Strength of ved; and a little reflection will fhow it to be impoffible, in Materials, confiftency with the equality of action and reaction. 35 A circum every conftruction re quiring treng h. Since all parts are thus equally stretched, it follows, that the train in any tranfverfe fection is the fame, as alfo in every point of that section. If therefore the body be fuppofed of a homogeneous texture, the cohesion of the parts is equable; and fnce every part is equally ftretched, the particles, are drawn to equal diftances from their quiefcent politions, and the forces which are thus excited, and now exerted in oppofition to the ftraining force, are equal. This external force may be increafed by degrees, which will gradually feparate the part of the body more and more from each other, and the connecting forces increase with this increase of distance, till at laft the cohesion of some particles is overcome. This must be immediately followed by a rupture, because the remaining forces are now weaker than before. It is the united force of cohesion, immediately before the disunion of the firit particles, that we call the STRENGTH of the fection. It may also be properly called its ABSOLUTE STRENGTH, being exerted in the fimplest form, and not modified by any relation to other circumstances. If the external force has not produced any permanent fance to be change on the body, and it therefore recovers its former dia tended to menfions when the force is withdrawn, it is plain that this frain may be repeated as often as we please, and the body which withstands it once will always withstand it. It is evident that this should be attended to in all conftructions, and that in all our investigations on this fubject this fhould be kept ftrictly in view. When we treat a piece of soft clay in this manner, and with this precaution, the force employed must be very small. If we exceed this, we produce a permanent change. The rod of clay is not indeed torn alunder; but it has become fomewhat more flender: the number of particles in a crofs fection is now fmaller; and therefore, although it will again, in this new form, fuffer, or allow an endless repetition of a certain ftrain without any farther permanent change, this ftrain is fmaller than the former. Something of the fame kind happens in all bodies which receive a SETT by the ftrain to which they are expofed. All ductile bodies are of this kind. But there are many bodies which are not ductile. Such bodies break completely whenever they are ftretched beyond the limit of their perfect elafticity. Bodies of a fibrous ftructure exhibit very great varieties in their cohesion. In some the fibres have no lateral cohesion, as in the cafe of a rope. The only way in which all the fibres can be made to unite their strength is, to twist them together. This causes them to bind each other fo faft, that any one of them will break before it can be drawn out of the bundle. In other fibrous bodies, fuch as timber, the fibres are held together by fome cement or gluten. This is feldom as ftrong as the fibre. Accordingly timber is much eafier pulled afunder in a direction tranfverfe to the fibres. There is, however, every poffible variety in this particular. 36 Great varieties in cohefiun, Dut fuddenly, but give warning by complaining, as the carpenters Strength of call it; that is, by giving vifible figns of a derangement of Materials. texture. Hard bodies of an uniform glaffy ftructure, or granulated like ftones, are elaftic through the whole extent of their cohesion, and take no fett, but break at once when overloaded. Notwithstanding the immenfe variety which nature exhibits in the ftructure and cohesion of bodies, there are certain general facts of which we may now avail ourfelves with advantage. In particular, 37 The abfolute cohesion is proportional to the area of The abfothe fection. This mult be the cafe where the texture is lute cobefion or perfectly uniform, as we have reason to think it is in glafs and the ductile metals. The cohefion of each particle proportionftrength being alike, the whole cohefion must be proportional al to the to their number, that is, to the area of the fection. The area of the felion per fame must be admitted with refpect to bodics of a granula- pendicular ted texture, where the granulation is regular and uniform. ro the exThe fame muit be admitted of fibrous bodies, if we fuppofe tensing their fibres equally ftrong, equally denfe, and fimilarly dif. force. pofed through the whole fection; and this we muff either suppose, or must state the diversity, and measure the cohefion accordingly. We may therefore affert, as a general propofition on this fubject, that the abfolute ftrength in any part of a body by which it refifts being pulled afunder, or the force which must be employed to tear it asunder in that part, is proportional to the area of the fection perpendicular to the extending force. Therefore all cylindrical or prifmatical rods are equally ftrong in every part, and will