covering of any building by which its inhabitants and contents are protected from the injuries and inclemencies of the weather. So essential is it, that the word is frequently used for the house itself. To "come under the roof" is a Hebrew phrase; and the word "tectum" had the same meaning among the Romans. It is derived from the Anglo-Saxon hrof, who thought so much of its importance, that they called the carpenter hrof-wyrhta, or the "roof-worker."

Roofs may be considered as to their covering, and the framing which carries such covering. The former is either of metal, as lead, copper, zinc, corrugated or galvanized iron, &c.; or of tile, either Italian, pan or Flemish tile, plain tile, &c.; or of slate, and sometimes of stone. The Greek temples were covered with long thin pieces of marble sunk or worked hollow by the mason, so that the wet could not run back under the next, and consequently these roofs shot off the water easily, and were very flat. Both in ancient and modern times, in all countries, the poorer classes of roofs are covered or thatched with straw, reed, heather, or some similar material. In most hot climates, and also in many parts of Italy, the roofs are flat, and covered with a sort of concrete or cement, which is carried on joists like a floor; the object being to form a sort of terrace to walk on early in the morning or late in the evening, to enjoy the cool air, which can only be felt in elevated situations.

The elevation of a roof, which governs the angle its rafters make with the horizon, is called its pitch. On this subject there has been a great deal of controversy. Some have considered, as they find the farther we go south the flatter the roofs are, that the pitch must be governed by climate; and most elaborate calculations have been made of certain angles at which it is proposed that roofs should be constructed in various latitudes. But it should be remembered that in hot climates the rains come all at once; in such floods our roofs could not resist; and it would be poor economy, because for months together there was no rain, if, when it does come, the house should be daily drenched. Others have considered the whole a mere matter of taste, and the pitch is chosen as we wish more or less of a roof to be seen. The Greeks made their roofs very flat, and placed large antefixes along the eaves, so that the roof could not be seen from below except from a great distance. (See the restored view of the Parthenon, Architecture, Plate L.) The angle is about 16°, the pitch or height at the apex being about a seventh of the width. Roman roofs average about 22°, or a fifth pitch. That the mediæval builders had no rule, is shown from the extreme variety of the height of roofs in their different edifices. In the Lombardic cathedral of Pisa, erected 1063 (Architecture, Plate LXX.), the roof is about 27°, or nearly quarter pitch. The Norman roofs are seldom more than 40°, or less than half pitch; while in the early English period they suddenly sprung up to whole pitch,—i.e., the height equal to the entire width (see Beverley Minster, Architecture, Plate LXVIII., fig. 1), being an angle of about 64°. They then gradually were less in height till the perpendicular period, when many roofs were nearly flat; that of Henry the Seventh chapel, for example, being but about 16°, or as flat as a Greek roof. Now, that this variety was matter of taste,—we had almost said caprice,—is evidenced by this fact: these examples are all covered with lead (which might have been laid quite flat, and yet have been perfectly sound), and all have a stone groined roof below them, which has nothing whatever to do with the upper covering, and which, after all, is the real roof or cover which protects the building from the weather. Much has been said of the propriety of always showing the roof of a building, and the Gothic architects have been eulogised for so doing. The facts stated above, however, prove this was not always the case. We cannot, however, justify the going out of the way to conceal a roof by false attics, slitted balustrades, &c.; and the screen wall at St Paul's at London must always be considered a defect in that fine building. Still, a wide expanse of plain roof is as ugly in itself as a bare wall; and we cannot approve of such roofs as some of the modern imitations of early English work are, where the wall is so low that we could touch the eaves with a walking-stick, and there is three or four times as much roof as wall. The roof of a house has not inaptly been likened to a man's hat. There is no need to try and hide or disguise it if you are obliged to wear it; and if the weather is warm, and you do not require it, it would be folly to wear one without a crown. If on board ship, you would wear as low a hat as possible to avoid striking it against the beams; but, above all, it should bear some reasonable proportion to the height of a man, as the roof should to the wall. It would be absurd to wear a hat as tall as the man himself.

After all, although much latitude must be given to taste, Pitch it is probable the pitch of a roof mainly depends on the pendent on material with which it is covered. The largest number of materials buildings are erected with a view to utility and strict economy, and without any regard to aesthetics. Everybody knows that if slates or tiles are laid at too flat a pitch, the wind will drive the rain up under them, and the roof will leak; and everybody also knows that if the same covering be taken off and re-laid to a steeper pitch, the roof will be sound. Practice teaches what is the safe minimum pitch. Let us suppose it to be quarter pitch, and for considerations of taste we make it three-quarter pitch. Now, it is quite clear we waste not only the rafters and covering, but our whole roof must be constructed of stronger timbers, and our walls also must be thicker and stronger, insomuch as they have more weight to bear. We therefore pay dear in more ways than one for our liking for high-pitched roofs.

Although it happens that both Greek, Roman, and Italian roofs are flatter than ours, and the climate is warmer, due to the same material used in our climates would answer perfectly well. An inspection of the elaborate plate 97, in the Architettura Antica Greca of the celebrated Canina will show this; and the frequent failure in our climate of Italian tiles (which are exactly like the ancient Roman) arises from the fact, that the tegole and imbrici only have been used, our builders being ignorant of the use of the mattoni, which in Italy are a very essential part of the soundness of those roofs. In one respect climate must be considered, and that is, where there are long winters, and the snow is likely to lie on them; in this case they should be sharper in pitch, and stronger in framing.

If covered with lead or other metals, roofs may be made nearly flat, with only so much fall, in fact, as to prevent the water flowing back under the drips. (See Building, Plumbers' Work, &c.) Italian tiles, to be sound, should have a fifth pitch, or 25°. Slates with extra lap may be laid at quarter pitch, about 27°; if it be necessary the roof should be flat; a third pitch (34°) is rather too much; the mean between a third and fourth (31°) is a good rule. Pan-tiles should be laid rather sharper still, and plain tiles from about 35° to 40°; but of course very much will depend on the gauge they are laid to, or the length of the part of the slate or tile which overlaps the other, as the larger this lap is, the less likely the rain is to drive under. Thatched roofs should be somewhat sharper in pitch than plain tiles.

Lead or copper, in an economical point of view, are the best materials for roofs. They may be laid nearly flat, and so save all the framing and roof timbers; and the metal, should it be worn into holes, is nearly as valuable as when first laid down; the only objection is, that the first expense is so great. Zinc, though very cheap and light, and though it can be laid flat, is apt to go into holes with the action of acids. Slating is both light and very cheap, and will lie at a flat pitch; and consequently requires much lighter walls and timbers than tiles. It will not decay with the weather. It is apt to break under the feet; and if not very well done will lift with heavy winds. Each slate should be nailed with two copper nails, as iron rusts and breaks them. (See Building, Slating, &c.) Pan-tiles are dearer than slates, but not much heavier; they also break if trodden on, and the snow will drift under if the pointing comes out. Plain tiles are very durable, but they require a steep pitch, and are very heavy; thus in two ways distressing the walls and the roof timbers.

This also depends on the gauge; but the following may be taken as the ordinary average:

- A square (100 feet superficial, or 11 yards superficial nearly) of zinc will weigh about... 1 cwt. - A square of lead, according to thickness, from... 6 to 7 " - A square of slating from... 6½ to 7½ " - A square of pan-tiling... 7½ " - A square of plain tiling... 14 to 16 "

All roofs, till very lately, except some which have been arched or domed, were framed with timber; no other material being known at that time which possessed such lengths with such qualities of tension. Later years, however, and more extended requirements, have developed the advantages of the use of iron. As everything must depend on the soundness of both design and execution of framing, whether in wood or iron, it is proposed to divide this subject, one of the most important in architecture, into the following sections:

I. Theory of Roof will comprehend the whole of the scientific part of the celebrated essay of Professor Robison, which was originally written for this work, and which is acknowledged to be the best yet given to the public.

II. Causes of Failure of Roofs, given in terms that are intelligible to those unacquainted with the higher branches of mathematical analysis.

III. Medieval Roofs.

IV. Account of Roofs of great span (à grande portée).

1. Those trussed with straight timbers (en bois plat). 2. Those trussed with curved timbers—a, With timbers side by side, breaking joint (système en planches de champ); b, Those bent in thickness (courbes sur leur plat).

V. Roofs constructed of Iron

The late Professor Robison's Theory of Roof.

We shall attempt in this article to give an account of the purpose of leading principles of this art, in a manner so familiar and this article palpable, that any person who knows the common properties of the lever, and the composition of motion, shall so far understand them as to be able, on every occasion, so to dispose his materials, with respect to the strains to which they are to be exposed, that he shall always know the effective strain on every piece, and shall, in most cases, be able to make the disposition such as to derive the greatest possible advantage from the materials which he employs.

It is evident that the whole must depend on the principles which regulate the strength of the materials, relative which refer to the manner in which this strength is exerted, and the galeate the manner in which the strain is laid on the piece of matter. With respect to the first, this is not the proper place for remarks considering it, and we must refer the reader to the article Strength of Materials in Mechanics. We shall just borrow from that article two or three propositions suited to our purpose.

The force with which the materials of our edifices, roofs, floors, machines, and framings of every kind, resist being broken or crushed, or pulled asunder, is immediately or ultimately the cohesion of their particles. When a weight hangs by a rope, it tends either immediately to break all the fibres, overcoming the cohesion amongst the particles of each, or it tends to pull one parcel of them from amongst the rest, with which they are joined. This union of the fibres is brought about by some kind of glue, or by twisting, which causes them to bind each other so hard that any one will break rather than come out, so much is it withheld by friction. The ultimate resistance is therefore the cohesion of the fibre; and the force or strength of all fibrous materials, such as timber, is exerted in much the same manner. The fibres are either broken or pulled out from among the rest. Metals, stone, glass, and the like, resist being pulled asunder by the simple cohesion of their parts.

The force which is necessary for breaking a rope or wire is a proper measure of its strength. In like manner, the force necessary for tearing directly asunder any rod of wood or metal, breaking all its fibres, or tearing them from amongst each other, is a proper measure of the united strength of all these fibres; and it is the simplest strain to which they can be exposed, being just equal to the sum of the forces necessary for breaking or disengaging each fibre. And, if the body is not of a fibrous structure, which is the case with metals, stones, glass, and many other substances, this force is still equal to the simple sum of the cohesive forces of each particle which is separated by the fracture. Let us distinguish this mode of exertion of the cohesion of the body by the name of its absolute strength.

When solid bodies are, on the contrary, exposed to great compression, they can resist only a certain degree. A piece of clay or lead will be squeezed out; a piece of freestone will be crushed to powder; a beam of wood will be crippled, swelling out in the middle, and its fibres lose their mutual cohesion, after which it is easily crushed by the load. A notion may be formed of the manner in which these strains are resisted, by conceiving a cylindrical pipe filled with small shot, well shaken together, so that each sphericle is lying in the closest manner possible, that is, in contact with six others in the same vertical plane, this being the position in which the shot will take the least room. Thus each touches the rest in six points. Now suppose them all united, in these six points only, by some cement. This assemblage will stick together and form a Now suppose this pillar standing upright, and loaded above. The supports arising from the cement act obliquely, and the load tends either to force them asunder laterally, or to make them slide on each other; either of these things happening, the whole is crushed to pieces. The resistance of fibrous materials to such a strain is a little more intricate, but may be explained in a way very similar.

A piece of matter of any kind may also be destroyed by wrenching or twisting it. We can easily form a notion of its resistance to this kind of strain by considering what would happen to the cylinder of small shot if treated in this way.

And, lastly, a beam, or a bar of metal, or piece of stone or other matter, may be broken transversely. This will happen to a rafter or joist supported at the ends when overloaded, or to a beam having one end stuck fast in a wall and a load laid on its projecting part. This is the strain to which materials are most commonly exposed in roofs; and, unfortunately, it is the strain which they are the least able to bear; or rather it is the manner of application which causes an external force to excite the greatest possible immediate strain on the particles. It is against this that the carpenter must chiefly guard, avoiding it when in his power, and in every case diminishing it as much as possible. It is necessary to give the reader a clear notion of the great weakness of materials in relation to this transverse strain. But we shall do nothing more, referring him to the articles STRAIN, and STRESS, and STRENGTH.

Let ABCD (fig. 1) represent the side of a beam projecting horizontally from a wall in which it is firmly fixed, and let it be loaded with a weight W appended to its extremity. This tends to break it; and the least reflection will convince any person, that if the beam is equally strong throughout, it will break in the line CD, even with the surface of the wall. It will open at D, while C will serve as a sort of joint, round which it will turn. The cross section through the line CD is for this reason called the section of fracture; and the horizontal line drawn through C on its under surface is called the axis of fracture. The fracture is made by tearing asunder the fibres, such as DE or FG. Let us suppose a real joint at C, and that the beam is really sawed through along CD, and that in place of its natural fibres, threads are substituted all over the section of fracture. The weight now tends to break these threads, and it is our business to find the force necessary for this purpose.

