1

118

Strain pro

beam.

We are not fufficiently fenfible of our principles to be confident that the correction of fhould be in the proportion of the section, although we think it most probable. It is quite empirical, founded on Buffon's experiments. Therefore the safe way of using this rule is to fuppofe the beam square, by increasing or diminishing its breadth till equal to the depth. Then find the ftrength by this rule, and diminish or increase it for the change which has been made in its breadth. Thus, there can be no doubt that the ftrength of the beam given as an example is double of that of a beam of the same depth and half the breadth.

The reader cannot but obferve that all this calculation relates to the very greatest weight which a beam will bear for a very few minutes. Mr Buffon uniformly found that two-thirds of this weight fenfibly impaired its strength, and frequently broke it at the end of two or three months. One-half of this weiht brought the beam to a certain bend, which did not increase after the first minute or two, and may be borne by the beam for any length of time. But the beam contracted a bend, of which it did not recover any confiderable portion. One-third feemed to have no permanent effect on the beam ; but it recovered its rectilineal shape completely, even after having been loaded feveral months, provided that the timber was feafoned when first loaded; that is to say, one-third of the weight which would quickly break a feasoned beam, or one-fourth of what would break one just felled, may lie on it for ever without giving the beam a lett.

We have no detail of experiments on the ftrength of other kinds of timber: only Mr Buffon fays, that fir has about ths of the ftrength of oak; Mr Parent makes it ths; Emerfon, ds, &c.

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We have been thus minute in our examination of the mechanism of this tranfverfe ftrain, because it is the greatest to which the parts of our machines are expofed. We wish to imprefs on the minds of artists the neceffity of avoiding this much as poffible. They are improving in this refpect, as may be seen by comparing the centres on which ftone arches of great pan are now turned with thofe of former times. They were formerly a load of mere joilts refting on a multitude of posts, which obstructed the navigation, and were frequently lofing their fhape by fome of the pofts finking into the ground. Now they are more generally truffes, where the beams abutt on each other, and are relieved from tranfverfe trains. But many performances of eminent artists are still very injudiciously expofed to cross ftrains. We may in tance one which is confidered as a fine work, viz. the bridge at Walton on Thames. Here every beam of the great arch is a joilt, and it hangs together by framing. The finett piece of carpentry that we have feen is the centre employed in turning the arches of the bridge at Orleans, defcribed by Perronet. In the whole there is not one crofs ftrain. The beam, too, of Hornblower's steam-engine, described in that article, is very scientifically conftructed.

IV. The last species of strain which we are to examine is duced by that produced by twifting. This takes place in all axles wifting. which connect the working parts of machines.

Although we cannot pretend to have a very diftinct conception of that modification of the cohesion or a body by which it relifts this kind of ftrain, we can have no doubt that, when all the particles act alike, the refiitance mult be VOL. XVIII. Part I.

proportional to the number. Therefore if we fuppofe the Strength of two parts ABCD, ABFE (fig. 24.), of the body EFCD Materials, to be of infuperable ftength, but cohering more weakly in 119 the common furface AB, and that one part ABCD is pufh-The refift ed laterally in the direction AB, there can be no doubt that a ce must it will yield only there, and that the refiftance will be pro-be propo portional to the surface.

tional to

the number

In like manner, we can conceive a thin cylindrical tube, of particles. of which KAH (fig. 25.) is the section, as cohering more weakly in that fection than anywhere elfe. Suppose it to be. grafped in both hands, and the two parts twisted round the axis in oppofite directions, as we would twift the two joints of a flute, it is plain that it will first fail in this section, which is the circumference of a circle, and the particles of the two parts which are contiguous to this circumference will be drawn from each other laterally. The total refistance will be as the number of equally refifting particles, that is, as the circumference (for the tube being supposed very thin, there can be no fenfible difference between the dilatation of the external and internal particles). We can now suppose another tube within this, and a third within the fecond, and fo on till we reach the centre. If the particles of each ring exerted the fame force (by suffering the fame dilatation in the direction of the circumference), the refiftance of each ring of the section would be as its circumference and its breadth (fuppofed indefinitely fmall), and the whole refiftance would be as the furface; and this would reprefent the refiftance of a folid cylinder. But when a cylinder is twisted in this manner by an external force applied to its circumference, the external parts will fuffer a greater circular extenfion than the internal; and it appears that this extenfion (like the extenfion of a beam ftrained tranfverfely) will be proportional to the distance of the ticles from the axis. We cannot fay that this is demonftrable, but we can affign no proportion that is more probable. This being the cafe, the forces fimultaneously exerted by each particle will be as its diftance from the axis. Therefore the whole force exerted by each ring will be as the fquare of its radius, and the accumulated force actually exerted will be as the cube of the radius; that is, the accumulated force exerted by the whole cylinder, whose radius is CA, is to the accumulated force exerted at the fame time by the part whofe radius is CE, as CA3 to CE3.