break alike in any part; and bodies which have unequal sections will always break in the flendereft part. The length of the cylinder or prifm has no effect on the firength; and the vulgar notion, that it is eafier to break a very long rope than a fhort one, is a very great mistake. Alfo the abfolute ftrengths of bodies which have fimilar sections are proportional to the fquares of their diameters or homologous fides of the section. The weight of the body itself may be employed to strain it and to break it. It is evident, that a rope may be fo long as to break by its own weight. When the rope is hanging perpendicularly, although it is equally ftrong in every part, it will break towards the upper end, because the train on any part is the weight of all that is below it. Its 1 RELATIVE STRENGTH in any part, or power of withstand-strength. ing the ftrain which is actually laid on it, is inversely as the quantity below that part. When the rope is ftretched horizontally, as in towing a fhip, the ftrain arifing from its weight often bears a very fenfible proportion to its whole strength. Let AEВ (fig. 3.) be any portion of fuch a rope, and AC, BC be tangents to the curve into which its gravity bends it. Complete the parallelogram ACBD. It is well known that the curve is a catenaria, and that DC is perpendicular to the horizon; and that DC is to AC as the weight of the rope AEB to the ftrain at A. In order that a fufpended heavy body may be equally able in every part to carry its own weight, the fection in that part must be proportional to the folid contents of all that is below it. Suppose it a conoidal spindle, formed by the revolution of the curve A ae (fig. 4.) round the axis CE. AC2; We must have AC2: ac2= AEB fol.: a Eb fol. This condition requires the logarithmic curve for A a e, of which Cc is the axis. In ftretching and breaking fibrous bodies, the vifible extension is frequently very confiderable. This is not solely the increafing of the distance of the particles of the cohering fibre: the greatest part chiefly arises from drawing the crooked fibre ftraight. In this, too, there is great diverfity; and it is accompanied with important differences in their power of withstanding a ftrain. In fome woods, fuch as fir, the fibres on which the ftrength moft depends are very Rraight. Such woods are commonly very elaftic, do not take a fett, and break abruptly when overftrained: others, fuch as oak and birch, have their refifting tibres very undulating and cooked, and ftretch very fenfibly by a ftrain. They are very liable to take a fet, and they do not break fo VOL. XVIII. Part I. These are the chief general rules which can be fafely deduced from our clearest notions of the cohefion of bodies. In order to make any practical use of them, it is proper to have fome measures of the cohefion of fuch bodies as are B com 38 ive 39 circumftan cea, Strength of commonly employed in our mechanics, and other structures Materials where they are expofed to this kind of ftrain. Thefe must be deduced folely from experiment. Therefore they The cohe- mult be confidered as no more than general values, or as fion of me- the averages of many particular trials. The irregularities tals depends are very great, becaufe none of the fubftances are conftant on various in their texture and firmnels. Metals differ by a thousand circumstances unknown to us, according to their purity, to the heat with which they were melted, to the moulds in which they were caft, and the treatment they have afterwards received, by forging, wire-drawing, tempering, &c. It is a very curious and inexplicable fact, that by forging a metal, or by frequently drawing it through a fmooth hole in a fteel plate, its cohefion is greatly increased. This operation undoubtedly deranges the natural fituation of the particles. They are fqueezed clofer together in one direction; but it is not in the direction in which they refift the fracture. In this direction they are rather separated to a greater diftance. The general denfity, however, is augmented in all of them except lead, which grows rather rarer by wire-drawing but its cohelion may be more than tripled by this operation. Gold, filver, and brass, have their cohefion nearly tripled; copper and iron have it more than doubled. In this operation they alfo grow much harder. It is proper to heat them to rednefs after drawing a little. This is called nealing or annealing. It foftens the metal again, and renders it fufceptible of another drawing without the rifk of cracking in the operation. 40 Cohetion and : We do not pretend to give any explanation of this remarkable and very important fact, which has fomething refembling it in woods and other fibrous bodies, as will be mentioned afterwards. The varieties in the cohesion of ftones and other minerals, and of vegetable and animal fubftances, are hardly fufceptible of any defcription or claffification. We fhall take for the measure of cohefion the number of pounds avoirdupois which are just sufficient to tear afunder a rod or bundle of one inch square. From this it will be of different easy to compute the ftrength correfponding to any other trength metals. 