How a strong

Strength of AAKB, not cohering, each of them will bend, and the exMaterials. tension of the fibres AB of the under beam will not hinder the compression of the adjoining fibres AB of the upper beam. The two together therefore will not be more than compound twice as strong as one of them (supposing DA = A▲), inDeam may stead of being four times as strong; and they will bend as be formed. much as either of them alone would bend by half the load. This may be prevented, if it were possible to unite the two Deams all along the seam AB, so that the one shall not slide on the other. This may be done in small works by gluing them together with a cement as strong as the natural lateral cohesion of the fibres. If this cannot be done (as it cannot in large works), the sliding is prevented by joggling the beams together, that is, by cutting down several rectangular notches in the upper side of the lower beam, and making similar notches in the under side of the upper beam, and filling up the square spaces with pieces of very hard wood firmly driven in, as represented in fig. 9. Some employ iron bolts by way of joggles. But when the joggle is much harder than the wood into which it is driven, it is very apt to work loose, by widening the hole into

How strength may be combined with phiableness.

Fig. 9.

Fig. 10.

which it is lodged. The same thing is sometimes done by scarphing the one upon the other, as represented in fig. 10; but this wastes more timber, and is not so strong, because the mutual hooks which this method form on each beam are very apt to tear each other up. By one or other of these

methods, or something similar, may a compound beam be formed, of any depth, which will be almost as stiff and strong as an entire piece.

On the other hand, we may combine strength with pliableness, by composing our beam of several thin planks laid on each other, till they make a proper depth, and leav. ing them at full liberty to slide on each other. It is in this manner that coach-springs are formed, as is represented in fig. 11. In this assemblage there must be no joggles nor bolts of any kind put through the planks or plates, for this would hinder their mutual sliding. They must be kept together by straps which surround them, or by something equivalent.

Fig. 11.

Maxims of The preceding observations show the propriety of some construc- maxims of construction, which the artists have derived from tion. long experience.

Thus, if a mortise is to be cut out of a piece which is exposed to a cross strain, it should be cut out from that side which becomes concave by the strain.

If a piece is to be strengthened by the addition of another, the added piece must be joined to the side which grows convex by the strain.

Before we proceed any farther, it will be convenient to recall the reader's attention to the analogy between the strain on a beam projecting from a wall and loaded at the extremity, and a beam supported at both ends and loaded in some intermediate point. It is sufficient on this occasion to read attentively what is delivered in the article RooF. We learn there that the strain on the middle point C (fig. 16 of the present article) of a rectangular beam AB, supported on props at A and B, is the same as if the part CA projected from a wall, and were loaded with the half of the weight W suspended at A. The momentum of the strain is therefore & W × 4B =WxAB = p 4, or 4. The

momentum of cohesion must be equal to this in every hy- Strength of pothesis. Materials.

Having now considered in sufficient detail the circumstances which affect the strength of any section of a solid body that is strained transversely, it is necessary to take notice of some of the chief modifications of the strain itself. We shall consider only those that occur most frequently in our constructions.

The strain depends on the external force, and also on the lever by which it acts.

It is evidently of importance, that since the strain is ex- The strain erted in any section by means of the cohesion of the parts depends on intervening between the section under consideration and the exterthe point of application of the external force, the body nal force, must be able in all these intervening parts to propagate or excite the strain in the remote section. In every part it must be able to resist the strain excited in that part. It should therefore be equally strong; and it is useless to have any part stronger, because the piece will nevertheless break where it is not stronger throughout; and it is useless to make it stronger (relatively to its strain) in any part, for it will nevertheless equally fail in the part that is too weak.

Suppose, then, in the first place, that the strain arises from a weight suspended at one extremity, while the other end is firmly fixed in a wall. Supposing also the cross sections to be all rectangular, there are several ways of shaping the beam so that it shall be equally strong throughout. Thus it may be equally deep in every part, the upper and under surfaces being horizontal planes. The condition will be fulfilled by making all the horizontal sections triangles, as in fig. 12. The two sides are vertical planes, meeting in an edge at the extremity L.

For the equation expressing the balance of strain and strength is pl=fbď2. Therefore, since d is the same throughout, and also p, we must have fb = 1, and b (the breadth AD of any section ABCD) must be proportional to (or AL), which it evidently is.

Fig. 12.

