (2) 
 
 1000 : 981 
 
 Specific gravity . 
 
 7295 (4) 
 
 7229 (2) 
 
 1000: 991 
 
 Buff try Iron, No. 1 (English). 
 
 
 Cold blast. Hot blast. 
 
 Ratio repre- 
 senting cold 
 blast by 1000. 
 
 Tensile strength per square inch . 
 
 
 17466 (1) I 33434 (1) 
 
 1000: 769 
 
 Compressive strength do. . 
 
 
 93366 (4) 8o397 (4) 
 
 1000 : 925 
 
 Transverse strength . i T ,' 
 
 
 (5) (5) 
 
 1000 : 931 
 
 Power to resist impact 
 
 
 (2) 
 
 (2) 
 
 1000: 962 
 
 Transverse strength of bars, 1 inch 
 square, aud 4 feet 6 inches 
 
 
 463 (3) 
 
 436 (3) 
 
 1000: 942 
 
 between the supports . . 
 Ultimate deflexion, do. 
 
 
 1-55 (3) 
 
 1-64 (3) 
 
 1000:1058 
 
 Modulus of elasticitv, do. . 
 
 15381200 (2) 
 
 13730500 (2) 
 
 1000: 893 
 
 Specific gravity .... 
 
 
 7079 
 
 6998 ' 
 
 1000 : 989 
 
STRENGTH OF HOT AND COLD BLAST IRON. 
 Coed.Talon Iron, No. 2 (Welsh). 
 
 305 
 
 
 Cold blast. 
 
 Hot blast. 
 
 Ratio repre- 
 senting cold 
 blast by 1000. 
 
 Tensile strength per square inch . 
 Compressive strength do. . 
 Specific gravity .... 
 
 18855 (2) 
 81770 (4) 
 6955 (4) 
 
 16676 (2) 
 82739 (4) 
 6968 (3) 
 
 1000: 884 
 1000: 1012 
 1000 : 1002 
 
 Carron Iron, No. 3 (Scotch). 
 
 
 Cold blast. 
 
 Hot blast. 
 
 Ratio repre- 
 senting cold 
 blast by 1000. 
 
 Tensile strength per square inch . 
 Compressive strength do. . . 
 Specific gravity . . . . 
 
 14200 (2) 
 115442 (4) 
 7135 (1) 
 
 17755 (2) 
 133440 (3) 
 7056 (1) 
 
 1000:1250 
 1000:1156 
 1000 : 989 
 
 183. Abstract of the transverse strengths, and powers to 
 bear impact, as obtained from the experiments on the three 
 irons first mentioned in the preceding Table. The bars, of 
 whatever form, were usually cast 5 feet long, and laid upon 
 supports 4 feet 6 inches asunder. Those of 1 inch square 
 being the bars from which the comparative powers to bear 
 impact were computed, had their deflexions, from different 
 weights, very carefully observed up to the time of fracture ; 
 and as the measured dimensions of the bar usually differed a 
 small quantity from those of the model, the results, both as 
 to strength and deflexion, were reduced by computation to 
 what they would have been if the bar had been exactly 1 inch 
 square. A reduction of this nature was made in the results 
 of all the experiments, except otherwise mentioned. The 
 comparative power to bear impact was obtained by multiplying 
 the breaking weight of a bar, 1 inch square, by its ultimate 
 deflexion, the length being always the same ; a mode which 
 is admissible, as appears from my experiments on the power 
 of beams to bear impact (British Association, 5th Report). 
 
STRENGTH OF HOT AND COLD BLAST IRON. 
 
 
 ** 
 
 1 
 
 
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 S 
 
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 Power to 1 
 
 Hot blast 
 iron. 
 
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 rH (M CO 
 
 
 Cold blast 
 iron. 
 
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 ipi 
 
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 rH rH rH ,-, rH 1 1 
 
 co- 
 
 
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 Strength of iron 
 
 Hot blast iron. 
 
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 ^^^^^gg^ ^ ^25.^ i 
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 ii . . -- 2 - a 1 141 
 
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STRENGTH OF HOT AND COLD BLAST IRON. 
 
 367 
 
 
 
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368 
 
 STRENGTH OF HOT AND COLD BLAST IRON. 
 
 GENERAL SUMMARY OF TRANSVERSE STRENGTHS, AND COM- 
 PUTED POWERS TO RESIST IMPACT. 
 
 184. Selecting, from the irons above, the results of the 
 experiments on the transverse strength, and power to resist 
 impact, of the different bars broken, and adding to them the 
 results of Mr. Fairbairn's experiments (Brit. Assoc. vol. vi.), 
 we have as below. 
 
 Distance between supports 4 feet 6 inches. 
 
 
 
 
 * ^ ttr? 
 
 
 
 
 Ratio of pS la"! 
 
 Description of iron. 
 
 Strength of 
 cold blast. 
 
 Strength of 
 hot blast. 
 
 strength, 
 cold blast 
 
 w C-Z " 
 
 
 
 
 = 1000. 
 
 -^ ^ ** rt -*J 
 
 
 
 
 
 1,1!* j 
 
 Carron, No. 2. 
 
 &*. 
 
 His. 
 
 
 
 Results from bars 1 inch square 
 
 476 (3) 
 
 463 (3) 
 
 1000 : 973 
 
 
 from all the experiments 
 
 (11) 
 
 (13) 
 
 1000 : 9'Jl 
 
 1000 : lOOf 
 
 Devon, No. 3. 
 
 
 
 
 
 Results from bars 1 inch square 
 
 448 (2) 
 
 537 (2) 
 
 1000 : 1199 
 
 
 from all the experiments 
 
 (5) 
 
 (5) 
 
 1000 : 1417 
 
 1000 :278t 
 
 Buffery, No. 1. 
 
 
 
 
 
 Results from bars 1 inch square 
 
 463 (3) 
 
 436 (3) 
 
 1000 : 942 
 
 
 from all the experiments 
 
 (5) 
 
 (5) 
 
 1000 : 931 
 
 1000 : 96i 
 
 Muirkirk, No. 1. 
 
 
 
 
 
 Results from bars 1 in. sq. ^ %! . 
 Coed-Talon, No. 2, ditto | J -g 
 
 454-2 (4) 
 412-6 (5) 
 
 418-9 (4) 
 416-8 (4) 
 
 1000 : 922 
 1000 : 1010 
 
 1000 : 82: 
 1000 : 1234 
 
 Coed-Talon, No. 3, ditto 1 J |j 553-2 (4) 
 
 513-1 (4) 
 
 1000 : 927 
 
 1000 : 92f 
 
 Carron, No. 3, ditto f -g 
 
 445-7 (5) 
 
 525-7 (5) 
 
 1000 : 1179 
 
 1000 : 120] 
 
 Elsicar, cold, and ) ,.. . | & g, 
 Milton, hot, No. 1, i d 3 ) g 
 
 451-5 (4) 
 
 369-4 (4) 
 
 1000 : 818 
 
 1000 : 875 
 
 185. These Tables contain the results of a very large 
 number of experiments, made with great care upon English, 
 Welsh, and Scotch iron, mostly supplied from the makers. 
 They show that the irons marked No. 1, which are softer and 
 richer than those of Nos. 2 and 3, are injured by the heated 
 blast ; since the hot blast irons of this description are less 
 capable than the cold blast ones to resist fracture, whether 
 the forces are tensile, compressive, transverse, or impulsive. 
 
 186. The irons marked No. 2, being harder than those of 
 No. 1, have much less difference in their strength than the 
 latter. In the Carron iron, No. 2, on which a great many 
 experiments were made, the transverse strength, of the hot 
 
STRENGTH OF HOT AND COLD BLAST IRON. 369 
 
 and cold blast specimens, was as 99 : 100, and their power of 
 bearing impact equal. The Coed-Talon iron of this No. gave 
 the transverse strength, of hot and cold blast, as 101 : 100, 
 and their power to bear impact as 123 : 100. In both the 
 Carron and the Coed-Talon irons, the hot blast castings were 
 of equal strength to the cold blast ones, to resist crushing ; 
 but, in both, the strength of the hot blast was less than that 
 of the cold blast, to resist tension, in a ratio of 8 or 9 to 10. 
 
 187. The No. 3 irons seem, both in aspect and strength, 
 to be generally benefited by the heated blast. In the Carron 
 iron, No. 3, the hot blast was superior to the cold, in the 
 power of resisting tension, compression, transverse strain, and 
 impact, in a ratio approaching, in each case, that of 12 to 10. 
 The Coed-Talon iron of this No. had, however, its hot blast 
 kind weaker than the cold, to resist transverse strain and 
 impact, in the ratio of about 93 to 100. 
 
 The iron, No. 3, from the Devon works in Scotland, was 
 weak and irregular in the cold blast castings ; but the hot 
 blast iron from the same works was among the strongest I 
 have tried. In this the ratio of the powers, of hot and cold 
 blast iron, to bear pressure, was as 14 : 10 ; and to bear 
 impact, as 28 : 10 nearly. 
 
 188. In these experiments, the hot blast irons usually 
 differed from the cold blast, only so far as a different mode 
 of manufacture the introduction of a heated blast with coal, 
 instead of a cold blast with coke would produce. The 
 difficulty we experienced in obtaining from the makers irons 
 of both kinds made from the same materials, rendered it 
 necessary to make the experiments on a smaller number of 
 irons than would otherwise have been tried ; but, from the 
 evidence adduced, we may perhaps conclude that the intro- 
 duction of a heated blast, into the manufacture of cast iron, 
 has injured the softer irons, whilst it has frequently mollified 
 and improved those of a harder nature ; and considering the 
 small deterioration which the irons of the quality No. 2 have 
 
 B B 
 
370 THEORETICAL INQUIRIES WITH REGARD TO BEAMS. 
 
 sustained, and the apparent benefit to those of No. 3, 
 together with the great saving effected by the heated blast, 
 there seems good reason for the process becoming as general 
 as it has done. It is, however, to be feared that the facilities 
 which the heated blast gives, of adulterating cast iron by 
 mixture, have introduced into use a species of metal very 
 inferior to that used in this comparison, or that from which 
 the formulae and leading results of this work have been 
 obtained. 
 
 THEORETICAL INQUIRIES WITH REGARD TO THE STRENGTH 
 
 OF BEAMS. 
 
 189. In the course of our remarks on the transverse 
 strength of cast iron, as deduced from experiment, it was 
 shown that the formulae given by Tredgold, in Part I. of this 
 work, were usually inapplicable to the computation of the 
 strength of that metal to resist fracture. That very ingenious 
 writer following in the track of Dr. Young, and himself 
 followed by numerous others considered bodies, when not 
 overstrained, to be perfectly elastic ; and to resist extension 
 and compression with equal energy. But theories deduced 
 from these suppositions, however elegant, and nearly correct 
 for small displacements of the fibres or particles, give the 
 breaking strength of cast iron, in some cases, not half what it 
 has been shown to bear by experiment (arts. 150 and 151). 
 A square bar, instead of having its neutral line in the centre 
 one-half being extended and the other compressed, according 
 to the suppositions above requires to be considered as 
 totally incompressible, the neutral line being close to the side, 
 or even beyond it, a matter practically impossible. This 
 defect in the received theories has been shown to arise from 
 the neglect by writers of an element which appears always to 
 be conjoined with elasticity, diminishing its power. This 
 element ductility, producing defective elasticity will be 
 
THEOEETICAL INQUIRIES WITH REGARD TO BEAMS. 371 
 
 attempted to be introduced into the following investigation 
 but as the formulae are generally complex, and require addi- 
 tional experiments to supply their constant co-efficients, the 
 reader may perhaps take for practical use the approximate 
 ones previously given in this Second Part. 
 
 190. To find the position of the neutral line and the 
 strength of a cast iron beam, supported at the ends, and 
 loaded in the middle ; the form of a section of the beam in 
 the middle being that of the figure A B D E, where B C, H E, 
 represent sections of the top and bottom ribs, F G that of the 
 vertical one connecting them, and N passes through the 
 neutral line. 
 
 Let W = weight necessary to break the beam, 
 Z = distance between the supports > 
 a, of = N" I, N K, respectively, 
 c, c' = D H, A C do. 
 
 6, 6' = DE, AB do. 
 
 /3 = the thickness of the vertical rib, 
 f,f tensile and compressive forces of the metal, in 
 
 a unity of section, as exerted at a distance 
 
 a on opposite sides of the neutral line, 
 <f> (#), <' (x 1 ) = quantities respectively proportional 
 
 to the forces of extension and compression 
 
 of a particle, at a distance x from the neutral line, 
 n, n' = constant quantities dependent on the destruction of the elasticity of the 
 
 material, by tensile and compressive forces. 
 
 191. The bottom rib will be in a state of tension, and the 
 top one in a state of compression ; and the parts of the section 
 generally will be extended or compressed according to their 
 distance from the line N 0. 
 
 1st. To find the position of the neutral line. 
 
 192. Since/. ^ = the force of the extended fibres or par- 
 ticles in a unity of section, at 'a distance x from the neutral 
 line ; therefore, multiplying this quantity by b d x, or by (3 dx, 
 we have the force of the particles in an area of the section of 
 which the breadth is b or , and depth d x. Now the forces 
 of tension, or those exerted by the particles below the line 
 
 BB 2 
 
372 THEORETICAL INQUIRIES WITH REGARD TO BEAMS. 
 
 N 0, may be expressed in two functions, the first represent- 
 ing the forces of the particles in the section N G, and the 
 other those in the section H E. 
 
 a s ac b /*<*> 
 
 /. /. / < (x) dx + /. - / ; * / <t> (x) dx S = sum of the forces of tension (I). 
 
 ( a >J o y \ ' J ac 
 
 Proceeding in the same manner with respect to the com- 
 pressed particles, we have 
 
 f . -.- f.-lt.lffl dx' + f. r&tt f^' 0*0 dx ' = s ' = sum of the forces of 
 4>' ( a ) J o $ W J a'c' 
 
 compression (2). 
 
 The weight acts in a direction parallel to the section of 
 fracture, and therefore the sum of the forces of extension and 
 compression, being the only horizontal forces, are equal to 
 each other. 
 
 /. S = S' (3). 
 
 193. It appears from my recent experiments that no rigid 
 body is perfectly elastic ; and it has been shown (art. 99 to 
 106), in extracts from communications which I made to the 
 York and late Cambridge meetings of the British Association, 
 that, in the flexure of bars of cast iron and stone, the defect 
 of elasticity was nearly as the square of the weight applied, 
 or of the deflexion, though the defect from the smaller 
 deflexions seemed to increase in a somewhat lower ratio. 
 Other experiments on the defect of elasticity, as exhibited in 
 the flexure of bars of cast iron, wrought iron, steel, timber, 
 and stone and on the defect of elasticity in the longitudinal 
 variations of bodies will appear in a future volume of the 
 British Association. As the latter experiments, those on 
 longitudinal variation, are not at present completed, I shall 
 assume < (x) x ^ and </>' (#') = #' ^; where v, v, n, n 
 
 are supposed to be constant, and v, v to represent the powers 
 of #, x , to which the defects of elasticity of extension and 
 compression are proportional. 
 
