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Given the Length and Weight of a Cylinder or Prism, fixed Horizontally as in the foregoing problem, and a weight which, when hung at a given point, Breaks the Prism: it is required to determine how much longer the Prism, of equal Diameter or of equal Breadth and Depth, may be extended before it Break, either by its own weight, or by the addition of any other adventitious weight.
Let / denote the length of the given prism, w its weight, and u a weight attached to it at the distance d from the fixed end; also let L. denote the required length of the other prism, and u the weight attached to it at the distance D. Now the strain occasioned by the weight of the first beam is wl, and that by the weight u at the distance d, is du, their sum + du being the whole strain. In like manner WL + DU is the strain on the second beam; but 7: L:: W: =w the
weight of this beam, theref. + DU its strain. But the strength of the beam, which is just sufficient to resist these strains, is the same in both cases; therefore
+ DU= wl+du, and hence, by reduction, the required length wl+2du-2Du -2DU).
Corol. 1. When the lengthened beam just breaks by its own weight, then u = 0 or vanishes, and the required length becomes L =
Corol. 2. Also when u vanishes, if d become = l, then 1 = W√ is the required length.
Let AB be a beam moveable about the end A, so as to make any angle BAC with the plane of the horizon AC: it is required to determine the position of a prop or supporter DE of a given length, which shall sustain it with the greatest ease in any given position; also to ascertain the angle BAC when · the least force which can sustain AB, is greater than the least force in any other position.
Let G be the centre of gravity of the ́ beam; and draw Gm perp. to AB, Gn to Ac, nm to Gm, and AFH to DE. Put r = AG, P =DE, w = the weight of the beam AB, and Anx. Then, by the nature of the parallelogram of forces, Gn: Gm, or by sim. triangles, AG =1':
An=x:: W: the force which acting
at G in the direction mg, is sufficient to sustain the beam;
and, by the nature of the lever, AE : AG=r::
quisite force at G':
1 wx AE
the force capable of supporting it at E in a direction perp. to AB or parallel to me; and again as
AF AE ::
: the force or pressure actually sustained by the given prop DE in a direction perp. to AF. And this latter force will manifestly be the least possible when the perp. AF upon DE is the greatest possible, whatever the angle BAC may be, which is when the triangle ADE is isosceles, or has the side AD AE, by an obvious corol.. from the latter part of prob. 6 pa. 171 of this volume.
Secondly, for a solution to the latter part of the problem, we have to find when is a maximum; the angles D and E being always equal to each other, while they vary in magnitude by the change in the position of AB. Let AF produced meet Gn in H: then, in the similar triangles Adf, ahn, it
But, by theor. 83 Geom. and
An = x :: Gn = √ (r2 − x2) :
r + x
xw, acting on the prop, is also truly expressed by
✓ Then the fluxion of this made to vanish gives
T + I
√51r the cos. angle BAC = 51°50′, the inclination
Suppose the Beam AB, instead of being moveable about the centre A, as in the last problem, to be supported in a given position by means of the given prop DE: it is required to determine the position of that prop, so that the prismatic beam AC, on which it stands, may be the least liable to break ing, this latter beam being only supported at its two ends a and c.
Put the base AC = b, the prop DE = P, AG=r, the weight of AB = w, s and c the sine and cosine of 4A, x = sin. E, y sin. D, and z = AE. Then, by
trigon. zy: p; §, or ==÷, and
at G in direction Gm. Let F denote the force sustaining the beam at E in the direction ED: then, because action and reaction are equal and opposite, the same force will be exerted at D in the direction DE: therefore AG. cwFzx, and Again, the vertical stress at D, will be as F x sine
DX AD.PC Fy. AD. DC = × PI (b − p) = (sub
stituting for its equal 2-)
= row x
a minimum by the problem.