Fig. 13.

L

Or, if the beam be of uniform breadth, we must have ď2 everywhere proportional to . This will be obtained by making the depths the ordinates of a common parabola, of which L is the vertex and the length is the axis. The upper or under side may be a straight line, as in fig. 13, or the middle line may be straight, and then both upper and under surfaces will be curved. It is almost indifferent what is the shape of the upper and under surfaces, provided the distances between them in every part be at the ordinates of a common parabola.

D

P

Or, if the sections are all similar, such as circies, squares, or any other similar polygons, we must have d or b3 proportional to 7, and the depths or breadths must be as the ordinates of a cubical parabola.

acts.

It is evident that these are also the proper forms for a and on the lever moveable round a fulcrum, and acted on by a force at form of the the extremity. The force comes in the place of the weight levers by suspended in the cases already considered; and as such which it levers always are connected with another arm, we readily see that both arms should be fashioned in the same manner. Thus in fig. 12 the piece of timber may be supposed a kind of steelyard, moveable round a horizontal axis in the front of the wall, and having the two weights P and in equilibrio. The strain occasioned by each at the section in which the axis OP is placed must be the same, and each

Strength of arm OL and O> must be equally strong in all its parts. Materials. The longitudinal sections of each arm must be a triangle, a common parabola, or a cubic parabola, according to the conditions previously given.

The exter

nal strain ing force may be dis.. tributed

over the

beam.

To make

a wall.

And, moreover, all these forms are equally strong; for any one of them is equally strong in all its parts, and they are all supposed to have the same section at the front of the wall or at the fulcrum. They are not, however, equally stiff. The first, represented in fig. 12, will bend least upon the whole, and the one formed by the cubic parabola will bend most. But their curvature at the very fulcrum will be the same in all.

It is also plain, that if the lever is of the second or third kind, that is, having the fulcrum at one extremity, it must still be of the same shape; for in abstract mechanics it is indifferent which of the three points is considered as the axis of motion. In every lever the two forces at the extremities act in one direction, and the force in the middle acts in the opposite direction, and the great strain is always

at that point. Therefore a lever such as fig. 12, moveable round an axis passing horizontally through A, and acting against an obstacle at OP, is equally able in all its parts to resist the strains excited in those parts.

The same principles and the same construction will apply to beams, such as joists, supported at the ends L and λ (fig. 12), and loaded at some intermediate part OP. This will appear evident by merely inverting the directions of the forces at these three points, or by recurring to the article Roof.

Hitherto we have supposed the external straining force as acting only in one point of the beam. But it may be uniformly distributed all over the beam. To make a beam in such circumstances equally strong in all its parts, the shape must be considerably different from the former.

Thus suppose the beam to project from a wall. If it be of equal breadth throughout, its sides being vertical a beain planes parallel to each other and to the length, the verstrong which protical section in the direction of its length must be a triects from angle instead of a common parabola; for the weight uniformly distributed over the part lying beyond any section, is as the length beyond that section: and since it may all be conceived as collected at its centre of gravity, which is the middle of that length, the lever by which this load acts or strains the section is also proportioned to the same length. The strain on the section (or momentum of the load) is as the square of that length. The section must have strength in the same proportion. Its strength being as the breadth and the square of the depth, and the breadth being constant, the square of the depth of any section must be as the square of its distance from the end, and the depth must be as that distance; and therefore the longitudinal vertical section must be a triangle.

The strain upon a beam sup

But if all the transverse sections are circles, squares, or any other similar figures, the strength of every section, or the cube of the diameter, must be as the square of the lengths beyond that section, or the square of its distance from the end; and the sides of the beam must be a semicubical parabola.

If the upper and under surfaces are horizontal planes, it is evident that the breadth must be as the square of the distance from the end, and the horizontal sections may be formed by arches of the common parabola, having the length for their tangent at the vertex.

By recurring to the analogy so often quoted between a projecting beam and a joist, we may determine the proper form of joists which are uniformly loaded through their whole length.

This is a frequent and important case, being the office of joists, rafters, &c.; and there are some circumstances which ported at must be particularly noticed, because they are not so obboth ends. vious, and have been misunderstood. When a beam AB

Fig. 14.

