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Strength of train; and this is almost the only cafe of a fimple tranfverfe
Materials. fracture. Being fo rare, we may content ourfelves with
faying, that in this cafe the ftrength of the picce is propor-

54 Experi

ments

mad to afcertain it.

55

Their re

ftrength

of a lever

tional to the area of the section.

Experiments were made for difcovering the rcfiftances made by bodies to this kind of frain in the following manner: Two iron bars were difpofed horizontally at an inch diftance; a third hung perpendicularly between them, being fupported by a pin made of the fubftance to be examined. This pin was made of a prifmatic form, so as to fit exactly the holes in the three bars, which were made very exact, and of the fame fize and shape. A scale was fufpended at the lower end of the perpendicular bar, and loaded till it tore out that part of the pin which filled the middle hole. This weight was evi dently the measure of the lateral cohefion of two fections. The fide bars were made to grasp the middle bar pretty ftrongly between them, that there might be no distance impofed between the oppofite preffures. This would have combined the energy of a lever with the purely tranfverfe preffure. For the fame reason it was neceffary that the internal parts of the holes fhould be no smaller than the edges. Great irregularities occurred in our firft experiments from this cause, because the pins were fomewhat tighter within than at the edges; but when this was corrected they were extremely regular. We employed three fets of holes, viz. a circle, a fquare (which was occafionally made a rectangle whofe length was twice its breadth), and an equilateral triangle. We found in all our experiments the ftrength exactly proportional to the area of the section, and quite independent of its figure or pofition, and we found it confiderably above the direct cohefion; that is, it took confiderably more than twice the force to tear out this middle piece than to tear the pin afunder by a direct pull. A piece of fine freeftone required 205 pounds to pull it directly alunder, and 575 to break it in this way. The difference was very conftant in any one fubftance, but varied from ds to 4ds in different kinds of matter, being fmalleft in bodies of a fibrous texture. But indeed we could not make the trial on any bodies of confiderable cohesion, because they required fuch forces as our apparatus could not fupport. Chalk, clay baked in the fun, baked fugar, brick, and freeftone, were the strongest that we could examine.

But the more common cafe, where the energy of a lever intervenes, demands a minute examination.

Let DABC (fig. 5.n° 1.) be a vertical section of a prifmatic folid (that is, of equal fize throughout), projecting horizontally from a wall in which it is firmly fixed; and let a weight P be hung on it at B, or let any power P act at B in a direction perpendicular to AB. Suppofe the body of infuperable ftrength in every part except in the vertical fection DA, perpendicular to its length. It muft break in this fection only. Let the cohefion be uniform over the whole of this fection; that is, let each of the adjoining particles of the two parts cohere with an equal force f.

There are two ways in which it may break. The part ABCD may fimply flide down along the furface of fracture, provided that the power acting at B is equal to the accumulated force which is exerted by every particle of the fection in the direction AD.

But fuppofe this effectually prevented by fomething that fupports the point A. The action at P tends to make the body turn round A (or round a horizontal line paffing thro' A at right angles to AB) as round a joint. This it cannot do without feparating at the line DA. In this cafe the adjoining particles at D or at E will be feparated horizontally. But their cohefion refifts this feparation. In order, therefore, that the fracture may happen, the en

erry or momentum of the power P, acting by means of the Strength of
lever AB, must be fuperior to the accumulated energies of. Materials,
the particles. The energy of each depends not only on its
cohesive force, but alfo on its fituation; for the supposed
infuperable firmness of the rest of the body makes it a lever
turning round the fulcrum A, and the cohesion of each par-
ticle, fuch as D or E, acts by means of the arm DA or
EA. The energy
EA. The energy of each particle will therefore be had by
multiplying the force exerted by it in the inftant of fracture
by the arm of the lever by which it acts.

Let us therefore firft fuppofe, that in the inftant of frac-
ture every particle is exerting an equal force f. The ener
gy of D will be fXDA, and that of E will be ƒXEA,
and that of the whole will be the fum of all these products.
Let the depth DA of the fection be called d, and let any
undetermined part of
undetermined part of it EA be called x, and then the space
occupied by any particle will be x. The cohefion of this
fpace may be reprefented by fx, and that of the whole by
fd. The energy by which each element x of the line DA,
or d, refifts the fracture, will be fx x, and the whole accu-
cumulated energies will be fxfxx. This we know to be
fX { d2, or ƒ d× 4 d. It is the fame therefore as if the co-
hefion fd of the whole fection had been acting at the point
G, which is in the middle of DA.

