We must begin by informing our readers, that the bulk of the present article was written by the late Professor Robison, in order to form, with those on Roof, and Strength of Materials, also written by him for this Encyclopedia, a uniform system of the most useful departments of practical mechanics, deduced, in the same familiar and elementary manner, from the simple principles of the composition of forces. In here reprinting his contribution, we shall premise some introductory observations, which may be considered as a retrospective summary of the doctrine of Passive Strength, accompanied by some of the most useful propositions respecting the resistance of elastic substances, derived from the principles which have been already laid down in our article Bridge; and subjoining a few notes on such passages as may appear to require further illustration or correction. Some of the demonstrations will be partly borrowed from a work which has been published since the death of Professor Robison, but others will be more completely original; and of the remarks, the most important will probably be those which relate to the form and direction of the abutment of rafters; a subject which seems to have been very incorrectly treated by former writers on Carpentry.

I.—ABSTRACT OF THE DOCTRINE OF PASSIVE STRENGTH.

The effects of forces of different kinds, on the materials employed in the mechanical arts, require to be minutely examined in the arrangement of every work dependent on them; and of these effects, as exhibited in a solid body at rest, we may distinguish seven different varieties; the extension of a substance acting simply as a tie; the compression of a block supporting a load above it; the detrusion of an axis resting on a support close to its wheel, and resisting by its lateral adhesion only; the flexure of a body bent by a force applied unequally to its different parts; the torsion or twisting, arising from a partial detrusion of the external parts in opposite directions, while the axis retains its place; the alteration or permanent change of a body which settles, so as to remain in a new form, when the force is withdrawn; and lastly, the fracture, which consists in a complete separation of parts before united, and which has been the only effect particularly examined by the generality of authors on the strength of materials.

The analogy of the laws of extension and compression has been demonstrated in a former article (Bridge), and their connection with flexure has been investigated; but it is not easy to compare them directly with the resistance opposed to a partial detrusion, the effects of which are only so far understood as they are exhibited in the phenomenon of twisting; and these appear to justify us in considering the resistance of lateral adhesion as a primitive force, deduced from the rigidity or solidity of the substance, and proportional to the deviation from the natural situation of the particles. The resistance exhibited by steel wire, when twisted, bears a greater proportion to that of brass than the resistance to extension or compression, but the forces agree in being independent of the hardness produced by tempering.

Flexure may be occasioned either by a transverse or by a longitudinal force. When the force is transverse, the extent of the flexure is nearly proportional to its magnitude; but when it is longitudinal, there is a certain magnitude which it must exceed in order to produce, or rather to continue, the flexure, if the force be applied exactly at the axis. But it is equally true that the slightest possible force applied at a distance from the axis, however minute, or with an obliquity however small, or to a beam already a little curved, will produce a certain degree of flexure; and this observation will serve to explain some of the difficulties and irregularities which have occurred in making experiments on beams that are exposed to longitudinal pressure.

Stiffness, or the power of resisting flexure, is measured by the force required to produce a given minute change of form. For beams similarly fixed, it is directly proportional to the breadth and the cube of the depth, and inversely to the cube of the length. Thus a beam or bar two yards long will be equally stiff with a beam one yard, provided that it be either twice as deep or eight times as broad. If the ends of a beam can be firmly fixed, by continuing them to a sufficient distance, and keeping them down by a proper pressure, the stiffness will be four times as great as if the ends were simply supported. A hollow substance, of given weight and length, has its stiffness

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1 These introductory observations to Professor Robison's article, and the notes subjoined to it, were written by the late Dr Thomas Young. nearly proportional to the square of the diameter; and hence arises the great utility of tubes when stiffness is required, this property being still more increased by the expansion of the substance than the ultimate strength. It is obvious that there are a multiplicity of cases in carpentry where stiffness is of more importance than any other property, since the utility as well as beauty of the fabric might often be destroyed by too great a flexibility of the materials.

If we wish to find how much a beam of fir will sink when it is loaded in the middle, we may multiply the cube of the length in inches by the given weight in pounds, and divide by the cube of the depth, and by ten million times the breadth; but, on account of the unequal texture of the wood, we must expect to find the bending somewhat greater than this in practice, besides that a large weight will often produce an alteration, or permanent settling, which will be added to it; a beam of oak will also sink a little more than a beam of fir with the same weight.

With respect to torsion, the stiffness of a cylindrical body varies directly as the fourth power of the diameter, and inversely in the simple proportion of the length: it does not appear to be changed by the action of any force tending to lengthen or to compress the cylinder; and it may very possibly bear some simple relation to the force of cohesion, which has not yet been fully ascertained; but it appears that, in an experiment of Mr Cavendish, the resistance of a cylinder of copper to a twisting force, acting at its surface, was about \( \frac{1}{100} \) of the resistance that the same cylinder would have opposed to direct extension or compression.

Alteration is often an intermediate step between a temporary change and a complete fracture. There are many substances which, after bending to a certain extent, are no longer capable of resuming their original form; and in such cases it generally happens that the alteration may be increased without limit, until complete fracture takes place, by the continued operation of the same force which has begun it, or by a force a little greater. Those substances which are the most capable of this change are called ductile; and the most remarkable are gold, and a spider's web. When a substance has undergone an alteration by means of its ductility, its stiffness, in resisting small changes on either side, remains little or not at all altered. Thus, if the stiffness of a spider's web, in resisting torsion, were sufficient at the commencement of an experiment to cause it to recover itself, after being twisted in an angle of ten degrees, it would return ten degrees, and not more, after having been twisted round a thousand times. The ductility of all substances capable of being annealed is greatly modified by the effects of heat. Hard steel, for example, is incomparably less subject to alteration than soft, although in some cases more liable to fracture; so that the degree of hardness requires to be proportioned to the uses for which each instrument is intended; although it was proved by Coulomb, and has since been confirmed by other observers, that the primitive stiffness of steel, in resisting small flexures, is neither increased nor diminished by any variation in its temper.

The strength of a body is measured by the force required completely to overcome the corpuscular powers concerned in the aggregation of its particles, and it is jointly proportional to the primitive stiffness and to the toughness of the substance, that is, to the degree in which it is capable of a change of form without permanent alteration. It becomes, however, of importance in some cases to consider the measure of another kind of strength, which has sometimes been called resilience, or the power of resisting a body in motion, and which is proportional to the strength and the toughness conjointly, that is, to the stiffness and the square of the toughness. Thus, if we double the length of a given beam, we reduce its absolute strength to one half, and its stiffness to one eighth; but since the toughness, or the space through which it will continue to resist, is quadrupled, the resilience will be doubled, and it would require a double weight to fall from the same height, or the same weight to fall from a double height, in order to overcome its whole resistance. If we wish to determine the resilience of a body from an experiment on its strength, we must measure the distance through which it recedes or is bent previously to its fracture; and it may be shown that a weight which is capable of breaking it by pressure, would also break it by impulse if it moved with the velocity acquired by falling from a height equal to half the deflection. Thus, if a beam or bar were broken by a weight of 100 pounds, after being bent six inches without alteration, it would also be broken by a weight of 100 pounds falling from a height of three inches, or moving in a horizontal direction with a velocity of four feet in a second, or by a weight of one pound falling from a height of 300 inches. This substitution of velocity for quantity of matter has, however, one limit, beyond which the velocity must prevail over the resistance, without regard to the quantity of matter; and this limit is derived from the time required for the successive propagation of the pressure through the different parts of the substance, in order that they may participate in the resistance. Thus, if a weight fell on the end of a bar or column with a velocity of 100 feet in a second, and the substance could only be compressed \( \frac{1}{100} \) of its length, without being crushed, it is obvious that the pressure must be propagated through the substance with a velocity of 20,000 feet in a second, in order that it might resist the stroke; and, in general, a substance will be crushed or penetrated by any velocity exceeding that which is acquired by a body falling from a height, which is to half that of the modulus of elasticity of the substance, as the square of the greatest possible change of length is to the whole length. From the consideration of the effect of rigidity in lessening the resilience of bodies, we may understand how a diamond, which is capable of resisting an enormous pressure, may be crushed with a blow of a small hammer, moving with a moderate velocity. It is remarkable that, for the same substance in different forms, the resilience is in most cases simply proportional to the bulk or weight, while almost every other kind of resistance is capable of infinite variation by change of form only.

The elaborate investigations of M. Lagrange, respecting the strength and the strongest forms of columns, appear to have been conducted upon principles not altogether unexceptionable; but it is much easier to confute the results than to follow the steps of the computations. One great error is the supposition that columns are to be considered as elastic beams, bent by a longitudinal force; while, in reality, a stone column is never slender enough to be bent by a force which it can bear without being crushed; and even for such columns as are capable of being bent by a longitudinal force, M. Lagrange's determinations are in several instances inadmissible. He asserts, for example, that a cylinder is the strongest of all possible forms, and that a cone is stronger than any conoid of the same bulk; but it appears to be demonstrable in a very simple manner, and upon incontestable principles, that a conoidal form may be determined, which shall be stronger than either a cone or a cylinder of the same bulk.

When a column is crushed, its resistance to compression seems to depend in great measure on the force of lateral adhesion, assisted by a kind of internal friction, dependent on the magnitude of the pressure; and it commonly gives way by the separation of a wedge in an Carpentry. oblique direction. If the adhesion were simply proportional to the section, it may be shown that a square column would be most easily crushed when the angle of the wedge is equal to half of a right angle; but if the adhesion is increased by pressure, this angle will be diminished by half the angle of repose appropriate to the substance. In a wedge separated by a direct force from a prism of cast iron, the angle was found equal to $32^\circ$, consequently the angle of repose was $2 \times 12^\circ = 25^\circ$, and the internal friction to the pressure as $1 : 466$, the tangent of this angle; there was, however, a little bubble in the course of the fracture, which may have changed its direction in a slight degree. The magnitude of the lateral adhesion is measured by twice the height of the wedge, whatever its angle may be. In this instance the height was to the depth as $1:57$ to $1$, consequently the surface, affording an adhesion equal to the force, was somewhat more than three times as great as the transverse section, and the lateral adhesion of a square inch of cast iron would be equal to about $46,000$ pounds; the direct cohesive force of the same iron was found by experiment equal to about $20,000$ pounds for a square inch. It is obvious that experiments on the strength of a substance in resisting compression ought to be tried on pieces rather longer than cubes, since a cube would not allow of the free separation of a single wedge so acute as was observed in this experiment; although, indeed, the force required to separate a shorter wedge on each side would be little or no greater than for a single wedge. The same consideration of the oblique direction of the plane of easiest fracture would induce us to make the outline of a column a little convex externally, as the common practice has been; for a circle cut out of a plank possesses the advantage of resisting equally in every section, and consequently of exhibiting the strongest form, when there is no lateral adhesion; and in the case of an additional resistance proportional to the pressure, the strongest form is afforded by an oval consisting of two circular segments, each containing twice the angle formed by the plane of fracture with the horizon. If we wish to obtain a direct measure of the lateral adhesion, we must take care to apply the forces concerned at a distance from each other not greater than one sixth of the depth of the substance, otherwise the fracture will probably be rather the consequence of flexure than of destruction. Professor Robison found this force in some instances twice as great as the direct cohesion, or nearly in the same proportion as it appears to have been in the experiment on the strength of cast iron; Mr Coulomb thinks it most commonly equal only to the cohesion; and in fibrous substances, especially where the fibres are not perfectly straight, the repulsive strength is generally much less than would be inferred from this equality, and sometimes even less than the cohesive strength.

It is well known that the transverse strength of a beam is directly as the breadth and as the square of the depth, and inversely as the length; and the variation of the results of some experiments from this law can only have depended on accidental circumstances. If we wish to find the number of hundredweights that will break a beam of oak supported at both ends, supposing them to be placed exactly on the middle, we may multiply the square of the depth in inches by $100$ times the breadth, and divide by the length; and we may venture in practice to load a beam with at least an eighth as much as this, or, in case of necessity, even a fourth. And if the load be distributed equally throughout the length of the beam, it will support twice as much; but for a beam of fir the strength is somewhat less than for oak. A cylinder will bear the same curvature as the circumscribing prism, and it may be shown that its strength, as well as its stiffness, is to that of the prism as one fourth of its bulk is to one third of the bulk of the prism. The strength of a beam supported at its extremities may be doubled by firmly fixing the ends where it is practicable; and we have already seen that the stiffness is quadrupled: but the resilience remains unaltered, since the resistance is doubled, and the space through which it acts is reduced to a half. It is therefore obviously of importance to consider the nature of the resistance that is required from the fabric which we are constructing. A floor, considered alone, requires to be strong; but in connection with a ceiling, its stiffness requires more particular attention, in order that the ceiling may remain free from cracks. A coach-spring requires resilience for resisting the relative motions of the carriage, and we obtain this kind of strength as effectually by combining a number of separate plates, as if we united them into a single mass, while we avoid the stiffness, which would render the changes of motion inconveniently abrupt.

In all calculations respecting stiffness, it is necessary to be acquainted with the modulus of elasticity, which may be found for a variety of substances in the annexed table.

| Height of the Modulus of Elasticity in Thousands of Feet. | |--------------------------------------------------------| | Iron and steel ........................................ | 10,000 | | Copper .................................................. | 5,700 | | Brass .................................................... | 5,000 | | Silver ................................................... | 3,240 | | Tin ........................................................ | 2,250 | | Crown glass .............................................. | 9,800 | | Fir wood .................................................. | 10,000 | | Elm ........................................................ | 8,000 | | Beech ..................................................... | 8,000 | | Oak ........................................................ | 5,060 | | Box ........................................................ | 5,050 | | Ice .......................................................... | 850 |

II.—PROPOSITIONS RELATING TO FLEXURE.

A. The stiffness of a cylinder is to that of its circumscribing rectangular prism, as three times the bulk of the cylinder is to four times that of the prism.

We may consider the different strata of the substance as acting on levers equal in length to the distance of each from the axis; for although there is no fixed fulcrum at the axis, yet the whole force is the same as if such a fulcrum existed, since the opposite actions of the opposite parts would relieve the fulcrum from all pressure. Then the tension of each stratum being also as the same distance $x$, and the breadth of the stratum being called $2y$, the fluxion of the force on either side of the axis will be $2xydx$, while that of the force of the prism, the radius being $r$, is $2rx^2dx$. Now $z$ being the area of half the portion included between the stratum and the axis, of which the fluxion is $ydx$, the fluxion of $z - \frac{y^2}{r^2}$ will be

$$ydx - \frac{y^2dx}{r^2} = \frac{3y^2xdy}{r^2}.$$

But $1 - \frac{y^2}{r^2} = \frac{x^2}{r^2}$, and $-ydy = xdx$; therefore the fluxion is

$$\frac{x^2ydx}{r^2} + \frac{3x^2ydx}{r^2} = \frac{4x^2ydx}{r^2};$$

consequently the fluent of $2x^2ydx$ is $\frac{1}{3}x^3z - \frac{1}{3}y^3x$, which, when $y = 0$, becomes $\frac{1}{3}x^3z$, or one fourth of the product of the square of the radius by the area of the section, while the fluent of $2rx^2dx$, that is, $\frac{2}{3}rx^3$, the force of the prism, becomes $\frac{2}{3}r^4$ or $\frac{1}{3}r^2 \times 2r^2$, one third of the product of the same square into the area of the section of the prism.