E e

P n

C C

B

Ff

Q

D

Materials

(fig. 14) is supported at the ends, and a weight is laid on any Strength of point P, a strain is excited in every part of the beam. The load on P causes the beam to press on A and B, A and the props re-act with forces equal and opposite to these pressures. The load at P is to the pressures at A and B as AB to PB and PA, and the pressure at A is to that at B as BP to PA; the beam therefore is in the same state, with respect to strain in every part of it, as if it were resting on a prop at P, and were loaded at the ends with weights equal to the two pressures on the props: and observe, these pressures are such as will balance each other, being inversely as their distances from P. Let P represent the weight or load at P. The pressure on the prop P PA must be PX This is therefore the re-action of the prop B, and is the weight which we may suppose suspended at B, when we conceive the beam resting on a prop at P, and carrying the balancing weights at A and B.

AB'

The strain occasioned at any other point C, by the load Pat P, is the same with the strain at C, by the weight PA PX AB

hanging at B, when the beam rests on P, in the manner now supposed; and it is the same if the beam, instead of being balanced on a prop at P, had its part AP fixed in a wall. This is evident. Now we have shown at PA length that the strain at C, by the weight PX hanging AB PA at B, is P x x BC. We desire it to be particularly reAB marked, that the pressure at A has no influence on the strain at C, arising from the action of any load between A and C; for it is indifferent how the part AP of the projecting beam PB is supported. The weight at A just performs the same office with the wall in which we suppose the beam to be fixed. We are thus particular, because we have seen even persons not unaccustomed to discussions of this kind puzzled in their conceptions of this strain.

Now let the load P be laid on some point p between C and B. The same reasoning shows us that the point is, with respect to strain, in the same state as if the beam were fixed in a wall, embracing the part pB, and a weight PB =PX were hung on at A, and the strain at C is P AB

In general, therefore, the strain on any point C, arising A general the rectangle of the distances of P and C from the ends tion. from a load P laid on another point P, is proportional to proposi PAX CB nearest to each. It is P x pBX CA or Px AB AB cording as the load lies between C and A or between C and B.

, ac

Cor. 1. The strains which a load on any point P occasions on the points C, c, lying on the same side of P, are as the distances of these points from the end B. In like manner the strains on E and e are as EA and eA.

Cor. 2. The strain which a load occasions in the part on which it rests is as the rectangle of the parts on each side. Thus the strain occasioned at C by a load is to that at D by the same load as AC x CB to AD × DB. It is therefore greatest in the middle.

Let us now consider the strain on any point C arising from The strain a load uniformly distributed along the beam. Let AP be rearising from presented by a, and Pp by dr, and the whole weight on the buted along beam by a. Then

a load distri

the beam.

Strain at C by the weight on AC = a

AB AC2 AC2 AB2 2AB AC2 x BC 2AB2 BC2 X AC 2AB2

2. If the sections be similar, then CD3: EF3— AC: AE. 3. If the upper and under surfaces be parallel, then breadth at C: breadth at E=AC: AE.

The same principles enable us to determine the strain The strain and strength of square or circular plates of different ex-and tent but equal thickness. This may be comprehended in strength of this general proposition.

square or

circular

extent but

Do. by whole weight on AB = a

= a

2AB2 AC X BC × AC + CB AC X BC 2AB2 2AB Thus we see that the strain is proportional to the rectangle of the parts, in the same manner as if the load a had been laid directly on the point C, and is indeed equal to one half of the strain which would be produced at Cby the load a laid on there.

Mistakes on It was necessary to be thus particular, because we see this subject in some elementary treatises on mechanics, published by committed authors of reputation, mistakes which are very plausible, by authors and mislead the learner. It is there said that the presof reputation. sure at B from a weight uniformly diffused along AB, is the same as if it were collected at its centre of gravity, which would be the middle of AB; and then the strain at C is said to be this pressure at B multiplied by BC. But surely it is not difficult to see the difference of these strains. It is plain that the pressure of gravity downwards on any point between the end A and the point C has no tendency to diminish the strain at C, arising from the upward re-action of the prop B; whereas the pressure of gravity between C and B is almost in direct opposition to it, and must diminish it. We may however avoid the fluxionary calculus with safety by the consideration of the centre of gravity, by supposing the weights of AC and BC to be collected at their respective centres of gravity; and the result of this computation will be the same as above: and we may use either method, although the weight be not uniformly distributed, provided only that we know in what manner it is distributed.