THEORETICAL INQUIRIES WITH REGARD TO BEAMS. 373 
 
 Substituting in equations (1) and (2), the values of < (#) 
 and $' (#'), as above, we have 
 
 = S' . .(5) 
 
 Performing the integrations of equations (4) and (5) gives 
 
 ng /V~ C &\ dx _ /* /(a--c) 2 (a-c)-H 
 
 J 'na-a*- l J\ na/ na - a*- 1 \ 2 (^+l)wa 
 
 ^\. /rt6 C /a 2 a^ 1 \ _ 
 
 X ~ na} dx ~na- a 1 ( \ 2 (v+l)na/ 
 
 /(-c) 2 __ (a-c 
 \ 2 (v + l 
 
 nb 
 
 n>y / ^'\^,_ A y y j /"' 2 ^ V+1 
 
 / Va-a*'-! / V i5>/ ~ w'a-a"'- 1 i V 2 (t/ + l)n' 
 
 \ 
 
 l)n'a/ 
 
 _/(a / -c / ) 2 _ (a > -</)^+ 1 \ ) 
 \ 2 " (v' + l)w'a/ ) 
 
 Collecting together the integrals of equation (4), we have 
 
 S= 
 
 
 2 2 
 
 . "- 1 
 
 2 
 
 In like manner, 
 
 S' = J*l- \ b'a'* ( l - - J^-\ - (6' -B) 
 
 'a-a'-ir \2 (t/+l)nW l W 2 
 
 Equating the values of S and S' gives, for the equation of 
 the neutral line, 
 
 _ 
 
 2 (v + l)a/ \ 2 v + la / ) TO' -a"' - 
 
 where r/ = D , D being the depth of the beam. 
 
374 THEORETICAL INQUIRIES WITH REGARD TO BEAMS. 
 
 Cor. 1st. If v = 2, and v' = 2, as would appear to be nearly 
 the case from the experiments on the transverse flexure of 
 cast iron bars, mentioned above, the equation of the neutral 
 line would be 
 
 If the beam is of the ^ form, having no top rib, then 
 0, U = 0, and equation (7) becomes 
 
 If b = V /3, then the section of the beam is rectangular, 
 as a joist ; and the equation of the neutral line (7) becomes 
 
 Cor. 2nd. If v = 1, and v'= 1, or the defect of elasticity 
 be as the extension and compression, then equation (6) will 
 become 
 
 ......... (10). 
 
 But this equation becomes 
 
 6V x - 6'- 
 
 where # a = area of the whole section of tension considered 
 as a rectangular surface of which the breadth is b and depth 
 
 a\ | = distance of centre of gravity of that section from the 
 neutral line; (b /3) (a c) = area of part necessary to 
 complete the rectangular surface above ; ^~ = distance of 
 
 centre of gravity of this defective part. In like manner 
 b' a ' = area of whole section of compression considered as a 
 
THEOEETICAL INQUIRIES WITH REGARD TO BEAMS. 375 
 
 rectangular surface of breadth V and depth a \ ^ = distance 
 of its centre of gravity; (b'P) (a -c'):narea of part want- 
 ing; and ~^ = the distance of its centre of gravity. 
 
 Whence it appears that if/=/', the neutral line will be in 
 the centre of gravity of the section. 
 
 Cor. 3rd. If the elasticity be considered as perfect, then 
 n, n will be infinitely great, and taking as usual /=/'> 
 equation (6) will become 
 
 j a 2 _(&__) (a_ c )2 = jy a '2- (&'_) (a'-c') 2 .... (11). 
 
 The curious circumstance of this equation being in agree- 
 ment with equation (10), is rendered obvious by other reason- 
 ing. For, in Cor. 2, we have 
 
 A(x} = x -- = ( 1 } x, and <J>' (of) =x'-^=( l-~ V 
 na \ na/ ' ria V ria / 
 
 the forces being as the extensions and compressions. 
 
 If, on the supposition of perfect elasticity, b'= 0, and c'= 0, 
 the beam being of the J_ form of section, having no top rib, 
 the last equation will become 
 
 la?-.(b-0)(a-c) 2 = pa' 2 ....... (12). 
 
 If the beam be rectangular and perfectly elastic, then 
 I = b'= p t and equation (11) becomes 
 
 = a 
 
 or the neutral line is in the middle. 
 
 2nd. To find the strength of the beam, the values of a, a, 
 and consequently the position of the neutral line, having been 
 previously determined, from one of the preceding equations, 
 or by other means. 
 
 194. Since, by equation (4) of the preceding article, the 
 sum of the forces of extension is 
 
376 THEORETICAL INQUIRIES WITH REGARD TO BEAMS. 
 
 And as the moment of each of these forces, with respect to 
 the neutral line, is equal to the product of the force by its 
 distance x from that line, we have for the sum of the moments 
 of the forces of tension, 
 
 .* 3 - fo = S/ . (13). 
 na 
 
 and 
 
 3 (-t-2)wa 
 
 3 (v -(- 2) n a 
 
 Whence we have, for the sum of the moments of the forces 
 of the extended particles, 
 
 In like manner we obtain, for the sum of the moments of 
 the forces of the compressed particles, 
 
 But S 7 + S f/ the sum of the moments of the forces of exten- 
 sion and compression, must be equal to the product of half 
 the weight laid on the middle of the beam by half the distance 
 between the supports ; for we may consider the beam as fixed 
 firmly in the middle, and loaded at one end with half the 
 weight laid on the middle. 
 
 .-.S. + S^iWx^^^ ....... (16). 
 
 Cor. 1. If c'= 0, and 6'= 0, the section of the beam being 
 of the JL form, having no top rib, we have from equation (15), 
 
 and S, as in equation (14), for substitution in equation (16). 
 
 Cor. 2. If c, c , b, b' are each = 0, or the beam is rectan- 
 gular, as a joist, we have from equation (14) 
 
THEORETICAL INQUIRIES WITH REGARD TO BEAMS. 377 
 
 from equation (15) 
 
 " 3a' 
 
 where 
 
 ^ = S, + S// .......... (17). 
 
 Cor. 3. If v, v are each = 2, as would appear from the 
 experiments (art. 106), then the values of S y , S y/ , for the 
 strength of a rectangular section in the last corollary, give 
 
 ^ 
 
 where ^ is the ratio of the depth of the compressed section to 
 that of the extended one. 
 
 On the supposition of this corollary, that v v'= 2, the 
 general formulae in equations (14) (15) give 
 
 Cor. 4. If t; = t; / = 1, or the defect of elasticity is as the 
 extension and compression, equations (14) and (15) give 
 
 Cor. 5. If the beam be supposed to be perfectly elastic, 
 then n and n are both infinite, and we have from equations 
 
 (14) and (15) 
 
 3 a 
 
 agreeing with the results of the last corollary, as previously 
 remarked with respect to equations (10) and (11). 
 
 But as the body is elastic we will assume as usual /==/ / , 
 / i } w^ i 
 
 /.S, + S.,= /- &a3 + 6'a'3 (b j8)(a c) 3 (6' ^8) (a'-c')H = -7- (!) 
 da ( ) 4 
 
 If ^ = j'j and c =c, or the top and bottom ribs are equal, 
 
378 RESISTANCE TO TORSION. 
 
 then, the neutral line being in the centre, a = a ; and the 
 last equation gives 
 
 W I 2 f C 
 
 ~ =^\ ba3 -( b -( a - c ^\ ....... (20), 
 
 a result in agreement with equation xix., art. 116 of Tredgold. 
 If in this case /3 = 0, or the part between the ribs was so 
 thin that it might be neglected, 
 
 ' 
 
 agreeing with art. 117 of Tredgold. 
 
 If p = b, or the beam is rectangular, then equation (20) 
 becomes 
 
 where D is the whole depth = 2 a, a result usually arrived 
 at by a much simpler process (Tredgold, art. 110). 
 
 For other investigations on the subject of the neutral line, 
 and the strength of beams variously fixed the elasticity being 
 supposed perfect see Professor Moseley's * Principles of 
 Engineering and Architecture/ 
 
 RESISTANCE TO TORSION. 
 
 195. If a prismatic body, fixed firmly at one end, have a 
 weight applied to twist it by means of a lever acting at the 
 other, perpendicular to the length of the body, to find the 
 resistance to twisting and to fracture. 
 
 196. The problem here proposed has been made the sub- 
 ject of an ingenious article by Tredgold in the 1st Part of 
 this work ; but as it has been subjected to more recent and 
 profound theoretical investigation by Cauchy and others on 
 the Continent, the formulae given by Navier (' Application de 
 Mecanique '), including those of Cauchy, will be inserted 
 here, referring for their demonstrations to the work itself, or 
 to M. Cauchy J s 'Exercices de Mathematiques/ 4 e annee. 
 Experimental results by Bevan, Rennie, Savart, and the 
 Author, will likewise be given or noticed. 
 
RESISTANCE TO TORSION. 
 
 379 
 
 197. Let 1 = the length of the prism from the fixed end to 
 the point of application of the lever used to twist it. 
 r = the radius of the prism, if round. 
 b y d= its breadth and thickness, if rectangular. 
 P = the weight acting by means of th.e lever to twist it. 
 R, = the length of the lever., 
 = the angle of torsion, at the point of application, con- 
 
 sidered as very small. 
 G = a constant for each species of body, representing the 
 
 specific resistance to flexure by torsion. 
 T = a constant weight expressing the resistance to tor- 
 sion, with regard to a unit of surface, at the time 
 of fracture. 
 We have then as in the following Table. 
 
 Form of section of 
 prism. 
 
 Resistance to angular flexure by a 
 force of torsion. 
 
 Eesistance to fracture by a force of 
 torsion. 
 
 Round 
 Square . . . 
 
 Rectangular 
 
 G- = PB.^- 
 irr*0 
 
 G~PR 6 '. 
 
 T = PR. 
 TT r 3 
 
 6 
 Tp-p 
 
 & We 
 
 3(&* + eP)Z_ 
 
 V2.d 3 
 
 sV^T^F 
 
 PK ' &>C*>0 
 
 * 6*.d> 
 
 The formulae for G are in art. 160, and those for T in 
 art. 168, of the 'Application de Mecanique/ l re partie; those 
 for the rectangular prism being from Cauchy. 
 
 198. If E be the force necessary to elongate or shorten a 
 prism, the transverse section of which is a superficial unity, 
 as one square inch, by a quantity equal to the length of the 
 prism, the elasticity being supposed perfect, and the force 
 applied in the direction of the length ; and if F be the force 
 necessary to break such a prism ; then E will be the modulus 
 of elasticity of the material, and F the modulus of its resist- 
 ance to fracture ; and the values of G and T above will be 
 connected with those of E and F by the following relations. 
 
 
 
380 RESISTANCE TO TORSIOK 
 
 See 'Application de Mecanique/ notes to articles 159 
 and 167. 
 
 The connexion is, however, but little in accordance with 
 the results of experiment, with respect to the values of T and 
 F, as might be expected from the elasticity being much 
 injured previous to the time of fracture. 
 
 190^ The angle 6 being measured by the arc due to a 
 radius equal unity, if the angle were expressed in degrees, and 
 
 represented by A, we should have 9 ~A'iJQ*ffiSyfi? where 
 57*29578, or ^> is the number of degrees in the arc whose 
 
 length is equal to radius. 
 
 200. To obtain the values of G and T in the preceding 
 Table, with respect to any particular material, as cast iron, 
 we must refer to experiment, and will next insert some 
 results which were kindly sent by the author, Mr. Geo. 
 Rennie, in 1842, as part of a more general inquiry; noticing 
 afterwards other experiments previously mentioned, besides 
 some of an earlier date, quoted by Tredgold. 
 
 201. Experiments, by Geo. Rennie, Esq., F.R.S., on the 
 strength of three bars of cast iron to resist fracture by torsion. 
 The bars were planed exactly one inch square, and were 
 firmly fixed, at one end, in a horizontal position, and broken 
 by weights acting at the other by means of an arched lever, 
 3 feet in length, perpendicular to the bar, and exactly 
 balanced by a counter weight. The bars were cast from the 
 cupola, the first vertically, the other two horizontally. The 
 former was broken with 191 fts., and the two latter with 
 231 fts. each. 
 
 202. To determine from the preceding experiments the 
 value of T, the modulus of resistance to fracture by torsion, 
 
 in the formula, T = PR.-4 3 for a square prism, taking 
 the weights in pounds, and the dimensions in inches, we 
 
 haveR = 36, c! = 1, and T = P x-- = 152'735 x P. 
 
 V2 
 
RESISTANCE TO TORSION. 381 
 
 Ja the horizontal casting P =191, .'. T =. 29172-4 ) Mean 
 In the vertical castings P =231, .'. T = 35281-8 j 32227 fts. 
 
 If R were taken in feet and the rest in inches, and pounds, 
 as before, the value of T would be 26S5'6. 
 
 203. The experiments of Messrs. Bramah on the torsion 
 of square bars of cast iron (Tredgold, art. 85) give for T, 
 taking the dimensions in inches, 
 
 No. 3 . . . T = 42020-4 
 
 - 
 
 7 . . . . T = 33296-4 
 
 6 
 
 204. Mr. Dunlop's experiments on the torsion of cylinders, 
 varying, in diameter, from 2 to 4^ inches, and in length from 
 2f to 6 inches, the leverage being 14 feet 2 inches (Tredgold, 
 
 art. 85), give, from the formula T = PR. 7 , as follows. 
 
 In Experiment No. 2, T = 27056'4^ 
 3, T = 29187-6 
 
 5, T = 29142-0 
 
 6, T = 29509-2 
 
 8, T = 27286-8 
 
 9, T = 26217-6 
 10, T = 24338-4 
 
 Mean 27534 Ibs. 
 
 Taking a mean from the three mean results last obtained 
 gives, T = 32503 fts. ; the dimensions being in inches. 
 
 205. Putting this value for T, in the formula (art. 197), 
 and transposing, we obtain the following value of PR. 
 
 In a cylinder . . . PR = 51055 . r 3 . 
 
 In a square prism . . . P R = 7661 . d 3 . 
 
 TO 72 
 
 In a rectangular prism . . PR = 10834 . ^_ 
 
 V 6 2 + d? 
 
 206. If R be taken in feet, as was supposed by Tredgold, 
 
 AQQ /73 
 
 art. 265, we shall have, for a square prism, P= -^ 
 
 Hence his co-efficient, 150, being less than ^th of that of 
 fracture, may be regarded as perfectly safe for practical 
 application. 
 
 207. Mr. Benjamin Bevan gave, 'in the 'Philosophical 
 Transactions' for 1829, a memoir containing numerous 
 experiments " on the modulus of torsion." 
 
 They were principally on timber, but contained a small 
 Table of the modulus of torsion of metals. Mr. Bevan 
 
382 RESISTANCE TO TORSION. 
 
 defined this modulus by the value of T in the following 
 equation for a square prism, twisted as before described : 
 
 R 2 IP 
 
 ' 
 
 where 5 is the deflexion, considered as very small, and the 
 rest of the notation as before ; the weights and the dimen- 
 sions being in pounds and inches. 
 We have from above KZP s 
 
 ~cPT = R' 
 
 and as ~ is the deflexion at a unity of distance, and very 
 small, it may be taken for the arc. 
 
 RZP i m PRZ 
 
 :.=0, where T= 
 
 But from the Table (art. 197), G = PR. = 
 
 G = 6T. 
 
 208. Mr. Bevan finds the modulus T in cast iron, whose 
 specific gravity is 7*163, to be as below. 
 
 940000 ) 
 
 963000 [ Mean 951600 Ibs. 
 
 952000 ) 
 
 The moduli of wrought iron and steel were nearly equal to 
 each other ; and a mean from the results of eight experiments 
 on iron and three on steel, gave, for T, 1779090 fts. 
 ' 209. Mr. Bevan found the modulus T to be ^th of the 
 modulus of elasticity in metallic substances. 
 
 But it was shown above that G = 6T, .. G=j^ of the 
 modulus of elasticity ; which differs from f , as computed by 
 Cauchy (art. 198), only as 16 to 15. 
 