It is evident that DCA may be considered as a bended lever, of which C is the fulcrum. If f be the force which will just balance the cohesion of a thread when hung on it so that the smallest addition will break it, we may find the weight which will be sufficient for this purpose when hung on at A, by saying AC : CD = f : ϕ, and ϕ will be the weight which will just break the thread, by hanging ϕ by the point A. This gives us ϕ = f × CD / CA. If the weight be hung on at a, the force just sufficient for breaking the same thread will be = f × CD / Ca. In like manner, the force ϕ, which must be hung on at A in order to break an equally strong or an equally resisting fibre at F, must be = f × CF / CA.

And so on of all the rest.

If we suppose all the fibres to exert equal resistances at the instant of fracture, we know, from the simplest elements of mechanics, that the resistance of all the particles in the line CD, each acting equally in its own place, is the same as if all the individual resistances were united in the middle point g. Now this total resistance is the resistance or strength f of each particle, multiplied by the number of particles. This number may be expressed by the line CD, because we have no reason to suppose that they are at unequal distances. Therefore, in comparing different sections together, the number of particles in each are as the sections themselves. Therefore DC may represent the number of particles in the line DC. Let us call this line the depth of the beam, and express it by the symbol d. And since we are at present treating of roofs whose rafters and other parts are commonly of uniform breadth, let us call AH or BI the breadth of the beam, and express it by b, and let CA be called its length l. We may now express the strength of the whole line CD by f × d, and we may suppose it all concentrated in the middle point g. Its mechanical energy, therefore, by which it resists the energy of the weight w, applied at the distance l, is f × CD × Cg, whilst the momentum of w is w × CA. We must therefore have f × CD × Cg = w × CA, or fd × ld = wl, or fd : w = l : ld, or fd : w = 2l : d. That is, twice the length of the beam is to its depth as the absolute strength of one of its vertical planes to its relative strength, or its power of resisting this transverse fracture.

It is evident, that what has been now demonstrated of the resistance exerted in the line CD, is equally true of every line parallel to CD in the thickness or breadth of the beam. The absolute strength of the whole section of fracture is represented by fbd, and we still have 2l : d = fbd : wl; or twice the length of the beam is to its depth as the absolute strength to the relative strength. Suppose the beam twelve feet long and one foot deep; then whatever be its absolute strength, the twenty-fourth part of this will break it if hung at its extremity.

But even this is too favourable a statement. All the fibres are supposed to act alike in the instant of fracture. But this is not true. At the instant that the fibre at D breaks, it is stretched to the utmost, and is exerting its whole force. But at this instant the fibre at g is not so much stretched, and it is not then exerting its utmost force. If we suppose the extension of the fibres to be as their distance from C, and the actual exertion of each to be as their extensions, it may easily be shown (see Strength and Strain), that the whole resistance is the same as if the full force of all the fibres were united at a point r distant from C by one third of CD. In this case we must say, that the absolute strength is to the relative strength as three times the length to the depth; so that the beam is weaker than by the former statement in the proportion of two to three.

Even this is more strength than experiment justifies, and we can see an evident reason for it. When the beam is strained, not only are the upper fibres stretched, but the lower fibres are compressed. This is very distinctly seen, if we attempt to break a piece of cork cut into the shape of a beam. This being the case, C is not the centre of fracture. There is some point e which lies between the fibres which are stretched and those that are compressed. This fibre is neither stretched nor squeezed, and this point is the real centre of fracture; and the lever by which a fibre D resists, is not DC, but a shorter one De, and the energy of the whole resistances must be less than by the second statement. Till we know the proportion between the dilatability and compressibility of the parts, and the relation between the dilatations of the fibres and the resistances which they exert in this state of dilatation, we cannot positively say where the point is situated, nor what is the sum of the actual resistances, or the point where their action may be supposed concentrated. The firmer woods, such as oak and chestnut, may be supposed to be but slightly compressible; we know that willow and other soft woods are very compressible. These last must therefore be weaker; for it is evident, that the fibres which are in a state of compression do not resist the fracture. It is well known, that a beam of willow may be cut through from C to g without weakening it in the least, if the cut be filled up by a wedge of hard wood stuck in.

We can only say, that very sound oak and red fir have the centre of effort so situated, that the absolute strength is to the relative strength in a proportion of not less than that of three and a half times the length of the beam to its depth. A square inch of sound oak will carry about 8000 pounds. If this bar be firmly fixed in a wall, and project twelve inches, and be loaded at the extremity with 200 pounds, it will be broken. It will just bear 190, its relative strength being $\frac{1}{4}$ of its absolute strength; and this is the case only with the finest pieces, so placed that their annual plates or layers are in a vertical position. A larger log is not so strong transversely, because its plates lie in various directions round the heart.

These observations are enough to give us a distinct notion of the vast diminution of the strength of timber when the strain is across it; and we see the justice of the maxim which we inculcated, that the carpenter, in framing roofs, should avoid as much as possible the exposing his timbers to transverse strains. But this cannot be avoided in all cases. Nay, the ultimate strain arising from the very nature of a roof is transverse. The rafters must carry their own weight, and this tends to break them across. An oak beam a foot deep will not carry its own weight if it project more than sixty feet. Besides this, the rafters must carry the lead, tiling, or slates. We must therefore consider this transverse strain a little more particularly, so as to know what strain will be laid on any part by an unavoidable load, imposed either at that part or at any other.

We have hitherto supposed, that the beam had one of its ends fixed in a wall, and that it was loaded at the other end. This is not an usual arrangement, and was taken merely as affording a simple application of the mechanical principles. It is much more usual to have the beam supported at the ends, and loaded in the middle. Let the beam FEGH (fig. 2) rest on the props E and G, and be loaded at its middle point C with a weight W. It is required to determine the strain at the section CD.

It is plain that the beam will receive the same support, and suffer the same strain, if, instead of the blocks E and G, we substitute the ropes Fe, HA, going over the pulleys f and g, and loaded with proper weights e and g. The weight e is equal to the support given by the block E; and g is equal to the support given by G. The sum of e and g is equal to W; and on whatever point W is hung, the weights e and g are to W in the proportion of DG and DE to GE.

Now, in this state of things, it appears that the strain on the section CD arises immediately from the upward action of the ropes Fe and HA, or the upward pressions of the blocks E and G; and that the office of the weight W is to oblige the beam to oppose this strain. Things are in the same state in respect of strain as if a block were substituted at D for the weight W, and the weights e and g were hung on at E and G, only the directions will be opposite.

The beam tends to break in the section CD, because the ropes pull it upwards at E and G, whilst a weight W holds it down at C. It tends to open at D, and C becomes the centre of fracture. The strain therefore is the same as if the half ED were fixed in the wall, and a weight equal to g, that is, to the half of W, were hung on at G.

Hence we conclude, that a beam supported at both ends, but not fixed there, and loaded in the middle, will carry four times as much weight as it can carry at its extremity, when the other extremity is fast in a wall.

The strain occasioned at any point L by a weight W, hung on at any other point D, is $W \times \frac{DE}{EG} \times LG$. For EG is to ED as W to the pressure occasioned at G. This would be balanced by some weight g acting over the pulley h; and this tends to break the beam at L, by acting on the lever GL. The pressure at G is $W \times \frac{DE}{EG}$, and therefore the strain at L is $W \times \frac{DE}{EG} \times LG$.

In like manner, the strain occasioned at the point D by the weight W hung on there, is $W \times \frac{DE}{EG} \times DG$; which is therefore equal to $\frac{1}{2} W$ when D is the middle point.

Hence we see that the general strain on the beam arising from one weight, is proportional to the rectangle of the parts of the beam (for $\frac{W \times DE \times DG}{EG}$ is $DE \times DG$), and is greatest when the load is laid on the middle of the beam.

We also see, that the strain at L, by a load at D, is equal to the strain at D by the same load at L. And the strain at L from a load at D is to the strain by the same load at L as DE to LE. These are all very obvious corollaries, and they sufficiently inform us concerning the strains which are produced on any part of the timber by a load laid on any other part.

If we now suppose the beam to be fixed at the two ends, that is, firmly framed or held down by blocks at I and K, placed beyond E and G, or framed into posts, it will carry twice as much as when its ends were free. For suppose it sawn through at CD, the weight W hung on there will be just sufficient to break it at E and G. Now restore the connection of the section CD, it will require another weight W to break it there at the same time.

Therefore, when a rafter, or any piece of timber, is firmly connected with three fixed points, G, E, I, it will bear a greater load between any two of them than if its connection with the remote point were removed; and if it be fastened in four points, G, E, I, K, it will be twice as strong in the middle part as without the two remote connections.

One is apt to expect from this that the joist of a floor will be much strengthened by being firmly built in the wall. It is a little strengthened; but the hold which can thus be given to it is much too short to be of any sensible service, and it tends greatly to shatter the wall, because, when it is bent down by a load, it forces up the wall with a momentum of a long lever. Judicious builders therefore take care not to bind the joints tight in the wall. But when the joists of adjoining rooms lie in the same direction, it is a great advantage to make them of one piece. They are then twice as strong as when made in two lengths.

It is easy to deduce from these premises the strain on any point which arises from the weight of the beam itself, or from any load which is uniformly diffused over the whole or any part. We may always consider the whole of the weight which is thus uniformly diffused over any part as united in the middle point of that part; and if the load is not uniformly diffused, we may still suppose it united at its centre of gravity. Thus, to know the strain at D arising from the weight of the whole beam, we may suppose the whole weight accumulated in its middle point D. Also the strain at L, arising from the weight of the part ED, is the same as if this weight were accumulated in the middle point d of ED; and it is the same as if half the weight of ED were hung on at D. For the real strain at L is the upward pressure at G, acting by the lever GL. Now, calling the weight of the part DE e, this upward pressure will be \( \frac{e \times DE}{EG} \), or \( \frac{\frac{1}{2} e \times DE}{EG} \).

Therefore the strain on the middle of a beam, arising from its own weight, or from any uniform load, is the weight of the beam or its load \( \times \frac{ED}{EG} \times DG \); that is, half the weight of the beam or load multiplied or acting by the lever DG; for \( \frac{ED}{EG} \) is \( \frac{1}{2} \).

Also the strain at L, arising from the weight of the beam, or the uniform load, is \( \frac{1}{2} \) the weight of the beam or load acting by the lever LG. It is therefore proportional to LG, and is greatest of all at D. Therefore a beam of uniform strength throughout, uniformly loaded, will break in the middle.

It is of importance to know the relation between the strains arising from the weights of the beams, or from any uniformly diffused load, and the relative strength. We have already seen, that the relative strength is \( \int \frac{d^2b}{ml} \), where m is a number to be discovered by experiment for every different species of materials. Leaving out every circumstance but what depends on the dimensions of the beam, viz. d, b, and l, we see that the relative strength is in the proportion of \( \frac{d^2b}{l} \), that is, as the breadth and the square of the depth directly, and the length inversely.

Now, to consider, first, the strain arising from the weight of the beam itself; it is evident that this weight increases in the same proportion with the depth, the breadth, and the length of the beam. Therefore its power of resisting this strain must be as its depth directly, and the square of its length inversely. To consider this in a more popular manner, it is plain that the increase of breadth makes no change in the power of resisting the actual strain, because the load and the absolute strength increase in the same proportion with the breadth. But, by increasing the depth, we increase the resisting section in the same proportion, and therefore the number of resisting fibres and the absolute strength; but we also increase the weight in the same proportion. This makes a compensation, and the relative strength is yet the same. But, by increasing the depth, we have not only increased the absolute strength, but also its mechanical energy. For the resistance to fracture is the same as if the full strength of each fibre was exerted at the point which we called the centre of effort; and we showed that the distance of this from the under side of the beam was a certain portion (a half, a third, a fourth, &c.) of the whole depth of the beam. This distance is the arm of the lever, by which the cohesion of the wood may be supposed to act. Therefore this arm of the lever, and consequently the energy of the resistance, increases in the proportion of the depth of the beam, and this remains uncompensated by any increase of the strain. On the whole, therefore, the power of the beam to sustain its own weight increases in the proportion of its depth. But, on the other hand, the power of withstanding a given strain applied at its extremity, or to any aliquot part of its length, is diminished as the length increases, or is inversely as the length; and the strain arising from the weight of the beam also increases as the length. Therefore the power of resisting the strain actually exerted on it by the weight of the beam is inversely as the square of the length. On the whole, therefore, the power of a beam to carry its own weight varies in the proportion of its depth directly and the square of its length inversely.

As this strain is frequently a considerable part of the whole, it is proper to consider it apart, and then to reckon only on what remains for the support of any extraneous load.

In the next place, the power of a beam to carry any load which is uniformly diffused over its length, must be taken inversely as the square of the length; for the power of carrying a load withstanding any strain applied to an aliquot part of the uniformly diffused length (which is the case here, because the load may be over its conceived as accumulated at its centre of gravity, the length, middle point of the beam) is inversely as the length; and the actual strain is as the length, and therefore its momentum is as the square of the length. Therefore the power of a beam to carry a weight uniformly diffused over it, is inversely as the square of the length.

It is here understood, that the uniform load is of some determined quantity for every foot of the length, so that a beam of double length carries a double load.