par

The whole cohefion now exerted is just two-thirds of what it would be if all the particles were exerting the same attractive forces which are just now exerted by the particles in the external circumference. This is plain to any perfon in the least familiar with the fluxionary calculus. But fuch as are not may easily fee it in this way.

Let the rectangle AC ca be fet upright on the surface of the circle along the line CA, and revolve round the axis Cc. It will generate a cylinder whofe height is Ce or A a, and having the circle KAH for its bafe. If the diagonal C a be supposed also to revolve, it is plain that the triangle e Ca will generate a cone of the fame height, and having for its bafe the circle defcribed by the revolution of co, and the point C for its apex. The cylindrical furface generated by Aa will exprefs the whole cohefion exerted by the circumference AHK, and the cylindrical surface nerated by Ee will reprefent the cohesion exerted by the circumference ELM, and the folid generated by the triangle CA a will reprefent the cohesion exerted by the whole circle AHK, and the cylinder generated by the rectangle ACca will reprefent the cohefion exerted by the fame furface if each particle had fuffered the extenfion A a.

ge

Now it is plain, in the first place, that the folid generated by the triangle e EC is to that generated by a AC as EC to AC 3. In the next place, the folid generated by E вас

a Strength of a AC is two-thirds of the cylinder, because the cone geneMaterials. rated by c Ca is one-third of it. We may now suppose the cylinder twisted till the parti

120

cles in the external circumference lofe their cohefion. There can be no doubt that it will now be wrenched asunder, all the inner circles yielding in fucceffion. Thus we obtain With what one useful information, viz. that a body of homogeneous force a bo- texture refifts a fimple twist with two-thirds of the force with dy of a ho- which it refifts an attempt to force one part laterally from the mogeneous other, or with one-third part of the force which will cut it fits a fim- afunder by a fquare-edged tool. For to drive a fquareple twift. edged tool through a piece of lead, for inftance, is the fame as forcing a piece of the lead as thick as the tool laterally away from the two pieces on each fide of the tool. Experiments of this kind do not feem difficult, and they would give us very useful information.

texture re

121 The forces

ameters.

When two cylinders AHK and BNO are wrenched aexerted in funder, we must conclude that the external particles of each breaking are just put beyond their limit of cohesion, are equally extwo cylinder: are a tended, and are exerting equal forces. Hence it follows, Hence it follows, the fquare that in the inftant of fracture the fum total of the forces acof the di- tually exerted are as the fquares of the diameters. For drawing the diagonal C e, it is plain that E e, = Aa, expreffes the diftenfion of the circumference ELM, and that the folid generated by the triangle CE e expreffes the cohefion exerted by the furface of the circle ELM, when the particles in the circumference suffer the extenfion E e equal to A a. Now the folids generated by CA a and CE e being refpectively two-thirds of the correfponding cylinders, are as the fquares of the diameters.

122 Relative trength of the fection to the ex

Having thus afcertained the real ftrength of the section, and its relation to its abfolute lateral ftrength, let us exa mine its strength relative to the external force employed to ternal force break it. This examination is very fimple in the cafe unemployed der eonfideration. The ftraining force must act by fome lever, and the cohesion muft oppofe it by acting on fome other lever. The centre of the fection may be the neutral point, whofe pofition is not disturbed.

to break it.