 210. Multiplying Mr. Bevan's mean values of T by six 
 we obtain, 
 
 In cast iron ..... G = 951600 x 6 = 5709600 fcs. 
 In wrought iron and steel . . G = 1779090 x 6 = 10674540 Ibs. 
 
 211. Substitute in the formulae (art. 197) the value of G, 
 as obtained from the above experiments on cast iron, and 
 transposing, we have as below. 
 
 For a cylinder .... P R = 8968620' -^- 
 
 /74 /5 
 
 For a square prism . . . . P R = 951600' - 
 
RESISTANCE TO TORSION. 383 
 
 For a rectangular prism . . T R = 1903200 1 ^ -=- 
 
 (p + d ) i 
 
 212. The experiments of Mr. Bevan seem to have been 
 very carefully made, extra precaution being used both to 
 prevent friction and to obtain correct measures ; but a source 
 of error may have arisen from computing, by a less accurate 
 formula than Cauchy's, the results from those rectangular 
 prisms which differed considerably from squares. 
 
 213. Some experiments of my own, upon the torsion of 
 cylinders of wrought iron and steel, made some years since, 
 at the request of Mr. Babbage, but not yet published, showed 
 that the angle of torsion was very nearly as the weight, as 
 had previously been shown by Savart (Annales de Chimie et 
 de Physique, Aug. 1829). 
 
 214. The object of M. Savart's able Memoir was to compare 
 the theory of torsion, as given by Poisson and Cauchy, with 
 the results of experiment ; and though his experiments were 
 made on brass, copper, steel, glass, oak, &c. and included 
 none on cast iron it may not be amiss to give here the gene- 
 ral laws which he deduces from them. They are as below. 
 
 1st. Whatever be the form of the transverse section of the 
 rods (subjected to torsion), the arcs of torsion are directly pro- 
 portional to the moment of the force and to the length. 
 
 2nd. When the sections of the rods are similar, whether 
 circular, triangular, square, or rectangular, much elongated, 
 the arcs of torsion are in the inverse ratio of the fourth power 
 of the linear dimensions of the section. 
 
 3rd. When the sections are rectangles, and the rods possess 
 an uniform elasticity in every direction, the arcs of torsion are 
 in the inverse ratio of the product of the cubes of the trans- 
 verse dimensions, divided by the sum of their squares ; from 
 whence it follows that, if the breadth is very great compared 
 with the thickness, the arcs of torsion will be sensibly in the 
 inverse ratio of the breadth and of the cube of the thickness. 
 
 These laws, M. Savart observes, are in exact agreement 
 with those of Cauchy, both for cylindrical and rectangular 
 
384 RESISTANCE TO TORSION. 
 
 sections; showing tliat his formulae (art. 197) are constructed 
 on principles which may be applied with safety. When the 
 elasticity is not uniform, the laws are somewhat modified. 
 
 215. In concluding this notice of experiments on the strength 
 and other properties of cast iron, which may, perhaps, on a 
 future occasion be extended in various ways, I would refer - 
 the reader desirous of information on the effects of expansion 
 and contraction, upon structures, by heat, to a Memoir ' on 
 the Expansion of Arches ' through the changes of ordinary 
 temperature, by Mr. George Rennie. This Memoir contains 
 experiments on the rise of the arches in the Southwark 
 Bridge, which is of cast iron, having three rows of arches in 
 length, containing in the whole about 5560 tons of iron. 
 Mr. Rennie made experiments upon the rise of the arches in 
 each row ; from which it appears that the rise of an arch, 
 whose span is 246 feet and versed sine 23 feet 1 inch, is ' 
 about 4*0 th of an inch for each degree of Fahrenheit, making 
 1^ inch for a difference of 50. Mr. Rennie gives a Table of 
 experiments of his own upon the expansion of iron and stone, 
 with others from M. Destigpy ; and concludes that there is 
 no more danger to the stability of iron bridges, from the 
 effects of expansion and contraction, than to those of stone ; 
 for when the abutments are firmly fixed, the arches have no 
 alternative but to rise or fall. 
 
 The effects of percussion and vibration upon bodies, par- 
 ticularly cast iron, have been much further inquired into 
 since the time of Tredgold ; and upon these subjects I beg to 
 refer the reader to a Memoir of my own on the effects of 
 " Impact upon Beams," in the 5th Report of the British 
 Association, 1835. The object of this Memoir was to com- 
 pare theory with experiment, deducing practical conclusions. 
 
 THE END. 
 
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LONDON, June, 1870. 
 
 OF 
 
 NEW & STANDARD WORKS 
 
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 ENGINEERING, ARCHITECTURE, 
 
 AGRICULTURE, MATHEMATICS, MECHANICS, 
 
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 I 
 
 Humbers New Work on Water-Supply. 
 
 A COMPREHENSIVE TREATISE on the WATER-SUPPLY 
 of CITIES and TOWNS. By WILLIAM HUMBER, Assoc. Inst. 
 C.E., and M. Inst. M.E. Author of "Cast and Wrought Iron 
 Bridge Construction," &c. &c. This work, it is expected, will con- 
 tain about 50 Double Plates, and upwards of 250 pages of Text. 
 Imp. 4to, half bound in morocco. [/ the press. 
 
 %* In accumulating information for this volume, the Author has 
 been very liberally assisted by several professional friends, who have 
 made this department of engineering their special study. He has thus 
 been in a position to prepare a work which, within the limits of a 
 single volume, will supply the reader with the most complete and 
 reliable information upon all subjects, theoretical and practical, con- 
 nected with water supply. Through the kindness of Messrs. Ander- 
 son, Bateman, Hawksley, Homersham, Baldwin Latham, Lawson, 
 Milne, Quick, Rawlinson, Simpson, and others, several works, con- 
 structed and in course of construction, from the designs of these gentle- 
 men, will be fully illustrated and described. 
 
 AMONGST OTHER IMPORTANT SUBJECTS THE FOLLOWING WILL BE TREATED 
 IN THE TEXT : 
 
 Historical Sketch of the means that have been proposed and adopted for the Supply 
 of Water. Water and the Foreign Matter usually associated with it. Rainfall and 
 Evaporation. Springs and Subterranean Lakes. Hydraulics. The Selection of 
 Sites for Water Works. Wells. Reservoirs. Filtration and Filter Beds. Reservoir 
 and Filter Bed Appendages. Pumps and Appendages. Pumping Machinery. 
 Culverts and Conduits, Aqueducts, Syphons, &c. Distribution of Water. Water 
 Meters and general House Fittings. Cost of Works for the Supply of Water. Con- 
 stant and Intermittent Supply. Suggestions for preparing Plans, &c. &c., together 
 with a Description of the numerous Works illustrated, viz : Aberdeen, Bideford, 
 Cockermouth, Dublin, Glasgow, Loch Katrine, Liverpool, Manchester, Rotherham, 
 Sunderland, and several others ; with copies of the Contract, Drawings, and Specifi- 
 cation in each case. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Number s Modern Engineering. First Series. 
 
 A RECORD of the PROGRESS of MODERN ENGINEER- 
 ING, 1863. Comprising Civil, Mechanical, Marine, Hydraulic, 
 Railway, Bridge, and other Engineering Works, &c. By WILLIAM 
 HUMBER, Assoc. lust. C.E., &c. Imp. 4to, with 36 Double 
 Plates, drawn to a large scale, and Photographic Portrait of John 
 Hawkshaw, C.E., F.R.S., &c. Price 3/. 3.?. half morocco. 
 
 List of the Plates. 
 
 NAME AND DESCRIPTION. PLATES. 
 
 Victoria Station and Roof L. B.& S. C. Rail. i to 8 
 Southport Pier o and 10 
 
 16 
 
 17 
 
 iSandig 
 
 Victoria Stationand Roof L. C. & D. & G. W. 
 
 Railways 
 
 Roof of Cremorne Music Hall 
 
 Bridge over G. N. Railway 
 
 Roof of Station Dutch Rhenish Railway . . 
 Bridge over the Thames West London Ex- 
 tension Railway 20 to 24 
 
 Armour Plates 25 
 
 Suspension Bridge, Thames 26 to 29 
 
 The Allen Engine 30 
 
 Suspension Bridge, Avon 31 to 33 
 
 NAME OF ENGINEER. 
 
 Mr. R. Jacomb Hood, C.E. 
 Mr. James Brunlees, C.E. 
 
 Mr. John Fowler, C.E. 
 Mr. William Humber, C.E. 
 Mr. Joseph Cubitt, C.E. 
 Mr. Euschedi, C.E. 
 
 Mr. William Baker, C.E. 
 Mr. James Chalmers, C.E. 
 Mr. Peter W. Barlow, C.E. 
 Mr. G. T. Porter, M.E. 
 Mr. John Hawkshaw, C.E. 
 andW. H. Barlow, C.E. 
 Mr. John Fowler, C.E. 
 
 Underground Railway 34 to 36 
 
 With copious Descriptive Letterpress, Specifications, &c. 
 
 " Handsomely lithographed and printed. It will find favour with many who desire 
 to preserve in a permanent form copies of the plans and specifications prepared for the 
 guidance of the contractors for many important engineering works." Engineer. 
 
 Humber s Modern Engineering. Second Series. 
 
 A RECORD of the PROGRESS of MODERN ENGINEER- 
 ING, 1864 ; with Photographic Portrait of Robert Stephenson, 
 C.E., M.P., F.R.S., &c. Price 3/. 3*. half morocco. 
 
 List of the Plates. 
 
 NAME AND DESCRIPTION. PLATES. NAME OF ENGINEER. 
 
 Birkenhead Docks, Low Water Basin i to 15 Mr. G. F. Lyster, C.E. 
 
 Charing Cross Station Roof C. C. Railway. 16 to 18 Mr. Hawkshaw, C.E. 
 
 T\; rii TT;_J M. /~ *. TWT_.^I T-> - -i * T r^ t?.^ /-* TT 
 
 20 
 
 zoa 
 
 Digswell Viaduct Great Northern Railway. 
 Robbery Wood Viaduct Great N. Railway. 
 
 Iron Permanent Way 
 
 Clydach Viaduct Merthyr, Tredegar, and 
 
 Abergavenny Railway 
 
 Ebbw Viaduct ditto ditto ditto 
 College Wood Viaduct Cornwall Railway . . 
 Dublin Winter Palace Roof 
 
 Mr. 
 Mr 
 
 -. J. Cubitt, C.E. 
 -. J. Cubitt, C.E. 
 
 21 Mr. Gardner, C.E. 
 
 22 Mr. Gardner, C.E. 
 
 23 Mr. Brunei. 
 
 24 to 26 Messrs. Ordish & Le Feuvre. 
 
 Bridge over the Thames L. C. & D. Railw. 27 to 32 Mr. J. CubUt, C.E. 
 Albert Harbour, Greenock 33 to 36 Messrs. Bell & Miller. 
 
 With copious Descriptive Letterpress, Specifications, &c. 
 
 ' ' A resume of all the more interesting and important works lately completed in Great 
 Britain ; and containing, as it does, carefully executed drawings, with full working 
 details, will be found a valuable accessory to the profession at large. " Engineer. 
 
 " Mr. Hnmber has done the profession good and true service, by the fine collection 
 of examples he has here brought before the profession and the public." Practical 
 Mechanic's Journal. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Number s Modern Engineering. Third Series. 
 
 A RECORD of the PROGRESS of MODERN ENGINEER- 
 ING, 1865. Imp. 4to, with 40 Double Plates, drawn to a large 
 scale, and Photographic Portrait of J. R. M 'Clean, Esq., late Pre- 
 sident of the Institution of Civil Engineers. Price 3/. 3.?. half 
 
 morocco. 
 
 List of Plates and Diagrams. 
 
 MAIN DRAINAGE, METROPOLIS. 
 
 NORTH SIDE. 
 
 Map showing Interception of Sewers. 
 Middle Level Sewer. Sewer under Re- 
 gent's Canal. 
 
 Middle Level Sewer. Junction with Fleet 
 Ditch. 
 
 Bridge over River Lea. 
 
 Outfall Sewer. 
 
 Elevation. 
 Outfall Sewer. 
 
 Details. 
 Outfall Sewer. 
 
 Details. 
 Outfall Sew 
 
 Bridge over River Lea. 
 Bridge over River Lea. 
 
 Bridges over Marsh Lane, 
 
 North Woolwich Railway, and Bow and 
 
 Barking Railway Junction. 
 Outfall Sewer. Bridge over Bow and 
 
 Barking Railway. Elevation. 
 Outfall Sewer. Bridge over Bow and 
 
 Barking Railway. Details. 
 Outfall Sewer. Bridge over Bow and 
 
 Barking Railway. Details. 
 Outfall Sewer. Bridge over East London 
 
 Waterworks' Feeder. Elevation. 
 Outfall Sewer. Bridge over East London 
 
 Waterworks' Feeder. Details. 
 Outfall Sewer. Reservoir. Plan. 
 Outfall Sewer. Reservoir. Section. 
 Outfall Sewer. Tumbling Bay and Outlet. 
 Outfall Sewer. Penstocks. 
 
 SOUTH SIDE. 
 
 Outfall Sewer. Bermondsey Branch. 
 Outfall Sewer. Bermondsey Branch. 
 Outfall Sewer. Reservoir and Outlet. 
 Plan. 
 
 MAIN DRAINAGE, METROPOLIS, 
 
 continued 
 Outfall Sewer. Reservoir and Outlet. 
 
 Details. 
 Outfall Sewer. Reservoir and Outlet. 
 
 Details. 
 Outfall Sewer. Reservoir and Outlet. 
 
 Details. 
 
 Outfall Sewer. Filth Hoist. 
 Sections of Sewers (North and South 
 
 Sides). 
 
 THAMES EMBANKMENT. 
 
 Section of River Wall. 
 
 Steam-boat Pier, Westminster. Elevation. 
 
 Steam-boat Pier, Westminster. Details. 
 
 Landing Stairs between Charing Cross 
 and Waterloo Bridges. 
 
 York Gate. Front Elevation. 
 
 York Gate. Side Elevation and Details. 
 
 Overflow and Outlet at Savoy Street Sewer. 
 Details. 
 
 Overflow and Outlet at Savoy Street Sewer. 
 Penstock. 
 
 Overflow and Outlet at Savoy Street Sewer. 
 Penstock. 
 
 Steam-boat Pier, Waterloo Bridge. Eleva- 
 tion. 
 
 Steam-boat Pier, Waterloo Bridge. De- 
 tails. 
 
 Steam-boat Pier, Waterloo Bridge. De- 
 tails. 
 
 Junction of Sewers. Plans and Sections. 
 
 Gullies. Plans and Sections. 
 
 Rolling Stock. 
 
 Granite and Iron Forts. 
 
 With copious Descriptive Letterpress, Specifications, &c. 
 
 Opinions of the Press. 
 
 " Mr. number's works especially his annual ' Record,' with which so many of our 
 
 readers are now familiar fill a void occupied by no other branch of literature 
 
 The drawings have a constantly increasing value, and whoever desires to possess clear 
 representations of the two great works carried out by our Metropolitan Board will 
 obtain Mr. Humbert last volume." Engineering. 
 
 " No engineer, architect, or contractor should fail to preserve these records of works 
 which, for magnitude, have not their parallel in the present day, no student in the 
 profession but should carefully study the details of these great works, which he may be 
 one day called upon to imitate." Mechanics' Magazine, 
 
 " A work highly creditable to the industry of its author The volume is quite 
 
 an encyclopaedia for the study of the student who desires to master the subject of 
 municipal drainage on its scale of greatest development." Practical Mechanic's 
 Journal. 
 
4 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 number's Modern Engineering, Fourth Series. 
 