We have hitherto supposed that the forces which tend to break a beam transversely are acting in a direction perpendicular to the beam. This is always the case in level floors loaded in any manner; but in roofs, the action of the load tending to break the rafters is oblique, because gravity always acts in vertical lines. It may also frequently happen, that a beam is strained by a force acting obliquely. This modification of the strain is easily discussed. Suppose that the external force, which is measured by the weight W in fig. 1, acts in the direction Aa' instead of AW. Draw Co' perpendicular to AE. Then the momentum of this external force is not to be measured by \( W \times AC \), but by \( W \times a'C \). The strain therefore by which the fibres in the section of fracture DC are torn asunder, is diminished in the proportion of CA to Ca', that is, in the proportion of radius to the sine of the angle CAa', which the beam makes with the direction of the external force.

To apply this to our purpose in the most familiar manner, let AB (fig. 3) be an oblique rafter of a building, loaded with a weight W suspended at any point C, and thereby occasioning a strain in some part D. We have already seen, that the immediate cause of the strain on D is the re-action of the support which is given to the point B. The rafter may at present be considered as a lever, supported at A, and pulled down by the line CW. This occasions a pressure on B, and the support acts in the opposite direction to the action of the lever, that is, in the direction Bb, perpendicular to BA. This tends to break the beam in every part. The pressure exerted at B is \( \frac{W \times AE}{AB} \), AE being a horizontal line. Therefore the strain at D will be \( \frac{W \times AE}{AB} \times BD \). Had the beam been lying horizontally, the strain at D, from the weight W suspended at C, would have been \( \frac{W \times AC}{AB} \times BD \). It is therefore diminished in the proportion of AC to AE, that is, in the proportion of radius to the cosine of the elevation, or in the proportion of the secant of elevation to the radius.

It is evident, that this law of diminution of the strain is the same whether the strain arises from a load on any part of the rafter, or from the weight of the rafter itself, or from any load uniformly diffused over its length, provided only that these loads act in vertical lines.

We can now compare the strength of roofs which have different elevations. Supposing the width of the building to be given, and that the weight of a square yard of covering is also given. Then, because the load on the rafter will increase in the same proportion with its length, the load on the slant-side BA of the roof will be to the load of a similar covering on the half AF of the flat roof, of the same width, as AB to AF. But the transverse action of any load on AB, by which it tends to break it, is to that of the same load on AF as AF to AB. The transverse strain therefore is the same on both, the increase of real load on AB being compensated by the obliquity of its action. But the strengths of beams to resist equal strains, applied to similar points, or uniformly diffused over them, are inversely as their lengths, because the momentum or energy of the strain is proportional to the length. Therefore the power of AB to withstand the strain to which it is really exposed, is to the power of AF to resist its strain as AF to AB. If, therefore, a rafter AG of a certain scantling is just able to carry the roofing laid on it, a rafter AB of the same scantling, but more elevated, will be too weak in the proportion of AG to AB. Therefore steeper roofs require stouter rafters, in order that they may be equally able to carry a roofing of equal weight per square yard. To be equally strong, they must be made broader, or placed nearer to each other, in the proportion of their greater length, or they must be made deeper in the subduplicate proportion of their length. The following easy construction will enable the artist not familiar with computation to proportion the depth of the rafter to the slope of the roof.

Let the horizontal line af (fig. 4) be the proper depth of a beam whose length is half the width of the building; that is, such as would make it fit for carrying the intended tiling laid on a flat roof. Draw the vertical line fb, and the line ab having the elevation of the rafter; make ag equal to af, and describe the semicircle bdg; draw ad perpendicular to ab, then ad is the required depth. The demonstration is evident.

We have now treated in sufficient detail what relates to the chief strain on the component parts of a roof, namely, what tends to break them transversely; and we have enlarged more on the subject than what the present occasion indispensably required, because the propositions which we have demonstrated are equally applicable to all framings of carpentry, and are even of greater moment in many cases, particularly in the construction of machines. These consist of levers in various forms, which are strained transversely; and similar strains frequently occur in many of the supporting and connecting parts. We shall give, in another article, an account of the experiments which have been made by different naturalists, in order to ascertain the absolute strength of some of the materials which are most generally framed together in buildings and engines. The house-carpenter will derive from them absolute numbers, which he can apply to his particular purposes by means of the propositions which we have now established.

We proceed, in the next place, to consider the other strains to which the parts of roofs are exposed, in consequence of the support which they mutually give each other, and the pressures, or thrusts, as they are called in the language of the house-carpenter, which they exert on each other, and on the walls or piers of the building.

Let a beam or piece of timber AB (fig. 5) be suspended Effect of other strains, &c.

by two lines AC, BD; or let it be supported by two props AE, BF, which are perfectly moveable round their remote extremities E, F, or let it rest on the two polished planes KAH, LBM. Moreover, let G be the centre of gravity of the beam, and let GN be a line through the centre of gravity perpendicular to the horizon. The beam will not be in equilibrium unless the vertical line GN either passes through P, the point in which the directions of the two lines AC, BD, or the directions of the two props EA, FB, or the perpendiculars to the two planes KAH, LBM intersect each other, or is parallel to these directions. For the supports given by the lines or props are unquestionably exerted in the direction of their lengths, and it is well known in mechanics that the supports given by planes are exerted in a direction perpendicular to those planes in the points of contact; and we know that the weight of the beam acts in the same manner as if it were all accumulated in its centre of gravity G, and that it acts in the direction GN perpendicular to the horizon. Moreover, when a body is in equilibrium between three forces, they are acting in one plane, and their directions are either parallel or they pass through one point.

The support given to the beam is therefore the same as if it were suspended by two lines which are attached to the single point P. We may also infer, that the points of suspension C, D, the points of support E, F, the points of contact A, B, and the centre of gravity G, are all in one vertical plane.

When this position of the beam is disturbed by any external force, there must either be a motion of the points A and B round the centres of suspension C and D, or of the props round these points of support E and F, or a sliding of the ends of the beam along the polished planes KAH and LBM; and in consequence of these motions the centre of gravity G will go out of its place, and the vertical line GN will no longer pass through the point where the directions of the supports intersect each other. If the centre of gravity rises by this motion, the body will have a tendency to recover its former position, and it will require force to keep it away from it. In this case the equilibrium may be said to be stable, or the body to have stability. But if the centre of gravity descends when the body is moved from the position of equilibrium, it will tend to move still farther; and so far will it be from recovering its former position, that it will now fall. This equilibrium may be called a tottering equilibrium. These accidents depend on the situations of the points A, B, C, D, E, F; and they may be determined by considering the subject geometrically. It does not much interest us at present; it is rarely that the equilibrium of suspension is tottering, or that of props is stable. It is evident, that if the beam were suspended by lines from the point P, it would have stability, for it would swing like a pendulum round P, and therefore would always tend towards the position of equilibrium. The intersection of the lines of support would still be at P, and the vertical line drawn through the centre of gravity, when in any other situation, would be on that side of P towards which this centre has been moved. Therefore, by the rules of pendulous bodies, it tends to come back. This would be more remarkably the case if the points of suspension C and D were on the same side of the point P with the points of attachment A and B; for in this case the new point of intersection of the lines of support would shift to the opposite side, and be still further from the vertical line through the new position of the centre of gravity. But if the points of suspension and of attachment are on opposite sides of P, the new point of intersection may shift to the same side with the centre of gravity, and lie beyond the vertical line. In this case the equilibrium is tottering. It is easy to perceive, too, that if the equilibrium of suspension from the points C and D be stable, the equilibrium on the props AE and BF must be tottering. It is not necessary for our present purpose to engage more particularly in this discussion.

It is plain that, with respect to the mere momentary equilibrium, there is no difference in the support by threads, props, or planes, and we may substitute the one for the other. We shall find this substitution extremely useful, because we easily conceive distinct notions of the support of a body by strings.

Observe farther, that if the whole figure be inverted, and strings be substituted for props, and props for strings, the equilibrium will still obtain. For by comparing fig. 5 with fig. 6, we see that the vertical line through the centre of gravity will pass through the intersection of the two strings or props; and this is all that is necessary for the equilibrium; only it must be observed in the substitution of props for threads, and of threads for props, that if it be done without inverting the whole figure, a stable equilibrium becomes a tottering one, and vice versa.

Examples. This is a most useful proposition, especially to the unlettered artisan, and enables him to make a practical use of problems which the greatest mechanical geniuses have found it no easy task to solve. An instance will show the extent and utility of it. Suppose it were required to make a mansard or kirk roof whose width is AB (fig. 7), and consisting of the four equal rafters AC, CD, DE, EB. There can be no doubt but that its best form is that which will put all the parts in equilibrium, so that no ties or stays may be necessary for opposing the unbalanced thrust of any part of it. Make a chain acdeb (fig. 8) of four equal pieces, loosely connected by pin-joints, round which the parts are perfectly moveable. Suspend this from two pins a, b, fixed in a horizontal line. This chain or festoon will arrange itself in such a form that its parts are in equilibrium. Then we know that if the figure be inverted, it will compose the frame or truss of a kirk-roof acdeb, which is also in equilibrium, the thrusts of the pieces balancing each other in the same manner that the mutual pulls of the hanging festoon acdeb did. If the proportion of the height of the width ab is not such as pleases, let the pins a, b be placed nearer or more distant, till a proportion between the width and height is obtained which pleases, and then make

the figure ACDEB, fig. 7, similar to it. It is evident that this proposition will apply in the same manner to the determination of the form of an arch of a bridge; but this is not a proper place for a further discussion.

We are now enabled to compute all the thrusts and other pressures which are exerted by the parts of a roof on each other and on the walls. Let AB (fig. 9) be a beam standing anyhow obliquely, and G its centre of gravity. Let us suppose that the ends of it are supported in any directions AC, BD, by strings, props, or planes. Let these directions meet in the point P of the vertical line PG passing through its centre of gravity. Through G draw lines Ga, Gb parallel to PB, PA. Then

The weight of the beam \( \frac{PG}{PA} \) The pressure or thrust at A \( \frac{PG}{PA} \) And the pressure at B \( \frac{PG}{PB} \)

For when a body is in equilibrium between three forces, these forces are proportional to the sides of a triangle which have their directions.

In like manner, if Ag be drawn parallel to Pb, we shall have

Weight of the beam \( \frac{PG}{PA} \) Thrust on A \( \frac{PG}{PA} \) And thrust on B \( \frac{PG}{PB} \)

Or, drawing By parallel to Pa,

Weight of the beam \( \frac{PG}{PA} \) Thrust at A \( \frac{PG}{PA} \) And thrust at B \( \frac{PG}{PB} \)

It cannot be disputed that, if strength alone be considered, the proper form of a roof is that which puts the whole in equilibrium, so that it would remain in that shape although all the joints were perfectly loose or flexible. If it has any other shape, additional ties or braces are necessary for preserving it, and the parts are unnecessarily strained-equilled. When this equilibrium is obtained, the rafters which compose the roof are all acting on each other in the direction of their lengths; and by this action, combined with their weights, they sustain no strain but that of compression, the strain of all others that they are the most able to resist. We may consider them as so many inflexible lines having their weights accumulated in their centres of gravity. But it will allow an easier investigation of the subject, if we suppose the weights to be at the joints, equal to the real vertical pressures which are exerted on these points. These are very easily computed; for it is plain, that the weight of the beam AB (fig. 9) is to the part of this weight that is supported at B as AB to AG. Therefore, if W represent the weight of the beam, the vertical pressure at B will be \( W \times \frac{AG}{AB} \) and the vertical pressure at A will be \( W \times \frac{BG}{AB} \).

In like manner, the prop BF being considered as another beam, and f as its centre of gravity and w as its weight, a part of this weight, equal to \( w \times \frac{fF}{BF} \), is supported at B, and the whole vertical pressure at B is \( W \times \frac{AG}{AB} + w \times \frac{fF}{BF} \). And thus we greatly simplify the consideration of the mutual thrusts of roof frames. We need hard- ly observe, that although these pressures by which the parts of a frame support each other in opposition to the vertical action of gravity, are always exerted in the direction of the pieces, they may be resolved into pressures acting in any other direction which may engage our attention.

All that we propose to deliver on this subject at present may be included in the following proposition.

Let ABCDE (fig. 10) be an assemblage of rafters in a

vertical plane, resting on two fixed points A and E in a horizontal line, and perfectly moveable round all the joints A, B, C, D, E; let it be further supposed to be in equilibrium, and let us investigate what adjustment of the different circumstances of weight and inclination of its different parts is necessary for producing this equilibrium.

Let F, G, H, I be the centres of gravity of the different rafters, and let these letters express the weights of each. Then, by what has been said above, the weight which presses B directly downwards is \( F \times \frac{AF}{AB} + G \times \frac{CG}{BC} \).

The weight on C is in like manner \( G \times \frac{BG}{BC} + D \times \frac{DH}{CD} \)

and that on D is \( H \times \frac{CH}{CD} + I \times \frac{EI}{DE} \).