123

The refift

the cube of ite diame

Let F be the force exerted laterally by an exterior particle. Let a be the radius of the cylinder, and x the inde. terminate distance of any circumference, and x the indefinitely fmall interval between the concentric arches; that is, The forlet x be the breadth of a ring and x its radius. ces being as the extenfions, and the extenfions as the diftances from the axis, the cohelion actually exerted at any part of any ring will be fx. The force exerted by the

3

I

2

The

a

that is, when xa, it will be ƒ f = = ƒ a3 Hence we learn that the ftrength of an axle, by which it ance of the refrits being wrenched afunder by a force acting at a given axle is as distance from the axis, is as the cube of its diameter. But farther, ƒ a 3 is = ƒ a2 × † a. f Now faz reprefents the full lateral cohesion of the fection. The momen. The momen. tum therefore is the fame as if the full lateral cohesion were accumulated at a point diftant from the axis by 4th of the radius or 4th of the diameter of the cylinder. Therefore let F be the number of pounds which measures

ter.

I

We fee allo that the internal parts are not acting so powerfully as the external. If a hole be bored out of the axle of half its diameter, the ftrength is diminished only 4th, while the quantity of matter is diminished th. Therefore Hollow hollow axles are ftronger than folid ones containing the axles more fame quantity of matter. Thus let the diameter be 5 and Proper than that of the hollow 4: then the diameter of another folid cylinder having the fame quantity of matter with the tube is 3. The ftrength of the folid cylinder of the diameter 5 may be expreffed by 53 or 125. Of this the internal part (of the diameter 4) exerts 64; therefore the ftrength of the tube is 125-64, 61. But the ftrength of the folid axle of the fame quantity of matter and diameter 3 is 33, or 27, which is not half of that of the tube.

ufed.

125

Engineers, therefore, have of late introduced this im And now provement in their machines, and the axles of cast iron are generally all made hollow when their fize will admit it. They have the additional advantage of being much stiffer, and of affording much better fixure for the flanches, which are used for connecting them with the wheels or levers by which they are turned and ftrained. The fuperiority of strength of hollow tubes over folid cylinders is much greater in this kind of frain than in the former or tranfverfe. laft cafe the ftrengh of this tube would be to that of the folid cylinder of equal weight as 61 to 32 nearly.

In this

126

The apparatus which we mentioned on a former occafion for trying the lateral ftrength of a square inch of folid matter, enabled us to try this theory of twist with all defirable accuracy. The bar which hung down from the pin in the former trials was now placed in a horizontal position, and loaded with a weight at the extremity. Thus it acted as a power-The ratio ful lever, and enabled us to wrench asunder specimens of the of refiftftrongest materials. ftrongest materials. We found the refults perfectly con-ance to twisting portional ftrength of different fizes and forms: but we formable to the theory, in as far as it determined the pro- to the fimfound the ratio of the refiftance to twifting to the fimpk refistance lateral resistance confiderably different; and it was fome appears time before we difcovered the caufe.

pie lateral

different

We had here taken the fimpleft view that is poffible of the action of cohesion in refifting a twitt. It is frequently exerted in a very different way. When, for instance, an iron axle is joined to a wooden one by being driven into one end of it, the extenfions of the different circles of particles are in a very different proportion. A little confideration will fhow that the particles in immedate contact with the iron axle are in a ftate of violent extenfion; fo are the particles of the exterior surface of the wooden part, and the intermediate parts are lefs ftrained. It is almost impoffible to affign the exact proportion of the cohesive forces exerted in the different parts. Numberless cafes can be pointed out where parts of the axle are in a state of compreffion, and where it is ftill more difficult to determine the ftate of the other particles. We must content ourfelves But when with the deductions made from this fimple cafe, which is the experifortunately the most common. In the experiments just now mentioned the centre of the circle is by no means the neu- was exact. tral point, and it is very difficult to afcertain its place: but ly the when this confideration occurred to us, we eafily freed the ex- .fame. periments from this uncertainty, by extending the lever to both fides, and by means of a pulley applied equal force

127

ment was

altere 1, it