 A RECORD of the PROGRESS of MODERN ENGINEER- 
 ING, 1866. Imp. 4to, with 36 Double Plates, drawn to a large 
 scale, and Photographic Portrait of John Fowler, Esq. , President 
 of the Institution of Civil Engineers. Price 3/. 3-r. half-morocco. 
 
 List of the Plates and Diagrams. 
 
 NAME AND DESCRIPTION. PLATES. NAME OF ENGINEER. 
 
 Abbey Mills Pumping Station, Main Drainage, 
 
 Metropolis i to 4 Mr. Bazalgette, C.E. 
 
 Barrow Docks 5 to 9 Messrs. M'Clean & Stillman, 
 
 Manquis Viaduct, Santiago and Valparaiso [C. E. 
 
 Railway 10, n Mr. W. Loyd, C.E. 
 
 Adams' Locomotive, St. Helen's Canal Railw. 12, 13 Mr. H. Cross, C.E. 
 Cannon Street Station Roof, Charing Cross 
 
 Railway 14 to 16 Mr. J. Hawkshaw, C.E. 
 
 Road Bridge over the River Moka 17, 18 Mr. H. Wakefield, C.E. 
 
 Telegraphic Apparatus for Mesopotamia 19 Mr. Siemens, C. E. 
 
 Viaduct over the River Wye, Midland Railw. 20 to 22 Mr. W. H. Barlow, C.E. 
 
 St. Germans Viaduct, Cornwall Railway 23, 24 Mr. Brunei, C.E. 
 
 Wrought- Iron Cylinder for Diving Bell 25 Mr. J. Coode, C.E. 
 
 Millwall Docks 26 to 31 Messrs. J. Fowler, C.E. , and 
 
 William Wilson, C.E. 
 
 Milroy's Patent Excavator 32 Mr. Milroy, C. E. 
 
 Metropolitan District Railway 33 to 38 Mr. J. Fowler, Engineer-in- 
 
 Chief, and Mr. T. M. 
 Johnson, C.E. 
 
 Harbours, Ports, and Breakwaters A to c 
 
 The Letterpress comprises 
 
 A concluding article on Harbours, Ports, and Breakwaters, with 
 Illustrations and detailed descriptions of the Breakwater at Cher- 
 bourg, and other important modern works ; an article on the 
 Telegraph Lines of Mesopotamia ; a full description of the Wrought- 
 iron Diving Cylinder for Ceylon, the circumstances under which it 
 was used, and the means of working it ; full description of the 
 Millwall Docks ; &c., &c., &c. 
 
 Opinions of the Press. 
 
 "Mr. Humber's 'Record of Modern Engineering' is a work of peculiar value, as 
 well to those who design as to those who study the art of engineering construction. 
 It embodies a vast amount of practical information in the form of full descriptions and 
 working drawings of all the most recent and noteworthy engineering works. The 
 plates are excellently lithographed, and the present volume of the ' Record ' is not a 
 whit behind its predecessors." Mechanic -s* Magazine. 
 
 "We gladly welcome another year's issue of this valuable publication from the able 
 pen of Mr. Humber. The accuracy and general excellence of this work are well 
 known, while its usefulness in giving the measurements and details of some of the 
 latest examples of engineering, as carried out by the most eminent men in the profes- 
 sion, cannot be too highly prized." Artizan. 
 
 " The volume forms a valuable companion to those which have preceded it, and 
 cannot fail to prove a most important addition to every engineering library." Mining 
 Journal. 
 
 " No one of Mr. Humber's volumes was bad ; all were worth their cost, from the 
 mass of plates from well-executed drawings which they contained. In this respect, 
 perhaps, this last volume is the most valuable that the author has produced. "Prac- 
 tical Mechanics' Journal. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 5 
 
 Number s Great Work on Bridge Construction. 
 
 A COMPLETE and PRACTICAL TREATISE on CAST and 
 WROUGHT-IRON BRIDGE CONSTRUCTION, including 
 Iron Foundations. In Three Parts Theoretical, Practical, and 
 Descriptive. By WILLIAM HUMBER, Assoc. Inst. C. E., and M. Inst. 
 M.E. Third Edition, revised and much improved, with 115 Double 
 Plates (20 of which now first appear in this edition), and numerous 
 additions to the Text. In 2 vols. imp. 4to., price 6/. i6s. 6d. half- 
 bound in morocco. \_Just ready. 
 
 "A very valuable contribution to the standard literature of civil engineering. In 
 addition to elevations, plans, and sections, large scale details are given, which very 
 much enhance the instructive worth of these illustrations. No engineer would wil- 
 lingly be without so valuable a fund of information. Civil Engineer and Architect s 
 Journal. 
 
 " The First or Theoretical Part contains mathematical investigations of the prin- 
 ciples involved in the various forms now adopted in bridge construction. These 
 investigations are exceedingly complete, having evidently been very carefully con- 
 sidered and worked out to the utmost extent that can be desired by the practical man. 
 The tables are of a very useful character, containing the results of the most recent 
 experiments, and amongst them are some valuable tables of the weight and cost of 
 cast and wrought-iron structures actually erected. The volume of text is amply illus- 
 trated by numerous woodcuts, plates, and diagrams': and the plates in the second 
 volume do great credit to both draughtsmen and engravers. In conclusion, we have 
 great pleasure in cordially recommending this work to our readers." Artizan. 
 
 " Mr. Humber's stately volumes lately issued in which the most important bridges 
 erected during the last five years, under the direction of the late Mr. Brunei, Sir W. 
 Cubitt, Mr. Hawkshaw, Mr. Page, Mr. Fowler, Mr. Hemans, and others among our 
 most eminent engineers, are drawn and specified in great detail." Engineer, 
 
 Weale's Engineer s Pocket-Book. 
 
 THE ENGINEER'S, ARCHITECT'S, and CONTRACTOR'S 
 POCKET-BOOK (LOCKWOOD & Co.'s; formerly WEALE'S). 
 Published Annually. In roan tuck, gilt edges, with 10 Copper- 
 Plates and numerous Woodcuts. Price 6s. 
 
 " There is no work published by or without authority, for the use of the scientific 
 branches of the services, which contains anything like the amount of admirably 
 arranged, reliable, and useful information. It is really a most solid, substantial, and 
 excellent work ; and not a page can be opened by a man of ordinary intelligence which 
 will not satisfy him that this praise is amply deserved." Army and Navy Gazette. 
 
 " A vast amount of really valuable matter condensed into the small dimen- 
 sions of a book which is, in reality, what it professes to be a pocket-book 
 
 We cordially recommend the book to the notice of the managers of coal and other 
 mines ; to them it will prove a handy book of reference on a variety of subjects more 
 or less intimately connected with their profession. It might also be placed with 
 advantage in the hands of the subordinate officers in collieries." Colliery Guardian. 
 
 " The assignment of the late Mr. Weale's 'Engineer's Pocket-Book' to Messrs. 
 Lockwood & Co. has by no means Ipwered the standard value of the work. It is too 
 well known among those for whom it is specially intended, to need more from us than 
 the observation that this continuation of Mr. Weale's series of Pocket Books well 
 sustains the reputation the work has so long enjoyed. Every branch of engineering 
 is treated of, and facts, figures, and data of every kind abound." Mechanics' Mag. 
 
 " It contains a large amount of information peculiarly valuable to those for whose 
 use it is compiled. We cordially commend it to the engineering and architectural 
 professions generally." Mining Journal. 
 
 "A multitude of useful tables, without reference to which the engineer, architect, 
 or contractor could scarcely get through a single day's work." Scientific Review. 
 
6 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Barlow on the Strength of Materials , enlarged. 
 
 A TREATISE ON THE STRENGTH OF MATERIALS, 
 with Rules for application in Architecture, the Construction of 
 Suspension Bridges, Railways, &c. ; and an Appendix on the 
 Power of Locomotive Engines, and the effect of Inclined Planes 
 and gradients. By PETER BARLOW, F.R.S., Mem. Inst. of France ; 
 of the Imp. and Royal Academies of St. Petersburgh and Brussels ; 
 of the Amer. Soc. Arts ; and Hon. Mem. Inst. Civil Engineers. 
 A New and considerably Enlarged Edition, revised by his Sons, 
 P. W. BARLOW, F.R.S., Mem. Inst. C.E., and W. H. BARLOW, 
 F.R.S., Mem. of Council Inst. C.E., to which are added a Sum- 
 mary of Experiments by EATON HODGKINSON, F.R.S., WILLIAM 
 FAIRBAIRN, F.R.S., and DAVID KIRKALDY ; an Essay (with 
 Illustrations) on the effect produced by passing Weights over 
 Elastic Bars, by the Rev. ROBERT WILLIS, M.A., F.R.S. And 
 Formulae for Calculating Girders, &c. The whole arranged and 
 edited by WILLIAM HUMBER, Assoc. Inst. C.E., and Mem. Inst. 
 M.E., Author of " A Complete and Practical Treatise on Cast and 
 Wrought-Iron Bridge Construction," &c. &c. Demy 8vo, 400 pp., 
 with 19 large Plates, and numerous woodcuts, price iSs. cloth. 
 
 Opinions of the Press. 
 
 " This edition has undergone considerable improvement, and has been brought down 
 to the present date. It is one of the first books of reference in existence." Artizan. 
 
 " Although issued as the sixth edition, the volume under consideration is worthy of 
 being regarded, for all practical purposes, as an entirely new work . . . the book 
 is undoubtedly worthy of the highest commendation, and of an honourable place in 
 the library of every engineer." Mining Journal. 
 
 "An increased value has been given to this very valuable work by the addition of 
 a large amount of information, which cannot prove otherwise than highly useful to 
 
 those who require to consult it The arrangement and editing of this 
 
 mass of information has been undertaken by Mr. Plumber, who has most ably fulfilled 
 a task requiring special care and ability to render it a success, which this edition most 
 certainly is. He has given the finishing touch to the volume by introducing into it 
 an interesting memoir of Professor Barlow, which tribute of respect, we are sure, will 
 be appreciated by the members of the engineering profession." Mechanic J Magazine. 
 
 "A book which no engineer of any kind can afford to be without. In its present 
 form its former value is much increased." Colliery Guardian. 
 
 " The best book on the subject which has yet appeared We know of 
 
 no work that so completely fulfils its mission As a scientific work of the 
 
 first class, it deserves a foremost place on the bookshelves of every civil engineer and 
 practical mechanic." English Mechanic. 
 
 " There is not a pupil in an engineering school, an apprentice in an engineer's or 
 architect's office, or a competent clerk of works, who will not recognise in the scientific 
 volume newly given to circulation, an old and valued friend. . . So far as the strength 
 of timber is concerned, there is no greater authority than Barlow." Building News. 
 
 " It is scarcely necessary for us to make any comment upon the first portion of 
 
 the new volume Valuable alike to the student, tyro, and experienced 
 
 practitioner, it will always rank in future, as it has hitherto done, as the standard 
 treatise upon this particular subject." Engineer. 
 
 " The present edition offers some important advantages over previous ones. The 
 additions are both extensive and valuable, comprising experiments by Hodgkinson on 
 the strength of cast-iron ; extracts from papers on the transverse strength of beams by 
 W. H. Barlow ; an article on the strength of columns ; experiments by Fairbairn, on 
 iron and steel plates, on the behaviour of girders subjected to the vibration of a 
 changing load, and on various cast and wrought-iron beams ; experiments by Kirkaldy, 
 on wrought-iron and steel bars, and a short appendix of formulae for ready application 
 in computing the strains on bridges." Engineering. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 7 
 
 Strains, Formula & Diagrams for Calculation of. 
 
 A HANDY BOOK for the CALCULATION of STRAINS 
 in GIRDERS and SIMILAR STRUCTURES, and their 
 STRENGTH ; consisting of Formulae and Corresponding Diagrams, 
 with numerous Details for Practical Application, &c. By WILLIAM 
 HUMBER, Assoc. Inst. C.E., &c. Fcap. 8vo, with nearly 100 
 Woodcuts and 3 Plates, price Js. 6d. cloth. [Recently published. 
 
 It is hoped that a small work, in a. handy form, devoted entirely to Bridge and 
 Girder Calculations, without giving more than is absolutely necessary for the complete 
 solution of practical problems, will meet with ready acceptance from the engineering 
 profession. One of the chief features of the present work is the extensive application 
 of simply constructed DIAGRAMS to the calculation of the strains onbridges and girders. 
 
 "To supply a universally recognised want of simple formulae, applicable to the 
 varied problems to be met with in ordinary practice, Mr. Humber, whose works on 
 modem engineering afford sufficient evidence of his qualifications for the task, has 
 compiled his 'Handy Book.' The arrangement of the matter in this little volume is 
 
 as convenient as it well could be The system of employing diagrams as a 
 
 substitute for complex computations is one justly coming into great favour, and in that 
 respect Mr. Humber's volume is fully up to the times. "Engineering. 
 
 " The formulae are neatly expressed, and the diagrams good.'' Athencsum. 
 
 " That a necessity existed for the book is evident, we think ; that Mr. Humber has 
 
 achieved his design is equally evident We heartily commend the really handy 
 
 book to our engineer and architect readers. " English Mechanic. 
 
 " It is, in fact, what its name indicates, a handy book, .... giving no more than 
 is absolutely necessary for the complete solution of practical problems." Colliery 
 G^^ardian. 
 
 "This capital little work will supply a want, often found by engineers, viz., of 
 having the requisite formulae for calculating strains in a complete form, and yet suffi- 
 ciently portable to be carried in the pocket Almost every formula that could 
 
 possibly be required, together with diagrams of strains, is put concisely, yet clearly, 
 in a work of considerably less size than an engineering pocket book." Artizan. 
 
 Strains. 
 
 THE STRAINS ON STRUCTURES OF IRONWORK; 
 with Practical Remarks on Iron Construction. By F. W. SHEILDS, 
 M. Inst. C.E. Second Edition, with 5 plates. Royal 8vo, 5-y. cloth. 
 CONTENTS. Introductory Remarks ; Beams Loaded at Centre ; Beams Loaded at 
 unequal distances between supports ; Beams uniformly Loaded ; Girders with triangu- 
 lar bracing Loaded at centre ; Ditto, Loaded at unequal distances between supports ; 
 Ditto, uniformly Loaded; Calculation of the Strains on Girders with triangular 
 Basings ; Cantilevers ; Continuous Girders ; Lattice Girders ; Girders with Vertical 
 Struts and Diagonal Ties; Calculation of the Strains on Ditto; Bow and String 
 Girders ; Girders of a form not belonging to any regular figure ; Plate Girders ; Ap- 
 portionments of Material to Strain ; Comparison of different Girders ; Proportion of 
 Length to Depth of Girders ; Character of the Work ; Iron Roofs. 
 
 Trigonometrical Surveying. 
 
 AN OUTLINE OF THE METHOD OF CONDUCTING A 
 TRIGONOMETRICAL SURVEY, for the Formation of Geo- 
 graphical and Topographical Maps and Plans, Military Recon- 
 naissance, Levelling, &c., with the most useful Problems in Geodesy 
 and Practical Astronomy, and Formulae and Tables for Facilitating 
 their Calculation. By MAJOR-GENERAL FROME, R.E., Inspector- 
 General of Fortifications, &c. Third Edition, revised and improved. 
 With 10 Plates and 113 Woodcuts. Royal 8vo, 12^ cloth. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Hydraulics. 
 
 HYDRAULIC TABLES, CO-EFFICIENTS, and FORMULA 
 for finding the Discharge of Water from Orifices, Notches, Weirs, 
 Pipes, and Rivers. By JOHN NEVILLE, Civil Engineer, M.R.I. A. 
 Second Edition, with extensive Additions, New Formulae, Tables, 
 and General Information on Rain-fall, Catchment-Basins, Drainage, 
 Sewerage, Water Supply for Towns and Mill Power. With nume- 
 rous Woodcuts, 8vo, i6s. cloth. 
 