To form a

This investigation is evidently of importance in the practice of the engineer and architect, informing them what support is necessary in the different parts of their constructions. We considered some cases of this kind in the article Roof.

It is now easy to form a joist so that it shall have the Joist which same relative strength in all its parts. inay have

relative

I. To make it equally able in all its parts to carry a the same given weight laid on any point C taken at random, or unistrength in formly diffused over the whole length, the strength of the all its parts. section at the point C must be as ACX CB. Therefore,

1. If the sides be parallel vertical planes, the square of the depth (which is the only variable dimension), or CD2, must be as ACX CB, and the depths must be ordinates of an ellipse.

2. If the transverse sections be similar, we must make CD3 as ACX CB.

3. If the upper and under surfaces be parallel, the breadth must be as ACX CB.

II. If the beam be necessarily loaded at some given point C, and we would have the beam equally able in all its parts to resist the strain arising from the weight at C, we must make the strength of every transverse section between C and either end as its distance from that end. Therefore,

thickness

from the

Suppose the plates supported at the ends only, and same prinresisting fracture transversely. The strains, being as the ciples. weights and lengths, are as WL and wl, but their cohesions are as the breadths; and since they are of equal relative strength, we have WL: wl B:b, and WLb = w/B, and L:/= wB: Wb; but since they are of similar shapes, L:= B: b, and therefore w = W.

The same reasoning holds again when they are also supported along the sides, and therefore holds when they are supported all round (in which case the strength is doubled).

And if the plates be of any other figure, such as circles or ellipses, we need only conceive similar rectangles inscribed in them. These are supported all around by the continuity of the plates, and therefore will sustain equal weights; and the same may be said of the segments which lie without them, because the strengths of any similar segments are equal, their lengths being as their breadths. Therefore the thickness of the bottoms of vessels holding heavy liquors or grains should be as their diameters and as the square root of their depths jointly.

Also the weight which a square plate will bear is to that which a bar of the same matter and thickness will bear as twice the length of the bar to its breadth.

There is yet another modification of the strain which The strain tends to break a body transversely, which is of very fre- of a beam quent occurrence, and in some cases must be very care- arising from fully attended to, viz. the strain arising from its own weight. weight.

When a beam projects from a wall, every section is strained by the weight of all that projects beyond it. This may be considered as all collected at its centre of gravity. Therefore the strain on any section is in the joint ratio of the weight of what projects beyond it, and the distance of its centre of gravity from the section.

its own

The determination of this strain, and of the strength ne- General cessary for withstanding it, must be more complicated than principle the former, because the form of the piece which results respecting from this adjustment of strain and strength influences the it. strain. The general principle must evidently be, that the strength or momentum of cohesion of every section must be as the product of the weight beyond it, multiplied by the distance of its centre of gravity. For example: Suppose the beam DLA (fig. 15) to project from the wall, and that its sides are parallel vertical planes, so that the depth is the only variable dimension. Let LB=x and Bb=y. The element BbcC is=ydr. Let G be the centre of gravity of the part lying without Bb, and g be its distance from the extremity L. Then -g is the arm of the lever by which the strain is excited in the section Bò.

D

BC

G

Fig. 15.

Strength of Let Bo or y be as some power m of LB; that is, let y=xm. Then the contents of LBb is The momentum of gravity round a horizontal axis at L is yxdx=x+1dx, and the whole momentum round the axis is The distance of the centre of gravity from L is had by dividing this momentum by the whole weight, which The quotient or g is

A conoid equally able in

every section to bear its own weight.

The more a beam

the less able it is to bear its own weight.

xm+2

m+2′

is

m+1

xx m + m+2

and the dis

an herb could not support it if it were increased to the size Strength of of a tree, nor could an oak support itself if forty or fifty Materials. times bigger; nor could an animal of the make of a longlations of its legs could not support it. legged spider be increased to the size of a man; the articu

remark.

able for

=

xm+1 by m+1 m+2'

..

fore the strain on the section Bb is had by multiplying
The product is
This
m + 2 x m +1
must be as the square of the depth, or as y2. But y is as
x", and y2 as x2m Therefore we have m+2=2m, and
m=2; that is, the depth must be as the square of the
distance from the extremity, and the curve LOA is a para-

bola touching the horizontal line in L.