Hence the radius of curvature of a cylindrical column, instead of $\frac{Maa}{12fy}$ (Art. BRIDGE, Prop. G), will be $\frac{Maa}{16fy}$ the weight of the modulus $M$ decreasing in the same propor- The force is supposed in this proposition to be either transverse or applied at a considerable distance from the axis; but the error will not be material in any other case.

B. When a longitudinal force \( f \) is applied to the extremities of a straight prismatic beam, at the distance \( b \) from the axis, the deflection of the middle of the beam will be

\[ b \left( \secant \left( \sqrt{\frac{3f}{M}} \cdot \frac{e}{a} \right) - 1 \right); \quad M \text{ being the weight of the modulus, } e \text{ the length of the beam, and } a \text{ its depth.} \]

The curvature being proportional to the distance from the line of direction of the force, or to the ordinate, when that line is considered as the absciss, the elastic curve must in this case initially coincide with a portion of the harmonic curve, well known for its utility in the resolution of a variety of problems of this kind. Now if the half length of the complete curve be called \( k \), corresponding to a quadrant of the generating circle, and the greatest ordinate \( y \), \( e \) being the quadrant of a circle of which the radius is unity, the radius of curvature \( r \) corresponding to \( y \) will be \( \frac{kk}{ecy} \), that is, a third proportional to \( y \) and \( \frac{k}{c} \) the radius of the generating circle; consequently

\[ \frac{Maa}{12fy} = \frac{kk}{ecy}, \quad kk = \frac{Maaec}{12f}, \quad \text{and} \quad k = \frac{1}{4} \sqrt{\frac{M}{3f}} \cdot ac; \]

but by the nature of the curve, \( y : b = 1 : \cos \frac{ec}{2k} = \sec \frac{ec}{2k} : 1 \), and

\[ y = b \sec \frac{ec}{2k} = b \sec \left( \sqrt{\frac{3f}{M}} \cdot \frac{e}{a} \right), \quad \text{which is the ordinate at the middle; and the deflection from the natural situation is } y - b. \]

It follows that, since the secant of the quadrant is infinite, when \( \sqrt{\frac{3f}{M}} \cdot \frac{e}{a} \) becomes equal to \( c \), the deflection will be infinite, and the resistance of the column will be overcome, however small the distance \( b \) may be taken, provided that it be of finite magnitude; and since in this case

\[ \frac{3f}{Maa} = ec, \quad f = \frac{Maaec}{Sec} = 8225 \frac{Maa}{ec}, \quad \text{which is the utmost force that the column will bear: and for a cylinder we find,} \]

by the same reasoning,

\[ f = \frac{Maaec}{4ce} = 6169 \frac{Maa}{ec}. \quad \text{If } b \text{ be supposed to vanish, we shall have in theory an equilibrium without flexure; but since it will be tottering, it cannot exist in nature.} \]

By applying this determination to the strength of wood and iron, compared with the modulus of elasticity, it appears that a round column or a square pillar of either of these substances cannot be bent by any longitudinal force applied to the axis, which it can withstand without being crushed, unless its length be greater than twelve or thirteen times its thickness respectively; nor a column or pillar of stone, unless it be forty or forty-five times as long as it is thick. Hence we may infer, as a practical rule, that every piece of timber or iron intended to withstand any considerable compressing force, should be at least as many inches in thickness as it is feet in length, in order to avoid the loss of force which necessarily arises from curvature.

C. When a beam, fixed at one end, is pressed by a force in a direction deviating from the original position of the axis in a small angle, of which the tangent is \( t \), the deflection becomes \( d = \frac{M}{12f} \tan \left( \sqrt{\frac{12f}{M}} \cdot \frac{e}{a} \right) \).

The inclination of the curve to the absciss being inconsiderable, it will not differ sensibly from a portion of a harmonic curve; and supposing the quadrantal length of the curve \( k \), we have again, as in the last proposition,

\[ k = \frac{1}{4} \sqrt{\frac{M}{3f}} \cdot ac, \quad \text{or, for a cylinder, } k = \frac{1}{4} \sqrt{\frac{M}{f}} \cdot ac. \]

Now, the tangent of the inclination of the harmonic curve varies as the sine of the angular distance from the middle; consequently, as \( \sin \frac{ec}{k} \cdot e \), or \( \cos \frac{ec}{k} \), is to the radius, so is the tangent \( t \), expressing the difference of inclination of the end of the beam and the direction of the force, which is also that of the middle of the supposed curve, to the tangent of the extreme inclination of the curve to its absciss, which will therefore be \( t \sec \frac{ec}{k} \); consequently the greatest ordinate will be \( \frac{kt}{c} \sec \frac{ec}{k} \), and since the ordinates are as the sines of the angular distances from the origin of the curve, the ordinate at the fixed end of the beam, corresponding to the angle \( \frac{ec}{k} \), that is, the deflection, will be \( \frac{kt}{c} \sec \frac{ec}{k} \sin \frac{ec}{k} = \frac{kt}{c} \tan \frac{ec}{k} = \frac{1}{4} \sqrt{\frac{M}{3f}} \cdot at \tan \frac{ec}{k} \).

TANG. \( \frac{2e}{a} \sqrt{\frac{3f}{M}} \), or, for a cylinder, \( \frac{1}{4} \sqrt{\frac{M}{f}} \cdot at \tan \frac{4e}{a} \).

By means of this proposition we may determine the effect of a small lateral force in weakening a beam or pillar which is at the same time compressed longitudinally by a much greater force, considering the parts on each side of the point to which the lateral force is applied, as portions of two beams, bent in the manner here described, by a single force slightly inclined to the axis.

D. A bar fixed at one end, and bent by a transverse force applied to the other end, assumes initially the form of a cubic parabola, and the deflection at the end is \( d = \frac{4ef}{Maa} \).

The ordinate of a cubic parabola varying as \( x^3 \), its second fluxion varies as \( 6x(dx)^2 \), or since the first fluxion of the absciss is constant, simply as the absciss \( x \), measured from the vertex of the parabola, which must therefore be situated at the end to which the force is applied, and the absciss must coincide with the tangent of the bar. But if we begin from the other end, we must substitute \( e-x \) for \( x \), and the second fluxion of the ordinate will be as \( 6(e-x)(dx)^2 \), the first as \( 6edx-3x^2dx \), and the fluent as \( 3x^3-x^3 \), which, when \( x=e \), becomes \( 2e^3 \), while it would have been \( 3e^3 \) if the curvature had been uniform, and the second fluxion had been everywhere \( 6e(dx)^2 \). Now the radius of curvature at the fixed end being \( r = \frac{Maa}{12ef} \), and the versed sine of a small portion of a circle being equal to \( \frac{ec}{2r} \), this versed sine will be expressed by \( \frac{6ef}{Maa} \); and two thirds of this, or \( \frac{4ef}{Maa} \), will be the actual deflection.

E. The depression of a bar, fixed horizontally at one end, and supporting only its own weight, is \( d = \frac{3e^4}{2maa} \); \( m \) being the height of the modulus of elasticity.

The curvature here varies as the square of the distance from the end, because the strain is proportional to the weight of the portion of the bar beyond any given point, and to the distance of its centre of gravity conjointly, that is, to \( (e-x) \frac{1}{2} (e-x) \), so that if the second fluxion Carpentry at the fixed end be as \( e^2 (dx)^2 \); it will elsewhere be as \((e-x)^2 (dx)^2 \); and the corresponding first fluxions being \( e^2 dx \) and \( e^2 dx - ex dx + \frac{1}{2} x^2 dx \), the fluents will be \( e^2 x \) and \( \frac{1}{2} e^2 x^2 - \frac{1}{3} ex^3 + \frac{1}{4} x^4 \), or, when \( x = e \), \( \frac{1}{2} e^4 \), and \( \frac{1}{2} e^4 + \frac{1}{12} e^4 = \frac{1}{6} e^4 \); consequently the depression must be half the versed sine in the circle of greatest curva- ture. Now the radius of curvature \( \frac{Maa}{12fy} \) becomes here \( \frac{Maa}{6e^2} \), the force being applied at the distance \( \frac{1}{2} e \); and since the weight of the bar is to that of the modulus of elasticity in the proportion of the respective lengths, we have \( \frac{f}{M} = \frac{e}{m} \), and \( r = \frac{maa}{6ce} \), and the versed sine for the ordinate \( e \) will be \( \frac{3e^4}{maa} \), half of which is the actual de- pression.

F. The depression of the middle of a horizontal bar, fixed at both ends, and supporting its own weight only, is \( d = \frac{5e^4}{32maa} \).

The transverse force at each point of such a bar, re- sisted by the lateral adhesion, is as the distance \( x \) from the middle (Art. Bridge, under Prop. L); but this force is proportional to the first fluxion of the strain or curva- ture, consequently the curvature itself must vary as the corrected fluent of \( \frac{1}{2} x^2 dx \), taking here the negative sign, because the curvature diminishes as \( x \) increases; and the corrected fluent will be \( \frac{1}{2} e^2 - x^2 \), since it must vanish when \( x = \frac{1}{2} e \); the first fluxion of the ordinate will then be \( \frac{1}{2} e^2 dx - \frac{1}{2} x^2 dx \), and the fluent \( \frac{1}{6} e^2 x^2 - \frac{1}{12} x^3 \), or for the whole length \( \frac{1}{2} e^2 - \frac{1}{12} e^4 \); instead of \( \frac{1}{2} e^2 \), or \( \frac{1}{12} e^4 \), which would have been its value if the curvature had been equal throughout. Now the strain at the middle is the differ- ence of the opposite strains produced by the forces act- ing on either side; and these are the half weight acting at the mean distance \( \frac{1}{2} e \), and the resistance of the support, which is equal to the same half weight, but acts at the distance \( \frac{1}{2} e \), the difference being equivalent to the half weight, acting at the distance \( \frac{1}{2} e \), so that the curvature at the middle is the same as if the bar were fixed there, and loose at the ends; that is, as in the last proposition, sub- stituting \( \frac{1}{2} e \) for \( e \), \( r = \frac{2maa}{3ce} \); and the versed sine at the distance \( \frac{1}{2} e \) being \( \frac{e^4}{8r} \) or \( \frac{3e^4}{16maa} \), \( \frac{1}{2} \) of this will be \( \frac{5e^4}{32maa} \).

This demonstration may serve as an illustration of two modes of considering the effect of a strain, which have not been generally known, and which are capable of a very extensive application.

It follows that where a bar is equally loaded through- out its length, the curvature at the middle is half as great as if the whole weight were collected there, the strain de- rived from the resistance of the support remaining in that case uncompensated. The depression produced by the divided weight will be \( \frac{5}{8} \) as great as by the single weight, since \( \frac{5}{8} \times \frac{1}{2} \) is to \( \frac{5}{8} \) as 5 to 8. M. Dupin found the propo- sition, by many experiments, between \( \frac{5}{8} \) and \( \frac{5}{8} \); and \( \frac{5}{8} \) is a very good mean for representing these results.

III.—ELEMENTS OF CARPENTRY.

Definition. "Carpentry is the art of framing timber for the pur- poses of architecture, machinery, and, in general, for all considerable structures."

It is not intended in this article to give a full account of carpentry as a mechanical art, or to describe the various ways of executing its different works, suited to the varie- ty of materials employed, the processes which must be followed for fashioning and framing them for our purposes, and the tools which must be used, and the manner in which they must be handled. This would be an occupa- tion for volumes, and, though of great importance, must be entirely omitted here. Our only aim at present will be to deduce, from the principles and laws of mechanics, and the knowledge which experience, and judicious inferences from it, have given us concerning the strength of timber, in relation to the strain laid on it, such maxims of con- struction as will unite economy with strength and efficacy.

This object is to be attained by a knowledge, 1st, of the strength of our materials, and of the absolute strain that is to be laid on them; 2dly, of the modifications of this strain, by the place and direction in which it is exerted, and the changes that can be made by a proper disposition of the parts of our structure; and, 3dly, having disposed every piece in such a manner as to derive the utmost advantage from its relative strength, we must know how to form the joints and other connections in such a manner as to secure the advantages derived from this disposition.

This is evidently a branch of mechanical science which makes carpentry a liberal art, constitutes part of the learn- ing of the Engineer, and distinguishes him from the me- chanic workman. Its importance in all times and states of civil- society is manifest and great. In the present condition of these kingdoms, raised by the active ingenuity and ener- gy of our countrymen to a pitch of prosperity and in- fluence unequalled in the history of the world, a condition which consists chiefly in the superiority of our manufac- tures, attained by prodigious multiplication of engines of every description, and for every species of labour, the Science (so to term it) of carpentry is of immense conse- quence. We regret therefore exceedingly that none of our celebrated artists have done honour to themselves and their country, by digesting into a body of consecutive doc- trines the results of their experience, so as to form a sys- tem from which their pupils might derive the first prin- ciples of their education. The many volumes called Com- plete Instructors, Manuals, &c. take a much humbler flight, and content themselves with instructing the mere workman; or sometimes give the master builder a few ap- proved forms of roofs and other framings, with the rules for drawing them on paper, and from thence forming the working draughts which must guide the saw and the chisel of the workman. Hardly any of them offer any thing that can be called a principle, applicable to many particular cases, with the rules for this adaptation. We are indebted for Princi- ples the greatest part of our knowledge of this subject to the indefatig- labours of literary men, chiefly foreigners, who have pub- lished in the memoirs of the learned academies disserta- tions on different parts of what may be termed the Science of Carpentry. It is singular that the members of the Royal Society of London, and even of that established and supported for the encouragement of the arts, have con- tributed so little to the public instruction in this respect. We have observed some beginnings of this kind, such as the last part of Nicholson's Carpenter's and Joiner's Assistant; and it is with pleasure we can say, that we were told by the editor this work was prompted in a great measure by what has been delivered in our articles Roof and Strength of Materials. It abounds more in important and new observations than any book of the kind that we are ac- quainted with. We again call on such as have given a scientific attention to this subject, and pray that they would render a meritorious service to their country by imparting the result of their researches. The very limited nature of this work does not allow us to treat the subject in de- try-tail; and we must confine our observations to the fundamental and leading propositions.

The theory, so to term it, of carpentry is founded on two distinct portions of mechanical science, namely, a knowledge of the strains to which framings of timber are exposed, and a knowledge of their relative strength.