3,800 English block grain 5,200 6,500 1 Two parts of gold with one of filver Five parts of Japan copper with one of Banca tin Six parts of Chili copper with one of Malacca tin Six parts of Swedish copper with one of Malac ca tin Brafs confifts of copper and zinc in an un- Eight parts of lead with one of zinc zinc 28,000 These numbers are of confiderable use in the arts. mixtures of copper and tin are particularly interefting in the fabric of great guns. We fee that, by mixing copper whofe greateft ftrength does not exceed 37,000 with tin which does not exceed 6,000, we produce a metal whofe tenacity is almost double, at the fame time that it is harder and more easily wrought. It is, however, more fufible, which is a great inconvenience. We also fee that a very fmall addition of zinc almoft doubles the tenacity of tin, and increases the tenacity of lead five times; and a small addition of lead doubles the tenacity of tin. These are economical mixtures. This is a very valuable information to the plumbers for augmenting the frength of waterpipes. 1 By having recourfe to these tables, the engineer can proportion the thickness of his pipes (of whatever metal) to the preffures to which they are expofed. 2d, WOODS. We may premife to this part of the table the following general obfervations : 42 wood. 1. The wood immediately furrounding the pith or heart Tenacity or of the tree is the weakeft, and its inferiority is fo much strength of. more remarkable as the tree is older. In this affertion, however, we speak with fome hefitation. Mufchenbroek's detail of experiments is decidedly in the affirmative. Mr Buffon, on the other hand, fays, that his experience has taught him that the heart of a found tree is the ftrongest; but he gives no inftances. We are certain, from many obfervations ダン (A) This was an experiment by Muschenbroek, to examine the vulgar notion that iron forged from old horfe-nails was ftronger than all others, and fhows its falfity. Strength of fervations of our own on very large caks and firs, that the 43 Abfolue Strength of different 2. The wood next the bark, commonly called the white or blea, is also weaker than the reft; and the wood gradually increases in ftrength as we recede from the centre to the blea. 3. The wood is ftronger in the middle of the trunk than at the springing of the branches or at the root; and the wood of the branches is weaker than that of the trunk. 4. The wood of the north fide of all trees which grow ner on that fide. In conformity with this, it is a general 5. All woods are more tenacious while green, and lose The only author who has put it in our power to judge of the propriety of his experiments is Mufchenbrock. He has defcribed his method of trial minutely, and it feems unexceptionable. The woods were all formed into flips fit for his apparatus, and part of the flip was cut away to a parallelopiped of 4th of an inch square, and therefore 2th of a fquare inch in fection. The abfolute ftrengths of a fquare inch were as follow: lib. kinds of Beech, oak 17,300 lib. The reader will furely obferve, than thefe numbers.cx-Strength of press fomething more than the utmost cohefion; for the Materials, weights are fuch as will very quickly, that is, in a minute 45 in architec. or two, tear the rods afunder. It may be faid in general, No fab- According to Mr Emerfon, the load which may be fafely 44 And of other fub. ftances. Elm Mulberry I'lum Elder I 2,000 11,800 10,000 Fir 9,250 5,500 4,880 Mr Mufchenbrock has given a very minute detail of the Mr Muschenbrock has given a very minute detail of the experiments on the afh and the walnut, ftating the weights which were required to tear afunder flips taken from the four fides of the tree, and on each fide in a regular progreffion from the centre to the circumference. The numbers of this table correfponding to these two timbers may therefore be confi dered as the average of more than 50 trials made of each; and he fays that all the others were made with the fame care. We cannot therefore fee any reafon for not confiding in the refults; yet they are confiderably higher than those given by fome other writers. Mr Pitot fays, on the authority of his own experiments, and of thofe of Mr Pa rent, that 60 pounds will just tear afunder a fquare line of found oak, and that it will bear 50 with fafety. This gives 8640 for the utmoft ftrength of a square inch, which is much inferior to Muschenbroek's valuation. We may add to these, Ivory Bone Horn Whalebone Tooth of fea-calf II. BODIES MAY BE CRUSHED. know what will crush bodies. 