Let AbcdE be the figure ABCDE inverted, in the manner already described. It may be conceived as a thread fastened at A and E, and loaded at b, c, and d with the weights which are really pressing on B, C, and D. It will arrange itself into such a form that all will be in equilibrium. We may discover this form by means of this single consideration, that any part bc of the thread is equally stretched throughout in the direction of its length. Let us therefore investigate the proportion between the weight \( \beta \), which we suppose to be pulling the point b in the vertical direction \( b\beta \), to the weight \( \delta \), which is pulling down the point d in a similar manner. It is evident, that since AE is a horizontal line, and the figures AbcdE and ABCDE equal and similar, the lines Bb, Cc, Dd, are vertical. Take \( b'f \) to represent the weight hanging at b. By stretching the threads bA and be it is set in opposition to the contractile powers of the threads, acting in the directions bA and be, and it is in immediate equilibrium with the equivalent of these two contractile forces. Therefore make \( bg = bf \), and make it the diagonal of a parallelogram \( bbgf \). It is evident that \( bb, bi \), are the forces exerted by the threads bA, be. Then, seeing that the thread bc is equally stretched in both directions, make \( ck = bi \); \( ck \) is the contractile force which is excited at c by the weight which is hanging there. Draw \( kd \) parallel to cd, and \( lm \) parallel to bc. The force \( le \) is the equivalent of the contractile forces \( ck, cm \), and is therefore equal and opposite to the force of gravity acting at C. In like manner, make \( dm = cm \), and complete the parallelogram ndpo, having the vertical line od for its diagonal. Then \( dn \) and \( dp \) are the contractile forces excited at d, and the weight hanging there must be equal to od.

Therefore, the load at b is to the load at d as \( bg \) to \( do \).

But we have seen that the compressing forces at B, C, D may be substituted for the extending forces at b, c, d. Therefore the weights at B, C, D which produce the compressions, are equal to the weights at b, c, d which produce the extensions. Therefore

\[ bg : do = F \times \frac{AF}{AB} + G \times \frac{CG}{BC} : H \times \frac{CH}{CD} + I \times \frac{EI}{DE}. \]

Let us inquire what relation there is between this proportion of the loads upon the joints at B and D, and the angles which the rafters make at these joints with each other, and with the horizon or the plumb-lines. Produce AB till it cut the vertical Ce in Q; then draw BR parallel to CD, and BS parallel to DE. The similarity of the figures ABCDE and AbcdE, and the similarity of their position with respect to the horizontal and plumb lines, show, without any further demonstration, that the triangles QCB and gbi are similar, and that QB : BC = gi : ib = hb : ib. Therefore QB is to BC as the contractile forces exerted by the thread Ab to that exerted by bc; and therefore QB is to BC as the compression on BA to the compression on BC. Then, because \( bi = ck \), and the triangles CBR and cdk are similar, CB : BR = ck : kd = ck : cm, and CB is to BR as the compression on CB to the compression on CD. And, in like manner, because cm = dn, we have BR to BS as the compression on DC to the compression on DE. Also BR : RS = nd : do, that is, as the compression on DC to the load on D. Finally, combining all these ratios,

\[ QC : CB = gb : bi = gb : ke, \] \[ CB : BR = ke : kd = ke : dn, \] \[ BR : BS = nd : no = dn : no, \] \[ BS : RS = no : do = no : do, \] we have finally

\[ QC : RS = gb : cd = \text{load at } B : \text{load at } D. \]

Now

\[ QC : BC = \sin QBC : \sin BQC = \sin ABC : \sin ABB, \] \[ BC : BR = \sin BRC : \sin BCR = \sin CDd : \sin bBC, \] \[ BR : BS = \sin BSR : \sin RBS = \sin dDE : \sin CDE. \]

Therefore

\[ QC : RS = \sin ABC \sin CDd \sin dDE : \sin CDE \sin ABB \sin bBC. \]

Or

\[ QC : RS = \frac{\sin ABC}{\sin ABB \sin CBb} : \frac{\sin CDE}{\sin dDC \sin dDE}. \]

That is, the loads on the different joints are as the sines of the angles at these joints directly, and as the products of the sines of the angles which the rafters make with the plumb-lines inversely.

Or, the loads are as the sines of the angles of the joints directly, and as the products of the cosines of the angles of elevation of the rafters inversely.

Or, the loads at the joints are as the sines of the angles at the joints, and as the products of the secants of the angles of elevation of the rafters jointly; for the secants of angles are inversely as the cosines.

Draw the horizontal line BT. It is evident, that if this be considered as the radius of a circle, the lines BQ, BC, BR, BS are the secants of the angles which these lines

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1 This proportion might have been shown directly without any use of the inverted figure, or consideration of contractile forces; but the substitution gives distinct notions of the mode of acting, even to persons not much conversant in such disquisitions; and we wish to make it familiar to the reader, because it gives an easy solution of the most complicated problems, and furnishes the practical carpenter, who has little science, with solutions of the most difficult cases by experiment. A festoon, as we called it, may easily be made; and we are certain that the forms into which it will arrange itself are models of perfect frames. make with the horizon; and they are also as the thrusts of those rafters to which they are parallel. Therefore, the thrust which any rafter makes in its own direction is as the secant of its elevation.

The horizontal thrust is the same at all the angles. For \( i = bx = m \mu = n = p \pi \). Therefore both walls are equally pressed out by the weight of the roof. We can find its quantity by comparing it with the load on one of the joints. Thus, \( QC : CB = \sin ABC : \sin ABb \)

\[ BC : BT = \text{rad.} : \sin BCT = \text{rad.} : \sin CBb. \]

Therefore, \( QC : BT = \text{rad.} \times \sin ABC : \sin \delta BA \times \sin \delta BC. \)

It deserves remark, that the lengths of the beams do not affect either the proportion of the load at the different joints, or the position of the rafters. This depends merely on the weights at the angles. If a change of length affects the weight, it indeed affects the form also; and this is generally the case. For it seldom happens, indeed it never should happen, that the weight on rafters of longer bearing is not greater. The covering alone increases nearly in the proportion of the length of the rafter.

If the proportion of the weights at B, C, and D is given, as also the position of any two of the lines, the position of all the rest is determined. If the horizontal distances between the angles are all equal, the forces on the different angles are proportional to the verticals drawn on the lines through these angles from the adjoining angle, and the thrusts from the adjoining angles are as the lines which connect them. If the rafters themselves are of equal lengths, the weights at the different angles are as these verticals and as the secants of the angles of elevation of the rafters jointly.

This proposition is very fruitful in its practical consequences. It is easy to perceive that it contains the whole theory of the construction of arches; for each stone of an arch may be considered as one of the rafters of this piece of carpentry, since all is kept up by its mere equilibrium.

We may have an opportunity in a future article of exhibiting some very elegant and simple solutions of the most difficult cases of this important problem; and we now proceed to make use of the knowledge we have acquired for the construction of roofs.

We mentioned by the by a problem which is not unfrequent in practice, to determine the best form of a kirb-roof. M. Couplet of the Royal Academy of Paris has given a solution of it in an elaborate memoir in 1726, occupying several lemmas and theorems.

Let AE (fig. 11) be the width, and CF the height; it is required to construct a roof ABCDE, whose rafters AB, BC, CD, DE, are all equal, and which shall be in equilibrium.

Draw CE, and bisect it perpendicularly in H by the line DHG, cutting the horizontal line AE in G. About the centre G, with the distance GE, describe the circle EKC. It must pass through C, because CH is equal to HE and the angles at H are equal. Draw HK parallel to FE, cutting the circumference in K; draw CK, cutting GH in D; and join CD, ED. These lines are the rafters of half of the roof required.

We prove this by showing that the loads at the angles C and D are equal; for this is the proportion which results from the equality of the rafters, and the extent of surface of the uniform roofing which they are supposed to support. Therefore produce ED till it meet the vertical FC in N; and having made the side CBA similar to CDE, complete the parallelogram BCDP, and draw DB, which will bisect CP in R, as the horizontal line KH bisects CF in Q. Draw KP, which is evidently parallel to DP. Make CS perpendicular to CF, and equal to FG; and about S, with the radius SF, describe the circle FKW. It must pass through K, because SF is equal to CG, and CQ = QF. Draw WK, WS, and produce BC, cutting ND in O.

The angle WKF at the circumference is one half of the angle WSF at the centre, and is therefore equal to WSC or CGF. It is therefore double of the angle CEF or ECS. But ECS is equal to ECD and DCS, and ECD is one half of NDC, and DCS is one half of DCO or CDP. Therefore the angle WKF is equal to NDP, and WK is parallel to ND, and CF is to CW as CP to CN; and CN is equal to CP. But it has been shown above that CN and CP are as the loads upon D and C. These are therefore equal, and the frame ABCDE is in equilibrium.

A comparison of this solution with that of M. Couplet will show its great advantage in respect of simplicity and perspicuity; and the intelligent reader can easily adapt the construction to any proportion between the rafters AB and BC, which other circumstances, such as garret-rooms, &c., may render convenient. The construction must be such that NC may be to CP as CD to \( \frac{CD + DE}{2} \). Whatever proportion of AB to BC is assumed, the point D' will be found in the circumference of a semicircle H'D'K', whose centre is in the line CE, and having \( AB : BC = CH : HE = ch : hE \). The rest of the construction is simple.

In buildings which are roofed with slate, tile, or shingles, the circumstance which is most likely to limit the construction is the slope of the upper rafters CB, CD. This must be sufficient to prevent the penetration of rain, and the stripping by the winds. The only circumstance left in our choice in this case is the proportion of the rafters AB and BC. Nothing is easier than making NC to CP in any desired proportion when the angle BCD is given.

We need not repeat that it is always a desirable thing to form a truss for a roof in such a manner that it shall be in equilibrium. When this is done, the whole force of the struts and braces which are added to it is employed in preserving this form, and no part is expended in unnecessary strains.

For we must now observe, that the equilibrium of which we have been treating is always of that kind which we call the tottering, and the roof requires stays, braces, or hanging timbers, to give it stiffness, or keep it in shape. We have also said enough to enable any reader acquainted with the most elementary geometry and mechanics, to compute the transverse strains and the thrusts to which the component parts of all roofs are exposed.

It only remains now to show the general maxims by which all roofs must be constructed, and the circumstances which determine their excellence. In doing this we shall all roots be exceedingly brief, and almost content ourselves with ex- hibiting the principal forms, of which the endless variety of roofs are only slight modifications. We shall not trouble the reader with any account of such roofs as receive part of their support from the interior walls, but confine ourselves to the more difficult problem of throwing a roof over a wide building, without any intermediate support; because when such roofs are constructed in the best manner, that is, deriving the greatest possible strength from the materials employed, the best construction of the others is necessarily included. For all such roofs as rest upon the middle walls are roofs of smaller bearing. The only exception deserving notice is the roofs of churches, which have aisles separated from the nave by columns. The roof must rise on these. But if it is of an arched form internally, the horizontal thrusts must be nicely balanced, that they may not push the columns aside.

The simplest notion of a roof-frame is, that it consists of two rafters AB and BC (fig. 12), meeting in the ridge.

Fig. 12.

But even this simple form is susceptible of better and worse. We have already seen, that when the weight of a square yard of covering is given, a steeper roof requires stronger rafters, and that when the scantling of the timbers is also given, the relative strength of a rafter is inversely as its length.

But there is now another circumstance to be taken into the account, viz. the support which one rafter leg gives to the other. The best form of a rafter will therefore be that in which the relative strength of the legs, and their mutual support, give the greatest product. Mr Muller, in his Military Engineer, gives a determination of the best pitch of a roof, which has considerable ingenuity, and has been copied into many books of military education both in this island and on the continent. Describe on the width AC (fig. 13) the semicircle AFC, and bisect it by the radius FD. Produce the rafter AB to the circumference in E, join EC, and draw the perpendicular EG. Now AB : AD = AC : AE, and AE = \(\frac{AD \times AC}{AB}\), and AE is inversely as AB, and may therefore represent its strength in relation to the weight actually lying on it. Also the support which CB gives to AB is as CE, because CE is perpendicular to AB. Therefore the form which renders AE \(\times\) EC a maximum seems to be that which has the greatest strength.

But AC : AE = EC : EG, and EG = \(\frac{AE \times EC}{AC}\), and is therefore proportional to AE \(\times\) EC. Now EG is a maximum when B is in F, and a square pitch is in this respect the strongest. But it is very doubtful whether this construction is deduced from just principles. There is another strain to which the leg AB is exposed, which is not taken into the account. This arises from the curvature which it unavoidably acquires by the transverse pressure of its load. In this state it is pressed in its own direction by the abutment and load of the other leg. The relation between this strain and the resistance of the piece is not very distinctly known. Euler has given a dissertation on this subject, which is of great importance, because it affects posts and pillars of all kinds; and it is very well known that a post of ten feet long and six inches square will bear with great safety a weight which would crush a post of the same scantling and twenty feet long in a minute; but his determination has not been acquiesced in by the first mathematicians. Now it is in relation to these two strains that the strength of the rafter should be adjusted. The firmness of the support given by the other leg is of no consequence, if its own strength is inferior to the strain. The force which tends to crush the leg AB, by compressing it in its curved state, is to its weight as AB to BD, as is easily seen by the composition of forces; and its incursion by this force has a relation to it, which is of intricate determination. It is contained in the properties demonstrated by Bernoulli of the elastic curve. This determination also includes the relation between the curvature and the length of the piece. But the whole of this seemingly simple problem is of much more difficult investigation than Mr Muller was aware of; and his rules for the pitch of a roof, and for the sally of a dock-gate, which depends on the same principles, are of no value. He is, however, the first author who attempted to solve either of these problems on mechanical principles susceptible of precise reasoning. Belidor's solutions, in his Architecture Hydraulique, are below notice.