It is easy to see that a conoid formed by the rotation of this figure round DL will also be equally able in every section to bear its own weight.

We need not prosecute this farther. When the figure of the piece is given, there is no difficulty in finding the strain; and the circumstance of equal strength to resist this strain is chiefly a matter of curiosity.

It is evident, from what has been already said, that a projecting beam becomes less able to bear its own weight as projects, it projects farther. Whatever may be the strength of the section DA, the length may be such that it will break by its own weight. If we suppose two beams A and B of the same substance and similar shapes, that is, having their lengths and diameters in the same proportion; and further suppose that the shorter can just bear its own weight; then the longer beam will not be able to do the same; for the strengths of the sections are as the cubes of the diameters, while the strains are as the biquadrates of the diameters; because the weights are as the cubes, and the levers by which these weights act in producing the strain are as the lengths or as the diameters.

Small bo

withstand

the ma

great bodies.

These considerations show us, that in all cases where dies more strain is affected by the weight of the parts of the machine able to or structure of any kind, the smaller bodies are more able the strain to withstand it than the greater; and there seem to be produced bounds set by nature to the size of machines constructed of by the any given materials. Even when the weight of the parts of weight of the machine is not taken into the account, we cannot enthin than large them in the same proportion in all their parts. Thus a steam-engine cannot be doubled in all its parts, so as to be still efficient. The pressure on the piston is quadrupled. If the lift of the pump be also doubled in height while it is doubled in diameter, the load will be increased eight times, and will therefore exceed the power. The depth of lift, therefore, must remain unchanged; and in this case the machine will be of the same relative strength as before, independent of its own weight. For the beam being doubled in all its dimensions, its momentum of cohesion is eight times greater, which is again a balance for a quadruple load acting by a double lever. But if we now consider the increase of the weight, of the machine itself, which must be supported, and which must be put in motion by the intervention of its cohesion, we see that the large machine is weaker and less efficient than the small one.

There is a similar limit set by nature to the size of plants and animals formed of the same matter. The cohesion of

The angular motions of small animals (in which consists large animals, supposing the force of the muscular fibre to their nimbleness or agility) must be greater than those of be the same in both. For supposing them similar, the number of equal fibres will be as the square of their linear dimensions; and the levers by which they act are as their linear dimensions. The energy therefore of the moving force is as the cube of these dimensions. But the momentum of inertia, or Spr2, is as the fourth power; therefore

the angular velocity of the greater animals is smaller. The number of strokes which a fly makes with its wings in a second is astonishingly great; yet, being voluntary, they are the effects of its agility.

We have hitherto confined our attention to the simplest form in which this transverse strain can be produced. This was quite sufficient for showing us the mechanism of nature by which the strain is resisted; and a very slight attention is sufficient for enabling us to reduce to this every other way in which the strain can be produced. We shall not take up the reader's time with the application of the same principles to other cases of this strain, but refer him to what has been said in the article Roof. In that arti cle we have shown the analogy between the strain on the section of a beam projecting from a wall and loaded at the extremity, and the strain on the same section of a beam simply resting on supports at the ends, and loaded at some intermediate point or points. The strain on the middle C of a beam AB (fig. 16) so supported, arising from a weight laid on there, is the same with the strain which half that weight hanging at B would produce on the same section C, if the other end of the beam were fixed in

a wall. If therefore 1000 pounds hung on the end of a beam projecting ten feet from a wall will just break it at the wall, it will require 4000 pounds on its middle to break the same beam resting on two props ten feet asunder. We have also shown in that article the additional strength which will be given to this beam by extending both ends beyond the props, and there framing it firmly into other pillars or supports. We Effects of can hardly add any thing to what has been said in that article, obliquity except a few observations on the effects of the obliquity of of the ex the external force. We have hitherto supposed it to act in force. the direction BP (fig. 6) perpendicular to the length of the beam. Suppose it to act in the direction BP', oblique to BA. In the article Roof we supposed the strain to be the same as if the force p acted at the distance AB', but still perpendicular to AB: so it is. But the strength of the section A▲ is not the same in both cases; for by the obliquity of the action the piece DCKA is pressed to the other. We are not sufficiently acquainted with the corpuscular forces to say precisely what will be the effect of the pressure arising from this obliquity; but we can clearly see in general, that the point A, which in the instant of fracture