 %* This work contains a vast number of different hydraulic 
 formulae, and the most extensive and accurate tables yet published 
 for finding the mean velocity of discharge from triangular, quadri- 
 lateral, and circular orifices, pipes, and rivers ; with experimental 
 results and co-efficients ; effects of friction ; of the velocity of 
 approach ; and of curves,- bends, contractions, and expansions ; the 
 best form of channel ; the drainage effects of long and short weirs, 
 and weir-basins ; extent of back-water from weirs ; contracted 
 channels ; catchment-basins ; hydrostatic and hydraulic pressure ; 
 water-power, &c. &c. 
 
 Levelling. 
 
 A TREATISE on the PRINCIPLES and PRACTICE of 
 LEVELLING ; showing its Application to Purposes of Railway 
 and Civil Engineering, in the Construction of Roads ; with Mr. 
 TELFORD'S Rules for the same. By FREDERICK W. SIMMS, 
 F.G.S., M. Inst. C.E. Fifth Edition, very carefully revised, with 
 the addition of Mr. LAW'S Practical Examples for Setting out 
 Railway Curves, and Mr. TRAUTWINE'S Field Practice of Laying 
 out Circular Curves. With 7 Plates and numerous Woodcuts. 8vo, 
 8s. 6d. cloth. \* TRAUTWINE on Curves, separate, price 5^. 
 
 " One of the most important text-books for the general surveyor, and there is 
 scarcely a question connected with levelling for which a solution would be sought but 
 that would be satisfactorily answered by consulting the volume." Mining Journal. 
 
 " The text-book on levelling in most of our engineering schools and colleges." 
 Engineer. 
 
 "The publishers have rendered a substantial service to the profession, especially to 
 the younger members, by bringing out the present edition of Mr. Simms's useful work." 
 Engineering. 
 
 Tunnelling. 
 
 PRACTICAL TUNNELLING ; explaining in Detail the Setting 
 out of the Works ; Shaft Sinking and Heading Driving ; Ranging 
 the Lines and Levelling Under-Ground ; Sub -Excavating, Timber- 
 ing, and the construction of the Brickwork of Tunnels ; with the 
 Amount of Labour required for, and the Cost of the various Por- 
 tions of the Work. By FREDK. W. SIMMS, F.R.A.S., F.G.S., 
 M. Inst. C.E., Author of "A Treatise on the Principles and 
 Practice of Levelling," &c. &c. Second Edition, revised by W. 
 DAVIS HASKOLL, Civil Engineer, Author of "The Engineer's 
 Field-Book," c. &c. With 16 large folding Plates and numerous 
 Woodcuts. Imperial 8vo r I/, is. cloth. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 9 
 
 Strength of Cast Iron, &c. 
 
 A PRACTICAL ESSAY on the STRENGTH of CAST IRON 
 and OTHER METALS ; intended for the Assistance of Engineers, 
 Iron-Masters, Millwrights, Architects, Founders, Smiths, and 
 others engaged in the Construction of Machines, Buildings, &c. ; 
 containing Practical Rules, Tables, and Examples, founded en a 
 series of New Experiments ; with an Extensive Table of the Pro- 
 perties of Materials. By the late THOMAS TREDGOLD, Mem. Inst. 
 C.E., Author of " Elementary Principles of Carpentry," " History 
 of the Steam-Engine," c. Fifth Edition, much improved. 
 Edited by EATON HODGKINSON, F.R.S. ; to which are added 
 EXPERIMENTAL RESEARCHES on the STRENGTH and 
 OTHER PROPERTIES of CAST IRON ; with the Develop- 
 ment of New Principles, Calculations Deduced from them, and 
 Inquiries Applicable to Rigid and Tenacious Bodies generally. By 
 the EDITOR. The whole Illustrated with 9 Engravings and nume- 
 rous Woodcuts. 8vo, 12s. cloth. 
 
 %* HODGKINSON'S EXPERIMENTAL RESEARCHES ON THE 
 STRENGTH AND OTHER PROPERTIES OF CAST IRON may be had 
 separately. With Engravings and Woodcuts. 8vo, price 6s. cloth. 
 
 The High-Pressure Steam Engine. 
 
 THE HIGH-PRESSURE STEAM ENGINE ; an Exposition 
 of its Comparative Merits, and an Essay towards an Improved 
 System of Construction, adapted especially to secure Safety and 
 Economy. By Dr. ERNST ALBAN, Practical Machine Maker, 
 Plau, Mecklenberg. Translated from the German, with Notes, by 
 Dr. POLE, F.R.S., M. Inst. C.E., &c. &c. With 28 fine Plates, 
 8vo, i6s. 6d. cloth. 
 " A work like this, which goes thoroughly into the examination of the high-pressure 
 
 engine, the boiler, and its appendages, &c. , is exceedingly useful, and deserves a place 
 
 in every scientific library." Steam Shipping Chronicle. 
 
 Tables of Curves. 
 
 TABLES OF TANGENTIAL ANGLES and MULTIPLES 
 for setting out Curves from 5 to 200 Radius. By ALEXANDER 
 BEAZELEY, M. Inst. C.E. Printed on 48 Cards, and sold in a 
 cloth box, waistcoat-pocket size, price 3-5-. 6d. 
 
 " Each table is printed on a small card, which, being placed on the theodolite, leaves 
 the hands free to manipulate the instrument no small advantage as regards the rapidity 
 of work. They are clearly printed, and compactly fitted into a small case for the 
 pocket an arrangement that will recommend them to all practical men." Engineer. 
 
 " Very handy : a man may know that all his day's work must fall on two of these 
 cards, which he puts into his own card-case, and leaves the rest behind." Athenceum. 
 
 Laying Out Curves. 
 
 THE FIELD PRACTICE of LAYING OUT CIRCULAR 
 CURVES for RAILROADS. By JOHN C. TRAUTWINE, C.E., 
 of the United States (extracted from SIMMS'S Work on Levelling). 
 8vo, 5-r. sewed. 
 
io WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Surveying (Land and Marine). 
 
 LAND AND MARINE SURVEYING, in Reference to the 
 Preparation of Plans for Roads and Railways, Canals, Rivers, 
 Towns' Water Supplies, Docks and Harbours ; with Description 
 and Use of Surveying Instruments. By W. DAVIS HASKOLL, C. E. , 
 Author of "The Engineer's Field Book," " Examples of Bridge 
 and Viaduct Construction," &c. Demy 8vo, price I2s. 6d. cloth, 
 with 14 folding Plates, and numerous Woodcuts. 
 
 " ' Land and Marine Surveying' is a most useful and well arranged book for the 
 
 aid of a student We can strongly recommend it as a carefully-written 
 
 and valuable text-book." Builder, July 14, 1868. 
 
 " So far as the general get-up of the work is concerned, it is much superior to either 
 of its predecessors ( ' The Practice of Engineering Field Work/ 2 vols.) ; the lettering 
 and figuring of the plans annexed to it are clear and intelligible, and its moderate 
 price will not fail to be a recommendation." Engineer, June 19, 1868. 
 
 " He only who is master of his subject can present it in such a way as to make it 
 intelligible to the meanest capacity. It is in this that Mr. Haskoll excels. He has 
 knowledge and experience, and can so give expression to it as to make any matter on 
 
 which he writes, clear to the youngest pupil in a surveyor's office The 
 
 work will be found a useful one to men of experience, for there are few such who will 
 not get some good ideas from it ; but it is indispensable to the young practitioner." 
 Colliery Guardian, May 9, 1868. 
 
 " A volume which cannot fail to prove of the utmost practical utility It 
 
 is one which may be safely recommended to all students who aspire to become clean 
 and expert surveyors ; and from the exhaustive manner in which Mr. Haskoll has 
 placed his long experience at the disposal of his readers, there will henceforth be no 
 excuse for the complaint that young practitioners are at a disadvantage, through the 
 neglect of their seniors to point out the importance of minute details, since they can 
 readily supply the deficiency by the study of the volume now under consideration." 
 Mining Journal, May 5, 1868. 
 
 " A very useful and thoroughly practical treatise We can confidently 
 
 recommend this work to the engineering student." Artizan, July, 1868. 
 
 Engineer's Office Almanack & Pocket Companion. 
 
 THE ENGINEER'S AND CONTRACTOR'S OFFICE 
 ALMANACK AND POCKET COMPANION, containing, 
 besides the usual Calendar and other Almanack information, 
 Memoranda relating to Standing Orders, Railway Construction 
 Regulations, Mensuration, Data and Formulae for Wrought- Iron 
 Girders, Steam Engines, Railways, Cranes, Roofs, Mill-Gearing, 
 Hydraulics, Weight of Iron, &c. Published annually, with Ruled 
 Paper for Memoranda, strongly bound in cloth, price is. 
 
 " Here are 48 well-filled pages of that kind of matter which is in most constant 
 
 reference by engineers and contractors We have checked many of the 
 
 formulae, and find them to have been selected from the latest and best authorities." 
 Engineering. 
 
 " Its contents are of the most handy kind, solving in a rapid manner many a problem 
 of daily occurrence with engineers. " English Mechanic. 
 
 " The mechanical engineer, the hydraulic engineer, the builder, and railway and 
 mining engineer will, each and all, find occasion throughout the year of referring to 
 this Almanack." Colliery Guardian. 
 
 " Undoubtedly the most useful engineering almanack yet issued. It comprises 
 information and formulae upon almost every subject connected with practical engi- 
 neering, upon which the man of business is likely to require to refresh his memory." 
 Mining Journal. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. u 
 
 Fire Engineering. 
 
 FIRES, FIRE-ENGINES, AND FIRE BRIGADES. With 
 a History of Manual and Steam Fire-Engines, their Construc- 
 tion, Use, and Management ; Remarks on Fire-Proof Build- 
 ings, and the Preservation of Life from Fire ; Statistics of the Fire 
 Appliances in English Towns ; Foreign Fire Systems ; Hints for 
 the formation of, and Rules for, Fire Brigades ; and an Account of 
 American Steam Fire-Engines. By CHARLES F. T. YOUNG, C.E., 
 Author of " The Economy of Steam Power on Common Roads," 
 &c. With numerous Illustrations, Diagrams, &c., handsomely 
 printed, 544 pp., demy 8vo, price I/. 4^. cloth. 
 
 " A large well-filled and useful book upon a subject which possesses a wide and 
 
 increasing public interest To such of our readers as are interested in the 
 
 subject of fires and fire apparatus we can most heartily commend this book 
 
 It is really the only English work we now have upon the subject." Engineering. 
 
 " Mr. Young has proved by his present work that he is a good engineer, and pos- 
 sessed of sufficient literary energy to produce a very readable and interesting volume." 
 Engineer. 
 
 " Fire, above all the elements, is to be dreaded in a great city, and Mr. Young 
 deserves hearty thanks for the elaborate pains, benevolent spirit, scientific knowledge, 
 and lucid exposition he has brought to bear upon the subject ; and his substantial book 
 should meet with substantial success, for it concerns every one who has even a skin 
 which is not fireproof." Illustrated London News. 
 
 "A volume which must be regarded as the text-book of its subject, and which in 
 point of interest and intrinsic value is second to no contribution to a special depart- 
 ment of history with which we are acquainted. ' Fires, Fire-Engines, and Fire 
 Brigades ' is the production of an earnest and diligent writer who comes to the task he 
 has undertaken with a thorough love of it, and a firm determination to do it justice. 
 
 The style of the work is admirable It has the surpassing 
 
 merit of being thoroughly reliable." Insurance Record. 
 
 " That Mr. Young's treatise is an exhaustive one will be admitted when we state 
 that there does not appear to be anything within the scope of his comprehensive title 
 that has been left unnoticed. An immense amount of the most varied information 
 relating to the subject has been collected from every conceivable source, and goes to 
 form a history full of abiding interest. Great credit is unquestionably due to Mr. 
 Young for having brought before the public the results of his exploration in this hitherto 
 untrodden field. We strongly recommend the book to the notice of all who are in 
 any way interested in fires, fire-engines, or fire-brigades.'' Mec/utnics' Magazine. 
 
 Earthwork, Measurement and Calculation of. 
 
 A MANUAL on EARTHWORK. By ALEX. J. S. GRAHAM, 
 C.E., Resident Engineer, Forest of Dean Central Railway. With 
 numerous Diagrams. i8mo, 2s. 6d. cloth. 
 
 " We can cordially recommend the work to the notice of our readers." Building 
 Neivs. 
 
 " As a really handy book for reference, we know of no work equal to it ; and the 
 railway engineers and others employed in the measurement and calculation of earth- 
 work will find a great amount of practical information very admirably arranged, and 
 available for general or rough estimates, as well as for the more exact calculations 
 required in the engineers' contractor's offices." Artizan. 
 
 " The object of this little book is an investigation of all the principles requisite for 
 the measurement and calculation of earthworks, and a consideration of the data neces- 
 sary for such operations. The author has evidently bestowed much care in effecting 
 this object, and points out with much clearness the results of his own observations, 
 derived from practical experience. The subjects treated of are accompanied by well- 
 executed diagrams and instructive examples." Army and Navy Gazette. 
 
12 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Field- Book for Engineers. 
 
 THE ENGINEER'S, MINING SURVEYOR'S, and CON- 
 TRACTOR'S FIELD-BOOK. By W. DAVIS HASKOLL, Civil 
 Engineer. Second Edition, much enlarged, consisting of a Series 
 of Tables, with Rules, Explanations of Systems, and Use of Theo- 
 dolite for Traverse Surveying and Plotting the Work with minute 
 accuracy by means of Straight Edge and Set Square only ; Levelling 
 with the Theodolite, Casting out and Reducing Levels to Datum, 
 and Plotting Sections in the ordinary manner; Setting out Curves 
 with the Theodolite by Tangential Angles and Multiples with Right 
 and Left-hand Readings of the Instrument ; Setting out Curve? 
 without Theodolite on the System of Tangential Angles by Sets of 
 Tangents and Offsets ; and Earthwork Tables to 80 feet deep cal- 
 culated for every 6 inches in depth. With numerous wood-cuts, 
 I2mo, price 12s. cloth. 
 
 "A very useful work for the practical engineer and surveyor. Every person 
 engaged in engineering field operations will estimate the importance of such a work 
 and the amount of valuable time which will be saved by reference to a set of reliable 
 tables prepared with the accuracy and fullness of those given in this volume." Rail- 
 way Nevus. 
 
 " The book is very handy, and the author might have added that the separate tables 
 of sines and tangents to every minute will make it useful for many other purposes, the 
 genuine traverse tables existing all the same." Athejiatum. 
 
 " The work forms a handsome pocket volume, and cannot fail, from its portability 
 and utility, to be extensively patronised by the engineering profession." Mining 
 Journal. 
 
 " We know of no better field-book of reference or collection of tables than that 
 Mr. Haskoll has given." Artizan. 
 
 " A series of tables likely to be very useful to many civil engineers." Building News. 
 
 "A very useful book of tables for expediting field-work operations. . . . The present 
 edition has been much enlarged." Mechanics' Magazine. 
 
 "We strongly recommend this second edition of Mr. HaskolFs ' Field Book' to all 
 classes of surveyors." Colliery Guardian. 
 
 Railway Engineering. 
 