Reasons of economy have made carpenters prefer a low pitch; and although this does diminish the support given by the opposite leg faster than it increases the relative strength of the other, it is not of material consequence, because the strength remaining in the opposite leg is still very great; for the supporting leg is acting against compression, in which case it is vastly stronger than the supported leg acting against a transverse strain.

But a roof of this simplicity will not do in most cases. Thrust on There is no notice taken, in its construction, of the thrust on the walls, which it exerts on the walls. Now this is the strain which is the most hazardous of all. Our ordinary walls, instead of being able to resist any considerable strain pressing them outwards, require, in general, some ties to keep them on foot. When a person thinks of the thinness and height of the walls of even a strong house, he will be surprised that they are not blown down by any strong puff of wind. A wall three feet thick, and sixty feet high, could not withstand a wind blowing at the rate of thirty feet per second (in which case it acts with a force considerably exceeding two pounds on every square foot), if it were not stiffened by cross walls, joists, and roof, which all help to tie the different parts of the building together.

A carpenter is therefore exceedingly careful to avoid every horizontal thrust, or to oppose them by other forces. And avoided. This introduces another essential part into the construction of a roof, namely, the tie or beam AC (fig. 14), laid from wall to wall, binding the feet A and C of the rafters together. This is the sole office of the beam; and it should be considered in no other light than as a string to prevent the roof from pushing out the walls. It is indeed used for carrying the ceiling of the apartments under it, and it is even made to support a flooring. But, considered as making part of a roof, it is merely a string; and the strain which it withstands tends to tear its parts asunder. It therefore acts with its whole absolute force, and a very small scantling would suffice if we could contrive to fasten it firmly enough to the foot of the rafter. If it is of oak, we may safely subject it to a strain of three tons for every square inch of its sec- And fir will safely bear a strain of two tons for every square inch. But we are obliged to give the tie-beam much larger dimensions, that we may be able to connect it with the foot of the rafter by a mortise and tenon. Iron straps are also frequently added. By attending to this office of the tie-beam, the judicious carpenter is directed to the proper form of the mortise and tenon, and of the strap. We shall consider both of these in a proper place, after we become acquainted with the various strains at the joints of a roof.

These large dimensions of the tie-beam allow us to load it with the ceilings without any risk, and even to lay floors on it with moderation and caution. But when it has a great bearing or span, it is very apt to bend downwards in the middle, or, as the workmen term it, to sway or swag; and it requires a support. The question is, where to find this support. What fixed points can we find with which to connect the middle of the tie-beam? Some ingenious carpenter thought of suspending it from the ridge by a piece of timber BD (fig. 15), called by our carpenters the king-post.

It must be acknowledged, that there was very great ingenuity in this thought. It was also perfectly just. For the weight of the rafters BA, BC tends to make them fly out at the foot. This is prevented by the tie-beam, and this excites a pressure, by which they tend to compress each other. Suppose them without weight, and that a great weight is laid on the ridge B. This can be supported only by the abutting of the rafters in their own directions AB and CB, and the weight tends to compress them in the opposite directions, and, through their intervention, to stretch the tie-beam. If neither the rafters can be compressed, nor the tie-beam stretched, it is plain that the triangle ABC must retain its shape, and that B becomes a fixed point very proper to be used as a point of suspension. To this point, therefore, is the tie-beam suspended by means of the king-post. A common spectator unacquainted with carpentry views it very differently, and the tie-beam appears to him to carry the roof. The king-post appears a pillar resting on the beam, whereas it is really a string; and an iron rod of one sixteenth of the size would have done just as well. The king-post is sometimes mortised into the tie-beam, and pins put through the joint, which gives it more the look of a pillar with the roof resting on it. This does well enough in many cases. But the best method is to connect them by an iron strap like a stirrup, which is bolted at its upper ends into the king-post, and passes round the tie-beam. In this way a space is commonly left between the end of the king-post and the upper side of the tie-beam. Here the beam plainly appears hanging in the stirrup; and this method allows us to restore the beam to an exact level, when it has sunk by the unavoidable compression or other yielding of the parts. The holes in the sides of the iron strap are made oblong instead of round; and the bolt which is drawn through all is made to taper on the under side; so that driving it farther draws the tie-beam upwards. A notion of this may be formed by looking at fig. 16, which is a section of that post and beam.

It requires considerable attention, however, to make this suspension of the tie-beam sufficiently firm. The top of the king-post is cut into the form of the arch-stone of a bridge, and the heads of the rafters are firmly mortised into this projecting part. These projections are called joggles, and are formed by working the king-post out of a much larger piece of timber, and cutting off the unnecessary wood from the two sides; and, lest all this should not be sufficient, it is usual in great works to add an iron plate or strap of three branches, which are bolted into the heads of the king-post and rafters.

The rafters, though not so long as the beam, seem to stand as much in need of something to prevent their bending, for they carry the weight of the covering. This cannot be done by suspension, for we have no fixed points above them. But we have now got a very firm point of support at the foot of the king-post.

Braces, or rather struts, ED, FD (fig. 17), are put under the middle of the rafters, where they are slightly mortised, and their lower ends are firmly mortised into joggles formed on the foot of the king-post.

As these braces are very powerful in their resistance to compression, and the king-post equally so to resist extension, the points E and F may be considered as fixed; and the rafters being thus reduced to half their former length, have now four times their former relative strength.

Roofs do not always consist of two sloping sides meeting in a ridge. They have sometimes a flat on the top, with two sloping sides. They are sometimes formed with a double slope, and are called hirb or mansarde roofs. They sometimes have a valley in the middle, and are then called M roofs. Such roofs require another piece, which may be called the truss-beam, because all such frames are called trusses; probably from the French word trousse, because such roofs are like portions of plain roofs troussés or shortened.

A flat-topped roof is thus constructed. Suppose the three rafters AB, BC, CD (fig. 18), of which AB and CD are equal, and BC horizontal. It is plain that they will be in equilibrio, and the roof have no tendency to go on either side. The tie-beam AD withstands the horizontal thrusts of the whole frame, and the two rafters AB and CD are each pressed in their own directions in consequence of their abutting with the middle rafter or truss-beam BC. It lies between them like the key-stone of an arch. They lean towards it, and it rests on them. The pressure which the truss-beam and its load excites on the two rafters is the very same as if the rafters were produced till they meet in G, and a weight were laid on these equal to that of BC and its load. If therefore the truss-beam is of a scantling sufficient for carrying its own load, and notwithstanding the compression from the two rafters, the roof will be equally strong, whilst it keeps its shape, as the plain roof AGD, furnished with the king-post and braces. We may conceive this another way. Suppose a plain roof AGD, without braces to support the middle B and C of the rafters. Then let a beam BC be put in between the rafters, abutting upon little notches cut in the rafters. It is evident that this must prevent the rafters from bending downwards, because the points B and C cannot descend, moving round the centres A and D, without shortening the distance BC between them. This cannot be without compressing the beam BC. It is plain that BC may be wedged in, or wedges driven in between its ends B and C and the notches in which it is lodged. These wedges may be driven in till they even force out the rafters GA and GD. Whenever this happens, all the mutual pressure of the heads of these rafters at G is taken away, and the parts GB and GC may be cut away, and the roof ABCD will be as strong as the roof AGD furnished with the king-post and braces, because the truss-beam gives a support of the same kind at B and C as the brace would have done.

But this roof ABCD would have no firmness of shape. Any addition of weight on one side would destroy the equilibrium at the angle, would depress that angle, and would cause the opposite one to rise. To give it stiffness, it must either have ties or braces, or something partaking of the nature of both. The usual method of framing is to make the heads of the rafters about on the joggles of two side-posts BE and CF, whilst the truss-beam, or strut as it is generally termed by the carpenters, is mortised square into the inside of the heads. The lower ends E and F of the side-posts are connected with the tie-beam either by mortises or straps.

This construction gives firmness to the frame; for the angle B cannot descend in consequence of any inequality of pressure, without forcing the other angle C to rise. This it cannot do, being held down by the post CF. And the same construction fortifies the tie-beam, which is now suspended at the points E and F from the points B and C, whose firmness we have just now shown.

But although this roof may be made abundantly strong, it is not quite so strong as the plain roof AGD of the same scantling. The compression which BC must sustain in order to give the same support to the rafters at B and C that was given by braces properly placed, is considerably greater than the compression of the braces. And this strain is an addition to the transverse strain which BC gets from its own load. This form also necessarily exposes the tie-beam to cross strains. If BE is mortised into the tie-beam, then the strain which tends to depress the angle ABC presses on the tie-beam at E transversely, whilst a contrary strain acts on F, pulling it upwards. These strains, however, are small; and this construction is frequently used, being susceptible of sufficient strength, without much increase of the dimensions of the timbers; and it has the great advantage of giving free room in the garrets.

Were it not for this, there is a much more perfect form represented in fig. 19. Here the two posts BE, CF are united below. All transverse action on the tie-beam is now entirely removed. We are almost disposed to say that this is the strongest roof of the same width and slope. For if the iron strap which connects the pieces BE, CF with the tie-beam have a large bolt G through it, confining it to one point of the beam, there are five points, A, B, C, D, G, which cannot change their places, and there is no transverse strain in any of the connections.

When the dimensions of the building are very great, so that the pieces AB, BC, CD, would be thought too weak for withstanding the cross strains, braces may be added as is expressed in fig. 18 by the dotted lines. The reader will observe, that it is not meant to leave the top flat externally; it must be raised a little in the middle, to shoot off the rain. But this must not be done by incavrating the beam BC. This would soon be crushed, and spring upwards. The slopes must be given by pieces of timber added above the strutting-beam.

And thus we have completed a frame of a roof. It consists of these principal members: the rafters, which are immediately loaded with the covering; the tie-beam, which withstands the horizontal thrust by which the roof tends to fly out below and push out the walls; the king-posts, which hang from fixed points and serve to uphold the tie-beam, and also to afford other fixed points on which we may rest the braces which support the middle of the rafters; and, lastly, the truss or strutting-beam, which serves to give mutual abutment to the different parts which are at a distance from each other. The rafters, braces, and trusses are exposed to compression, and must therefore have not only cohesion, but stiffness. For if they bend, the prodigious compressions to which they are subjected would quickly crush them in this bended state. The tie-beams and king-posts, if performing no other office but supporting the roof, do not require stiffness; and their places might be supplied by ropes, or by rods of iron of one-tenth part of the section that even the smallest oak stretcher requires. These members require no greater dimensions than what is necessary for giving sufficient joints, and any more is a needless expense. All roofs, however complicated, consist of these essential parts; and if pieces of timber are to be seen which perform none of these offices, they must be pronounced useless, and they are frequently hurtful, by producing cross strains in some other piece. In a roof properly constructed there should be no such strains. All the timbers, excepting those which immediately carry the covering, should be either pushed or drawn in the direction of their length. And this is the rule by which a roof should always be examined.

These essential parts are susceptible of numberless combinations and varieties. But it is a prudent maxim to make the construction as simple, and consisting of as few parts less combined as possible. We are the less exposed to the imperfections of masonry and workmanship, such as loose joints, &c. Another essential variety, harm arises from many pieces, by the compression and the shrinking of the timber in the cross direction of the fibres. The effect of this is equivalent to the shortening of the piece which abuts on the joint. This alters the proportions of the sides of the triangle on which the shape of the whole depends. Now, in a roof such as fig. 18, there is twice as much of this as in the plain pent-roof, because there are two posts. And when the direction of the abutting pieces is very oblique to the action of the load, a small shrinking permits a great change of shape. Thus, in a roof of what is called pediment pitch, where the rafters make an angle of thirty degrees with the horizon, half an inch compression of the king-post will produce a sagging of an inch, and occasion a great strain on the tie-beam, if the posts are mortised into it.

This method of including a truss within the rafters of a pent-roof is a very considerable addition to the art of carpentry. But to insure its full effect, it should always be executed with abutting rafters under the principal ones, abutting on joggles in the heads of the posts. Without this the strut-beam is hardly of any service. We would therefore recommend fig. 20 as a proper construction of a trussed roof; and the king-post which is placed in it may be employed to support the upper part of the rafters, and also for preventing the strut-beam from bending in their direction in consequence of its great compression. It will also give a suspension for the great burdens which are sometimes necessary in a theatre. The machinery has no other firm points to which it can be attached; and the portions of the single rafters which carry this king-post are but short, and therefore may be considerably loaded with safety.

We observe in the drawings which we sometimes have of Chinese buildings, that the trussing of roofs is understood by them. Indeed they must be very experienced carpenters. We see wooden buildings run up to a great height, which can be supported only by such trussing. One of these is sketched in fig. 21. There are some very excellent specimens to be seen in the buildings at Deptford, be- longing to the victualling-office, commonly called the Redhouse, which were erected about the year 1788, and we believe are the performance of Mr James Arrow of the Board of Works, one of the most intelligent artists in this kingdom.