46 It is therefore most defirable to have fome general know- tact, and difpofed in lines which are in the direction of the Materials, Strength of tervals, being in fituations that are oblique with refpect to the preffures, it must follow, that by fqueezing them together in one direction, they are made to bulge out or feparate in other directions. This may proceed fo far that some may be thus pushed laterally beyond their limits of cohefion. The moment that this happens the refiftance to compreffion is diminished, and the body will now be crushed together. We may form some notion of this by fuppofing a number of fpherules, like fmall fhot, fticking together by means of a cement. Compreffing this in fome particular direction causes the fpherules to act among each other like fo many wedges, each tending to penetrate through between the three which lie below it: and this is the fimpleft, and perhaps the only distinct, notion we can have of the matter. We have reason to think that the conflitution of very homogeneous bodies, such as glass, is not very different from this. The particles are certainly arranged fymmetrically in the angles of fome regular folids. It is only fuch an arrangement that is confiftent with transparency, and with the free paffage of light in every direction. 47 Their ftrength or power of refift ance to fuch a force If this be the conftitution of bodies, it appears probable that the ftrength, or the refiftance which they are capable of making to an attempt to crush them to pieces, is proportional to the area of the section whofe plane is perpendicular to the external force; for each particle being fimilarly and equally acted on and refifted, the whole refiftance must be as their number; that is, as the extent of the fection. Accordingly this principle is affumed by the few writers who have confidered this fubject; but we confefs that it appears to us very doubtful. Suppofe a number of brittle or friable balls lying on a table uniformly arranged, but not cohéring nor in contact, and that a board is laid over them and loaded with a weight; we have no hesitation in faying, that the weight necessary to crush the whole collection is proportional to their number or to the area of the fection. But when they are in contact (and still more if they cohere), we imagine that the cafe is materially altered. Any individual ball is crufhed only in confequence of its being bulged outwards in the direction perpendicular to the preffure employed. If this could be prevented by a hoop put round the ball like an equator, we cannot fee how any force can cruth it. Any thing therefore which makes this bulging outwards more difficult, makes a greater force neceffa. ry. Now this effect will be produced by the mere contact of the balls before the preffure is applied; for the central ball cannot fwell outward laterally without pushing away the balls on all fides of it. This is prevented by the friction on the table and upper board, which is at leaft equal to one third of the preffure. Thus any interior ball becomes ftronger by the mere vicinity of the others; and if we farther fuppofe them to cohere laterally, we think that ite ftrength will be still more increased. The analogy between these balls and the cohering particles of a friable body is very perfect. We should therefore expect that the ftrength by which it refifts being crushed will increase in a greater ratio than that of the fection, or the fquare of the diameter of fimilar fections; and that a fquare inch of any matter will hear a greater weight in proportion as it makes a part of a greater fection. Accordingly this appears in many experiments, as will be noticed afterwards. Mufchenbroek, Euler, and fome others, have fuppofed the ftrength of columns to be as the biquadrates of their diameters. But Euler deduced this from formula which occurred to him in the course of his algebraic analyfis; and he boldly adopts it as a principle, without looking for its foundation in the phyfical affumptions which he had made in the beginning of his inveftigation. But fome of his original affumptions were as paradoxical, or at Strength of leaft as gratuitous, as these results: and thofe, in parti- Materials.. cular, from which this proportion of the ftrength of columns was deduced, were almost foreign to the cafe; and therefore the inference was of no value. Yet it was recei ved as a principle by Mufchenbroek and by the academicians of St Petersburgh. We make these very few obfervations, because the subject is of great practical importance; and it is a great obftacle to improvements when deference to a great name, joined to incapacity or indolence, caufes authors to adopt his careless reveries as principles from which they are afterwards to draw. important confequences. It mult. be acknowledged that we have not as yet eftablished the relation between the dimenfions and the strength of a pillar on. folid mechanical principles. Experience plainly contradicts the general opinion, that the ftrength is proportional to the area of the fection; but it is ftill more inconsistent with the opinion, that it is in the quadruplicate ratio of the diameters of fimilar fections. It would feem that the ratio depends much on the internal ftructure of the body; and ex- ont by