 THE PRACTICAL RAILWAY ENGINEER. A concise 
 Description of the Engineering and Mechanical Operations and 
 Structures which are combined in the Formation of Railways for 
 Public Traffic ; embracing an Account of the Principal Works exe- 
 cuted in the Construction of Railways ; with Facts, Figures, and 
 Data, intended to assist the Civil Engineer in designing and executing 
 the important details required. By G. DRYSDALE DEMPSEY, C.E. 
 Fourth Edition, revised and greatly extended. With 71 double 
 quarto Plates, 72 Woodcuts, and Portrait of GEORGE STEPHENSON. 
 One large vol. 4to., 2/. 12s. 6d. cloth. 
 
 Harbours. 
 
 THE DESIGN and CONSTRUCTION of HARBOURS. By 
 THOMAS STEVENSON, F.R.S.E., M.I. C.E. Reprinted and en- 
 larged from the Article "Harbours," in the Eighth Edition of " The 
 Encyclopedia Britannica." With 10 Plates and numerous Cuts. 
 8vo, IOJ-. 6d. cloth. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 13 
 
 Bridge Construction in Masonry, Timber, and 
 Iron. 
 
 EXAMPLES OF BRIDGE AND VIADUCT CONSTRUC- 
 TION OF MASONRY, TIMBER, AND IRON ; consisting of 
 46 Plates from the Contract Drawings or Admeasurement of select 
 Works. By W. DAVIS HASKOLL, C.E. Second Edition, with 
 the addition of 554 Estimates, and the Practice of Setting out Works, 
 illustrated with 6 pages of Diagrams. Imp. 4to, price 2/. 12s. 6d. 
 half-morocco. 
 
 " One of the very few works extant descending to the level of ordinary routine, and 
 treating on the common every-day practice of the railway engineer. ... A work of 
 the present nature by a man of Mr. Haskoll's experience, must prove invaluable to 
 hundreds. The tables of estimates appended to this edition will considerably enhance 
 its value." Engineering. 
 
 " We must express our cordial approbation of the work just issued by Mr. Haskoll. 
 .... Besides examples of the best and most economical forms of bridge construction, 
 the author has compiled a series of estimates which cannot fail to be of service to the 
 practical man. . . . The examples of bridges are selected from those of the most notable 
 construction on the different lines of the kingdom, and their details may consequently 
 be safely followed." Railway News. 
 
 " A very valuable volume, and may be added usefully to the library of every young 
 engineer. " Builder. 
 
 "An excellent selection of examples, veiy carefully drawn to useful scales of pro- 
 portion. " A rtizan. 
 
 Mathematical and Drawing Instruments. 
 
 A TREATISE ON THE -PRINCIPAL MATHEMATICAL 
 AND DRAWING INSTRUMENTS employed by the Engineer, 
 Architect, and Surveyor. By FREDERICK W. SIMMS, F.G.S., M. 
 Inst. C.E., Author of "Practical Tunnelling," &c. &c. Third 
 Edition, with a Description of the Theodolite, together with Instruc- 
 tions in Field Work, compiled for the use of Students on commenc- 
 ing practice. With "numerous Cuts. I2mo, price 3^. 6d. cloth. 
 
 Oblique Arches. 
 
 A PRACTICAL TREATISE ON THE CONSTRUCTION of 
 OBLIQUE ARCHES. By JOHN HART. Third Edition, with 
 Plates. Imperial 8vo, price Ss. cloth. 
 
 %* The small remaining stock of this work, which has been un- 
 obtainable for some ti?ne, has j^lst been purchased by LOCK WOOD & Co. 
 
 Oblique Bridges. 
 
 A PRACTICAL and THEORETICAL ESSAY on OBLIQUE 
 BRIDGES, with 13 large folding Plates. By GEO. WATSON 
 BUCK, M. Inst. C.E. Second Edition, corrected by W. If. 
 BARLOW, M. Inst. C.E. Imperial 8vo, I2j. cloth. 
 
 "The standard text-book for all engineers regarding skew arches, is Mr. Buck's 
 treatise, and it would be impossible to consult a better." Engineer. 
 
 "A very complete treatise on the subject, re-edited by Mr. Barlow, who has added 
 to it a method of making the requisite calculations without the use of trigonometrical 
 formulae." Builder. 
 
I 4 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Rudimentary Works in Engineering, &c. 
 
 WEALE'S SERIES OF RUDIMENTARY WORKS IN 
 ENGINEERING, ARCHITECTURE, MECHANICS, &c. &c. 
 At prices varying from is. to 5-r. 
 
 ** The following are inchided in this excellent and cheap Series 
 of Books (numbering upwards 0/200 different Works in almost every 
 department of Science, Art, d^.)> a complete List of which may be 
 had on application to Messrs. LOCKWOOD & Co. 
 
 STEAM ENGINE. By DR. LARDNEB. is. 
 
 TUBULAR AND IRON GIRDER BRIDGES, including the Britannia and 
 Conway Bridges. By G. D. DEMPSEV. is. 6af. 
 
 STEAM BOILERS, their Construction and Management. By R. ARMSTRONG. 
 With Additions by R. MALLET, is. 6d. 
 
 RAILWAY CONSTRUCTION. By SIR M. STEPHENSON. New Edition, is. 6d. 
 
 STEAM AND LOCOMOTION, on the Principle of connecting Science with Prac- 
 tice. By J. SEWELL. zs. 
 
 THE LOCOMOTIVE ENGINE. By G. D. DEMPSEY. is. 6rf. 
 ILLUSTRATIONS TO THE ABOVE. 4 to. 4*. bt. 
 STEAM ENGINE, Mathematical Theory of. By T. BAKER, is. 
 
 ENGINEER'S GUIDE TO THE ROYAL AND MERCANTILE NAVIES. 
 By a Practical Engineer. Revised by D. F. MCCARTHY. 3*. 
 
 LIGHTHOUSES, their Construction and Illumination. By ALAN STEVENSON. 3*. 
 
 CRANES AND MACHINERY FOR RAISING HEAVY BODIES, the Art of 
 Constructing. By J. GLYNN. is. 
 
 CIVIL ENGINEERING. By H. LAW and G. R. BURNELL. New Edition, 5 s. 
 DRAINING DISTRICTS AND LANDS. By G. D. DEMPSEY. is.6<t. ) The 
 DRAINING AND SEWAGE OF TOWNS AND BUILDINGS. By f zvols.inr, 
 G. D. DEMPSEY. 2 s. -> 3*- 6 *- 
 
 WELL-SINKING, BORING, AND PUMP WORK. By J. G. SWINDELL ; 
 Revised by G. R. BURNELL. is. 
 
 ROAD-MAKING AND MAINTENANCE OF MACADAMISED ROADS. 
 By GEN. SIR J. BURGOYNE. is. (>d. 
 
 AGRICULTURAL ENGINEERING, BUILDINGS, MOTIVE POWERS, 
 FIELD MACHINES, MACHINERY AND IMPLEMENTS. By G. H. 
 ANDREWS, C.E. y. 
 
 ECONOMY OF FUEL. By T. S. PRIDEAUX. is. 6d. 
 
 EMBANKING LANDS FROM THE SEA. By J. WIGGINS, as. 
 
 WATER POWER, as applied to Mills, &c. By J. GLYNN. 25. 
 
 A TREATISE ON GAS WORKS, AND THE PRACTICE OF MANUFAC- 
 TURING AND DISTRIBUTING COAL GAS. By S. HUGHES, C.E. 3*. 
 
 WATERWORKS FOR THE SUPPLY OF CITIES AND TOWNS. By S. 
 
 HUGHES, C.E. 3*. 
 SUBTERRANEOUS SURVEYING, AND THE MAGNETIC VARIATION 
 
 OF THE NEEDLE. By T. FENWICK, with Additions by T. BAKER, zs. 6d. 
 
 CIVIL ENGINEERING OF NORTH AMERICA. By D. STEVENSON. 3* 
 
 HYDRAULIC ENGINEERING. By G. R. BURNELL. 3*. 
 
 RIVERS AND TORRENTS, with the Method of Regulating their Course and 
 
 Channels, Navigable Canals, &c., from the Italian of PAUL FRISI. 2*. 6d. 
 COMBUSTION OF COAL AND THE PREVENTION OF SMOKE. By 
 
 C. WYE WILLIAMS, M.I.C.E. 3 *. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 15 
 
 ARCHITECTURE. 
 
 Construction. 
 
 THE SCIENCE of BUILDING : an Elementary Treatise on 
 the Principles of Construction. Especially adapted to the Re- 
 quirements of Architectural Students. By E. WYNDHAM TARN, 
 M.A., Architect. Illustrated with 47 Wood Engravings. Demy 
 8vo, price 8j. 6d. cloth. [Just published. 
 
 " No architectural student should be without this hand-book of constructional 
 knowledge." Architect, April 9, 1870. 
 
 "The book is very far from being a mere compilation; it is an able digest of 
 information which is only to be found scattered through various works, and contains 
 more really original writing than many puttmg.forth far stronger claims to originality. 
 .... Mr. Tarn has done his work exceedingly well, and he has produced a book 
 which ought to earn him the thanks of all architectural students. The book is clearly 
 printed in bold type, the wood-cuts are all well executed, and the work is made of a 
 very convenient size for reference." Engineering, May 5, 1870. 
 
 "The quiet, neat, scientific appropriateness of the language employed, distinguishes 
 the master inind.'' Mining Journal, April 9, 1870. 
 
 Chambers Civil Architecture, by Gwilt. 
 
 A TREATISE ON THE DECORATIVE PART of CIVIL 
 ARCHITECTURE. By Sir WILLIAM CHAMBERS, K.P.S., 
 F.R.S., F.S.A.,F.R.S.S. With Illustrations, Notes, and an Exami- 
 nation of Grecian Architecture, by JOSEPH GWILT, F.S. A. Newand 
 Cheap Edition, revised and edited by W. H. LEEDS. With 65 
 Plates, and Portrait of the Author. Royal 4to, I/. is. cloth. 
 
 \* A new edition of this standard architectitral work (which has 
 already passed through several high-priced issues], so cheap as to place 
 it 'within the reach of the humbler classes of students and practical 
 men, and at the same time so carefully edited and well executed as to 
 make it worthy of a place on the shelves of the more opulent^ cannot 
 fail to be received as a boon by the professional public. 
 
 Villa Architecture. 
 
 A HANDY BOOK of VILLA ARCHITECTURE ; being a 
 Series of Designs for Villa Residences in various Styles. With 
 Detailed Specifications and Estimates. By C. WICKES, Architect, 
 Author of " The Spires and Towers of the Mediaeval Churches of 
 England," &c. First Series, consisting of 30 Plates ; Second 
 Series, 31 Plates. Complete in I vol., 4to, price 2/. IQJ. half 
 morocco. Either Series separate, price I/. 7-r. each, half morocco. 
 
 " The whole of the designs bear evidence of their being the work of an artistic 
 architect, and they will prove very valuable and suggestive to architects, students, and 
 amateurs." Building News. 
 
 "Suburban builders, who are now so largely employed all round the metropolis, 
 may get a few valuable hints by consulting Mr. Wickes's publication." Art Journal. 
 
 "There is so much elegance and good taste manifested in these designs for villa 
 residences, and so much practical knowledge is displayed in the accompanying working 
 plans, that we have pleasure in recommending the work to the profession as well as to 
 persons about to build." News of the World. 
 
16 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 The Young Architect's Book. 
 
 HINTS TO YOUNG ARCHITECTS ; comprising Advice to 
 those who, while yet at school, are destined to the Profession ; to 
 such as, having passed their pupilage, are about to travel ; and to 
 those who, having completed their education, are about to practise. 
 By GEORGE WIGHTWICK, Architect, Author of " The Palace of 
 Architecture," &c. &c. Second Edition. With numerous Wood- 
 cuts. 8vo, 7-r., extra cloth. 
 
 Drawing for Builders and Students. 
 
 PRACTICAL RULES ON DRAWING for the OPERATIVE 
 BUILDER and YOUNG STUDENT in ARCHITECTURE. 
 By GEORGE PYNE, Author of a " Rudimentary Treatise on Per- 
 spective for Beginners." With 14 Plates. 4to, 'js. 6d. t boards. 
 CONTENTS. I. Practical Rules on Drawing Outlines. II. Ditto the Grecian 
 and Roman Orders. III. Practical Rules on Drawing Perspective. IV. Practical 
 Rules on Light and Shade. V. Practical Rules on Colour, &c. c. 
 
 Ventilation. 
 
 A TREATISE ON VENTILATION, NATURAL and ARTI- 
 FICIAL. By ROBERT RITCHIE, C.E., Author of " Railways, 
 their Rise, Progress, and Construction," &c. &c. With numerous 
 Plates and Woodcuts. 8vo, 8s. 6d. cloth. 
 
 "This must continue to be for some time the text-book upon one of the chief diffi- 
 culties of domestic architectural construction and of social hygienics." Lancet. 
 
 Cottages, Villas, and Country Houses. 
 
 DESIGNS and EXAMPLES of COTTAGES, VILLAS, and 
 COUNTRY HOUSES ; being the Studies of several eminent 
 Architects and Builders ; consisting of Plans, Elevations, and Per- 
 spective Views ; with approximate Estimates of the Cost of each. 
 In 410, with 67 plates, price I/, u., cloth. 
 
 Wealds Builder s and Contractor s Price Book. 
 
 THE BUILDER'S AND CONTRACTOR'S PRICE BOOK 
 (LOCKWOOD & Co.'s, formerly WEALE'S). Published Annually. 
 Containing Prices for Work in all branches of the Building Trade, 
 with Items numbered for easy reference, and an Appendix of 
 Tables, Notes, and Memoranda, arranged to afford detailed infor- 
 mation, commonly required in preparing Estimates, &c. Originally 
 Edited by the late GEO. R. BURNELL, C.E., &c. I2mo, 4-5-., cloth. 
 
 " A multitudinous variety of useful information for builders and contractors 
 
 With its aid the prices for all work connected with the building trade may be esti- 
 mated. " Building News. 
 
 " Carefully revised, admirably arranged, and clearly printed, it offers at a glance a 
 ready method of preparing an estimate or specification upon a basis that is unquestion- 
 able. A reliable book of reference in the event of a dispute between employer and 
 employed." Engineer. 
 
 " Well done and reliable. It is the duty of a just critic to point out where any 
 improvement can be made in any work, but Mr. Burnell has anticipated all objections 
 in his clearly-printed book. We therefore recommend it to all branches of the pro- 
 fession. " English Mechanic. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 17 
 
 NAVAL ARCHITECTURE, &C- 
 
 Ship-Building and Steam Ships. 
 
 SHIP-BUILDING IN IRON AND WOOD. By ANDREW 
 MURRAY, M.I.C.E., Chief Engineer and Inspector of Machinery 
 of H.M.'s Dockyard, Portsmouth; and STEAM SHIPS, by 
 ROBERT MURRAY, C.E., Engineer Surveyor to the Board of 
 Trade. Second Edition. In I vol., 4to, with 28 Plates and 
 numerous Woodcuts, price iqs. cloth. 
 
 " Indispensable in the office of the naval architect, whether employing wood or 
 iron." Practical Mechanic 's Journal. 
 
 " Ought to be in the hands of every shipbuilder or shipwright." Sunderland 
 Herald. 
 
 Naval Architecture. 
 
 RUDIMENTS OF NAVAL ARCHITECTURE. Compiled 
 for the Use of Beginners. By JAMES PEAKE, formerly of the 
 School of Naval Architecture, H.M. Dockyard, Portsmouth. 
 Third Edition, with many Illustrations. I2mo, cloth limp, price y. 
 
 Principles of Ship Construction. 
 