We shall therefore attempt to bring into one point of view the propositions of mechanical science that are more immediately applicable to the art of carpentry, and are to be found in various articles of our work, particularly Roof and Strength of Materials. From these propositions we hope to deduce such principles as shall enable an attentive reader to comprehend distinctly what is to be aimed at in framing timber, and how to attain this object with certainty; and we shall illustrate and confirm our principles by examples of pieces of carpentry which are acknowledged to be excellent in their kind.

The most important proposition of general mechanics to the carpenter is that which exhibits the composition and resolution of forces; and we beg our practical readers to endeavour to form very distinct conceptions of it, and to make it very familiar to their minds. When accommodated to their chief purposes, it may be thus expressed:

1. If a body, or any part of a body, be at once pressed in the two directions AB, AC (fig. 1, Plate CXLVII), and if the intensity or force of those pressures be in the proportion of these two lines, the body is affected in the same manner as if it were pressed by a single force acting in the direction AD, which is the diagonal of the parallelogram ABDC formed by the two lines, and whose intensity has the same proportion to the intensity of each of the other two that AD has to AB or AC.

Such of our readers as have studied the laws of motion, know that this is fully demonstrated. Such as wish for a very accurate view of this proposition will do well to read the demonstration given by D. Bernoulli, in the first volume of the Comment. Petropol., and the improvement of this demonstration by D'Alembert in his Opuscules and in the Comment. Turinensis. The practitioner in carpentry will get more useful confidence in the doctrine, if he will shut his book, and verify the theoretical demonstrations by actual experiments. They are remarkably easy and convincing. Therefore it is our request that the artist, who is not so habitually acquainted with the subject, do not proceed further till he has made it quite familiar to his thoughts. Nothing is so conducive to this as the actual experiment; and since this only requires the trifling expense of two small pulleys and a few yards of whalecord, we hope that none of our practical readers will omit it; they will thank us for this injunction.

2. Let the threads Ad, Ab, and AEc (fig. 2), have the weights d, b, and c, appended to them, and let two of the threads be laid over the pulleys F and E. By this apparatus the knot A will be drawn in the directions AB, AC, and AK. If the sum of the weights b and c be greater than the single weight d, the assemblage will of itself settle in a certain determined form: if you pull the knot A out of its place, it will always return to it again, and will rest in no other position. For example, if the three weights are equal, the threads will always make equal angles, of 120 degrees each, round the knot. If one of the weights be three pounds, another four, and the third five, the angle opposite to the thread stretched by five pounds will always be square, &c. When the knot A is thus in equilibrio, we must infer that the action of the weight d, in the direction Ad, is in direct opposition to the combined action of b in the direction AB, and of c in the direction AC. Therefore, if we produce dA to any point D, and take AD to represent the magnitude of the force, or pressure exerted by the weight d, the pressures Carpentery exerted on A by the weights b and c, in the directions AB, AC, are in fact equivalent to a pressure acting in the direction AD, whose intensity we have represented by AD. If we now measure off by a scale on AF and AE the lines AB and AC, having the same proportions to AD that the weights b and c have to the weight d, and if we draw DB and DC, we shall find DC to be equal and parallel to AB, and DB equal and parallel to AC; so that AD is the diagonal of the parallelogram ABDC. We shall find this always to be the case, whatever are the weights made use of; only we must take care that the weight which we cause to act without the intervention of a pulley be less than the sum of the other two; if any one of the weights exceeds the sum of the other two, it will prevail, and drag them along with it.

Now since we know that the weight d would just balance an equal weight g, pulling directly upwards by the intervention of the pulley G; and since we see that it just balances the weights b and c, acting in the directions AB, AC; we must infer that the knot A is affected in the same manner by those two weights, or by the single weight g; and therefore that two pressures, acting in the directions and with the intensities AB, AC, are equivalent to a single pressure having the direction and proportion of AD. In like manner, the pressures AB, AK, are equivalent to AH, which is equal and opposite to AC. Also AK and AC are equivalent to AI, which is equal and opposite to AB.

We shall consider this combination of pressures a little more particularly.

Suppose an upright beam BA (fig. 3), pushed in the direction of its length by a load B, and abutting on the ends of two beams AC, AD, which are firmly resisted at their extreme points C and D, which rest on two blocks, but are nowise joined to them; these two beams can resist no way but in the directions CA, DA, and therefore the pressures which they sustain from the beam BA are in the directions AC, AD. We wish to know how much each sustains: Produce BA to E, taking AE from a scale of equal parts, to represent the number of tons or pounds by which BA is pressed. Draw EF and EG parallel to AD and AC; then AF measured on the same scale, will give us the number of pounds by which AC is strained or crushed, and AG will give the strain on AD.

It deserves particular remark here, that the length of AC or AD has no influence on the strain arising from the thrust BA, while the directions remain the same. The effects, however, of this strain are modified by the length of the piece on which it is exerted. This strain compresses the beam, and will therefore compress a beam of double length twice as much. This may change the form of the assemblage. If AC, for example, be very much shorter than AD, it will be much less compressed: the line CA will turn about the centre C, while DA will hardly change its position; and the angle CAD will grow more open, the point A sinking down. The artist will find it of great consequence to pay a very minute attention to this circumstance, and to be able to see clearly the change of shape which necessarily results from these mutual strains. He will see in this the cause of failure in many very great works. By thus changing shape, strains are often produced in places where there were none before, and frequently of the very worst kind, tending to break the beams across.

The dotted lines of this figure show another position of the beam AD. This makes a prodigious change, not only in the strain on AD, but also in that on AC. Both of them are much increased; AG is almost doubled, and AF is four times greater than before. This addition was Carpentry made to the figure to show what enormous strains may be produced by a very moderate force, AE, when it is exerted on a very obtuse angle.

The fourth and fifth figures will assist the most uninstructed reader in conceiving how the very same strains, AF, AG, are laid on these beams, by a weight simply hanging from a billet resting on A, pressing hard on AD, and also leaning a little on AC; or by an upright piece, AE, joggled on the two beams AC, AD, and performing the office of an ordinary king-post. The reader will thus learn to call off his attention from the means by which the strains are produced, and learn to consider them abstractedly merely as strains, in whatever situation he finds them, and from whatever cause they arise.

We presume that every reader will perceive, that the proportions of these strains will be precisely the same if every thing be inverted, and each beam be drawn or pulled in the opposite direction. In the same way that we have substituted a rope and weight in fig. 4, or a king-post in fig. 5, for the loaded beam BA of fig. 3, we might have substituted the framing of fig. 6, which is a very usual practice. In this framing, the batten DA is stretched by a force AG, and the piece AC is compressed by a force AF. It is evident that we may employ a rope or an iron rod hooked on at D, in place of the batten DA, and the strains will be the same as before.

This seemingly simple matter is still full of instruction; and we hope that the well-informed reader will pardon us, though we dwell a little longer on it for the sake of the young artist.

By changing the form of this framing, as in fig. 7, we produce the same strains as in the disposition represented by the dotted lines in fig. 3. The strains on both the battens AD, AC, are now greatly increased.

The same consequences result from an improper change of the position of AC. If it is placed as in fig. 8, the strains on both are vastly increased. In short, the rule is general, that the more open we make the angle against which the push is exerted, the greater are the strains which are brought on the struts or ties which form the sides of the angle.

The reader may not readily conceive the piece AC of fig. 8 as sustaining a compression; for the weight B appears to hang from AC as much as from AD. But his doubts will be removed by considering whether he could employ a rope in place of AC. He cannot; but AD may be exchanged for a rope. AC is therefore a strut, and not a tie.

In fig. 9, Plate CXLVIII. AD is again a strut, butting on the block D, and AC is a tie; and the batten AC may be replaced by a rope. While AD is compressed by the force AG, AC is stretched by the force AF.

If we give AC the position represented by the dotted lines, the compression of AD is now AG', and the force stretching AC' is now AF'; both much greater than they were before. This disposition is analogous to fig. 8, and to the dotted lines in fig. 3. Nor will the young artist have any doubts of AC' being on the stretch, if he consider whether AD can be replaced by a rope. It cannot, but AC' may; and it is therefore not compressed, but stretched.

In fig. 10 all the three pieces, AC, AD, and AB, are ties, on the stretch. This is the complete inversion of fig. 3; and the dotted position of AC induces the same changes in the forces AF, AG', as in fig. 3.

Thus have we gone over all the varieties which can happen in the bearings of three pieces on one point. All calculations about the strength of carpentry are reduced to this case; for when more ties or braces meet in a point (a thing that rarely happens), we reduce them to three, by substituting for any two the force which results from their combination, and then combining this with another; and so on.

The young artist must be particularly careful not to mistake the kind of strain that is exerted on any piece of the framing, and suppose a piece to be a brace which is really a tie. It is very easy to avoid all mistakes in this matter by the following rule, which has no exception. (See Note AA.)

Take notice of the direction in which the piece acts from which the strain proceeds. Draw a line in that direction from the point on which the strain is exerted, and let its length (measured on some scale of equal parts) express the magnitude of this action in pounds, hundreds, tons. From its remote extremity draw lines parallel to the pieces on which the strain is exerted. The line parallel to one piece will necessarily cut the other, or its direction produced. If it cut the piece itself, that piece is compressed by the strain, and it is performing the office of a strut or brace; if it cut its direction produced, the piece is stretched, and it is a tie. In short, the strains on the pieces AC, AD, are to be estimated in the direction of the points F and G from the strained point A. Thus, in fig. 3, the upright piece BA, loaded with the weight B, presses the point A in the direction AE; so does the rope AB in the other figures, or the batten AB in fig. 5.

In general, if the straining piece is within the angle formed by the pieces which are strained, the strains which they sustain are of the opposite kind to that which it exerts. If it be pushing, they are drawing; but if it be within the angle formed by their directions produced, the strains which they sustain are of the same kind. All the three are either drawing or pressing. If the straining piece lie within the angle formed by one piece and the produced direction of the other, its own strain, whether compression or extension, is of the same kind with that of the most remote of the other two, and opposite to that of the nearest. Thus, in fig. 9, where AB is drawing, the remote piece AC is also drawing, while AD is pushing or resisting compression.

In all that has been said on this subject, we have not spoken of any joints. In the calculations with which we are occupied at present, the resistance of joints has no share; and we must not suppose that they exert any force which tends to prevent the angles from changing. The joints are supposed perfectly flexible, or to be like compass joints, the pin of which only keeps the pieces together when one or more of the pieces draws or pulls. The carpenter must always suppose them all compass joints when he calculates the thrusts and draughts of the different pieces of his frames. The strains on joints, and their power to produce or balance them, are of a different kind, and require a very different examination.

Seeing that the angles which the pieces make with each other are of such importance to the magnitude and the proportion of the excited strains, it is proper to find out some way of readily and compendiously conceiving and expressing this analogy.

In general the strain on any piece is proportional to the straining force. This is evident.

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1 The reader is requested to add accents to the extreme letters D and F of fig. 3, which correspond to the position of the beam AGD indicated by the dotted lines. Accents are also wanted to the upper F and the lower C and G in fig. 9; also to the upper F and lower G in fig. 10; and in this & should be C. In fig. 11, the i towards the left should be t, and an accent is wanting over the upper f. In fig. 12, the dotted line CK should be continued upward and marked L. In fig. 16, the letters should stand thus, A C E & D f F B. Secondly, the strain on any piece AC is proportional to the sine of the angle which the straining force makes with the other piece directly, and to the sine of the angle which the pieces make with each other inversely.

For it is plain that the three pressures AE, AF, and AG, which are exerted at the point A, are in the proportion of the lines AE, AF, and FE (because FE is equal to AG). But because the sides of a triangle are proportional to the sines of the opposite angles, the strains are proportional to the sines of the angles AFE, AEF, and FAE. But the sine of AFE is the same with the sine of the angle CAD, which the two pieces AC and AD make with each other; and the sine of AEF is the same with the sine of EAD, which the straining piece BA makes with the piece AC. Therefore we have this analogy, Sin. CAD : Sin. EAD = AE : AF, and AF = AE × Sin. CAD. Now the sines of angles are most conveniently conceived as decimal fractions of the radius, which is considered as unity. Thus, Sin. 30° is the same thing with 0.5, or \( \frac{1}{2} \); and so of others. Therefore, to have the strain on AC, arising from any load AE acting in the direction AE, multiply AE by the sine of EAD, and divide the product by the sine of CAD.

This rule shows how great the strains must be when the angle CAD becomes very open, approaching to 180 degrees. But when the angle CAD becomes very small, its sine (which is our divisor) is also very small; and we should expect a very great quotient in this case also. But we must observe, that in this case the sine of EAD is also very small; and this is our multiplier. In such a case, the quotient cannot exceed unity.

But it is unnecessary to consider the calculation by the tables of sines more particularly. The angles are seldom known any otherwise but by drawing the figure of the frame of carpentry. In this case, we can always obtain the measures of the strains from the same scale, with equal accuracy, by drawing the parallelogram AFCG.

Hitherto we have considered the strains excited at A only as they affect the pieces on which they are exerted. But the pieces, in order to sustain, or be subject to, any strain, must be supported at their ends C and D; and we may consider them as mere intermediums, by which these strains are made to act on those points of support: Therefore AF and AG are also measures of the forces which press or pull at C and D. Thus we learn the supports which must be found for these points. These may be infinitely various. We shall attend only to such as somehow depend on the framing itself.

Such a structure as fig. II very frequently occurs, where a beam BA is strongly pressed to the end of another beam AD, which is prevented from yielding, both because it lies on another beam HD, and because its end D is hindered from sliding backwards. It is indifferent from what this pressure arises: we have represented it as owing to a weight hung on at B, while B is withheld from yielding by a rod or rope hooked to the wall. The beam AD may be supposed at full liberty to exert all its pressure on D, as if it were supported on rollers lodged in the beam HD; but the loaded beam BA presses both on the beam AD and on HD. We wish only to know what strain is borne by AD.

All bodies act on each other in the direction perpendicular to their touching surfaces; therefore the support given by HD is in a direction perpendicular to it. We may therefore supply its place at A by a beam AC, perpendicular to HD, and firmly supported at C. In this case, therefore, we may take AE, as before, to represent the pressure exerted by the loaded beam, and draw EG perpendicular to AD, and EF parallel to it, meeting the perpendicular AC in F. Then AG is the strain compressing AD, and AF is the pressure on the beam HD.