The reader who is not familiarly acquainted with this
fluxionary calculus may arrive at the fame conclusion in an-
other way. Suppofe the beam, inftead of projecting hori-
zontally from a wall, to be hanging from the ceiling, in
which it is firmly fixed. Let us confider how the equal co-
hefton of every part operates in hindering the lower part
from leparating from the upper by opening round the joint
A. The equal cohelion operates juft as equal gravity would
do, but in the oppofite direction. Now we know, by the
molt elementary mechanics, that the effect of this will be the
fame as if the whole weight were concentrated in the centre
of gravity G of the line DA, and that this point G is in
the middle of DA. Now the number of fibres being as
the length of the line, and the cohelion of each bre
being f, the cohefion of the whole line is ƒX d or fd.

The accumulated energy therefore of the cohefion in the inftant of fracture is fdXd. Now this mult be equal or juft inferior to the energy of the power employed to break it. Let the length AB be called ; then Pis the correfponding energy of the power. This gives us ƒ d1⁄2 d=pl for the equation of equilibrium correfponding to the verti cal fection ADCB.

Suppose now that the fracture is not permitted at DA, but at another fection a more remote from B. The body being prifmatic, all the vertical fections are equal; and therefore fd is the fame as before. But the energy of the power is by this means increafed, being now = Px Bo, inftead of Px BA: Hence we fee that when the prifmatic body is not infuperably ftrong in all its parts, but equally ftrong throughout, it must break clofe at the wall, where the ftrain or energy of the power is greatest. We fee, too, that a power which is just able to break it at the wall is unable to break it anywhere elfe; alfo an abfolute cohefion fd, which can withstand the powerp in the fection DA, will not withstand it in the fection, and will withftand more in the fection d' a'.

This teaches us to diftinguish between abfolute and relative strength. The relative strength of a fection has a reference to the ftrain actually exerted on that fection. This relative ftrength is properly measured by the power which is juft able to balance or overcome it, when applied at its

Strength of proper place. Now fince we had ƒdid=pl, we have proportion each fibre is extended. It feems most probable Strength of Materials. fd + d for the measure of the strength of the fection We fhall suppose this to be really the cafe. Now recollect that the extentions are proportional to the distances from A. Materials. DA, in relation to the power applied at B. the general law which we formerly faid was obferved in all moderate extenfions, viz. that the attractive forces exerted by the dilated particles were proportional to their dilatations. Suppofe now that the beam is fo much bent that the particles at D are exerting their utmost force, and that this fibre is just ready to break or actually breaks. It is plain that a total fracture muft immediately enfue; because the force which was fuperior to the full cohesion of the particle at D, and a certain portion of the cohesion of all the reft, will be more than fuperior to the full cohesion of the particle next within D, and a finaller portion of the cohefion of the remainder.

In the folid is a rectangular beam, whose breadth is b, it is plain that all the vertical fections are equal, and that AG ord is the fame in all. Therefore the equation expreffing the equilibrium between the momentum of the external force and the accumulated momenta of cohesion will be pl = fdbx 4 d.

The product db evidently expreffes the area of the fec-
tion of fracture, which we may calls, and we may exprefs
the equilibrium thus, plfsd, and 21: dfs: p.

Now f's is a proper expreffion of the abfolute cohesion of
the fection of fracture, and p is a proper measure of its
ftrength in relation to a power applied at B.
We may
therefore say, that twice the length of a retangular beạm is to
the depth as the absolute cohesion to the relative ftrength.

Since the action of equable cohesion is fimilar to the ac-
tion of equal gravity, it follows, that whatever is the
figure of the lection, the relative ftrength will be the fame
as if the abfolute cohesion of all the fibres were acting at
the centre of gravity of the fection. Let g be the diflance
Between the centre of gravity of the fection and the axis of
fracture, we shall have pl=fsg, and l:g=fsp. It will
be very useful to recollect this analogy in words: "The
"The
length of a prifmatic beam of any shape is to the height of the
centre of gravity above the lower fide, as the abfoiute cobefion to
the frength relative to this length."

or

Because the relative ftrength of a rectangular beam is
fbd ÷ d fb ď2
b d 1⁄2 d
it follows, that the relative ftrengths of
219
different beams are proportional to the abfolute cohefion of
the particles, to the breadth, and to the square of the depth
directly, and to the length inverfely; alfo in prifms whofe
fections are fimilar, the ftrengths are as the cubes of the dia-

Now let F represent, as before, the full force of the ex-
terior fibre D, which is exerted by it in the inftant of its
breaking, and then the force exerted at the fame inftant by
the fibre E will be had by this analogy AD:AE, or
fx
dix=f:, and the force really exerted by the fibre E.
isfx