 ELEMENTARY AND PRACTICAL PRINCIPLES of the 
 CONSTRUCTION OF SHIPS for OCEAN AND RIVER 
 SERVICE. By HAKON A. SOMMERFELDT, Surveyor to the 
 Royal Norwegian Navy. I2mo, cloth limp, price is. 
 V ATLAS OF FIFTEEN PLATES to the above. 4to, price 7*. 6d. 
 
 Masting, Rigging of Ships, &c. 
 
 RUDIMENTARY TREATISE ON MASTING, MAST- 
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 N.A. loth Edition, Illustrated. I2mo, cloth limp, price is. 6tt. 
 
 Sail-Making. 
 
 ELEMENTARY TREATISE ON SAIL-MAKING ; with 
 Draughting, and the Centre of Effort of the Sails. Also Weights 
 and Sizes of Ropes, Masting, Rigging, and Sails of Steam Vessels, 
 c. By ROBERT KIPPING, N.A. Seventh Edition. With 
 numerous Woodcuts. I2mo, cloth limp, price 2s. 6d. 
 
 Navigation, &c. 
 
 RUDIMENTARY TREATISE ON NAVIGATION ; the 
 Sailor's Sea- Book. In Two Parts. PART I. How to Keep the 
 Log and Work it off, &c. PART II. On Finding the Latitude 
 and Longitude. By JAMES GREENWOOD, Esq., B.A. Fifth 
 Edition. With several Engravings and Illustrations (in colours) 
 of the Flags of Maritime Nations. I2mo, cloth limp, price 2s. 
 
i8 
 
 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Granthams Iron Ship- Building, enlarged. 
 
 ON IRON SHIP-BUILDING; with Practical Examples and 
 Details. Fifth Edition. Imp. 4to, boards, enlarged from 24 to 40 
 Plates (21 quite new), including the latest Examples. Together 
 with separate Text, I2mo, cloth limp, also considerably enlarged, 
 By JOHN GRANTHAM, M. Inst. C.E., &c. Price 2/. 2s. complete. 
 
 Description of Plates. 
 
 Hollow and Bar Keels, Stem and 
 Stern Posts. [Pieces. 
 
 Side Frames, Floorings, and Bilge 
 
 Floorings continued Keelsons, Deck 
 Beams, Gunwales, and Stringers. 
 
 Gunwales continued Lower Decks, 
 
 and Orlop Beams. 
 . Gunwales and Deck Beam Iron. 
 
 Angle-Iron, T Iron, Z Iron, Bulb 
 Iron, as Rolled for Building. 
 
 Rivets, shown in section, natural size ; 
 Flush, and Lapped Joints, with 
 Single and Double Riveting. 
 
 Plating, three plans ; Bulkheads and 
 Modes of Securing them. 
 
 Iron Masts, with Longitudinal and 
 Transverse Sections. 
 
 Sliding Keel, Water Ballast, Moulding 
 the Frames in Iron Ship Building, 
 Levelling Plates. 
 
 Longitudinal Section, and Half- 
 breadth Deck Plan of Large Vessels 
 on a reduced Scale. 
 
 Midship Sections of Three Vessels. 
 
 Large Vessel, showing Details Fore 
 End in Section, and End View, 
 with Stern Post, Crutches, &c. 
 
 Large F^j^/,showing Details After 
 End in Section, with End View, 
 Stern Frame for Screw, and Rudder. 
 
 Large Vessel, showing Details Mid- 
 ship Section, half breadth. 
 
 Machines for Punching and Shearing 
 Plates and Angle-Iron, and for 
 Bending Plates ; Rivet Hearth. 
 . Beam-Bending Machine, Indepen- 
 dent Shearing, Punching and Angle- 
 Iron Machine. 
 
 15^. Double Lever Punching and Shearing 
 Machine, arranged for cutting 
 Angle and T Iron, with Dividing 
 Table and Engine. 
 
 16. Machines. Garforth's Riveting Ma- 
 
 chine, Drilling and Counter-Sinking 
 Machine. 
 i6tf. Plate Planing Machine. 
 
 17. Air Furnace for Heating Plates and 
 
 Angle-Iron : Various Tools used in 
 Riveting and Plating. 
 
 18. Gunwale ; Keel and Flooring ; Plan 
 
 for Sheathing with Copper. 
 i8rt. Grantham's Improved Planof Sheath- 
 ing Iron Ships with Copper. 
 
 19. Illustrations of the Magnetic Condi- 
 
 tion of various Iron Ships. 
 
 20. Gray's Floating Compass and Bin- 
 
 nacle, with Adjusting Magnets, &c. 
 
 21. Corroded Iron Bolt in Frame of 
 
 Wooden Ship ; Jointing Plates. 
 
 22-4. Great Eastern Longitudinal Sec- 
 tions and Half-breadth Plans Mid- 
 ship Section, with Details Section 
 in Engine Room, and Paddle Boxes. 
 
 25-6. Paddle Steam Vessel of Steel. 
 
 27. Scarbrough Paddle Vessel of Steel. 
 
 28-9. Proposed Passenger Steamer. 
 
 30. Persian Iron Screw Steamer. 
 
 31. Midship Section of H.M. Steam 
 
 Frigate, Warrior. 
 
 32. Midship Section of H.M. Steam 
 
 Frigate, Hercules. 
 
 33. Stem, Stern, and Rudder of H.M. 
 
 Steam Frigate, Bellerophon. 
 
 34. Midship Section of H. M. Troop Ship, 
 
 Serapis. 
 
 35. Iron Floating Dock. 
 
 "An enlarged edition of an elaborately illustrated work." Builder, July n, 1868. 
 
 "This edition of Mr. Grantham's work has been enlarged and improved, both with 
 respect to the text and the engravings being brought clown to the present period . . . 
 The practical operations required in producing a ship are described and illustrated with 
 care and precision." Mechanics' Magazine, July 17, 1868. 
 
 "A thoroughly practical work, and every question of the many in relation to iron 
 shipping which admit of diversity of opinion, or have various and conflicting personal 
 interests attached to them, is treated with sober and impartial wisdom and good sense. 
 As good a volume for the instruction of the pupil or student of iron naval 
 architecture as can be found in any language." Practical Mechanic s Journal, 
 August, 1868. 
 
 " A very elaborate work. . '. . It forms a most valuable addition to the history 
 of iron shipbuilding, while its having been prepared by one who has made the subject 
 his study for many years, and whose qualifications have been repeatedly recognised, 
 will recommend it as one of practical utility to all interested in shipbuilding." Army 
 and Navy Gazette, July 11, 1868. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 19 
 
 CARPENTRY, TIMBER, &c. 
 
 * 
 
 Tredgold's Carpentry, new & enlarged Edition. 
 
 THE ELEMENTARY PRINCIPLES OF CARPENTRY : 
 
 a Treatise on the Pressure and Equilibrium of Timber Framing, the 
 Resistance of Timber, and the Construction of Floors, Arches, 
 Bridges, Roofs, Uniting Iron and Stone with Timber, &c. To which 
 is added an Essay on the Nature and Properties of Timber, &c., 
 with Descriptions of the Kinds of Wood used in Building ; also 
 numerous Tables of the Scantlings of Timber for different purposes, 
 the Specific Gravities of Materials, &c. By THOMAS TREDGOLD, 
 C.E. Edited by PETER BARLOW, F.R.S. Fifth Edition, cor- 
 rected and enlarged. With 64 Plates (n of which now first appear 
 in this edition), Portrait of the Author, and several Woodcuts. In 
 I large vol., 4to, 2/. 2s. extra cloth. \Just published. 
 
 "'Tredgold's Carpentry' ought to be in every architect's and every builder's 
 library, and those who do not already possess it ought to avail themselves of the new 
 issue." Builder, April 9, 1870. 
 
 "A work whose monumental excellence must commend it wherever skilful car- 
 pentry is concerned. The Author's principles are rather confirmed than impaired by 
 time, and, as now presented, combine the surest base with the most interesting display 
 of progressive science. The additional plates are of great intrinsic value." Building 
 News, Feb. 25, 1870. 
 
 " 'Tredgold's Carpentry' has ever held a high position, and the issue of the fifth 
 edition, in a still more improved and enlarged form, will give satisfaction to a very 
 large number of artisans who desire to raise themselves in their business, and who 
 seek to do so by displaying a greater amount of knowledge and intelligence than their 
 fellow-workmen. It is as complete a work as need be desired. To the superior 
 workman the volume will prove invaluable ; it contains treatises written in language 
 which he will readily comprehend." Mining Journal, Feb. 12, 1870. 
 
 Grandys Timber Tables. 
 
 THE TIMBER IMPORTER'S, TIMBER MERCHANT'S, 
 and BUILDER'S STANDARD GUIDE. By RICHARD E. 
 GRANDY. Comprising : An Analysis of Deal Standards, Home 
 and Foreign, with comparative Values and Tabular Arrangements 
 for Fixing Nett Landed Cost on Baltic and North American Deals, 
 including all intermediate Expenses, Freight, Insurance, Duty, &c., 
 &c. ; together with Copious Information for the Retailer and 
 Builder. I2mo, price 7-r. 6d. cloth. 
 
 " Everything it pretends to be : built up gradually, it leads one from a forest to a 
 trenail, and throws in, as a makeweight, a host of material concerning bricks, columns, 
 cisterns, &c. all that the class to whom it appeals requires." English Mechanic. 
 
 " The only difficulty we have is as to what is NOT in its pages. What we have tested 
 of the contents, taken at random, is invariably correct." Illustrated Builder' 5 Journal. 
 
 Tables for Packing-Case Makers. 
 
 PACKING-CASE TABLES ; showing the number of Superficial 
 Feet in Boxes or Packing-Cases, from six inches square and 
 upwards. Compiled by WILLIAM RICHARDSON, Accountant. 
 Oblong 4to, cloth, price $s. 6d. 
 
 "Will save much labour and calculation to packing-case makers and those who use 
 packing-cases." Grocer. " Invaluable labour-saving tables." Ironmonger. 
 
20 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 Nicholsons Carpenter s Guide. 
 
 THE CARPENTER'S NEW GUIDE ; or, BOOK of LINES 
 for CARPENTERS : comprising all the Elementary Principles 
 essential for acquiring a knowledge of Carpentry. Founded on the 
 late PETER NICHOLSON'S standard work. A new Edition, revised 
 by ARTHUR ASHPITEL, F.S.A., together with Practical Rules on 
 Drawing, by GEORGE PYNE. With 74 Plates, 4to, \l. is. cloth. 
 
 Dowsing 1 s Timber Merchant's Companion. 
 
 THE TIMBER MERCHANT'S AND BUILDER'S COM- 
 PANION ; containing New and Copious Tables of the Reduced 
 Weight and Measurement of Deals and Battens, of all sizes, from 
 One to a Thousand Pieces, and the relative Price that each size 
 bears per Lineal Foot to any given Price per Petersburg!! Standard 
 Hundred ; the Price per Cube Foot of Square Timber to any given 
 Price per Load of 50 Feet ; the proportionate Value of Deals and 
 Battens by the Standard, to Square Timber by the Load of 50 Feet ; 
 the readiest mode of ascertaining the Price of Scantling per Lineal 
 Foot of any size, to any given Figure per Cube Foot. Also a 
 variety of other valuable information. By WILLIAM DOWSING, 
 Timber Merchant. Second Edition. Crown 8vo, 3-r. cloth. 
 
 " Everything is as concise and clear as it can possibly be made. There can be no 
 doubt that every timber merchant and builder ought to possess it, because such possession 
 would, with use, unquestionably save a very great deal of time, and, moreover, ensure 
 perfect accuracy in calculations. There is also another class besides these who ought 
 to possess it ; we mean all persons engaged in carrying wood, where it is requisite to 
 ascertain its weight. Mr. Dowsing's tables provide an easy means of doing this. 
 Indeed every person who has to do with wood ought to have it." Hull Advertiser. 
 
 MECHANICS, &c. 
 
 Mechanics Workshop Companion. 
 
 THE OPERATIVE MECHANIC'S WORKSHOP COM- 
 PANION, and THE SCIENTIFIC GENTLEMAN'S PRAC- 
 TICAL ASSISTANT ; comprising a great variety of the most 
 useful Rules in Mechanical Science ; with numerous Tables of Prac- 
 tical Data and Calculated Results. By W. TEMPLETON, Author 
 of "The Engineer's, Millwright's, and Machinist's Practical As- 
 sistant." Ninth Edition, with the addition of Mechanical Tables 
 for Operative Smiths, Millwrights, Engineers, &c. ; together with 
 several Useful and Practical Rules in Hydraulics and Hydrody- 
 namics, a variety of Experimental Results, and an Extensive Table 
 of Powers and Roots, n Plates. I2mo, 5-r. bound. 
 
 " As a text-book of reference, in which mechanical and commercial demands are 
 judiciously met, TEMPLETON'S COMPANION stands unrivalled." Mechanics' Magazine. 
 
 " Admirably adapted to the wants of a very large class. It has met with great 
 success in the engineering workshop, as we can testify ; and there are a great many 
 men who, in a great measure, owe their rise in life to this little work. " Building News. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 21 
 
 Engineer s Assistant. 
 
 THE ENGINEER'S, MILLWRIGHT'S, and MACHINIST'S 
 PRACTICAL ASSISTANT ; comprising a Collection of Useful 
 Tables, Rules, and Data. Compiled and Arranged, with Original 
 Matter, by W. TEMPLETON, Author of " The Operative Mechanic's 
 Workshop Companion." 4th Edition. iSrno, 2s.6d. cloth. . 
 
 " A perfect vade mecum for all who are practically engaged in mechanical pursuits. 
 So much varied information compressed into so small a space, and published at a price 
 which places it within the reach of the humblest mechanic, cannot fail to command the 
 sale which it deserves. With the utmost confidence we stamp this book with the appro- 
 bation of the ' Mechanics' Magazine,' and having done so, commend it to the attention 
 of our readers." Mechanics' Magazine, 
 
 "To a practical engineer it must be of incalculable value." Morning Herald. 
 
 " This work contains, in a connected and convenient form, admirably arranged and 
 lucidly explained, all the information likely to be immediately and lastingly useful to 
 those whose interests it is designed to subserve." Manufacturer and Inventor. 
 
 " Every mechanic should become the possessor of the volume, and a more suitable 
 present to an apprentice to any of the mechanical trades could not possibly be made." 
 Building News, 
 
 Designing, Measuring, and Valuing. 
 
 THE STUDENT'S GUIDE to the PRACTICE of DESIGN- 
 ING, MEASURING, and VALUING ARTIFICERS' WORKS; 
 
 containing Directions for taking Dimensions, Abstracting the same, 
 and bringing the Quantities into Bill, with Tables of Constants, 
 and copious Memoranda for the Valuation of Labour and Materials 
 in the respective Trades of Bricklayer and Slater, Carpenter and 
 Joiner, Sawyer, Stonemason, Plasterer, Smith and Ironmonger, 
 Plumber, Painter and Glazier, Paperhanger. With 43 Plates and 
 Woodcuts. The measuring, &c., edited by EDWARD DOBSON, 
 Architect and Surveyor. Second Edition, with the Additions on 
 Design, by E. LACY GARBETT, Architect ; together with Tables 
 for Squaring and Cubing. In one vol., 8vo, gs. extra cloth. 
 
 Superficial Measurement. 
 
 THE TRADESMAN'S GUIDE TO SUPERFICIAL MEA- 
 SUREMENT. Tables calculated from I to 200 inches in length, 
 by I to 108 inches in breadth. Particularly recommended to Archi- 
 tects, Surveyors, Engineers, Timber Merchants, Builders, Car- 
 penters, Upholsterers, Coach Makers, Looking and Crown Glass 
 Dealers, Painters, Stonemasons, &c. By JAMES HAWKINGS. 
 Fcp. 3-r. 6d. cloth. 
 