It may be thought, that since we assume as a principle the form that the mutual pressures of solid bodies are exerted perpendicular to their touching surfaces, this balance of pressures, in framings of timbers, depends on the directions of joint of no their butting joints; but it does not, as will readily appear by considering the present case. Let the joint or abutment of the two pieces BA, AD, be mitred in the usual manner, in the direction fA'. Therefore, if Ac be drawn perpendicular to Af, it will be the direction of the actual pressure exerted by the loaded beam BA on the beam AD. But the re-action of AD, in the opposite direction At, will not balance the pressure of BA; because it is not in the direction precisely opposite. BA will therefore slide along the joint, and press on the beam HD. AE represents the load on the mitre joint A. Draw Ec perpendicular to Ac, and Ef parallel to it. The pressure AE will be balanced by the re-actions eA and fA'; or, the pressure AE produces the pressures Ac and Af, of which Af must be resisted by the beam HD, and Ac by the beam AD. The pressure Af not being perpendicular to HD, cannot be fully resisted by it; because (by our assumed principle) it re-acts only in a direction perpendicular to its surface. Therefore draw fp, fi, parallel to HD, and perpendicular to it. The pressure Af will be resisted by HD with the force pA; but there is required another force iA, to prevent the beam BA from slipping outwards. This must be furnished by the re-action of the beam DA. (See Note BB.) In like manner, the other force Ac cannot be fully resisted by the beam AD, or rather by the prop D, acting by the intervention of the beam; for the action of that prop is exerted through the beam in the direction DA. The beam AD, therefore, is pressed to the beam HD by the force Ac, as well as by Af. To find what this pressure on HD is, draw eg perpendicular to HD, and eo parallel to it, cutting EG in r. The forces gA and oA will resist, and balance Ac.

Thus we see that the two forces Ac and Af, which are equivalent to AE, are equivalent also to Ap, At, Ao, and Ag. But because Af and eE are equal and parallel, and Er and fi are also parallel, as also cr and fn, it is evident, that if is equal to xe, or to oP, and iA is equal to ro, or to Gg. Therefore the four forces Ag, Ao, Ap, Ai, are equal to AG and AF. Therefore AG is the compression of the beam AD, or the force pressing it on D; and AF is the force pressing it on the beam HD. The proportion of these pressures, therefore, is not affected by the form of the joint.

This remark is important; for many carpenters think the form and direction of the butting joint of great importance; and even the theorist, by not prosecuting the general principle through all its consequences, may be led into an error. The form of the joint is of no importance, in as far as it affects the strains in the direction of the beams; but it is often of great consequence, in respect to its own firmness, and the effect it may have in bruising the piece on which it acts, or being crippled by it.

The same compression of AB, and the same thrust on the point D by the intervention of AD, will obtain, in whatever way the original pressure on the end A is produced. Thus, supposing that a cord is made fast at A, beam, and pulled in the direction AE, and with the same force, the beam AD will be equally compressed, and the prop D must re-act with the same force.

But it often happens that the obliquity of the pressure on AD, instead of compressing it, stretches it; and we desire to know what tension it sustains. Of this we have a familiar example in a common roof. Let the two rafters AC, AD (fig. 12), press on the tie-beam DC. We may Carpentry, suppose the whole weight to press vertically on the ridge A, as if a weight B were hung on there. (See Note C.) We may represent this weight by the portion Ab of the vertical or plumb line, intercepted between the ridge and the beam. Then drawing bf and bg parallel to AD and AC, Ag and Af will represent the pressures on AC and AD. Produce AC till CH be equal to Af. The point C is forced out in this direction, and with a force represented by this line. As this force is not perpendicular-ly across the beam, it evidently stretches it; and this extending force must be withstood by an equal force pulling it in the opposite direction. This must arise from a similar oblique thrust of the opposite rafter on the other end D. We concern ourselves only with this extension at present; but we see that the cohesion of the beam does nothing but supply the balance to the extending forces. It must still be supported externally, that it may resist, and by resisting obliquely, be stretched. The points C and D are supported on the walls, which they press in the directions CK and DO, parallel to Ab. If we draw HK parallel to DC, and HI parallel to CK (that is to Ab), meeting DC produced in I, it follows from the composition of forces, that the point C would be supported by the two forces KC and IC. In like manner, making DN = Ag, and completing the parallelogram DMNO, the point D would be supported by the forces OD and MD. If we draw go and fe parallel to DC, it is plain that they are equal to NO and CI, while Ao and Ak are equal to DO and CK, and Ab is equal to the sum of DO and CK (because it is equal to Ao + Ak). The weight of the roof is equal to its vertical pressure on the walls.

Thus we see, that while a pressure on A, in the direction Ab, produces the strains Af and Ag, on the pieces AC and AD, it also excites a strain CI or DM in the piece DC. And this completes the mechanism of a frame; for all derive their efficacy from the triangles of which they are composed, as will appear more clearly as we proceed.

But there is more to be learned from this. The consideration of the strains on the two pieces AD and AC, by the action of a force at A, only showed them as the means of propagating the same strains in their own direction to the points of support. But, by adding the strains exerted in DC, we see that the frame becomes an intermediate, by which exertions may be made on other bodies in certain directions and proportions, so that this frame may become part of a more complicated one, and, as it were, an element of its constitution. It is worth while to ascertain the proportion of the pressures CK and DO, which are thus exerted on the walls. The similarity of triangles gives the following analogies:

\[ \frac{DO}{DM} = \frac{Ab}{bD} \]

\[ CI \text{ or } DM : CK : Cb : Ab \]

Therefore \( DO : CK : Cb : bD \).

Or, the pressures on the points C and D, in the direction of the straining force Ab, are reciprocally proportional to the portions of DC intercepted by Ab.

Also, since Ab is = DO + CK, we have

\[ Ab : CK = Cb + bD \text{ (or CD) : bD, and } Ab : DO = CD : bC. \]

In general, any two of the three parallel forces Ab, DO, CK, are to each other in the reciprocal proportion of the parts of CD, intercepted between their directions and the direction of the third.

And this explains a still more important office of the frame ADC. If one of the points, such as D, be supported, an external power acting at A, in the direction Ab, and with an intensity which may be measured by Ab, may be set in equilibrio with another acting at C, in the direction CL, opposite to CK or Ab, and with an intensity represented by CK; for since the pressure CH is partly withstood by the force IC, or the firmness of the beam DC supported at D, the force KC will complete the balance. When we do not attend to the support at D, we conceive the force Ab to be balanced by KC, or KC to be balanced by Ab. And, in like manner, we may neglect the support or force acting at A, and consider the force DO as balanced by CK.

Thus our frame becomes a lever, and we are able to trace the interior mechanical procedure which gives it its efficacy: it is by the intervention of the forces of cohesion, which connect the points to which the external forces are applied with the supported point or fulcrum and with each other.

These strains or pressures Ab, DO, and CK, not being in the directions of the beams, may be called transverse. We see that by their means a frame of carpentry may be considered as a solid body: but the example which brought this to our view is too limited for explaining the efficacy which may be given to such constructions. We shall therefore give a general proposition, which will more distinctly explain the procedure of nature, and enable us to trace the strains as they are propagated through all the parts of the most complicated framing, finally producing the exertion of its most distant points.

We presume that the reader is now pretty well habituated to the conception of the strains as they are propagated along the lines joining the points of a frame, and we shall therefore employ a very simple figure.

Let the strong lines ACBD (fig. 13) represent a frame of carpentry. Suppose that it is pulled at the point A by a force acting in the direction AE, but that it rests on a fixed point C, and that the other extreme point B is held back by a power which resists in the direction BF: It is required to determine the proportion of the strains excited in its different parts, the proportion of the external pressures at A and B, and the pressure which is produced on the obstacle or fulcrum C.

It is evident that each of the external forces at A and B tend one way, or to one side of the frame, and that each would cause it to turn round C if the other did not prevent it; and that if, notwithstanding their action, it is turned neither way, the forces in actual exertion are in equilibrio by the intervention of the frame. It is no less evident that these forces concur in pressing the frame on the prop C. Therefore, if the piece CD were away, and if the joints C and D be perfectly flexible, the pieces CA, CB, would be turned round the prop C, and the pieces AD, DB, would also turn with them, and the whole frame change its form. This shows, by the way, and we desire it to be carefully kept in mind, that the firmness or stiffness of framing depends entirely on the triangles bounded by beams which are contained in it. An open quadrilateral may always change its shape, the sides revolving round the angles. A quadrilateral may have an infinity of forms, without any change of its sides, by merely pushing two opposite angles towards each other, or drawing them asunder. But when the three sides of a triangle are determined, its shape is also invariably determined; and if two angles be held fast, the third cannot be moved. It is thus that, by inserting the bar CD, the figure becomes unchangeable; and any attempt to change it by applying a force to an angle A, immediately excites forces of attraction or repulsion between the particles of the stuff which form its sides. Thus it happens, in the present instance, that a change of shape is prevented by the bar CD. The power at A presses its end against the prop; and in doing this it puts the bar AD on the stretch, and also the bar DB. Their places might therefore be supplied by cords or metal wires. Hence it is evident that DC is compressed, as is also AC; and, for the same reason, CB is also in a state of compression; for either A or B may be considered as the point that is impelled or withheld. Therefore DA and DB are stretched, and are resisting with attractive forces. DC and CB are compressed, and are resisting with repulsive forces; and thus the support of the prop, combined with the firmness of DC, puts the frame ADBC into the condition of the two frames in fig. 8 and fig. 9. Therefore the external force at A is really in equilibrium with an attracting force acting in the direction AD, and a repulsive force acting in the direction AK. And since all the connecting forces are mutual and equal, the point D is pulled or drawn in the direction DA. The condition of the point B is similar to that of A, and D is also drawn in the direction DB. Thus the point D, being urged by the forces in the directions DA and DB, presses the beam DC on the prop, and the prop resists in the opposite direction. Therefore the line DC is the diagonal of the parallelogram, whose sides have the proportion of the forces which connect D with A and B. This is the principle on which the rest of our investigation proceeds. We may take DC as the representation and measure of their joint effect. Therefore draw CH, CG, parallel to DA, DB. Draw HL, GO, parallel to CA, CB, cutting AE, BF, in L and O, and cutting DA, DB, in I and M. Complete the parallelograms IIKA, MONB. Then DG and AI are the equal and opposite forces which connect A and D; for GD = CH = AI. In like manner DH and BM are the forces which connect D and B.

The external force at A is in immediate equilibrium with the combined forces, connecting A with D and with C. AI is one of them, therefore AK is the other; and AL is the compound force with which the external force at A is in immediate equilibrium. This external force is therefore equal and opposite to AL. In like manner, the external force at B is equal and opposite to BO; and AL is to BO as the external force at A to the external force at B. The prop C resists with forces equal to those which are propagated to it from the points D, A, and C. Therefore it resists with forces CH, CG, equal and opposite to DG, DH; and it resists the compressions KA, NB, with equal and opposite forces CK, CN. Draw M, no, parallel to AD, BD, and draw CQ, CoP: It is plain that KCHI is a parallelogram equal to KAIH, and that CI is equal to AL. In like manner CO is equal to BO. Now the forces CK, CH, exerted by the prop, compose the force CI; and CM, CG, compose the force CO. These two forces CI, CO, are equal and parallel to AL and BO; and therefore they are equal and opposite to the external forces acting at A and B. But they are, primitively, equal and opposite to the pressures, or at least the compounds of the pressures, exerted on the prop, by the forces propagated to C from A, D, and B. Therefore the pressures exerted on the prop are the same as if the external forces were applied there in the same directions as they are applied to A and B. Now if we make Cr, Cz, equal to CI and CO, and complete the parallelogram Ceyz, it is plain that the force yC is in equilibrium with IC and oC. Therefore the pressures at A, C, and B are such as would balance if applied to one point.

Lastly, in order to determine their proportions, draw CS and CR perpendicular to DA and DB. Also draw Ad, If, perpendicular to CQ and CP; and draw Cg, Ci, perpendicular to AE, BF.

The triangles CPR and BPf are similar, having a common angle P, and a right angle at R and f.

In like manner, the triangles CQS and AQd are similar. Also the triangles CHR, CGS, are similar, by reason of the equal angles at H and G, and the right angles at R and S. Hence we obtain the following analogies:

\[ \begin{align*} \text{Co} : \text{CP} &= \text{on} : \text{PB}, = \text{CG} : \text{PB} \\ \text{CP} : \text{CR} &= \text{PB} : fB \\ \text{CR} : \text{CS} &= \text{CH} : \text{CG} \\ \text{CS} : \text{CQ} &= \text{Ad} : \text{AQ} \\ \text{CQ} : \text{CI} &= \text{AQ} : \text{M}, = \text{AQ} : \text{CH}. \end{align*} \]

Therefore, by equality,

\[ \begin{align*} \text{Co} : \text{CI} &= \text{Ad} : fB \\ \text{or BO} : \text{AL} &= \text{Cg} : \text{Ci}. \end{align*} \]

That is, the external forces are reciprocally proportional to the perpendiculars drawn from the prop on the lines of their direction.

This proposition, sufficiently general for our purpose, is extensive fertile in consequences, and furnishes many useful instructions to the artist. The strains LA, OB, CY, that are excited, occur in many, we may say in all, framings of carpentry, whether for edifices or engines, and are the sources of their efficiency. It is also evident that the doctrine of the transverse strength of timber is contained in this proposition; for every piece of timber may be considered as an assemblage of parts, connected by forces which act in the direction of the lines which joined the strained points on the matter which lies between those points, and also act on the rest of the matter, exciting those lateral forces which produce the inflexibility of the whole. See Strength of Materials.

Thus it appears that this proposition contains the principles which direct the artist to frame the most powerful levers; to secure uprights by shores or braces, or by tiers and ropes; to secure scaffoldings for the erection of spires; and many other more delicate problems of his art. He also learns from this proposition how to ascertain the strains that are produced, without his intention, by pieces which he intended for other offices, and which, by their transverse action, puts his work in hazard. In short, this proposition is the key to the science of this art.

We would now counsel the artist, after he has made the tracing of the strains and thrusts through the various parts of a frame familiar to his mind, and even amused himself with some complicated fancy framings, to read over with care the articles Strength of Materials and Roof. He will now conceive its doctrine much more clearly than when he was considering them as abstract theories. The mutual action of the woody fibres will now

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1. "The learned reader will perceive that this analogy is precisely the same with that of forces which are in equilibrium by the intervention of a lever. In fact, this whole frame of carpentry is nothing else than a built or framed lever in equilibrium. It is acting in the same manner as a solid, which occupies the whole figure compressed in the frame, or as a body of any size and shape whatever that will admit the three points of application A, C, and B. It is always in equilibrium in the case first stated; because the pressure produced at B by a force applied to A is always such as balances it. The reader may also perceive, in this proposition, the analysis or tracing of those internal mechanical forces which are indispensably requisite for the functions of a lever. The mechanicians have been extremely puzzled to find a legitimate demonstration of the equilibrium of a lever ever since the days of Archimedes. Mr Vince has the honour of first demonstrating, most ingeniously, the principle assumed by Archimedes, but without sufficient ground for his demonstration; but Mr Vince's demonstration is only a putting the mind into that perplexed state which makes it acknowledge the proposition, but without a clear perception of its truth. The difficulty has proceeded from the abstract notion of a lever, conceiving it as a mathematical line—inflexible, without reflecting how it is inflexible; for the very source of this indispensable quality furnishes the mechanical connection between the remote pressures and the fulcrum; and this supplies the demonstration (without the least difficulty) of the desperate case of a straight lever urged by parallel forces." See the article Rotation. There is a proposition (see article Roof) which has been called in question by several very intelligent persons; and they say that Belidor has demonstrated, in his Science des Ingenieurs, that a beam firmly fixed at both ends is not twice as strong as when simply lying on the props; and that its strength is increased only in the proportion of two to three; and they support this determination by a list of experiments recited by Belidor, which agree precisely with it. Belidor also says that Pitot had the same result in his experiments. These are respectable authorities, but Belidor's reasoning is any thing but demonstration, and his experiments are described in such an imperfect manner that we cannot build much on them. It is not said in what manner the battens were secured at the ends, any further than that it was by chevrelats. If by this word is meant a trestle, we cannot conceive how they were employed; but we see it sometimes used for a wedge or key. If the battens were wedged in the holes, their resistance to fracture may be made what we please; they may be made loose, and therefore resist little more than when simply laid on props. They may be (and probably were) wedged very fast, and bruised or crippled.