experiment feems the only method for ascertaining its general perment. laws. If we suppose the body to be of a fibrous texture, having the fibres fituated in the direction of the preffure, and flightly adhering to each other by fome kind of cement, fuch a body will fail only by the bending of the fibres, by which they will. break the cement and be detached from each other. Something like this may be fuppofed in wooden pillars. In fuch cafes, too, it would appear that the refiftance must be as the number of equally refifting fibres, and as their mutual fupport, jointly; and, therefore, as fome function of the area of the fection. The fame thing muft happen if the fibres are naturally crooked or undulated, as is obferved in many woods, &c. provided we fuppofe fome fimilarity in their form. Similarity of fome kind muft always be fuppofed, otherwise we need never aim at any general inferences. In all cafes therefore we can hardly refuse admitting that the ftrength in oppofition to compreffion is proportional to a function of the area of the fection. As the whole length of a cylinder or prifm is equally preffed, it does not appear that the ftrength of a pillar is at all affected by its length. If indeed it be fupposed to bend under the preffure, the cafe is greatly changed, because it is then exposed to a tranfverfe ftrain; and this increases with the length of the pillar. But this will be considered with due attention under the next clafs of strains. way, Few experiments have been made on this fpecies of ftrength and ftrain. Mr Petit fays, that his experiments, and those of Mr Parent, fhow that the force neceflary for crushing a body is nearly equal to that which will tear it afunder. He fays that it requires fomething more than 60 pounds on every fquare line to cruth a piece of found oak. But the rule is by no means general: Glass, for inftance,. will carry a hundred times as much as oak in this that is, refting on it; but will not fufpend above four or five times as much. Oak will suspend a great deal more than fir; but fir will carry twice as much as a pillar. Woods of a foft texture, although confifting of very tenacious fibres, are more easily crushed by their load. This foftness of texture is chiefly owing to their fibres not being straight but undulated, and there being confiderable vacuities between them, fo that they are easily bent laterally and crushed.. When a poft is overstrained by its load, it is obferved to. Twell fenfibly in diameter. Increafing the load caufes longitudinal cracks or fhivers to appear, and it presently after gives way. This is called crippling. In all cafes where the fibres lie oblique to the ftrain the ftrength is greatly diminished, because the parts can then be made 48 To be af certained Materials.. other, that little use can be made of them. The fubject is Strength of of great importance, and well deferves the attention of the patriotic philofopher. Streng h of made to slide on each other, when the cohesion of the ceMate als menting matter is overcome. Mufchenbrock has given fome experiments on this fubject; but they are cases of long pillars, and therefore do not belong to this place. They will be confidered afterwards. The only experiments of which we have feen any detail (and it is ufelefs to infert mere affertions) are thofe of Mr Gauthey, in the 4th volume of Rozier's Journal de Physique. This engineer expofed to great preffures fmall rectangular parallelopiper's, cut from a great variety of ftones, and noted the weights which crushed them. The following table exhibits the medium refults of many trials on two very uni form kinds of freestone, one of them among the hardest and the other among the fofteft ufed in building. 49 Expe imen's for this burp fe made on free ftone 30 Not fatisfactory. Column ift expreffes the length AB of the fection in French lines or 12ths of an inch; column zd expreffes the breadth BC; column 3d is the area of the fection in fquare lines; column 4th is the number of ounces required to crush the piece; column 5th is the weight which was then borne by each fquare line of the section; and column 6th is the round numbers to which Mr Gauthey imagines that thofe in column 5th approximate. 5296 12,2 +567 4 4,5 12 18 24 432 Little can be deduced from these experiments: The 1ft and 3d, compared with the 5th and 6th, fhould furnish fimilar refults; for the 1ft and 5th are refpectively half of the 3d and 6th: but the 3d is three times ftronger (that is, a line of the 3d) than the firft, whereas the 6th is only twice as Arong as the 5th. It is evident, however, that the ftrength increases much faster than the area of the section, and that a square line can carry more and more weight, according as it makes a part of a larger and larger fection. In the series of experiments on the foft ftone, the individual strength of a fquare line feems to increase nearly in the proportion of the section of which it makes a part. Mr Gauthey deduces, from the whole of his numerous