 Manufacture of Iron. 
 
 IRON : its History, Properties, and Processes of Manufacture. 
 
 By WILLIAM FAIRBAIRN, C.E., LL.D., F.R.S., &c. With 
 
 numerous Woodcuts. New Edition, revised and enlarged. 8vo, 
 
 price IOJ. 6d. cloth. 
 
 " A scientific work of the first class, whose chief merit lies in bringing the more 
 important facts connected with iron into a small compass, and within the comprehen- 
 sion and the means of all persons engaged in its manufacture, sale, or use." Mechanics' 
 Magazine. 
 
22 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 MATHEMATICS, &c. 
 
 Gregory s Practical Mathematics. 
 
 MATHEMATICS for PRACTICAL MEN ; being a Common- 
 place Book of Pure and Mixed Mathematics. Designed chiefly 
 for the Use of Civil Engineers, Architects, and Surveyors. Part I. 
 PURE MATHEMATICS comprising Arithmetic, Algebra, Geometry, 
 Mensuration, Trigonometry, Conic Sections, Properties of Curves. 
 Part II. MIXED MATHEMATICS comprising Mechanics in general, 
 Statics, Dynamics, Hydrostatics, Hydrodynamics, Pneumatics, 
 Mechanical Agents, Strength of Materials. With an Appendix of 
 copious Logarithmic and other Tables. By OLINTHUS GREGORY, 
 LL.D., F.R.A.S. Enlarged by HENRY LAW, C.E. 4th Edition, 
 carefully revised and corrected by J. R. YOUNG, formerly Profes- 
 sor of Mathematics, Belfast College; Author of " A Course of 
 Mathematics," &c. With 13 Plates. Medium 8vo, I/, u. cloth. 
 " As a standard work on mathematics it has not been excelled. " A rtizan. 
 " The engineer or architect will here find ready to his hand, rules for solving nearly 
 every mathematical difficulty that may arise in his practice. As a moderate acquaint- 
 ance with arithmetic, algebra, and elementary geometry is absolutely necessary to the 
 proper understanding of the most useful portions of this book, the author very wisely 
 has devoted the first three chapters to those subjects, so that the most ignorant may be 
 enabled to master the whole of the book, without aid from any other. The rules are in 
 all cases explained by means of examples, in which every step of the process is clearly 
 worked out." Builder. 
 
 " One of the most serviceable books to the practical mechanics of the country. . 
 The edition of 1847 was fortunately entrusted to the able hands of Mr. Law, who 
 revised it thoroughly, re-wrote many chapters, and added several sections to those 
 which had been rendered imperfect by advanced knowledge. On examining the various 
 and many improvements which he introduced into the work, they seem almost like a 
 new structure on an old plan, or rather like the restoration of an old ruin, not only to 
 its former substance, but to an extent which meets the larger requirements of modern 
 
 times In the edition just brought out, the work has again been revised by 
 
 Professor Young. He has modernised the notation throughout, introduced a few 
 paragraphs here and there, and corrected the numeious typographical errors which 
 nave escaped the eyes of the former Editor. The book is now as complete as it is 
 
 possible to make it We have carried our notice of this book to a greater 
 
 length than the space allowed us justified, but the experiments it contains are so 
 interesting, and the method of describing them so clear, that we may be excused for 
 overstepping our limit. It is an instructive book for the student, and a Text- 
 book for him who having once mastered the subjects it treats of, needs occasionally to 
 refresh his memory upon them." Building New s. 
 
 The Metric System. 
 
 A SERIES OF METRIC TABLES, in which the British 
 Standard Measures and Weights are compared with those of the 
 Metric System at present in use on the Continent. By C. H. 
 DOWLING, C. E. 8vo, lew. 6d. strongly bound. 
 
 "Mr. Dowling's Tables, which are well put together, come just in time as a ready 
 reckoner for the conversion of one system into the other." Athetueum. 
 
 "Their accuracy has been certified by Professor Airy, the Astronomer Royal." 
 Builder. 
 
 " Resolution 8. That advantage will be derived from the recent publication of 
 Metric Tables, by C. H. Dowling, C.E." Report of Section F, British Association, 
 Bath. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 23 
 
 Inwoods Tables, greatly enlarged and improved. 
 
 TABLES FOR THE PURCHASING of ESTATES, Freehold, 
 Copyhold, or Leasehold ; Annuities, Advowsons, &c. , and for the 
 Renewing of Leases held under Cathedral Churches, Colleges, or 
 other corporate bodies ; for Terms of Years certain, and for Lives ; 
 also for Valuing Reversionary Estates, Deferred Annuities, Next 
 Presentations, &c., together with Smart's Five Tables of Compound 
 Interest, and an Extension of the same to lower and Intermediate 
 Rates. By WILLIAM INWOOD, Architect. The i8th edition, with 
 considerable additions, and new and valuable Tables of Logarithms 
 for the more Difficult Computations of the Interest of Money, Dis- 
 count, Annuities, &c., by M. FEDOR THOMAN, of the Societe 
 Credit Mobilier of Paris. I2mo, 8s. cloth. 
 
 %* This edition (the l8///) differs in many important particulars 
 from former ones. The changes consist, first, in a more convenient 
 and systematic arrangement of the original Tables, and in the removal 
 of certain numerical errors which a very careful revision of the whole 
 has enabled the present editor to discover ; and secondly, in the 
 extension of practical titility conferred on the work by the introduction 
 of Tables now inserted for the first time. This new and important 
 matter is all so much actually added to INWOOD'S TABLES ; nothing 
 has been abstracted from the original collection : so that those who have 
 been long in the habit of consulting INWOOD for any special profes- 
 sional piirpose will, as heretofore, find the information sought still in 
 its pages. 
 
 " Those interested in the purchase and sale of estates, and in the adjustment of 
 compensation cases, as well as in transactions in annuities, life insurances, &c., will 
 find the present edition of eminent service." Engineering. 
 
 " More than half a century has elapsed since the first edition was published, yet 
 ' Inwood's Tables' still maintain a most enviable reputation ; and when it is considered 
 that the new issue, the Eighteenth edition, has been enriched by large additional 
 contributions by Mr. Fedor Thoman, of the French Credit Mobilier, whose carefully 
 arranged tables of logarithms for the more difficult computations of the interest of 
 money, discount, annuities, &c., cannot fail to be of the utmost utility, its value will 
 be appreciated. The introduction contains an admirable epitome of the principles of 
 decimals, and an explanation of all that is necessary to render the elaborate tables in 
 the book of thorough utility to all consulting it. This new edition will certainly be 
 referred to with quite as much confidence as its predecessors." Mining Journal. 
 
 Compound Interest and Annuities. 
 
 THEORY of COMPOUND INTEREST and ANNUITIES ; 
 with Tables of Logarithms for the more Difficult Computations of 
 Interest, Discount, Annuities, &c., in all their Applications and 
 Uses for Mercantile and State Purposes. With an elaborate Intro- 
 duction. By FEDOR THOMAN, of the Societe Credit Mobilier, 
 Paris. I2mo, cloth, 5-r. 
 
 " A very powerful work, and the Author has a very remarkable command of his 
 subject." Professor A. de Morgan. 
 
 " No banker, merchant, tradesman, or man of business, ought to be without Mr. 
 Thoman's truly 'handy-book.'" Review. 
 
 " The author of this ' handy-book ' deserves our thanks." Insurance Gazette. 
 
 " We recommend it to the notice of actuaries and accountants." Athenanm. 
 
24 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 SCIENCE AND ART. 
 
 The Military Sciences. 
 
 AIDE-MEMOIRE to the MILITARY SCIENCES. Framed 
 from Contributions of Officers and others connected with the dif- 
 ferent Services. Originally edited by a Committee of the Corps of 
 Royal Engineers. Second Edition, most carefully revised by an 
 Officer of the Corps, with many additions ; containnig nearly 350 
 Engravings and many hundred Woodcuts. 3 vols. royal 8vo, extra 
 cloth boards, and lettered, price 4/. IGJ-. 
 
 "A compendious encyclopaedia of military knowledge, to which we are greatly in- 
 debted." Edinburgh Review. 
 
 " The most comprehensive work of reference to the military and collateral sciences. 
 Among the list of contributors, some seventy-seven in number, will be found names of 
 the highest distinction in the services. . . . The work claims and possesses the great 
 merit that by far the larger portion of its subjects have been treated originally by the 
 practical men who have been its contributors. Volunteer Service Gazette. 
 
 Field Fortification. 
 
 A TREATISE on FIELD FORTIFICATION, the ATTACK 
 of FORTRESSES, MILITARY, MINING, and RECON- 
 NOITRING. By Colonel I. S. MACAULAY, late Professor of 
 Fortification in the Royal Military Academy, Woolwich. Sixth 
 Edition, crown 8vo, cloth, with separate Atlas of 12 Plates, sewed, 
 price I2s. complete. {Recently published. 
 
 Dye- Wares and Colours. 
 
 THE MANUAL of COLOURS and DYE-WARES: their 
 Properties, Applications, Valuation, Impurities, and Sophistica- 
 tions. For the Use of Dyers, Printers, Dry Salters, Brokers, &c. 
 By J. W. SLATER. Post 8vo, cloth. [Just Ready. 
 
 %* The object of this Manual is to furnish, in brief space, an 
 account of the chemical prodttcts and natural wares used in dyeing, 
 printing, and the accessory arts, their properties, their applications, 
 the means of ascertaining their respective values, and of detecting the 
 impurities which may be present. 
 
 Electricity. 
 
 A MANUAL of ELECTRICITY ; including Galvanism, Mag- 
 netism, Diamagnetism, Electro-Dynamics, Magno- Electricity, and 
 the Electric Telegraph. By HENRY M. NOAD, Ph.D., F.C.S., 
 Lecturer on Chemistry at St. George's Hospital. Fourth Edition, 
 entirely rewritten. Illustrated by 500 Woodcuts. 8vo, I/. 4^-. cloth. 
 
 " This publication fully bears out its title of ' Manual.' It discusses in a satisfactory 
 manner electricity, frictional and voltaic, thermo-electricity, and electro-physiology." 
 A thencEum. 
 
 " The commendations already bestowed in the pages of the Lancet on the former 
 editions of this work are more than ever merited by the present. The accounts given 
 of electricity and galvanism are not only complete in a scientific sense, but, which is a 
 rarer thing, are popular and interesting." Lancet. 
 
WORKS PUBLISHED BY LOCKWOOD & CO. 25 
 
 Text- Bo ok of Electricity. 
 
 THE STUDENT'S TEXT-BOOK OF ELECTRICITY: in- 
 
 eluding Magnetism, Voltaic Electricity, Electro-Magnetism, Dia- 
 magnetism, Magneto-Electricity, Thermo-Electricity, and Electric 
 Telegraphy. Being a Condensed Resume of the Theory and Ap- 
 plication of Electrical Science, including its latest Practical Deve- 
 lopments, particularly as relating to Aerial and Submarine Tele- 
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WORKS PUBLISHED BY LOCKWOOD & CO. 27 
 
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WORKS PUBLISHED BY LOCKWOOD & CO. 
 
 29 
 
 AGRICULTURE, &c. 
 
 Youatt and Burris Complete Grazier. 
 
 THE COMPLETE GRAZIER, and FARMER'S and CATTLE- 
 BREEDER'S ASSISTANT. A Compendium of Husbandry. 
 By WILLIAM YOUATT, ESQ., V.S. nth Edition, enlarged by 
 ROBERT SCOTT BURN, Author of "The Lessons of My Farm," &c. 
 One large 8vo volume, 784 pp. with 215 Illustrations. I/, is. half-bd. 
 
 On the Breeding, Rearing, Fattening, 
 and General Management of Neat Cattle. 
 
 Introductory View of the different Breeds 
 of Neat Cattle in Great Britain. Com- 
 parative View of the different Breeds of 
 Neat Cattle. General Observations on 
 Buying and Stocking a Farm with Cattle. 
 The Bull. The Cow. Treatment and 
 Rearing of Calves. Feeding of Calves for 
 Veal. Steers and Draught Oxen. Graz- 
 ing Cattle. Summer Soiling Cattle. 
 Winter Box and Stall-feeding Cattle. 
 Artificial Food for Cattle. Preparation 
 of Food. Sale of Cattle. 
 
 On the Economy and Management of 
 the Dairy. Milch Kine. Pasture and 
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 it regards their Milk. Situation and 
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 On the Breeding, Rearing, and Ma- 
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 and Comparative View of the different 
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 Asses and Mules. 
 
 On the Breedi?ig, Rearing, and Fat- 
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 Feeding and Fattening of Swine. Curing 
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 CONTENTS. 
 
 On the Diseases of Cattle. Diseases 
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 Diseases of Horses. Diseases of Sheep. 
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 the Eradication of 
 
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 Scott Burns Introduction to Farming. 
 
 THE LESSONS of MY FARM : a Book for Amateur Agricul- 
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WORKS PUBLISHED BY LOCKWOOD & CO. 31 
 
 Text-Book for Architects, Engineers, Surveyors, 
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 A GENERAL TEXT-BOOK for ARCHITECTS, ENGI- 
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 CONTENTS. 
 
 ARITHMETIC. 
 
 PLANE AND SOLID GEOMETRY. 
 
 MENSURATION. 
 
 TRIGONOMETRY. 
 
 CONIC SECTIONS. 
 
 LAND MEASURING. 
 
 LAND SURVEYING. 
 
 LEVELLING. 
 
 PLOTTING. 
 
 COMPUTATION OF AREAS. 
 
 COPYING MAPS. 
 
 RAILWAY SURVEYING. 
 
 COLONIAL SURVEYING. 
 
 HYDRAULICS IN CONNECTION 
 
 WITH DRAINAGE, SEWERAGE, 
 
 AND WATER SUPPLY. 
 
 TIMBER MEASURING. 
 
 ARTIFICERS' WORK. 
 
 VALUATION OF ESTATES. 
 
 VALUATION OF TILLAGE AND TENANT 
 RIGHT. 
 
 VALUATION OF PARISHES. 
 
 BUILDERS' PRICES. 
 
 DILAPIDATIONS AND NUISANCES. 
 
 THE LAW RELATING TO APPRAISERS AND 
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 LANDLORD AND TENANT. 
 
 TABLES OF NATURAL SINES AND CO- 
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 STAMP LAWS. 
 
 EXAMPLES OF VILLAS, &c. 
 
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 Chap. VII. Meadows and Embankments, Beds of Rivers, Water Courses, 
 
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 Chap. VIII. Land Draining, Opened and Covered : Plan, Execution, and 
 
 Arrangement between Landlord and Tenant. 
 IX. Minerals, Working, and Value. 
 X. Expenses of an Estate. 
 
 XL Valuation of Landed Property ; of the Soil, of Houses, of Woods, 
 of Minerals, of Manorial Rights, of Royalties, and of Fee 
 Farm Rents. 
 
 Chap. XII. Land Steward and Farm Bailiff: Qualifications and Duties. 
 Chap. XIII. Manor Bailiff, Woodrieve, Gardener, and Gamekeeper: their 
 
 Position and Duties. 
 
 Chap. XIV. Fixed Days of Audit : Half-yearly Payments of Rents, Form of 
 Notices, Receipts, and of Cash Books, General Map of Es- 
 tates, &c. 
 
 Chap. 
 Chap. 
 
 Chap. 
 Chap. 
 
 Chap. 
 Chap. 
 
 Chap. 
 Chap. 
 Chap. 
 
32 WORKS PUBLISHED BY LOCKWOOD & CO. 
 
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 LOAN 
 
 DEC 9 
 
Y