Our proposition mentioned distinctly the security given to the ends of the beams. They were mortised into remote posts. Our precise meaning was, that they were simply kept from rising by these mortises, but at full liberty to bend up at E and I, and between G and K. Our assertion was not made from theory alone (although we think the reasoning incontrovertible), but was agreeable to numerous experiments made in those precise circumstances. Had we mortised the beams firmly into two very stout posts which could not be drawn nearer to each other by bending, the beam would have borne a much greater weight, as we have verified by experiments. We hope that the following mode of conceiving this case will remove all doubts.

Let LM be a long beam (fig. 14) divided into six equal parts, in the points D, B, A, C, E. Let it be firmly supported at L, B, C, M. Let it be cut through at A, and have compass joints at B and C. Let FB, GC, be two equal uprights, resting on B and C, but without any connection. Let AH be a similar and equal piece, to be occasionally applied at the seam A. Now let a thread or wire AGE be extended over the piece GC, and made fast at A, G, and E. Let the same thing be done on the other side of A. If a weight be now laid on at A, the wires AFD, AGE, will be strained, and may be broken. In the instant of fracture we may suppose their strains to be represented by Af and Ag. Complete the parallelogram, and Aa is the magnitude of the weight. It is plain that nothing is concerned here but the cohesion of the wires; for the beam is sawed through at A, and its parts are perfectly movable round B and C.

Instead of this process, apply the piece AH below A, and keep it there by straining the same wire BHC over it. Now lay on a weight. It must press down the ends of BA and CA, and cause the piece AH to strain the wire BHC. In the instant of fracture of the same wire, its resistances H6 and He must be equal to Af and Ag, and the weight Ah which breaks them must be equal to Aa.

Lastly, employ all the three pieces FB, AH, GC, with the same wire attached as before. There can be no doubt but that the weight which breaks all the four wires must be = aA + Ah, or twice Aa.

The reader cannot but see that the wires perform the very same office with the fibres of an entire beam LM held fast in the four holes D, B, C, and E, of some upright posts.

In the experiments for verifying this, by breaking slender bars of fine deal, we get complete demonstration, by measuring the curvatures produced in the parts of the beam thus held down, and comparing them with the curvature of a beam simply laid on the props B and C; and there are many curious inferences to be made from these observations, but we have not room for them in this place.

We may observe by the way, that we learn from this The best case that purlins are able to carry twice the load when mortised into the rafters that they carry when mortised framing into them, which is the most usual manner of framing them. So would the bending joists of floors; but this would double the thickness of the flooring. But this method should be followed in every possible case, such as breast-sambers, lintels over several pillars, &c. These should never be cut off and mortised into the sides of every upright; numberless cases will occur which show the importance of the maxim.

We must here remark, that the proportion of the spaces BC and CM, or BC and LB, has a very sensible effect on the strength of the beam BC; but we have not yet satisfied our minds as to the rationale of this effect. It is undoubtedly connected with the serpentine form of the curve of the beam before fracture. This should be attended to in the construction of the springs of carriages. These are frequently supported at the middle point (and it is an excellent practice); and there is a certain proportion which will give the easiest motion to the body of the carriage. We also think that it is connected with that deviation from the best theory observable in Buffon's experiments on various lengths of the same scantling. The force of the beams diminished much more than in the inverse proportion of their lengths.

We have seen that it depends entirely on the position of the pieces in respect of their points of ultimate support, and of the direction of the external force which produces the strains, whether any particular piece is in a state of tension or of compression. The knowledge of this circumstance may greatly influence us in the choice of the construction. In many cases we may substitute slender iron rods for massive beams, when the piece is to act the part of a tie. But we must not invert this disposition; for when a piece of timber acts as a strut, and is in a state of compression, it is next to certain that it is not equally compressible in its opposite sides through the whole length of the piece, and that the compressing force on the abutting joint is not acting in the most equable manner all over the joint. A very trifling inequality in either of these circumstances (especially in the first) will compress the beam more on one side than on the other. This cannot be without the beam's bending, and becoming concave on that side on which it is most compressed. When this happens, the frame is in danger of being crushed, and soon going to ruin. It is, therefore, indispensably necessary to make use of beams in all cases where struts are required of considerable length, rather than of metal rods of slender dimensions, unless in situations where we can effectually prevent their bending, as in trussing a girder internally, where a cast-iron strut may be firmly cased in it, so as not to bend in the smallest degree. In cases where the pressures are enormous, as in the very oblique struts of a centre or arch frame, we must be particularly cautious to do nothing which can facilitate the compression of either side. No mortises should be cut near to one side; no lateral pressures, even the slightest, should be allowed to touch it. We have seen a pillar of fir twelve inches long, and one inch in section, when loaded with three tons, snap in an instant when pressed on one side by sixteen pounds, while another bore four and a half tons without hurt, because it was inclosed (loosely) in a stout pipe of iron. (See Note DD.) In such cases of enormous compression it is of great importance that the compressing force bear equally on the whole abutting surface. The German carpenters are accustomed to put a plate of lead over the joint. This prevents, in some measure, the penetration of the end fibres. M. Perrotet, the celebrated French architect, formed his abutments into arches of circles, the centre of which was the remote end of the strut. By this contrivance the unavoidable change of form of the triangle made no partial bearing of either angle of the abutment. This always has a tendency to splinter off the heel of the beam where it presses strongest. It is a very judicious practice. (See Note EE.)

When circumstances allow it, we must rather employ ties than struts for securing a beam against lateral strains. When an upright pillar, such as a flag-staff, a mast, or the uprights of a very tall scaffolding, are to be shoared up, the dependence is more certain on those braces that are stretched by the strain than on those which are compressed. The scaffolding of the iron bridge near Sunderland had some ties very judiciously disposed, and others with less judgment.

We should proceed to consider the transverse strains as they affect the various parts of a frame of carpentry; but we have very little to say here in addition to what will be found in the articles Strength of Materials and Roof. What we shall add in this article will find a place in our occasional remarks on different works. It may, however, be of use to recall to the reader's memory the following propositions.

1. When a beam AB (fig. 15) is firmly fixed at the end A, and a straining force acts perpendicularly to its length at any point B, the strain occasioned at any section C between B and A is proportional to CB, and may therefore be represented by the product \( w \times CB \); that is, by the product of the number of tons, pounds, &c., which measure the straining force, and the number of feet, inches, &c., contained in CB. As the loads on a beam are easily conceived, we shall substitute this for any other straining force.

2. If the strain or load is uniformly distributed along any part of the beam lying beyond C (that is, farther from A), the strain at C is the same as if the load were all collected at the middle point of that part; for that point is the centre of gravity of the load.

3. The strain on any section D of a beam AB (fig. 16) resting freely on two props A and B, is \( w \times \frac{AD \times DB}{AB} \).

(See Roof, No. 19, and Strength of Materials, No. 92, &c.) Therefore,

4. The strain on the middle point, by a force applied there, is one fourth of the strain which the same force would produce if applied to one end of a beam of the same length having the other end fixed.

5. The strain on any section C of a beam, resting on two props A and B, occasioned by a force applied perpendicularly to another point D, is proportional to the rectangle of the exterior segments, or is equal to \( w \times \frac{AC \times DB}{AB} \).

Therefore,

The strain at C occasioned by the pressure on D is the same with the strain at D occasioned by the same pressure on C.

6. The strain on any section D, occasioned by a load uniformly diffused over any part EF, is the same as if the two parts ED, DF, of the load were collected at their middle points e and f. Therefore,

The strain on any part D, occasioned by a load uniformly distributed over the whole beam, is one half of the strain that is produced when the same load is laid on at D; and

The strain on the middle point C, occasioned by a load uniformly distributed over the whole beam, is the same which half that load would produce if laid on at C.

7. A beam supported at both ends on two props B and C (fig. 14), will carry twice as much when the ends beyond the props are kept from rising, as it will carry when it rests loosely on the props.

8. Lastly, the transverse strain on any section, occasioned by a force applied obliquely, is diminished in the proportion of the sine of the angle which the direction of the force makes with the beam. Thus, if it be inclined to it in an angle of thirty degrees, the strain is one half of the strain occasioned by the same force acting perpendicularly.

On the other hand, the relative strength of a beam, or its power in any particular section to resist any transverse strain, is proportional to the absolute cohesion to the section directly, to the distance of its centre of effort from the axis of fracture directly, and to the distance from the strained point inversely.

Thus, in a rectangular section of the beam, of which \( b \) is the breadth, \( d \) the depth (that is, the dimension in the direction of the straining force), measured in inches, and \( f \) the number of pounds which one square inch will just support without being torn asunder, we must have \( f \times b \times d^2 \), proportional to \( w \times CB \) (fig. 15). Or, \( f \times b \times d^2 \), multiplied by some number \( m \), depending on the nature of the timber, must be equal to \( w \times CB \). Or, in the case of the section C of fig. 16, that is strained by the force \( w \) applied at D, we must have \( m \times \frac{fbd^2}{9} = w \times \frac{AC \times DB}{AB} \).

Thus, if the beam is of sound oak, \( m \) is very nearly \( \frac{1}{2} \) (see Strength of Materials, No. 116.) Therefore we have \( \frac{fbd^2}{9} = w \times \frac{AC \times CB}{AB} \). (See Note FF.)

Hence we can tell the precise force \( w \) which any section C can just resist when that force is applied in any way whatever; for the above-mentioned formula gives \( w = \frac{fbd^2}{9CB} \) for the case represented by fig. 15. But the case represented in fig. 16, having the straining force applied at D, gives the strain at C (\( = w \)) \( = f \times \frac{bd^2 \times AB}{9AC \times CB} \).

Example. Let an oak beam, four inches square, rest freely on the props A and B, seven feet apart, or eighty-four inches. What weight will it just support at its middle point C, on the supposition that a square inch rod will just carry 16,000 pounds, pulling it asunder?

The formula becomes \( w = \frac{16000 \times 4 \times 16 \times 84}{9 \times 42 \times 42} \),

or \( w = \frac{86016000}{15876} = 5418 \) pounds. This is very near what was employed in Buffon's experiment, which was 5312.

Had the straining force acted on a point D, half way between C and B, the force sufficient to break the beam at C would be \( \frac{16000 \times 4 \times 16 \times 84}{9 \times 42 \times 21} = 10836 \) lbs.

Had the beam been sound red fir, we must have taken \( f = 10,000 \) nearly, and \( m \) nearly 8; for although fir be less cohesive than oak in the proportion of five to eight nearly, it is less compressible, and its axis of fracture is therefore nearer to the concave side.

Having considered at sufficient length the strains of joints, different kinds which arise from the form of the parts of a frame of carpentry, and the direction of the external forces which act on it, whether considered as impelling or as supporting its different parts, we must now proceed to con... Carpentry. The means by which this form is to be secured, and the connections by which those strains are excited and communicated.

The joinings practised in carpentry are almost infinitely various, and each has advantages which make it preferable in some circumstances. Many varieties are employed merely to please the eye. We do not concern ourselves with these; nor shall we consider those which are only employed in connecting small works, and can never appear on a great scale; yet even in some of these, the skill of the carpenter may be discovered by his choice; for in all cases, it is wise to make every, even the smallest, part of his work as strong as the materials will admit. He will be particularly attentive to the changes which will necessarily happen by the shrinking of timber as it dries, and will consider what dimensions of his framings will be affected by this, and what will not; and will then dispose the pieces which are less essential to the strength of the whole, in such a manner that their tendency to shrink shall be in the same direction with the shrinking of the whole framing. If he do otherwise, the seams will widen, and parts will be split asunder. He will dispose his boardings in such a manner as to contribute to the stiffness of the whole, avoiding at the same time the giving them positions which will produce lateral strains on truss beams which bear great pressures; recollecting, that although a single board has little force, yet many united have a great deal, and may frequently perform the office of very powerful struts.

Our limits confine us to the joinings which are most essential for connecting the parts of a single piece of a frame when it cannot be formed of one beam, either for want of the necessary thickness or length; and the joints for connecting the different sides of a trussed frame.

Much ingenuity and contrivance has been bestowed on the manner of building up a great beam of many thicknesses, and many singular methods are practised as great nostrums by different artists; but when we consider the manner in which the cohesion of the fibres performs its office, we will clearly see that the simplest are equally effected with the most refined, and that they are less apt to lead us into false notions of the strength of the assemblage.

Thus, were it required to build up a beam for a great lever or a girder, so that it may act nearly as a beam of the same size of one log, it may either be done by plain joggling, as in Plate CXLIX. fig. 17, A, or by scarfings, as in fig. 17, B or C. If it is to act as a lever, having the gudgeon on the lower side at C, we believe that most artists will prefer the form B and C; at least this has been the case with nine tenths of those to whom we have proposed the question. The best informed only hesitated; but the ordinary artists were all confident in its superiority, and we found their views of the matter very coincident. They considered the upper piece as grasping the lower in its hooks; and several imagined, that by driving the one very tight on the other, the beam would be stronger than an entire log; but if we attend carefully to the internal procedure in the loaded lever, we shall find the upper one clearly the strongest. If they are formed of equal logs, the upper one is thicker than the other by the depth of the joggling or scarfing, which we suppose to be the same in both; consequently, if the cohesion of the fibres in the intervals is able to bring the uppermost filaments into full action, the form A is stronger than B, in the proportion of the greater distance of the upper filaments from the axis of the fracture. This may be greater than the difference of the thickness if the wood is very compressible. If the gudgeon be in the middle, the effect, both of the joggles and the scarfings, is considerably diminished; and if it is on the upper side the scarfings act in a very different way. In this situation, if the loads on the arms are also applied to the upper side, the joggled beam is still more superior to the scarfed one. This will be best understood by resolving it in imagination into a trussed frame. But when a gudgeon is thus put on that side of the lever which grows convex by the strain, it is usual to connect it with the rest by a powerful strap, which embraces the beam, and causes the opposite point to become the resisting point. This greatly changes the internal actions of the filaments, and in some measure brings it into the same state as the first, with the gudgeon below. Were it possible to have the gudgeon on the upper side, and to bring the whole into action without a strap, it would be the strongest of all; because in general the resistance to compression is greater than to extension. In every situation the joggled beam has the advantage, and it is the easiest executed. (See Note GG.)