experiments, that a pillar of hard tone of Givry, whofe fection is a fquare foot, will bear with perfect safety 664,000 pounds, and that its extreme ftrength is 871,000, and the fmalleft ftrength observed in any of his experiments was 460,000. The foft bed of Givry ftone had for its smallest ftrength 187,000, for its greatest 311,000, and for its fafe load 249,000. Good brick will carry with fafety 320,000; chalk will carry only 9000. The boldeft piece of architecture in this refpect which he has feen is a pillar in the church of All-Saints at Angers. It is 24 feet long and 11 inches fquare, and is loaded with 60,000, which is not 4th of what is neceffary for crushing it. We may obferve here by the way, that Mr Gauthey's measure of the fufpending ftrength of flone is vaftly fmall in proportion to its power of fupporting a load laid above it. He finds that a prifm of the hard bed of Givry, of a foot fection, is torn afunder by 46c0 pounds; and if it be firmly fixed horizontally in a wall, it will be broken by a weight of 56,000 fufpended a foot from the wall. If it reft on two props at a foot diftance, it will be broken by 206,oco laid on its middle. Thefe experiments agree fo ill with each Auch ·wanted. A fet of good experiments would be very valuable, be- Good ex cause it is against this kind of ftrain that we muft guard by periments judicious conftruction in the most delicate and difficult problems which come through the hands of the civil and military engineer. The construction of stone arches, and the conftruction of great wooden bridges, and particularly the conftruction of the frames of carpentry called centres in the erection of stone bridges, are the most difficult jobs that occur. In the centres on which the arches of the bridge of Orleans were built fome of the pieces of oak were carrying upwards of two tons on every fquare inch of their scantling. All who faw it faid that it was not able to carry the fourth part of the intended load. But the engineer understood the principles of his art, and ran the risk and the refult completely juftified his confidence; for the centre did not complain in any part, only it was found too fupple; so that it went out of shape while the haunches only of the arch were laid on it. The engineer corrected this by loading it at the crown, and thus kept it completely in shape during the progrefs of the work. In the Memoirs (old) of the Academy of Petersburgh for 17-8, there is a differtation by Euler on this subject, but particularly limited to the ftrain on columns, in which the bending is taken into the account. Mr Fufs has treated the fame fubject with relation to carpentry in a subsequent volume. But there is little in these papers befides a dry mathematical difquifition, proceeding on affumptions which (to fpeak favourably) are extremely gratuitous. The most important confequence of the compreffion is wholly overlooked, as we shall prefently fee. Our knowledge of the mechanism of cohesion is as yet far too imperfect to entitle us to a confident application of mathematics. Experiments. fhould be multiplied. 52 made ule The only way we can hope to make thefe experiments How they ufeful is to pay a careful attention to the manner in which are to be the fracture is produced. By difcovering the general re-ful. femblances in this particular, we advance a step in our power of introducing mathematical measurement. Thus, when a cubical piece of chalk is flowly crushed between the chaps of a vice, we see it uniformly split in a surface oblique to the preffure, and the two parts then flide along the furface of fracture. This thould lead us to examine mathematically what relation there is between this furface of fracture and the neceffary force; then we should endeavour to determine experimentally the pofition of this furface. Having difcovered fome general law or resemblance in this circumitance, we fhould try what mathematical hypothefis will agree with this. Having found one, we may then apply our fimpleft notions of cohesion, and compare the refult of our computations with experiment. We are authorised to say, that a feries. of experiments have been made in this way, and that their refults have been very uniform, and therefore fatisfactory, and. that they will foon be laid before the public as the founda tians of fuccessful practice in the conftruction of arches. III. A BODY MAY BE BROKEN ACROSS. 53 to know The most ufual, and the greatest ftrain, to which mate- It is of im. rials are expofed, is that which tends to break them trani- Fortance verfely. It is feldom, however, that this is done in a man- what ftrain ner perfectly fimple; for when a beam projects horizontally will break from a wall, and a weight is fufpended from its extremity, a body the beam is commonly broken near the wall, and the inter- tranfverfe mediate part has performed the functions of a lever. It fometimes, though rarely, happens that the pin in the joint of a pair of pincers or fciffars is cut through by the ftrain ; ly. |