Thus have we given an elementary, but a rational or scientific, account of this important part of the art of carpentry. It is such, that any practitioner, with the trouble of a little reflection, may always proceed with confidence, and without resting any part of his practice on the vague notions which habit may have given him of the strength and supports of timbers, and of their manner of acting. That these frequently mislead, is proved by the mutual criticisms which are frequently published by the rivals in the profession. They have frequently sagacity enough, for it seldom can be called science, to point out glaring blunders; and any person who will look at some of the performances of Mr Price, Mr Wyatt, Mr Arrow, and others of acknowledged reputation, will readily see them distinguishable from the works of inferior artists by simplicity alone. A man without principles is apt to consider an intricate construction as ingenious and effectual; and such roofs sometimes fail merely by being ingeniously loaded with timber, but still more frequently by the wrong action of some useless pieces, which produces strains that are transverse to other pieces, or which, by rendering some points too firm, cause them to be deserted by the rest in the general subsiding of the whole. Instances of this kind are pointed out by Price in his British Carpenter. Nothing shows the skill of a carpenter more than the distinctness with which he can foresee the changes of shape which must take place in a short time in every roof. A knowledge of this will often correct a construction which the mere mathematician thinks unexceptionable, because he does not reckon on the actual compression which must obtain, and imagines that his triangles, which sustain no cross strains, invariably retain their shape till the pieces break. The sagacity of the experienced carpenter is not, however, enough without science for perfecting the art. But when he knows how much a particular piece will yield to compression in one case, science will then tell him, and nothing but science can do it, what will be the compression of the same piece in another very different case. Thus he learns how far it will now yield, and then he proportions the parts so to each other, that when all have yielded according to their strains, the whole is of the shape he wished to produce, and every joint is in a state of firmness. It is here that we observe the greatest number of improprieties. The iron straps are frequently in positions not suited to the actual strain on them; and they are in a state of violent twist, which both tends strongly to break the straps, and to cripple the pieces which they surround.

In like manner, we frequently see joints or mortises in a state of violent strain on the tenons, or on the heels and shoulders. The joints were perhaps properly shaped for the primitive form of the truss; but by its settling, the bearing of the push is changed. The brace, for example, in a very low-pitched roof, comes to press with the upper part of the shoulder, and, acting as a powerful lever on the tenon, breaks it. In like manner, the lower end of the brace, which at first abutted firmly and squarely on the joggle of the king-post, now presses with one corner in prodigious force, and seldom fails to splinter off on that side. We cannot help recommending a maxim of M. Perronet, the celebrated hydraulic architect of France, as a golden rule, viz., to make all the shoulders of abutting pieces in the form of an arch of a circle, having the opposite end of the piece for its centre. Thus, in fig. 18, if the joggle-point B be of this form, having A for its centre, the sagging of the roof will make no partial bearing at the joint; for in the sagging of the roof the piece AB turns or bends round the centre A; and the counter-pressure of the joggle is still directed to A, as it ought to be. We have just now said bends round A. This is too frequently the case, and it is always very difficult to give the tenon and mortise in this place a true and invariable bearing. The rafter pushes in the direction BA, and the beam resists in the direction AD. The abutment should be perpendicular to neither of these, but in an intermediate direction, and it ought also to be of a curved shape. But the carpenters perhaps think that this would weaken the beam too much to give it this shape in the shoulder; they do not even aim at it in the heel of the tenon. The shoulder is commonly even with the surface of the beam. When the bearing therefore is on this shoulder, it causes the foot of the rafter to slide along the beam till the heel of the tenon bears against the outer end of the mortise (See Price's British Carpenter, plate C, fig. IK). This abutment is perpendicular to the beam in Price's book; but it is more generally pointed a little outwards below, to make it more secure against starting. The consequence of this construction is, that when the roof settles, the shoulder comes to bear at the inner end of the mortise, and it rises at the outer, and the tenon, taking hold of the wood beyond it, either tears it out or is itself broken. This joint therefore is seldom trusted to the strength of the mortise and tenon, and is usually secured by an iron strap, which lies obliquely to the beam, to which it is bolted by a large bolt quite through, and then embraces the outside of the rafter foot. This strap is very frequently not made sufficiently oblique, and we have seen some made almost square with the beam. When this is the case, it not only keeps the foot of the rafter from flying out, but it binds it down. In this case, the rafter acts as a powerful lever, whose fulcrum is in the inner angle of the shoulder, and then the strap never fails to cripple the rafter at the point. All this can be prevented only by making the strap very long and very oblique, and by making its outer end (the stirrup part) square with its length, and making a notch in the rafter foot to receive it. It cannot now cripple the rafter, for it will rise along with it, turning round the bolt at its inner end. We have been thus particular on this joint, because it is here that the ultimate strain of the whole roof is exerted, and its situation will not allow the excavation necessary for making it a good mortise and tenon.

Similar attention must be paid to some other straps, such as those which embrace the middle of the rafter, and connect it with the post or truss below it. We must attend to the change of shape produced by the sagging of the roof, and place the strap in such a manner as to yield to it by turning round its bolt, but so as not to become loose, and far less to make a fulcrum for any thing acting as a lever. The strains arising from such actions, in framings of carpentry which change their shape by sagging, are enormous, and nothing can resist them.

We shall close this part of the subject with a simple method, by which any carpenter, without mathematical calculating, may calculate with sufficient precision the strains or thrusts which are produced on any point of his work, whatever be the obliquity of the pieces.

Let it be required to find the horizontal thrust acting on the tie-beam AD of fig. 18. This will be the same as if the weight of the whole roof were laid at G on the two rafters GA and GD. Draw the vertical line GH. Then, having calculated the weight of the whole roof that is supported by this single frame ABCD, including the weight of the pieces AB, BC, CD, BE, CF themselves, take the number of pounds, tons, &c. which expresses it from any scale of equal parts, and set it from G to H. Draw HK, HL parallel to GD, GA, and draw the line KL, which will be horizontal when the two sides of the roof have the same slope. Then ML measured on the same scale will give the horizontal thrust, by which the strength of the tie-beam is to be regulated. GL will give the thrust which tends to crush the rafters, and LM will also give the force which tends to crush the strut-beam BC.

In like manner, to find the strain of the king-post BD of fig. 17, consider that each brace is pressed by half the weight of the roofing laid on BA or BC, and this pressure, or at least its hurtful effect, is diminished in the proportion of BA to DA, because the action of gravity is vertical, and the effect which we want to counteract by the braces is in a direction Ee perpendicular to BA or BC. But as this is to be resisted by the brace fE acting in the direction fE, we must draw fe perpendicular to Ee, and suppose the strain augmented in the proportion of Ee to fE.

Having thus obtained in tons, pounds, or other measures, the strains which must be balanced at f by the cohesion of the king-post, take this measure from the scale of equal parts, and set it off in the directions of the braces to G and H, and complete the parallelogram GfHK; and fK measured on the same scale will be the strain on the king-post.

The artist may then examine the strength of his truss upon this principle, that every square inch of oak will bear at an average 7000 pounds compressing or stretching it, and may be safely loaded with 3500 for any length of time; and that a square inch of fir will in like manner securely bear 2500. And, because straps are used to resist some of these strains, a square inch of well-wrought tough iron may be safely strained by 50,000 pounds. But the artist will always recollect, that we cannot have the same confidence in iron as in timber. The faults of this last are much more easily perceived; and when the timber is too weak, it gives us warning of its failure by yielding sensibly before it breaks. This is not the case with iron; and much of its service depends on the honesty of the blacksmith.

In this way may any design of a roof be examined. We shall here give the reader a sketch of two or three trussed roofs, which have been executed in the chief varieties of circumstances which occur in common practice.

Fig. 22 is the roof of St Paul's Church, Covent Garden, London, the work of Inigo Jones. Its construction is singular. The roof extends to a considerable distance beyond the building, and the ends of the tie-beams support the Tuscan corniche, appearing like the mutules of the Doric order. Such a roof could not rest on the tie-beam. Inigo Jones has therefore supported it by a truss below it; and the height has allowed him to make this extremely strong with very little timber. It is accounted the highest roof of its width in London. But this was not difficult, by reason of the great height which its extreme width allowed him to employ without hurting the beauty of it by too high a pitch. The supports, however, are disposed with judgment.

Fig. 23 is a kirb or mansarde roof by Price, and supposed to be of large dimensions, having braces to carry the middle of the rafters. It will serve exceedingly well for a church having pillars. The middle part of the tie-beams being taken away, the strains are very well balanced, so that there is no risk of its pushing aside the pillar on which it rests.

Fig. 24 is the celebrated roof of the Theatre of the University of Oxford, by Sir Christopher Wren. The span between the walls is 75 feet. This is accounted a very ingenious, and is a singular performance. The middle part of it is almost unchangeable in its form; but from this circumstance it does not distribute the horizontal thrust with the same regularity as the usual construction. The horizontal thrust on the tie-beam is about twice the weight of the roof, and is withstood by an iron strap below the beam, which stretches the whole width of the building in the form of a rope, making part of the ornament of the ceiling.

In all the roofs which we have considered hitherto, the cases in thrust is discharged entirely from the walls by the tie-beam, which the But this cannot always be done. We frequently want great thrust-cancellation within, and arched ceilings. In such cases, it is not charged a much more difficult matter to keep the walls free of all from the pressure outwards, and there are few buildings where it is walls by completely done. Yet this is the greatest fault of a roof: the tie-beam. We shall just point out the methods which may be most successfully adopted.

We have said that a tie-beam just performs the office of a string. We have said the same of the king-post. Now suppose two rafters AB, BC (fig. 25), moveable about the point B, and resting on the top of the walls. If the line BD be suspended from B, and the two lines DA, DC be fastened to the feet of the rafters, and if these lines be incapable of extension, it is plain that all thrust is removed from the walls as effectually as by a common tie-beam; and by shortening BD to B'd, we gain a greater inside height, and more room for an arched ceiling. Now if we substitute a king-post BD (fig. 26), and two stretchers or hammer-beams DA, DC for the other strings, and connect them firmly by means of iron straps, we obtain our purpose.

Let us compare this roof with a tie-beam roof in point of strain and strength. Recur to fig. 25, and complete the parallelogram ABCF, and draw the diagonals AC, BF, crossing in E. Draw BG perpendicular to CD. We have

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1 This church was burnt down after the present article was written. seen that the weight of the roof, which we may call $W$, is to the horizontal thrust at $C$ as $BF$ to $EC$; and if we express this thrust by $T$, we have $T = \frac{W \times EC}{BF}$. We may at present consider $BC$ as a lever moveable round the joint $B$, and pulled at $C$ in the direction $EC$ by the horizontal thrust, and held back by the string pulling in the direction $CD$. Suppose that the forces in the directions $EC$ and $CD$ are in equilibrium, and let us find the force $S$ by which the string $CD$ is strained. These forces must, by the property of the lever, be inversely as the perpendiculars drawn from the centre of motion on the lines of their direction.

Therefore $BG : BE = T : S$, and $S = T \times \frac{BE}{BG} = W \times \frac{BE}{BF \times BG}$.

Therefore the strain upon each of the ties $DA$ and $DC$ is always greater than the horizontal thrust or the strain on a simple tie-beam. This would be no great inconvenience, because the smallest dimensions that we could give to these ties, so as to procure sufficient fixtures to the adjoining pieces, are always sufficient to withstand this strain. But although the same may be said of the iron straps which make the ultimate connections, there is always some hazard of imperfect work, cracks, or flaws, which are not perceived. We can judge with tolerable certainty of the soundness of a piece of timber, but cannot say so much of a piece of iron. Moreover, there is a prodigious strain excited on the king-post when $BG$ is very short in comparison of $BI$, namely, the force compounded of the two strains $S$ and $S$ on the ties $DA$ and $DC$.

But there is another defect from which the straight tie-beam is entirely free. All roofs settle a little. When this roof settles, and the points $B$ and $D$ descend, the legs $BA$, $BC$ must spread further out, and thus a pressure outwards is excited on the walls. It is seldom therefore that this kind of roof can be executed in this simple form, and other contrivances are necessary for counteracting this supervening action on the walls. Fig. 27 is one of the best which we have seen, and is executed with great success in the circus or equestrian theatre (now, 1809, a concert-room) in Edinburgh, the width being sixty feet. The pieces $EF$ and $ED$ help to take off some of the weight, and by their greater uprightness they exert a smaller thrust on the walls. The beam $Dd$ is also a sort of truss-beam, having something of the same effect. Mr Price has given another very judicious one of this kind (British Carpenter, plate IX, fig. C), from which the tie-beam may be taken away, and there will remain very little thrust on the walls. Those which he has given in the following plate, $K$, are, in our opinion, very faulty. The whole strain in these last roofs tends to break the rafters and ties transversely, and the fixtures of the ties are also not well calculated to resist the strain to which the pieces are exposed. We hardly think that these roofs could be executed.