We may frequently gain a considerable accession of strength by this building up of a beam, especially if the part which is stretched by the strain be of oak, and the other part be fir. Fir being so much superior to oak as a pillar (if Muschenbroeck's experiments may be confided in), and oak so much preferable as a tie, this construction seems to unite both advantages. But we shall see much better methods of making powerful levers, girders, &c., by trussing.

Observe that the efficacy of both methods depends entirely on the difficulty of causing the piece between the cross joints to slide along the timber to which it adheres. Therefore, if this be moderate, it is wrong to make the notches deep; for as soon as they are so deep that their ends have a force sufficient to push the slice along the line of junction, nothing is gained by making them deeper; and this requires a greater expenditure of timber.

Scarfings are frequently made oblique, as in fig. 18; but we imagine that this is a bad practice. It begins to yield at a point where the wood is crippled and splintered off, or at least bruised out a little. As the pressure increases, this part, by squeezing broader, causes the solid parts to rise a little upwards, and gives them some tendency, not only to push their antagonists along the base, but even to tear them up a little. For similar reasons, we disapprove of the favourite practice of many artists to make the angles of their scarfings acute, as in fig. 19. This often causes the two pieces to tear each other up. The abutments should always be perpendicular to the directions of the pressures. Lest it should be forgotten in its proper place, we may extend this injunction also to the abutments of different pieces of a frame, and recommend it to the artist even to attend to the shrinking of the timbers by drying. When two timbers abut obliquely, the joint should be most full at the obtuse angle of the end; because, by drying, that angle grows more obtuse, and the beam would then be in danger of splintering off at the acute angle.

It is evident that the nicest work is indispensably necessary in building up a beam. The parts must shut on not weld each other completely, and the smallest play or void takes too hard away the whole efficacy. It is usual to give the butting joints a small taper to one side of the beam, so that they may require moderate blows of a maul to force them in; and the joints may be perfectly close when the external surfaces are even on each side of the beam. But we must not exceed in the least degree, for a very taper wedge has great force; and if we have driven the pieces together by very heavy blows, we leave the whole in a state of violent strain, and the abutments are perhaps ready to splinter off by a small addition of pressure. This is like too severe a proof for artillery; which, though not sufficient to burst the pieces, has weakened them to such a Carpentry.

The strain of ordinary service is sufficient to complete the fracture. The workman is tempted to exceed in this, because it smooths off and conceals all uneven seams; but he must be watched. It is not unusual to leave some abutments open enough to admit a thin wedge reaching through the beam. Nor is this a bad practice, if the wedge is of material which is not compressed by the driving or the strain of service. Iron would be preferable for this purpose, and for the joggles, were it not that, by its too great hardness, it cripples the fibres of timber some distance. In consequence of this it often happens, that in beams which are subjected to de- sultory and sudden strains (as in the levers of reciprocating engines), the joggles or wedges widen the holes, and work themselves loose; therefore skilful engineers never admit them, and indeed as few bolts as possible, for the same reason; but when resisting a steady or dead pull, they are not so improper, and are frequently used.

Beams are built up, not only to increase their dimensions in the direction of the strain (which we have hitherto called their depth), but also to increase their breadth, or the dimensions perpendicular to the strain. We sometimes double the breadth of a girder which is thought too weak for its load, and where we must not increase the thickness of the flooring.

The mast of a great ship of war must be made bigger athwartship, as well as fore and aft. This is one of the nicest problems of the art; and professional men are by no means agreed in their opinions about it. We do not presume to decide; and shall content ourselves with exhibiting the different methods.

The most obvious and natural method is that shown in fig. 20. It is plain that (independent of the connection of cross bolts, which are used in them all when the beams are square) the piece C cannot bend in the direction of the plane of the figure without bending the piece D along with it. This method is much used in the French navy; but it is undoubtedly imperfect. Hardly any two great trees are of equal quality, and swell or shrink alike. If C shrinks more than D, the feather of C becomes loose in the groove wrought in D to receive it; and when the beam bends, the parts can slide on each other like the plates of a coach-spring; and if the bending is in the direction ef, there is nothing to hinder this sliding but the bolts, which soon work themselves loose in the bolt-holes.

Fig. 21 exhibits another method. The two halves of the beam are tabled into each other in the same manner as in fig. 17. It is plain that this will not be affected by the unequal swelling or shrinking, because this is insensible in the direction of the fibres; but when bent in the direction ab, the beam is weaker than fig. 20 bent in the direction ef. Each half of fig. 20 has, in every part of its length, a thickness greater than half the thickness of the beam. It is the contrary in the alternate portions of the halves of fig. 21. When one of them is bent in the direction AB, it is plain that it drags the other with it by means of the cross buttments of its tables, and there can be no longitudinal sliding. But unless the work is accurately executed, and each hollow completely filled up by the table of the other piece, there will be a lateral slide along the cross joints sufficient to compensate for the curvature; and this will hinder the one from compressing or stretching the other in conformity to this curvature.

The imperfection of this method is so obvious that it has seldom been practised; but it has been combined with the other, as is represented in fig. 22, where the beams are divided along the middle, and the tables in each half are alternate, and alternate also with the tables of the other half. Thus 1, 3, 4, are prominent, and 5, 2, 6, are depressed. This construction evidently puts a stop to both slides, and obliges every part of both pieces to move together. ab and cd show sections of the built-up beam corresponding to AB and CD.

No more is intended in this practice by any intelligent artist, than the causing the two pieces to act together in all their parts, although the strains may be unequally distributed on them. Thus, in a built-up girder, the binding joists are frequently mortised into very different parts of the two sides. But many seem to aim at making the beam stronger than if it were of one piece; and this inconsiderate project has given rise to many whimsical modes of tabbling and scarfing, which we need not regard.

The practice in the British dock-yards is somewhat different from any of these methods. The pieces are tabled as in fig. 22, but the tables are not thin parallelopipeds, but thin prisms. The two outward joints or visible seams are straight lines, and the table No. 1 rises gradually to its greatest thickness in the axis. In like manner, the hollow, 5, for receiving the opposite table, sinks gradually from the edge to its greatest depth in the axis. Fig. 23, No. 1, represents a section of a round piece of timber built up in this way, where the full line EFGH is the section corresponding to AB of fig. 22, and the dotted line EGFH is the section corresponding to CD.

This construction, by making the external seam straight, leaves no lodgment for water, and looks much fairer to the eye; but it appears to us that it does not give so firm a hold when the mast is bent in the direction EH. The exterior parts are most stretched and most compressed by this bending; but there is hardly any abutment in the exterior parts of these tables. In the very axis, where the abutment is the firmest, there is little or no difference of extension and compression.

But this construction has an advantage, which, we imagine, much more than compensates for these imperfections, at least in the particular case of a round mast; it will draw together by hooping incomparably better than any of the others. If the cavity be made somewhat too shallow for the prominence of the tables, and if this be done uniformly along the whole length, it will make a somewhat open seam; and this opening can be regulated with the utmost exactness from end to end by the plane. The heart of those vast trunks is very sensibly softer than the exterior circles; therefore, when the whole is hooped, and the hoops hard driven, and at considerable intervals between each spell, we are confident that all may be compressed till the seam disappears; and then the whole makes one piece, much stronger than if it were an original log of that size, because the middle has become, by compression, as solid as the crust, which was naturally firmer, and resisted farther compression. We verified this beyond a doubt by hooping a built stick of a timber which has this inequality of firmness in a remarkable degree, and it was nearly twice as strong as another of the same size.

Our mast-makers are not without their fancies and whims; and the manner in which our masts and yards are generally built up is not near so simple as fig. 23; but it consists of the same essential parts, acting in the very same manner, and derives all its efficacy from the principles which are here employed.

This construction is particularly suited to the situation and office of a ship's mast. It has no bolts; or, at least, with peculiarity of magnitude, or that make very important parts of its construction. The most violent strains perhaps that it is exposed to, is that of twisting, when the lower yards are close braced by the force of many men acting by a long lever. This form resists a twist with peculiar energy; it is therefore an excellent method for building up a great shaft for a mill. The way in which they are usually built up is by reducing a central log to a poly- Carpentry, gonal prism, and then filling it up to the intended size by planting pieces of timber along its sides, either spiking them down, or cocking them into it by a feather, or joggling them by slips of hard wood sunk into the central log and into the slips. N.B. Joggles of elm are sometimes used in the middle of the large tables of masts; and when sunk into the firm wood near the surface, they must contribute much to the strength. But it is very necessary to employ wood not much harder than the pine, otherwise it will soon enlarge its bed, and become loose, for the timber of these large trunks is very soft.

The most general reason for piecing a beam is to increase its length. This is frequently necessary, in order to procure tie-beams for very wide roofs. Two pieces must be scarfed together. Numberless are the modes of doing this, and almost every master carpenter has his favourite nostrum. Some of them are very ingenious; but here, as in other cases, the most simple are commonly the strongest. We do not imagine that any, the most ingenious, is equally strong with a tie consisting of two pieces of the same scantling laid over each other for a certain length, and firmly bolted together. We acknowledge that this will appear an artless and clumsy tie-beam, but we only say that it will be stronger than any that is more artificially made up of the same thickness of timber. This, we imagine, will appear sufficiently certain.

The simplest and most obvious scarfing, after the one now mentioned, is that represented in fig. 24, No. 1 and 2. If considered merely as two pieces of wood joined, it is plain that, as a tie, it has but half the strength of an entire piece, supposing that the bolts (which are the only connections) are fast in their holes. No. 2 requires a bolt in the middle of the scarf to give it that strength, and in every other part is weaker on one side or the other. (See Note HHL.)

But the bolts are very apt to bend by the violent strain, and require to be strengthened by uniting their ends by iron plates; in which case it is no longer a wooden tie. The form of No. 1 is better adapted to the office of a pillar than No. 2, especially if its ends be formed in the manner shown in the elevation No. 3. By the sally given to the ends, the scarf resists an effort to bend it in that direction. Besides, the form of No. 2 is unsuitable for a post; because the pieces, by sliding on each other by the pressure, are apt to splinter off the tongue which confines their extremity.

Fig. 25 and 26 exhibit the most approved form of a scarf, whether for a tie or for a post. The key represented in the middle is not essentially necessary; the two pieces might simply meet square there. This form, without a key, needs no bolts (although they strengthen it greatly); but, if worked very true and close, and with square abutments, will hold together, and will resist bending in any direction. But the key is an ingenious and a very great improvement, and will force the parts together with perfect tightness. The same precaution must be observed that we mentioned on another occasion, not to produce a constant internal strain on the parts by overdriving the key. The form of fig. 25 is by far the best; because the triangle of 26 is much easier splintered off by the strain, or by the key, than the square wood of 25. It is far preferable for a post, for the reason given when speaking of fig. 24, No. 1 and No. 2. Both may be formed with a sally at the ends equal to the breadth of the key. In this shape fig. 25 is vastly well suited for joining the parts of the long corner posts of spires and other wooden towers. Fig. 25, No. 2, differs from No. 1 only by having three keys. The principal and the longitudinal strength are the same. The long scarf of No. 2, tightened by the three keys, enables it to resist a bending much better.

None of these scarfed tie-beams can have more than one third of the strength of an entire piece, unless with the assistance of iron plates; for if the key be made thinner than one third, it has less than one third of the fibres to pull by.

We are confident, therefore, that when the heads of the bolts are connected by plates, the simple form of fig. 24, No. 1, is stronger than those more ingenious scarfings. It may be strengthened against lateral bending by a little tongue, or by a sally, but cannot have both.

The strongest of all methods of piecing a tie-beam would be to set the parts end to end, and grasp them between other pieces on each side, as in fig. 27, Plate CL. This is what the ship-carpenter calls fishing a beam, and this is a frequent practice for occasional repairs. M. Perronet used it for the tie-beams or stretchers, by which he connected the opposite feet of a centre, which was yielding to its load, and had pushed aside one of the piers above four inches. Six of these not only withstood a strain of 1800 tons, but, by wedging behind them, he brought the feet of the truss 2½ inches nearer. The stretchers were 14 inches by 11 of sound oak, and could have withstood three times that strain. M. Perronet, fearing that the great length of the bolts employed to connect the beams of these stretchers would expose them to the risk of bending, scarfed the two side pieces into the middle piece. The scarfing was of the triangular kind (Trait de Jupiter), and only an inch deep, each face being two feet long, and the bolt passed through close to the angle.

In piecing the pump-rods and other wooden stretchers of great engines, no dependence is had on scarfing; and the engineer connects every thing by iron straps. We doubt the propriety of this, at least in cases where the bulk of the wooden connection is not inconvenient. These observations must suffice for the methods employed for connecting the parts of a beam; and we now proceed to consider what are more usually called the joints of a piece of carpentry.

Where the beams stand square with each other, and the strains are also square with the beams, and in the plane of joints of the frame, the common mortise and tenon is the most perfect junction. A pin is generally put through both, in order to keep the pieces united, in opposition to any force which tends to part them. Every carpenter knows how to bore the hole for this pin, so that it shall draw the tenon tight into the mortise, and cause the shoulder to butt close, and make neat work; and he knows the risk of tearing out the bit of the tenon beyond the pin, if he draw it too much. We may just observe, that square holes and pins are much preferable to round ones for this purpose, bringing more of the wood into action, with less tendency to split it. The ship-carpenters have an ingenious method of making long wooden bolts, which do not pass completely through, take a very fast hold, though not nicely fitted to their holes, which they must not be, lest they should be crippled in driving. They call it fox-tail wedging. They stick into the point of the bolt a very thin wedge of hard wood, so as to project a proper distance; when this reaches the bottom of the hole by driving the bolt, it splits the end of it, and squeezes it hard to the side. This may be practised with advantage in carpentry. If the ends of the mortise are widened inwards, and a thin wedge be put into the end of the tenon, it will have the same effect, and make the joint equal to a dove-tail. But this risks the splitting the piece beyond the shoulder of the tenon, which would be unsightly. This may be avoided as follows: Let the tenon T, fig. 28, have two very thin wedges a and c struck in near its angles, projecting equally; at a very small distance within these, put in two shorter ones b, d, and more within these if necessary. In driving this tenon, the wedges \(a\) and \(c\) will take first, and split off a thin slice, which will easily bend without breaking. The wedges \(b\), \(d\), will act next, and have a similar effect, and the others in succession. The thickness of all the wedges taken together must be equal to the enlargement of the mortise towards the bottom.