The force exerted by a fibre whofe thickness is x is therefore-fxx but this force refifts the ftrain by acting d by means of the lever EA or x. Its energy or momentum is therefore ƒx2x and the accumulated momenta. of all the fibres in the line AE will be ƒX sum of d This, when x is taken equal to d, will exprefs the momentum of the whole fibres in the line AD. This, therefore, ÷ 43 is f, or fi d', or fdxd. Now fd expreffes the abfolute cohesion of the whole line AD. The accumulated Such are the more general refults of the mechanifm of momentum is therefore the fame as if the abfolute cohesion this tranfverfe ftrain, in the hypothefis that all the particles of the whole line were exerted at 4d of AD from A. hypothefis are exerting equal forces in the inftant of fracture. We are From thefe premifes it follows that the equation expref- The indebted for this doctrine to the celebrated Galileo; and it fing the equilibrium of the ftrain and cohetion is p/fdtrength "As thrice the was one of the firft fpecimens of the application of mathe-Xd; and hence we deduce the analogy, length is to the depth, fo is the abfolute cohesion to the relative principles. strength."

57 Aftert.in

cd on ile

clequal cobefion;

58

But that

not conformable

to nature.

meters.

matics to the fcience of nature.

We have not included in the preceding investigation that action of the external force by which the folid is drawn fidewife, or tends to flide along the furface of fracture. We have fuppofed a particle E to be pulled only in the direction Ee, perpendicular to the section of fracture, by the action of the crooked lever BAE. Dut it is alfo pulled in the direction EA; and its reaction is in fome direction E, compounded off, by which it refifts being pulled outwards; and e, by which it refifts being pulled downwards. We are but imperfe&ly acquainted with the force e, and only know that their accumulated fum is equal to the force p: but in all important cafes which occur in practice, it is unneceffary to attend to this force; because it is fo fmall in comparison of the forces in the direction E e, as we eafily conclude from the usual smallness of AD in comparison of AB.

The hypothesis of equal cohesion, exerted by all the parhypothefis ticles in the inftant of fracture, is not conformable to nature: for we know, that when a force is applied tranfverfely at B, the beam is bent downwards, becoming convex on the upper fide; that fide is therefore on the ftretch. ticles at D are farther removed from each other than those at E, and are therefore adually exerting greater cohesive for We cannot fay with certainty and precision in what

ces.

'The par

This equation and this proportion will equally apply to rectangular beams whole breadth is b; for we fhall then have pl/ b d x ÷ d.

=

We alfo fee that the relative ftrength is proportional to
the abfolute cohefion of the particles, to the breadth, and
to the fquare of the depth directly, and to the length in-
verfely: for p is the measure of the force with which it is
fb d z d
ƒ bd2
refifted, and =
31. In this respect there-
fore this hypothefis agrees with the Galilean; but it affigus
to every beam a fmaller proportion of the abfolute cohelion
of the fection of fracture, in the proportion of 3 to 2. In
the Galilean hypothesis this fection has a momentun equal-
to of its abfolute strength, but in the other hypothefis it
is only d. In beams of a different form the proportion.
may be different.

As this is a most important propofition, and the founda
tion of many practical maxims, we are anxious to have it
clearly comprehended, and its evidence perceived by all.
Our better informed readers will therefore indulge us while
we endeavour to prefent it in another point of view, where
it will be better feen by those who are not familiarly ac-
quainted with the fluxionary calculus.
Figs.

4

59

afcertained un other

Strength of Call their diftances from the axis of fracture o and g.

Then

og
d

Materials.

thefis.

- AI or i =

curve Add' will be a common parabola, having AB for its Strength of
axis and AD for the tangent at its vertex. The area

, and the momentum of cohesion is fs X ADd will be d ADXDd; and in the cafe of a rectan
gular beam, AO will be ths AD, and AI will be 4th of
AD.

2, where s is the area of fracture.

This index is easily determined in all the cafes which generally occur in practice. In a rectangular beam A I is +d of AD; in a cylinder (circular or elliptic) AI is ths of AD, &c. &c.

ठ

In this hypothesis, that the cohesion actually exerted by each fibre is as its extenfion, and that the extenfions of the fibres are as their distances from A (fig. 5. no 1.), it is plain that the forces exerted by the fibres D, E, &c. will be reprefented by the ordinates D d, Ee, &c. to a ftraight line A d. And we learn from the principles of ROTATION that the centre of percuffion O is in the ordinate which paffes through the centre of gravity of the triangle A D d, or (if we confider the whole fection having breadth as well as depth) through the centre of gravity of the folid bounded by the planes DA, dA; and we found that this point O was the centre of effort of the cohesions adually exerted in the inftant of fracture, and that I was the centre of an equal momentum, which would be produced if all the fibres were accumulated there and exerted their full cohesion.

This confideration enables us to determine, with equal facility and neatnefs, the ftrength of a beam in any hypothefis of forces. The above hypothefis was introduced with a cautious limitation to moderate ftrains, which produced no permanent change of form, or no fett as the artists call it: and this fuffices for all purposes of practice, feeing that it would be imprudent to expose materials to more violent ftrains. But when we compare this theory with experiments in which the pieces are really broken, confiderable deviations may be expected, because it is very probable that in the vicinity of rupture the forces are no longer proportional to the extenfions.