It is scarcely necessary to remind the reader, that in all that we have delivered on this subject, we have attended only to the construction of the principal rafters or trusses. In small buildings all the rafters are of one kind; but in great buildings the whole weight of the covering is made to rest on a few principal rafters, which are connected by beams placed horizontally, and either mortised into them or scarfed on them. These are called purlins. Small rafters are laid from purlin to purlin; and on these the laths for tiles, or the skirting-boards for slates, are nailed. Thus the covering does not immediately rest on the principal frames. This allows some more liberty in their construction, because the garrets can be so divided that the principal rafters shall be in the partitions, and the rest left unencumbered. This construction is so far analogous to that of floors which are constructed with girders, binding, and bridging joists.

It may appear presuming in us to question the propriety of this practice. There are situations in which it is unavoidable, as in the roofs of churches, which can be allowed to rest on some pillars. In other situations, where partition-walls intervene at a distance not too great for a stout purlin, no principal rafters are necessary, and the whole may be roofed with short rafters of very slender scantling. But in a great uniform roof, which has no intermediate supports, it requires at least some reasons for preferring this method of carcass-roofing to the simple method of making all the rafters alike. The method of carcass-roofing requires the selection of the greatest logs of timber, which are seldom of equal strength and soundness with thinner rafters. In these the outside planks can be taken off, and the best part alone worked up. It also exposes to all the defects of workmanship in the mortising of purlins, and the weakening of the rafters by this very mortising; and it brings an additional load of purlins and short rafters. A roof thus constructed may surely be compared with a floor of similar construction. Here there is not a shadow of doubt, that if the girders were sawed into planks, and these planks laid as joists sufficiently near for carrying the flooring boards, they will have the same strength as before, except so much as is taken out of the timber by the saw. This will not amount to one-tenth part of the timber in the binding, bridging, and ceiling joists, which are an additional load, and all the mortises and other joinings are so many diminutions of the strength of the girders; and as no part of a carpenter's work requires more skill and accuracy of execution, we are exposed to many chances of imperfection. But, not to rest on these considerations, however reasonable they may appear, we shall relate an experiment made by one on whose judgment and exactness we can depend.

Two models of floors were made, eighteen inches square, confirmed of the finest uniform deal, which had been long seasoned, by experience. One consisted of simple joists, and the other was framed with girders, binding, bridging, and ceiling joists. The plain joists of the one contained the same quantity of timber with the girders alone of the other, and both were made by a most accurate workman. They were placed in wooden trunks eighteen inches square within, and rested on a strong projection on the inside. Small shot was gradually poured in upon the floors, so as to spread uniformly over them. The plain joisted floor broke down with 487 pounds, and the carcass floor with 327. The first broke without giving any warning, and the other gave a violent crack when 294 pounds had been poured in. A trial had been made before, and the loads were 341 and 482; but the models having been made by a less accurate hand, it was not thought a fair specimen of the strength which might be given to a carcass floor.

The only argument of weight which we can recollect in favour of the compound construction of roofs is, that the plain method would prodigiously increase the quantity of work, would admit nothing but long timber, which would greatly add to the expense, and would make the garrets a mere thicket of planks. We admit this in its full force; but we continue to be of the opinion that plain roofs are greatly superior in point of strength, and therefore should be adopted in cases where the main difficulty is to insure this necessary circumstance.

It would appear very neglectful to omit an account of the roofs put on round buildings, such as domes, cupolas, and the like. They appear to be the most difficult tasks in the carpenter's art. But the difficulty lies entirely in the mode of framing, or what the French call the *trait de charpente*. The view which we are taking of the subject, as a part of mechanical science, has little connection with this. It is plain, that whatever form of a truss is excellent in a square building, must be equally so as one of the frames of a round one; and the only difficulty is how to manage their mutual intersections at the top. Some of them must be discontinued before they reach that length, and common sense will teach us to cut them short alternately, and always leave as many, that they may stand equally thick as at their first springing from the base of the dome. Thus the length of the purlins, which reach from truss to truss, will never be too great.

The truth is, that a round building which gathers in at top, like a glass-house, a potter's kiln, or a spire steeple, instead of being the most difficult to erect with stability, is of all others the easiest. Nothing can show this more forcibly than daily practice, where they are run up without centres and without scaffoldings; and it requires gross blunders indeed in the choice of their outline to put them in much danger of falling from a want of equilibrium. In like manner, a dome of carpentry can hardly fail, give it what shape or what construction you will. It cannot fall, unless some part of it flies out at the bottom. An iron hoop round it, or straps at the joinings of the trusses and purlins, which make an equivalent to a hoop, will effectually secure it. And as beauty requires that a dome shall spring almost perpendicularly from the wall, it is evident that there is hardly any thrust to force out the walls. The only part where this is to be guarded against is where the tangent is inclined about forty or fifty degrees to the horizon. Here it will be proper to make a course of firm horizontal joinings.

We doubt not but that domes of carpentry will now be raised of great extent. The Halle du Bled at Paris, of two hundred feet in diameter, was the invention of an intelligent carpenter, the Sieur Moulineau. He was not by any means a man of science, but had much more mechanical knowledge than artisans usually have, and was convinced that a very thin shell of timber might not only be so shaped as to be nearly in equilibrio, but that, if hooped or firmly connected horizontally, it would have all the stiffness that was necessary; and he presented his project to the magistracy of Paris. The grandeur of it pleased them, but they doubted of its possibility. Being a great public work, they prevailed on the Academy of Sciences to consider it. The members who were competent judges were instantly struck with the justness of M. Moulineau's principles, and astonished that a thing so plain had not been long familiar to every house-carpenter. It quickly became an universal topic of conversation, dispute, and cabal, in the polite circles of Paris. But the academy having given a very favourable report of their opinion, the project was immediately carried into execution, and soon completed; and now stands as one of the great exhibitions of Paris.

The construction of this dome is the simplest thing that can be imagined. The circular ribs which compose it consist of planks nine feet long, thirteen inches broad, and three inches thick; and each rib consists of three of these planks bolted together in such a manner that two points meet. A rib is begun, for instance, with a plank of three feet long standing between one of six feet and another of nine; and this is continued to the head of it. No machinery was necessary for carrying up such small pieces, and the whole went up like a piece of bricklayer's work. At various distances these ribs were connected horizontally by purlins and iron straps, which made so many hoops to the whole. When the work had reached such a height that the distance of the ribs was two thirds of the original distance, every third rib was discontinued, and the space was left open and glazed. When carried so much higher that the distance of the ribs is one third of the original distance, every second rib, now consisting of two ribs very near each other, is in like manner discontinued, and the void is glazed. A little above this the heads of the ribs are framed into a circular ring of timber, which forms a wide opening in the middle; over which is a glazed canopy or umbrella, with an opening between it and the dome for allowing the heated air to get out. All who have seen this dome say that it is the most beautiful and magnificent object they have ever beheld.

The only difficulty which occurs in the construction of wooden domes is when they are unequally loaded, by carrying a heavy lanthorn or cupola in the middle. In such a case, if the dome were a mere shell, it would be crushed in at the top, or the action of the wind on the lanthorn might tear it out of its place. Such a dome must therefore consist of trussed frames. Mr Price has given a very good one in his plate OP, though much stronger in the trusses than there was any occasion for. This causes a great loss of room, and throws the lights of the lanthorn too far up. It is evidently copied from Sir Christopher Wren's dome of St Paul's Church in London; a model of propriety in its particular situation, but by no means a general model of a wooden dome. It rests on the brick cone within it; and Sir Christopher has very ingeniously made use of it for stiffening this cone, as any intelligent person will perceive by attending to its construction.

Fig. 28 presents a dome executed in the Register Office in Edinburgh by James and Robert Adam, and is very agreeable to mechanical principles. The span is fifty feet clear, and the thickness is only four and a half feet. (J. R.)

Causes of Failure in Roofs.

It has been shown in the preceding treatise that the simplest form of a roof is that given in fig. 12. Let us now inquire the method in which such a roof would fail, as deduced from the former treatise, and given there in scientific language. Let the dotted lines (fig. 29) show the original line of roof. If an undue weight be put on this, it has been shown the point B will descend, and A and C will spread or open to the right and left, just as pressing on the top of a pair of compasses makes them open. The rafters must then either slip off the top of the walls, or, if properly secured to them, which we ought to suppose, must push the walls over; or if they be very strong, the rafters must bend or sag in the middle at D and E (fig. 30). Now, to prevent the walls being thrust over, an easy remedy, as our author shows, is fig. 14, to tie them together with either a piece of string or a rod of timber, or, as Mr Robert Stephenson (art. IRON BRIDGES, fig. 13) shows, by a chain. But whether the walls be kept upright by this tie or by their own size and strength, still the same bending at D and E will take place if the timbers be not strong enough to bear the weight. Now, in small roofs we have a ready remedy; we put a collar beam DE (fig. 31) between them, which has a double effect—it not only keeps these points from coming towards each other, which they must do if the rafters bend, but it also assists very much to prevent the rafters AB, BC, going out, as is shown in fig. 29. If, however, the roof be too large and the timbers too weak, or, which is the same thing, the load be too heavy, the roof, though it cannot bend (fig. 31) between BD, BE, will yield because failure in the early mediaeval roofs (see figs. 35-38), where they are all weak between A and B. However trusses may be braced together at the apex of the triangle, it is clear nothing except excessive thickness of timber will prevent one of these results. We will now go back to fig. 14, and for the present lay aside the consideration of a collar-beam. We are here liable to fall into this contingency—the beam AC must have considerable weight if strong enough to act as an efficient tie. Now wood is always weakest in horizontal position; it is therefore liable to sink or sag in the middle, and the effect of this will be to bring the points AC (fig. 32) closer together, and to pull the walls in. It would interfere with our space to prop this beam up from below; but some ingenious carpenter at some time has thought of hanging it up to the ridge by another string or tie, BD (fig. 15), usually called a king-post, though it does not act as a post to prop up, as is shown above, but as a string or tie, to hang up the tie-beam and keep it straight. We will now suppose the rafters AB, BC (fig. 15) too weak for their weight. We must bear in mind, in all questions of carpentry, there are two difficulties to contend with. First, timber itself is limited in size; and next, if it be made too strong for its length, its own weight will cause it to bend. So we are between two difficulties. But we have several remedies: either we may resort to a collar-beam in addition to a tie-beam, as in fig. 31; or, what is better, we may employ two struts, DE, DF (fig. 17). An inspection of this diagram will show why, although a chain or rod would answer every purpose at BD, we prefer a post of wood. The struts FD, ED are supports and not ties, and require a good buttment, which is best got by framing them into the king-post, as shown at D (fig. 17), and also in the various diagrams of the article CARPENTRY. Still referring to the same diagram (fig. 17), we will suppose the span still further increased; then the tie-beam may be too weak, and may sag between AD and DC. This must be remedied by again suspending the weak parts, which may be done by two rods or posts BE, CF (fig. 18), which are generally called queen-posts, as BD (figs. 15 and 17) is called a king. This system divides the tie into three parts instead of two, or if a king-post also be used, into four parts, each of which is suspended. Fig. 18, however, shows what is generally called a queen-post roof, and is framed with a collar BC, in the points of which the purlins are usually seated, and the common rafters run up, as AG, DG, unless, as in the figure, it is intended to make the roof flat between B and C. In the same manner, an inspection of Roof.

The diagram following will show (as well as those in Carpentry), and infra fig. 51, how the principle of using queen-posts and struts may be multiplied almost to any extent.

Roofs sometimes fail in consequence of the trusses being placed too far apart; the purlins are then unable to sustain the weight, and the surface undulates between truss and truss, bringing down the ridge, and producing the most unpleasant and sometimes pernicious effects. The medieval architects braced the purlins upon the principles as shown in fig. 40. A better plan, however, has been devised at the Lambeth Baths, which will be hereafter described.

Roofs also frequently fail from the weakness of walls, or the want of extra thickness under the principals; but this is rather matter for consideration under the article Stone Masonry. They, however, often fail from a simpler cause. The plate or template is generally of timber, and bedded into the wet brickwork. It swells with the moisture, and afterwards shrinks, which causes it to lose its hold in the walls; so that in case of settlement it is easily pulled out; and the walls, which should be kept upright by the tension of the tie-beams, are unsupported, and settle outwards. The best remedy is to make the templates of stone, and pin down the tie-beams to them by strong iron pins, going some depth into the walls. There is another benefit about this system, that air may be allowed to come sideways to the rafter feet, or ends of the tie-beams, and so prevent their becoming rotten.

Medieval Roofs.

Those of the Italian basilicas, erected before the tenth century, are framed much according to the present methods: of the same pitch, and covered with tiles like those which have been in use from the Roman period down to the present day. Fig. 33 shows the roof of San Paolo Fuori le Mura, lately consumed by fire at Rome. It is upwards of 84 feet in span, and consists of a king-post and two queens, without struts, but with a collar. The principal rafter is doubled from the head of the queen to the plate, which adds immensely to its strength. The kings and queens are not framed into the tie-beam, but the latter is hung up to them by iron straps. This perhaps is the earliest instance known where iron has formed one of the chief features in the construction of a roof, and this is said to have stood upwards of 1400 years. A curious instance of a roof is found in the tomb of Theodosius at Ravenna, supposed to have been erected shortly after the year 526; this is composed of a circular dome of white marble 36 feet in diameter, surrounded by a number of ears or lugs, by which no doubt it was hoisted to its place. It is of one huge single block, all chiselled out of the solid.