When the strain is transverse to the plane of the two beams, the principles laid down in No. 85, 86, of the article Strength of Materials, will direct the artist in placing his mortise. Thus the mortise in a girder for receiving the tenon of a binding joist of a floor should be as near the upper side as possible, because the girder becomes concave on that side by the strain. But as this exposes the tenon of the binding-joist to the risk of being torn off, we are obliged to mortise farther down. The form (fig. 29) generally given to this joint is extremely judicious. The sloping part \(a b\) gives a very firm support to the additional bearing \(e d\), without much weakening of the girder. This form should be copied in every case where the strain has a similar direction.

The joint that most of all demands the careful attention of the artist, is that which connects the ends of beams, one of which pushes the other very obliquely, putting it into a state of extension. The most familiar instance of this is the foot of a rafter pressing on the tie-beam, and thereby drawing it away from the other wall. When the direction is very oblique (in which case the extending strain is the greatest), it is difficult to give the foot of the rafter such a hold of the tie-beam as to bring many of its fibres into the proper action. There would be little difficulty if we could allow the end of the tie-beam to project to a small distance beyond the foot of the rafter; but, indeed, the dimensions which are given to tie-beams for other reasons, are always sufficient to give enough of abutment when judiciously employed. Unfortunately this joint is much exposed to failure by the effects of the weather. It is much exposed, and frequently perishes by rot, or becomes so soft and friable that a very small force is sufficient either for pulling the filaments out of the tie-beam, or for crushing them together. We are therefore obliged to secure it with particular attention, and to avail ourselves of every circumstance of construction.

One is naturally disposed to give the rafter a deep hold by a long tenon; but it has been frequently observed in old roofs that such tenons break off. Frequently they are observed to tear up the wood that is above them, and push their way through the end of the tie-beam. This in all probability arises from the first sagging of the roof, by the compression of the rafters and of the head of the king-post. The head of the rafter descends; the angle with the tie-beam is diminished by the rafter revolving round its step in the tie-beam. By this motion the heel or inner angle of the rafter becomes a fulcrum to a very long and powerful lever much loaded. The tenon is the other arm, very short; and being still fresh, it is therefore very powerful. It therefore forces up the wood that is above it, tearing it out from between the cheeks of the mortise, and then pushes it along. Carpenters have therefore given up long tenons, and give to the toe of the tenon a shape which abuts firmly, in the direction of the thrust, on the solid bottom of the mortise, which is well supported on the under side by the wall-plate. This form has the further advantage of having no tendency to tear up the end of the mortise. This form is represented in fig. 30. The tenon has a small portion \(ab\) cut perpendicular to the surface of the tie-beam, and the rest \(be\) is perpendicular to the rafter. (See Note CC.)

But if the tenon is not sufficiently strong (and it is not so strong as the rafter, which is thought not to be stronger than is necessary), it will be crushed, and then the rafter will shade out along the surface of the beam. It is therefore necessary to call in the assistance of the whole rafter. It is in this distribution of the strain among the various abutting parts that the varieties of joints and their merits chiefly consist. It would be endless to describe every nostrum, and we shall only mention a few that are most generally approved of.

The aim in fig. 31 is to make the abutments exactly perpendicular to the thrusts. (See Note CC.) It does prove this very precisely; and the share which the tenon forms, the shoulder have of the whole may be what we please, by the portion of the beam that we notch down. If the wall-plate lie duly before the heel of the rafter, there is no risk of straining the tie across or breaking it, because the thrust is made to direct to that point where the beam is supported. The action is the same as against the joggle on the head or foot of a king-post. We have no doubt but that this is a very effectual joint. It is not, however, much practised. It is said that the sloping seam at the shoulder lodges water; but the great reason seems to be a secret notion that it weakens the tie-beam. If we consider the direction in which it acts as a tie, we must acknowledge that this form takes the best method for bringing the whole of it into action.

Fig. 32 exhibits a form that is more general, but certainly worse. Such part of the thrust as is not borne by the tenon acts obliquely on the joint of the shoulder, and gives the whole a tendency to rise up and slide outward.

The shoulder joint is sometimes formed like the dotted line \(abcde\) of fig. 32. This is much more agreeable to the true principle, and would be a very perfect method, were it not that the intervals \(bd\) and \(df\) are so short that the little wooden triangles \(bed\), \(def\), will be easily pushed off their bases \(bd\), \(df\).

Fig. 33, No. 1, seems to have the most general approbation. It is the joint recommended by Price, and copied into all books of carpentry as the true joint for a rafter foot. The visible shoulder-joint is flush with the upper surface of the tie-beam. The angle of the tenon at the tie nearly bisects the obtuse angle formed by the rafter and the beam, and is therefore somewhat oblique to the thrust. The inner shoulder \(ac\) is nearly perpendicular to \(bd\). The lower angle of the tenon is cut off horizontally, as at \(ed\). Fig. 34 is a section of the beam and rafter foot, showing the different shoulders.

We do not perceive the peculiar merit of this joint. The effect of the three oblique abutments, \(ab\), \(ac\), \(ed\), is undoubtedly to make the whole bear on the outer end of the mortise, and there is no other part of the tie-beam that makes immediate resistance. Its only advantage over a tenon extending in the direction of the thrust is, that it will not tear up the wood above it. Had the inner shoulder had the form \(ee\), having its face \(ie\) perpendicular, it would certainly have acted more powerfully in stretching many filaments of the tie-beam, and would have had much less tendency to force out the end of the mortise. The little bit \(ci\) would have prevented the sliding upwards along \(ee\). At any rate, the joint \(ab\) being flush with the beam, prevents any sensible abutment on the shoulder \(ac\).

Fig. 33, No. 2, is a simpler, and in our opinion a preferable, joint. We observe it practised by the most eminent carpenters for all oblique thrusts; but it surely employs less of the cohesion of the tie-beam than might be used without weakening it, at least when it is supported on the other side by the wall-plate.

Fig. 33, No. 3, is also much practised by the first carpenters.

Fig. 35, No. 1, is proposed by Mr Nicholson as preferable to fig. 33, No. 3, because the abutment of the inner Carpentry part is better supported. This is certainly the case; but it supposes the whole rafter to go to the bottom of the socket, and the beam to be thicker than the rafter. Some may think that this will weaken the beam too much, when it is no broader than the rafter is thick; in which case they think that it requires a deeper socket than Nicholson has given it. Perhaps the advantages of Nicholson's construction may be had by a joint like fig. 35, No. 2.

Whatever is the form of these butting joints, great care should be taken that all parts bear alike; and the artist will attend to the magnitude of the different surfaces. In the general compression, the greater surfaces will be less compressed, and the smaller will therefore change most. When all has settled, every part should be equally close. Because great logs are moved with difficulty, it is very troublesome to try the joint frequently to see how the parts fit; therefore we must expect less accuracy in the interior parts. This should make us prefer those joints whose efficacy depends chiefly on the visible joint.

It appears from all that we have said on this subject, that a very small part of the cohesion of the tie-beam is sufficient for withstanding the horizontal thrust of a roof, even though very low pitched. If therefore no other use is made of the tie-beam, one much slenderer may be used, and blocks may be firmly fixed to the ends, on which the rafters might abut, as they do on the joggles on the head and foot of a king-post. Although a tie-beam has commonly floors or ceilings to carry, and sometimes the workshops and store-rooms of a theatre, and therefore requires a great scantling, yet there frequently occur in machines and engines very oblique stretchers, which have no other office, and are generally made of dimensions quite inadequate to their situation, often containing ten times the necessary quantity of timber. It is therefore of importance to ascertain the most perfect manner of executing such a joint. We have directed the attention to the principles that are really concerned in the effect. In all hazardous cases, the carpenter calls in the assistance of iron straps; and they are frequently necessary, even in roofs, notwithstanding this superabundant strength of the tie-beam. But this is generally owing to bad construction of the wooden joint, or to the failure of it by time. Straps will be considered in their place.

There needs but little to be said of the joints at a joggle worked out of solid timber; they are not near so difficult as the last. When the size of a log will allow the joggle to receive the whole breadth of the abutting brace, it ought certainly to be made with a square shoulder; or, which is still better, an arch of a circle, having the other end of the brace for its centre. (See Note EE.) Indeed this in general will not sensibly differ from a straight line perpendicular to the brace. By this circular form, the settling of the roof makes no change in the abutment; but when there is not sufficient stuff for this, we must avoid bevel joints at the shoulders, because these always tend to make the brace slide off. The brace in fig. 36, No. 1, must not be joined as at b, but as at a, or in some equivalent manner. Observe the joints at the head of the main posts of Drury Lane theatre, fig. 44, Plate CLII.

When the very oblique action of one side of a frame of carpentry does not extend, but compress, the piece on which it abuts (as in fig. 11), there is no difficulty in the joint. Indeed a joining is unnecessary, and it is enough that the pieces abut on each other; and we have only to take care that the mutual pressure be equally borne by all the parts, and that it do not produce lateral pressures, which may cause one of the pieces to slide on the butting joint. A very slight mortise and tenon is sufficient at the joggle of a king-post with a rafter or straining beam. It is best, in general, to make the butting plain, bisecting the angle formed by the sides, or else perpendicular to one of the pieces. In fig. 36, No. 2, where the straining beam, ab, cannot slip away from the pressure, the joint a is preferable to b, or indeed to any uneven joint, which never fails to produce very unequal pressures on the different parts, by which some are crippled, others are splintered off, &c.

When it is necessary to employ iron straps for strengthening a joint, considerable attention is necessary, that we may place them properly. The first thing to be determined is the direction of the strain. This is learned by the observations in the beginning of this article. We must then resolve this strain into a strain parallel to each piece, and another perpendicular to it. Then the strap which is to be made fast to any of the pieces must be so fixed that it shall resist in the direction parallel to the piece. Frequently this cannot be done; but we must come as near to it as we can. In such cases we must suppose that the assemblage yields a little to the pressures which act on it. We must examine what change of shape a small yielding will produce. We must now see how this will affect the iron strap which we have already supposed attached to the joint in some manner that we thought suitable. This settling will perhaps draw the pieces away from it, leaving it loose and unserviceable (this frequently happens to the plates which are put to secure the obtuse angles of butting timbers, when their bolts are at some distance from the angles, especially when these plates are laid on the inside of the angles); or it may cause it to compress the pieces harder than before, in which case it is answering our intention. But it may be producing cross strains, which may break them, or it may be crippling them. We can hardly give any general rules; but the reader will do well to read what is written in No. 36 and 41 of the article Roof. In No. 36 he will see the nature of the strap or stirrup, by which the king-post carries the tie-beam. The strap that we observe most generally ill placed is that which connects the foot of the rafter with the beam. It only binds down the rafter, but does not act against its horizontal thrust. It should be placed farther back on the beam, with a bolt through it, which will allow it to turn round. It should embrace the rafter almost horizontally near the foot, and should be notched square with the back of the rafter. Such a construction is represented in fig. 37. By moving round the eye-bolt, it follows the rafter, and cannot pinch and cripple it, which it always does in its ordinary form. We are of opinion that straps which have eye-bolts in the very angles, and allow all motion round them, are of all the most perfect. A branched strap, such as may at once bind the king-post and the two braces which butt on its foot, will be more serviceable if it have a joint. When a roof warps, those branched straps frequently break the tenons, by affording a fulcrum in one of their bolts. An attentive and judicious artist will consider how the beams will act on such occasions, and will avoid giving rise to these great strains by levers. A skilful carpenter never employs many straps, considering them as auxiliaries foreign to his art, and subject to imperfections in workmanship which he cannot discern or amend. We must refer the reader to Nicholson's Carpenter and Joiner's Assistant for a more particular account of the various forms of stirrups, screwed rods, and other iron work for carrying tie-beams, &c.

As for those that are necessary for the turning joints of great engines constructed of timber, they make no part of the art of carpentry. (See Note II.)

After having attempted to give a systematic view of the principles of framing carpentry, we shall conclude by giving some examples which will illustrate and confirm the foregoing principles. Fig. 38, Plate CLI. is the roof of the chapel of the Royal Hospital at Greenwich, constructed by Mr S. Wyatt.

AA is the tie-beam, 57 feet long, spanning 51 feet clear ........................................... 14 by 12 CC, queen-posts .................................................. 9 × 12 D, braces .......................................................... 9 × 7 E, straining beam .................................................. 10 × 7 F, straining piece .................................................. 6 × 7 G, principal rafters .................................................. 10 × 7 H, a cambered beam for the platform ......................... 9 × 7 R, an iron string, supporting the tie-beam ................. 2 × 2

The trusses are seven feet apart, and the whole is covered with lead, the boarding being supported by horizontal ledgers h, h, of six by four inches.

This is a beautiful roof, and contains less timber than most of its dimensions. The parts are all disposed with great judgment. Perhaps the iron rod is unnecessary, but it adds great stiffness to the whole.

The iron straps at the rafter feet would have had more effect if not so oblique. Those at the head of the post are very effective.

We may observe, however, that the joints between the straining beam and its braces are not of the best kind, and tend to bruise both the straining beam and the truss beam above it.

Fig. 39, the roof of St Paul's, Covent Garden, designed by Mr Hardwick, and constructed by Mr Wapshot in 1796.

AA, tie-beam spanning fifty feet two inches ........................................... 16 × 12 BB, queen-posts .................................................. 9 × 8 C, straining beam .................................................. 10 × 8 D, king-post (fourteen at the joggle) ......................... 9 × 8 EE, struts .......................................................... 8 × 7½ FF, auxiliary rafters (at bottom) .................................. 10 × 8½ HH, principal rafter (at bottom) .................................. 10 × 8½ gg, studs supporting the rafter .................................. 8 × 8

The trusses are about ten feet six inches apart, and the dotted lines in the middle compartment show the manner in which the roof is framed under the cupola.

This roof far excels the original one put up by Inigo Jones. One of its trusses contains 198 feet of timber. One of the old roof had 273, but had many inactive timbers, and others ill disposed. The internal truss PCF is admirably contrived for supporting the exterior rafters, without any pressure on the far projecting ends of the tie-beam. The former roof had bent them greatly, so as to appear ungraceful. (See Note KK.)

We think that the camber (six inches) of the tie-beam is rather hurtful, because, by settling, the beam lengthens; and this must be accompanied by a considerable sinking of the roof. This will appear by calculation. (See Note LL.)

Fig. 43, Plate CLIII, the roof of Birmingham theatre, constructed by Mr George Saunders. The span is eighty feet clear, and the trusses are ten feet apart.