Conseq. — is a minimum, or a maximum, that is,
x=1, and the angle E is a right angle. Hence the point E is easily found by this proportion, sin. A: cos. A ¦ ¦ ED : EA,
To explain the Disposition of the Parts of Machines. When several pieces of timber, iron, or any other materials, are employed in a machine or structure of any kind, all the parts, both of the same piece, and of the different pieces in the fabric, ought to be so adjusted with respect to magnitude, that the strength in every part may be, as near as possible, in a constant proportion to the stress or strain to which they will be subjected. Thus, in the construction of any engine, the weight and pressure on every part should be investigated, and the strength apportioned accordingly. All levers, for instance, should be made strongest where they are most strained: viz, levers of the first kind, at the fulcrum; levers
of the second kind, where the weight acts; and those of the third kind, where the power is applied. The axles of wheels and pulleys, the teeth of wheels, also ropes, &c, must be made stronger or weaker, as they are to be more or less acted on. The strength allotted should be more than fully competent to the stress to which the parts can ever be liable; but without allowing the surplus to be extravagant: for an over excess of strength in any part, instead of being serviceable, would be very injurious, by increasing the resistance the machine has to overcome, and thus encumbering, impeding, and even preventing the requisite motion; while, on the other hand, a defect of strength in any part will cause a failure there, and either render the whole useless, or demand very frequent repairs.
To ascertain the Strength of Various Substances.
The proportions that we have given on the strength and stress of materials, however true, according to the principles assumed, are of little or no use in practice, till the comparative strength of different substances is ascertained: and even then they will apply more or less accurately to different substances. Hitherto they have been applied almost exclusively to the resisting force of beams of timber; though probably no materials whatever accord less with the theory than timber of all kinds. In the theory, the resisting body is supposed to be perfectly homogeneous, or composed of parallel fibres, equally distributed round an axis, and presenting uniform resistance to rupture. But this is not the case in a beam of timber: for, by tracing the process of vegetation, it is readily seen that the ligneous coats of a tree, formed by its annual growth, are almost concentric; being like so many hollow cylinders thrust into each other, and united by a kind of medullary substance, which offers but little resistance: these hollow cylinders therefore furnish the chief strength and resistance to the force which tends to break them.
Now, when the trunk of a tree is squared, in order that it may be converted into a beam, it is plain that all the ligneous cylinders greater than the circle inscribed in the square or rectangle, which is the transverse section of the beam, are cut off at the sides; and therefore almost the whole strength or resistance arises from the cylindric trunk inscribed in the solid part of the beam; the portions of the cylindric coats, situated towards the angles, adding but little comparatively to the strength and resistance of the beam. Hence it follows that we cannot, by legitimate comparison, accurately deduce
the strength of a joist, cut from a small tree, by experiments on another which has been sawn from a much larger tree or block. As to the concentric cylinders above mentioned, they are evidently not all of equal strength: those nearest the centre, being the oldest, are also the hardest and strongest ; which again is contrary to the theory, in which they are supposed uniform throughout. But yet, after all however, it is still found that, in some of the most important problems, the results of the theory and well-conducted experiments coincide, even with regard to timber: thus, for example, the experiments on rectangular beams afford results deviating but in a very slight degree from the theorem, that the strength is proportional to the product of the breadth and the square of the depth.
Experiments on the strength of different kinds of wood, are by no means so numerous as might be wished: the most useful seem to be those made by Muschenbroek, Buffon, Emerson, Parent, Banks, and Girard. But it will be at all times highly advantageous to make new experiments on the same subject; a labour especially reserved for engineers who possess skill and zeal for the advancement of their profession. It has been found by experiments, that the same kind of wood, and of the same shape and dimensions, will bear or break with very different weights: that one piece is much stronger than another, not only cut out of the same tree, but out of the same rod; and that even, if a piece of any length, planed equally thick throughout, be separated into three or four pieces of an equal length, it will often be found that these pieces require different weights to break them. Emerson observes that wood from the boughs and branches of trees is far weaker than that of the trunk or body; the wood of the large limbs stronger than that of the smaller ones; and the wood in the heart of a sound tree strongest of all; though some authors differ on this point. It is also observed that a piece of timber which has borne a great weight for a short time, has broke with a far less weight, when left upon it for a much longer time. Wood is also weaker when green, and strongest when thoroughly dried, in the course of two or three years, at least. Wood is often very much weakened by knots in it; also when cross-grained, as often happens in sawing, it will be weakened in a greater or less degree, according as the cut runs more or less across the grain. From all which it follows, that a considerable allowance ought to be made for the various strength of wood, when applied to any use where strength and durability are required.
Iron is much more uniform in its strength than wood. Yet experiments