AD

q+2

We may obferve here in general, that if the forces
actually exerted in the inftant of fracture be as any power q
of the distance from A, the index A I will be =
for a rectangular beam, and the momentum of cohesion will
always be (cæteris paribus) as the breadth and as the fquare
of the depth; nay, this will be the case whenever the action
of the fibres D and E is expreffed by any fimilar fundions
of d and x. This is evident to every reader acquainted with
the fluxionary calculus.

As far as we can judge from experience, no fimple algebraic
power of the distance will exprefs the actual cohesions of the
fibres. No curve which has either AD or AB for its
tangent will fuit. The observations which we made in the
beginning fhow, that although the curve of fig. 2. muft be
fenfibly straight in the vicinity of the points of interfection
with the axis, in order to agree with our obfervations which
fhow the moderate extenfions to be as the extending forces,
the curve must be concave towards the axis in all its attractive
branches, because it cuts it again. Therefore the curve A eď
of fig. 5. (n° 1.) must make a finite angle with AD or AB,
and it must, in all probability, be also concave towards AD
in the neighbourhood of d. It may however be convex in
We have made ex-
fome part of the intermediate arch.
periments on the extenfions of different bodies, and find
great diverfities in this refpect: But in all, the moderate
extenfions were as the forces, and this with great accuracy
till the body took a fett, and remained longer than formerly
when the extending force was removed.

Materials.

We must now remark, that this correction of the Galilean hypothefis of equal forces was fuggefted by the bending which is obferved in all bodies which are ftrained transverfeHow the That no doubt may remain as to the juftnefs and comrelative pleteness of the theory, we muft fhow how the relative ly. Because they are bent, the fibres on the convex fide have ft,ength may be de- ftrength may be determined in any other hypothefis. There- been extended. We cannot fay in what proportion this obtains in the different fibres. Our moft diftinct notions of termined fore fuppofe that it has been established by experiment on by any o- any kind of folid matter, that the forces actually exerted in the internal equilibrium between the particles render it highther hypo- the inftant of fracture by the fibres at D, E, &c. are as the ly probable that their extenfion is proportional to their ordinates Dd, E é', &c. of any curve line A ed. . We are distance from that fibre which retains its former dimensions. supposed to know the form of this curve, and that of the But by whatever law this is regulated, we fee plainly that folid which is bounded by the vertical plane through AD, the actions of the ftretched fibres muft follow the proporand by the surface which paffes through this curve A edtions of some function of this distance, and that therefore perpendicularly to the length of the beam. We know the the relative ftrength of a beam is in all cafes susceptible of place of the centre of gravity of this curve farface or folid, mathematical determination. and can draw a line through it parallel to A B, and cutting the surface of fracture in fome point O. This point is alfo the centre of effort of all the cohefions actually exerted and the product of AO and of the folid which expreffes the actual cohesions will give the momentum of cohetion equivalent to the former ƒs 3. Or we may find an index fs A I, by making A I a fourth proportional to the full cohe. fion of the furface of fracture, to the accumulated actual cohefions, and to AO; and then ƒs Xi (AI) will be the momentum of cohefion; and we fhall Itill have I for the point in which all the fibres may be fuppofed to exert their full cohesion ƒ, and to produce a momentum of cohefion equal to the real momentum of the cohesions actually exerted, fsi and the relative strength of the beam will still be por

62 We also fee an intimate connection between the strain and Bernoulli's

og

the elastic

curve.

the curvature. This fuggefted to the celebrated James problem of Bernoulli the problem of the ELASTIC CURVE, i. e. the ; curve into which an extenfible rigid body will be bent by a tranfverse strain. His folution in the Ata Lipfie 1694 and 1695 is a very beautiful specimen of mathematical dif cuffion; and we recommend it to the perufal of the curious reader. He will find it very perfpicuously treated in the first volume of his works, publifhed after his death, where the wide steps which he had taken in his investigation are explained fo as to be eafily comprehended. His nephew Dan. Bernoulli has given an elegant abridgment in the Peterburg Memoirs tor 1729. The problem is too inalfo too intimately connected with our present subject to be tricate to be fully difcuffed in a work like ours; but it is entirely omitted. We mult content ourselves with fhowing the leading mechanical property of this curve, from which the mathematician may deduce all its geometrical properties.

Thus, if the forces be as the fquares of the exten frons (ftill fuppofed to be as the distances from A), the VOL. XVIII. Part I.

When a bar of uniform depth and breadth, and of a given length, is bent into an arch of a circle, the extenfion of the C

outer