A is an oak corbel .................................................. 9 × 5 B, inner plate ...................................................... 9 × 9 C, wall-plate .......................................................... 8 × 5½ D, pole-plate .......................................................... 7 × 5 E, tie-beam .......................................................... 15 × 15 F, straining beam .................................................. 12 × 9 G, oak king-post (in the shaft) .................................. 9 × 9 H, oak queen-post (in the shaft) .................................. 7 × 9 I, principal rafters .................................................. 9 × 9 K, common ditto .................................................. 4 × 2½ L, principal braces .................................................. 9 and 6 × 9 M, common ditto .................................................. 6 × 9 N, purlins ............................................................ 7 × 5 Q, straining sill ..................................................... 5½ × 9 S, ridge piece .......................................................... 9

This roof is a fine specimen of British carpentry, and is one of the boldest and lightest roofs in Europe. The straining sill Q gives a firm abutment to the principal braces, and the space between the posts is 19½ feet wide, affording roomy workshops for the carpenters and other workmen connected with a theatre. The contrivance for taking double hold of the wall, which is very thin, is excellent. There is also added a beam (marked R), bolted down to the tie-beams. The intention of this was to prevent the total failure of so bold a trussing, if any of the tie-beams should fail at the end by rot.

Akin to this roof is fig. 44, Plate CLII, the roof of Drury-Lane theatre, eighty feet three inches in the clear, and Lane the trusses fifteen feet apart, constructed by Edward Grey Saunders.

A, beams .......................................................... 10 by 7 B, rafters ............................................................ 7 × 7 C, king-posts .......................................................... 12 × 7 D, struts .............................................................. 5 × 7 E, purlins ............................................................. 9 × 5 G, pole-plates .......................................................... 5 × 5 H, gutter plates framed into the beams ....................... 12 × 6 I, common rafters .................................................. 5 × 4 K, tie-beam to the main truss .................................. 15 × 12 L, posts to ditto ..................................................... 15 × 12 M, principal braces to ditto ...................................... 14 and 12 × 12 N, struts .............................................................. 8 × 12 P, straining beams .................................................. 12 × 12

The main beams are trussed in the middle space with oak trusses five inches square. This was necessary for its width of thirty-two feet, occupied by the carpenters, painters, &c. The great space between the trusses afford good store-rooms, dressing-rooms, &c.

It is probable that this roof has not its equal in the world for lightness, stiffness, and strength. The main truss is so judiciously framed, that each of them will safely bear a load of three hundred tons; so it is not likely that they will ever be quarter loaded. The division of the whole into three parts makes the exterior roofings very light. The strains are admirably kept from the walls, and the walls are even firmly bound together by the roof. They also take off the dead weight from the main truss one third.

The intelligent reader will perceive that all these roofs are on one principle, depending on a truss of three pieces and a straight tie-beam. This is indeed the great principle of a truss, and is a step beyond the roof with two rafters and a king-post. It admits of much greater variety of forms, and of greater extent. We may see that even the middle part may be carried to any space, and yet be flat at top; for the truss-beam may be supported in the middle by an inverted king-post (of timber, not iron), carried by iron or wooden ties from its extremities; and the same ties may carry the horizontal tie-beam K; for till K be torn asunder, or M, M, and P be crippled, nothing can fail.

The roof of St Martin's church in the Fields is constructed on good principles, and every piece properly disposed. But although its span does not exceed forty feet from column to column, it contains more timber in a truss than there is in one of Drury-Lane theatre. The roof of the chapel at Greenwich, that of St Paul's, Covent-Garden, those of Birmingham and Drury-Lane theatres, form a series gradually more perfect. Such specimens afford excellent lessons to the artist. We therefore account them a useful present to the public.

There is a very ingenious project offered to the public Project by Mr P. Nicholson. (Carpenter's Assistant, p. 68.) He Mr Ni proposes iron rods for king-posts, queen-posts, and all cholson. Carpentry.

He receives the feet of the braces and struts in a socket very well connected with the foot of his iron king-post; and he secures the feet of his queen-posts from being pushed inwards, by interposing a straining sill. He does not even mortise the foot of his principal rafter into the end of the tie-beam, but sets it in a socket like a shoe, at the end of an iron bar, which is bolted into the tie-beam a good way back. All the parts are formed and disposed with the precision of a person thoroughly acquainted with the subject; and we have not the smallest doubt of the success of the project, and the complete security and durability of his roofs. We abound in iron; but we must send abroad for building timber. This is therefore a valuable project; at the same time, however, let us not overrate its value. Iron is about twelve times stronger than red fir, and is more than twelve times heavier; nor is it cheaper, weight for weight, or strength for strength.

Our illustrations and examples have been chiefly taken from roofs, because they are the most familiar instances of the difficult problems of the art. We could have wished for more room even on this subject. The construction of dome roofs has been, we think, mistaken, and the difficulty is much less than is imagined: we mean in respect of strength; for we grant that the obliquity of the joints, and a general intricacy, increases the trouble of workmanship exceedingly. Wooden bridges form another class equally difficult and important; but our limits are already overpassed, and will not admit them. The principle on which they should all be constructed, without exception, is that of a truss, avoiding all lateral bearings on any of the timbers. In the application of this principle we must further remark, that the angles of our truss should be as acute as possible; therefore we should make it of as few and of as long pieces as we can, taking care to prevent the bending of the truss beams by bridles, which embrace them, but without pressing them to either side. When the truss consists of many pieces, the angles are very obtuse, and the thrusts increase nearly in the duplicate proportion of the number of angles.

With respect to the frames of carpentry which occur in engines and great machines, the varieties are such that it would require a volume to treat of them properly. The principles are already laid down; and if the reader be really interested in the study, he will engage in it with seriousness, and cannot fail of being instructed. We recommend to his consideration, as a specimen of what may be done in this way, the working beam of Hornblower's steam-engine. (See Steam-Engine.) When the beam must act by chains hung from the upper end of arch-heads, the framing there given seems very scientifically constructed; at the same time we think that a strap of wrought iron reaching the whole length of the upper bar (see the figure) would be vastly preferable to those partial plates which the engineer has put there, for the bolts will soon work loose.

But when arches are not necessary, the form employed by Mr Watt is vastly preferable, both for simplicity and for strength. It consists of a simple beam, AB (fig. 45, Plate CLII.), having the gudgeon, C, on the upper side. The two piston rods are attached to wrought-iron joints, A and B. Two strong struts, DC, EC, rest on the upper side of the gudgeon, and carry an iron string, ADEB, consisting of three pieces, connected with the struts by proper joints of wrought iron. A more minute description is not needed for a clear conception of the principle. No part of this is exposed to a cross strain; even the beam AB might be sawed through at the middle. The iron string is the only part which is stretched; for AC, DC, EC, BC, are all in a state of compression. We have made the angles equal, that all may be as great as possible, and the pressure on the struts and strings a minimum. Mr Watt makes them much lower, as AdeH, or AdeB. But this is for economy, because the strength is almost insuperable. It might be made with wooden strings; but the workmanship of the joints would more than compensate the cheapness of the materials.

We offer this article to the public with deference, and we hope for an indulgent reception of our essay on a subject which is in a manner new, and would require much study. We have bestowed our chief attention on the strength of the construction, because it is here that persons of the profession have the most scanty information. We beg them not to consider our observations as too refined, and that they will study them with care. One principle runs through the whole; and when that is clearly conceived and familiar to the mind, we venture to say that the practitioner will find it of easy application, and that he will improve every performance by a continual reference to it.

IV.—Notes.

AA, p. 158. This rule may be somewhat more accurately expressed in these words: From the point at which any three forces meet and balance each other, draw a line in the actual direction of any one of them, and from the extremity of this line draw two others, parallel to the directions of the other two forces respectively; then supposing the pieces affording these two forces to be produced indefinitely at their remotest ends, either of them which is cut by one of the two lines will be compressed, and act as a brace, and either of them which is not cut will be stretched, and act as a tie.

BB, p. 159. It is, however, difficult to imagine how the beam DA can furnish a force iA, to prevent the force Af from carrying the beam BA towards H, when DA only affords a repulsive abutment. The true resolution of the force AE is found by considering the intersection of GE with Ae, which are the directions of the separate forces composing it: these lines meeting in a point a little above r, we may call their intersection r*: then in the triangle AE*r*, the side Ar* will represent the pressure on the mitred joint, and r*E the pressure on the beam HD; and the former being again resolved into AG and Gr*, we have ultimately AG and Gr* + r*E = GE = AF, for the horizontal and vertical forces, however they may be modified by intermediate combinations.

CC, p. 160. The reasoning contained in this and some of the subsequent articles may serve as an approximation to the truth in many cases of common occurrence; but the supposition on which it is founded is by no means generally admissible as affording a result mathematically accurate; for, in reality, the distribution of the weight of a roof over the whole extent of the rafters, or the concentration of the whole weight in the point where they meet, is far from being an indifferent alternative, either with respect to the magnitude of the thrusts, or to the proper directions of the abutments or joints. In the case here discussed, where there is no king-post, it is clear that the centre of gravity of the whole roof must be much nearer to the middle of the figure than the angular point, and that consequently the weights supported by the two walls will be very different from those which would be support-

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1 See figures 40, 41, 42, Plate CLI. and Mr Nicholson's work, p. 68, where these figures are particularly described. must have been reduced in the proportion of 1:207 to 1. Carpentry.

(Art. Bridge, Prop. E.) But considering how near the

are thus determined approaches to a quadrant, it is obvi-

ous that any slight variations of the quantities concerned

in the calculation must have greatly affected the magnitude

of the tangent; so that the loss of strength may easily

have been considerably greater than this, as it appears to

have been found in the experiment. It would, however,

scarcely have been expected that such a pillar, however

supported, could withstand the pressure of ninety hun-

dredweight, since Emerson informs us that the cohesive

strength of a pillar of fir an inch in diameter is only about

thirty-five; but supposing the facts correct, the coincidence

tends to show the near approach to equality of the forces

of cohesion and lateral adhesion, as explained in the in-

troduction to this article.

EE, p. 163. A similar remark of the author has already

been noticed in the article Barlow, at the end of the fifth

section. In the form in which it is here expressed, it be-

comes still more objectionable; for whatever part of a

circular abutment a rafter equal to the radius may be

brought into contact, it is very plain that its opposite end

can never be either higher or lower than the original cen-

tre of curvature; and even if the curvature were made

twice as great, so that the rafter might be equal to the di-

ameter of the circle, it would be necessary that the lower end

should slide upwards on the abutment as much as the upper

end fell, in order to preserve the contact; and there

would obviously be no force in the structure capable of

producing such a change as this. Any general curvature

of the joint must therefore be totally useless; but a judi-

cious workman will make it somewhat looser below than

above, when there is any probability that the rafters will

sink, taking care, however, to avoid all bearing too near

the surface, lest it should splinter, and, for these reasons

combined, making the end a little prominent somewhat

above the middle of the surface which rests on the abut-

ment.

With this precaution, the direction of the joint between

a rafter and a tie-beam ought to be made precisely perpen-

dicular to the true thrust of the rafter, determined as al-

ready explained (Note CC): for, in the first place, unless

we trust either to the friction, or to straps, the bearing

cannot be more nearly horizontal than this, without danger

of the rafters sliding outwards; and, in the second place, if

we made it more nearly vertical, we should lessen the verti-

cal pressure on the end of the tie-beam, immediately beyond

the joint; a pressure which gives firmness to the wood,

by pressing its fibres more closely together, and increas-

ing their lateral adhesion, or rather internal friction. If,

however, the tie-beam were not deep enough to receive

the whole of the rafter so terminated, without too great a

reduction of its depth, it would be proper to make the joint

a little flatter, or more horizontal, and to restrain the end

from sliding upwards by an iron strap fixed in a proper di-

rection. We should preserve the end of the rafter as little

diminished in breadth as possible, when the tie-beam is

wide enough to receive it; a moderate thickness, left on

each side of the mortise in the tie-beam, being sufficient to

assist in securing the connection of the ends of the beam

with the intermediate parts.

FF, p. 163. The doctrine of the initial equality of the

resistances to compression and extension, as stated in the

article Bridge, enables us to demonstrate that the trans-

verse strength can never exceed one sixth of that which

would be derived from the resistance of all the fibres,

co-operating at the distance of the whole depth from a fix-

ed fulcrum, and acting with the weaker of the two powers

appropriate to the body. It is true that the results of

some direct experiments seem to favour the opinion that the cohesive power is the weaker; but where the flexure is already considerable, it is probable that this circumstance materially diminishes the primitive power of resisting compression, so that the principles on which the calculation proceeds are by no means strictly applicable to the case of a bar so broken.

GG, p. 164. There seems to be a little confusion in the idea of the possibility of altering the nature of the action of the fibres of a beam by altering the place of the gudgeon in this manner; but the author has very properly abstained from making any practical application of the supposed modification thus introduced. With respect to the strength required for scarfing or jogging, it may be observed, that the whole of the compressed fibres of the concave side may be considered as abutting against the whole of the extended fibres on the convex side; and this abutment is equally divided throughout the length of the beam; so that if the scarfings or joggles in the whole length of the arm of a lever, taken together, are as strong as one half of the depth of the lever, exerting half its powers, from the inequality of tension, there will be no danger of the failing of these joints; and from this principle it will be easy to determine the depth to which the joints ought to extend in any particular case. Hence also we may understand how a beam may become so short as to be incapable of transverse fracture in its whole extent; for the lateral adhesion between the different fibres of wood is generally far inferior to the longitudinal strength of the fibres; and if, for example, it were only one fourth as great, a beam less than twice as long as it is deep would separate, if urged in the middle by a transverse force, into two strata, from its incapacity of affording sufficient abutment, before its longitudinal fibres would give way.

HHI, p. 166. If the bolts were sufficiently numerous and sufficiently firm, so as to produce a great degree of adhesion or of friction between the parts, this joint might be made almost as strong as the entire beam, since there is nothing to prevent the co-operation of each side with the other throughout its extent; but much of the strength would be lost if the bolts became loose, even in an inconsiderable degree.

II, p. 169. The author has reasoned upon the direction of straps, as if it were universally necessary to economize their immediate strength only, without regard to the effect produced on the tightness of the joint; but it may happen that the principal purpose of the strap will be answered by its pressing the rafter firmly upon the beam, and this effect may be produced by a certain deviation from the horizontal position, with but little diminution of the strength of the strap; a deviation which has also the advantage of allowing the strap to embrace the whole of the beam, without weakening it by driving a bolt through it. We must not, however, run the risk of crippling the end of the beam, and the straps represented in fig. 38 may be allowed to be somewhat too erect.

KK, p. 169. It does not appear to be desirable that the ends of the rafters should be supported without any pressure on the ends of the beams, since these ends would bear a small weight without any danger of bending, and would thus lessen the pressure on the king-post.

LL, p. 169. The half length being 25 feet, and the camber 6 inches, the excess of the oblique length will be $\sqrt{625-25} = 25$, or $\frac{1}{5}$ of a foot, that is, $\frac{1}{15}$ of an inch, which is all that the beam would appear to lengthen in sinking; nor would the settling of the roof be more "considerable" than about a quarter of an inch. But, there seems to be no advantage in this deviation of the tie-beam from the rectilinear direction; and the idea, which appears to be entertained by some workmen, that a bent beam partakes of the nature of an arch, is one of the many mischievous fallacies which it is the business of the mathematical theory of carpentry to dispel.