Of Saxon roofs we have no remains; but from the illuminations in the MSS. of the Elfric Pentateuch, and to Cædmon, now in the British Museum, we should suppose they formed an angle of 45°. They seem to be covered with a sort of tiles rounded at the end, or they probably may have been of shingles. A very curious instance, taken from the Harleian MS. (fig. 34), shows not only a gabled roof, but also a sort of dome. The subject is Lot entertaining the two angels.

Few Norman roofs now remain, if any, and those which are supposed to be so are evidently much altered. Judging from the string courses up the gables, and the water- so the aspiring roofs of the refined orientals suggested those of the early English period.

Year by year, as styles changed, the roofs became of less pitch, till in the latter styles, as has been stated, many, if covered with lead, became almost flat. A very curious illustration is found in the tower of St Regulus' church at St Andrews, on which are the marks of the lines of three roofs which have covered the building at three different periods. The lower is probably the line of the original Norman roof; the upper that of the early English period, and the middle that of the decorated. Much of course depended at all periods on the covering, tiles, for instance, using those of the Norman period, which are in fact mere shallow pilasters. The truth is, the buttresses, which were originally intended simply to stiffen the walls, were afterwards enlarged to that extent that they became struts to it (A, A, fig. 42), especially the flying buttresses, which continued the line of the principal rafters down to the ground, making the earth, as it were, the tie-beam. To such a degree was this system carried, that in many continental churches, and in some of our own, of which Henry VII's chapel is a known instance, there is literally no support derived from the walls, the windows filling the whole space between buttress and buttress, which last, from their vast mass and projection, sustain both roof and groining.

Medieval roofs may be considered,—1st, As those in which every pair of common rafters is framed together, and forms of itself a separate truss, or, as it was called in those days, a "couple;" 2d, As those with common rafters and trusses framed with collar-beams; 3d, As those with hammer-beams; 4th, As those with tie-beams.

But before going into this subject, we must warn our readers who are accustomed to roofs framed of fir, that the medieval timbers are almost invariably of oak or some hard wood; so the strength or the scantling of the timber must not be judged by our modern notions. That of oak to fir is assumed by Professor Robison to be as three to two. An inch of oak may be safely subjected to a strain of three tons for every square inch, while fir will bear but two.

The most common form of the first is a simple St An drew's cross, as at fig. 35; a cross and collar, as fig. 36; a collar and struts, as fig. 37; or two collars and struts, as fig. 38. All these are taken from examples of the thirteenth and fourteenth centuries. As has been before explained, these roofs are liable to spread, as there is nothing to tie the plates together, and they are all weak at the point between the end of the brace A, and the top of the ashler-piece B. To obviate this, various contrivances were used; one of the most elegant is the addition of curved braces, which not only strengthened each rafter, but tended to brace the whole together. Fig. 39 is a common example in England, particularly in Somersetshire.

As those roofs with simple couples could not be erected of any great span, they were commonly framed with trusses, purlins, and common rafters; and then the defect of these open roofs became more felt, as each truss had not only to bear its own load, but that of the adjacent common rafters. The usual means of strengthening the roofs last described, that of introducing curved braces to the couples, was adopted for the truss collar-beam roofs, till at length they assumed the form of an arch, filling in the space between the principal rafter and collar like a rib of solid timber. Fig. 40 is from St Mary's, Leicester, and is a very good example. But the most extraordinary mediæval roof, on the principle of a collar-beam and curved braces, is that over the Salle des Pas Perdus at Rouen, which was built in 1493, and is of inconceivable lightness and boldness (see fig. 41). It is 54 feet 5 inches English in span, and covers a hall 155 feet long; the trusses being not quite 4 feet apart. It is close-boarded inside, so as to show as one long, pointed vault below. The thrust, which is immense, is resisted by walls 6 feet thick, with huge buttresses. The upper part of the construction is very good; but how the part between AB can sustain the weight of the upper part without bending has puzzled everybody. It can, in effect, be only due to the excellence of the workmanship and strength of the materials.

On looking back to fig. 40, it will be seen not only that the curved brace was intended to strengthen the rafters, but also to relieve the thrust by conveying it lower down the wall, and distributing it over a much larger portion of its surface, as well as bringing it more in a line with the buttress AA (fig. 42). Fig. 43 shows the beautiful roof over the church at Wymondham in Norfolk. Here the brace at the foot of the rafter is composed of two curved pieces, which meet together in a horizontal piece carved as the figure of an angel. This construction, at once so elegant and useful, soon became developed as the hammer-beam roof. Figure 42 is from a church in Suffolk. In this A, A are the wall buttresses; B the hammer-beams; C the wall-pieces, or, as some call them, the pendant-posts; D the hammer-braces; E the collar; F the collar-braces; G the side-posts; H the ashler-pieces; I, I the purlins; K, K the principal rafters. There is an infinite variety of those roofs in England, chiefly of the perpendicular period. As they were required of a larger span, the roof became a double hammer-beam roof. Figure 44 shows the general section of these: the nomenclature is the same, with the addition that A is the upper hammer-beam, B the upper hammer-brace, and C the upper side-post. Of these,

---

Most Gothic curved braces are cut out of boughs of large trees bent by nature. the most celebrated are those over Westminster Hall, Hampton Court, the palace at Eltham, and very many of our college halls. It will be seen by the section, the use of the hammer-beam framing was to deepen, as it were, the principal rafter, and thereby prevent its bending between the collar and the plate; besides which it got better hold of the wall; but there was nothing to tie the walls together, nor to keep the truss from spreading; the consequence is, in spite of the buttresses, many of these roofs have opened, and the walls have gone over. A very curious roof, and one which is much stronger in point of construction, is that over the Parliament House at Edinburgh, which was begun in 1632. In this roof the pieces which act as hammer-beams are not horizontal but incline, or rather radiate towards a centre; they are filled in with arched pieces bearing pendants, and have a very original and pleasing effect.

The use of level tie-beams in roofs is of two kinds: one where the rafters are in couples, and they seem merely introduced to tie the plates together, and probably were inserted afterwards; the other where they form parts of the trusses themselves. In large roofs, we have already an example in fig. 85. In smaller roofs of the fourteenth century, fig. 45 is a very common and pleasing example. From

Fig. 44.

Fig. 45.

the tie-beam springs a king-post, from which branch four curved struts, one pair serving to support the principal rafters, the other pair doing the same for the ridge. This is called a tree-post roof. Figure 46 shows a very curious combination of a level tie-beam and curved struts. It is from the church of St Mary the Virgin at Pulham in Norfolk. But it was reserved for the perpendicular period to design the most elegant roofs with level tie-beams. These are of infinite variety in design, and are generally filled in with tracery, and ornamented with carving. Figure 47 shows a very beautiful example, that of St Martin's at Leicester. In many instances these roofs are richly adorned with painting and gilding of the most brilliant description.

Another sort of mediæval roof is yet to be noticed, and that is where, instead of timber principals, arches of stone are thrown across from wall to wall, and carry the purlins and common rafters; among these may be named the great hall at Mayfield; but they are of very rare occurrence.

Account of Roofs of Large Span (à grand portées) and some modern Roofs.

For many years the roofs of the basilicas at Rome were the largest spans that had been covered by the carpenter's art; but about three-quarters of a century ago there was a great desire to roof over much larger spaces without internal supports, for the purpose of military and other riding-schools. The vast roof over the Salle d'Exercice at Darmstadt, erected by M. Schubknecht in 1771, is 228 feet long and 154 feet in the clear of the walls, or 2 feet more in span than that roof lately erected over the railway station at Lime Street, Liverpool, and was the largest roof in the world till that we are about shortly to describe was erected over the New Street station at Birmingham. It appears to have stood very well, although its construction is certainly not on the best principles. The thickening out of the tie-beam by packing beam after beam, one on the other, must have caused considerable shrinkage; and the struts would be much better if placed the reverse way. This roof attracted considerable attention; and about ten years after its erection the Emperor of Russia, Paul I., happened to travel through Darmstadt, and visited the building. He expressed great astonishment at its vast proportions, and determined on his return that one should be erected at Moscow which should entirely eclipse it in magnitude. Accordingly the design was prepared. This gigantic roof was intended to have covered a hall 852 feet in length, by 308 feet in width from out to out. The walls were double, and formed a system of arcaded galleries round the building about 25 feet in depth; so that the span of the roof in the clear was reduced to about 230 feet. The main support of this roof was the curved rib of three thicknesses of timber, notched on to each other en cremaillère. Krafft says it was executed, and was used in 1790 for the exercise of the Cossack cavalry and infantry. But M. de Bétancourt, of whom we shall speak presently, affirms that it never was finished. The dotted lines show a method proposed by Rondelet and Krafft to strengthen this roof and make it effective. The chief defect, however, seems to be, that the principal rafters are too weak, and receive no direct support from the cross struts. The troubles in Russia seem to have caused the matter to drop, till the year 1817, when the Emperor Alexander, being at Moscow, resolved on carrying out a roof that should rival the one at Darmstadt. A great number of designs were prepared, none exceeding spans of from 110 to 115 feet. They were referred to General de Béancourt, before named, who was then chief director of public roads. After some study, this officer prepared the design of it. It is for a hall 501 feet long and 160 in width, which was executed in the short space of five months. On striking the scaffold, which had supported the roof during its erection, the tie-beams went down about two inches. This depression increased in three months to eight inches, when the tie-beam of the twenty-fourth truss gave way, in consequence of the existence of a large knot. The roof was shored up and strengthened, and afterwards stood perfectly well. In justice to M. Béancourt, it should be stated that such was the hurry in which the work was done that the greater part of the timbers were cut down and floated on the river only a few days before they were framed, which was done by 400 carpenters, or rather woodmen, who hewed the wood with axes—they being ignorant of the use of any other tool. The writer of this article has designed a roof for the first-class swimming bath in the Westminster Road. This is an entirely new principle, being, in fact, the adaptation of the trellis or lattice principle to a roof. The trusses are about 18 feet apart, or nearly double the usual distance. The purlins, however, are prevented from bending by a series of light longitudinal trusses, likewise on the lattice principle. The roof is, in truth, trussed fore and aft, so that no part can move. It is of extraordinary lightness and cheapness, the trusses being so few and the timbers so slender. Its deflection, after receiving the load of slates, skylights, &c., was scarcely perceptible.

(a) Roofs Trussed with Curved Timbers.—These are (a) with timbers side by side, flatwise, the ends breaking joint, or, as the system is called by French writers, "en bois plat." It is the invention of Philibert de Lorme, a French architect, who published it in 1561, in a work called Traité sur la Manière de Bien Bâtir, et Petits Frais. In this system the rafters are in effect curved ribs, of several thicknesses of timber, nailed side by side, care being taken that the ends do not come in the same place; in other words, that they break joint. The rib then resembles a beam cut out of a crooked limb of a tree, and owes its strength simply to cohesion of the particles. These beams are often used in pairs joined together by cross pieces, which are keyed on the outside, to prevent their shifting apart, and are of vast strength, and particularly adapted to rooms where the ceiling is intended to represent a large arch or barrel vault, or domes intended to be covered with lead or other metal. One of the most ingenious adaptations of this principle is at the Hotel Legion d'Honneur at Paris, where it forms a dome, with an inner half-dome or cove which carries a gallery. The roof over the great hall of the Pantheon in Oxford Street is a long vault on this principle, the middle thickness or flitch of each rib being of teak wood. That lately erected over the Surrey Music Hall is also a long vaulted ceiling; but here the main ribs are formed of four thicknesses of 1½-inch deal, 1 foot 3 inches wide at bottom and 1 foot at top, in the middle of which is a rib of rolled 4½ boiler-plate, 6 inches wide, the whole of which are bolted together. The main ribs are 17 feet apart, between which are two intermediate ribs of lighter construction.

(b) Roofs constructed of Timber bent on the flat "en bois plat."—This is the reverse of the former system; the boards being bent over a centre, and thickness added upon thickness till the beam is thought strong enough, when the whole is bolted together, care being taken, as in the former system, that the ends of the boards break joint. The difficulty is, to prevent the natural tendency of the wood to spring back to the straight, as well as to counteract the weight of the roof covering, which would cripple the curve either at the haunch or crown, as the pressure might be exercised. In this respect something analogous to the laws of arches will apply to explain the result.

The roof over the great riding-house at Libourne was designed by the celebrated Colonel Emry, and executed in 1826. Every rib is composed of five thicknesses of deal, each nearly 2 inches thick, about 6 inches wide and 40 feet long. The rib is semicircular, the springing about 24 feet from the ground, and the span about 70 feet. The ribs are not only bolted together, but clipped with a sort of stirrup-iron and bolts. From these ribs a number of struts radiate to and support the principal rafters, purlins, &c., and at the same time prevent the rib from being crippled at either the haunch or crown. The worst of this roof is, there is nothing to prevent their spreading at the foot. The consequence is, the wall was not only of unusual thickness (nearly 5 feet), but large buttresses were added to counteract the thrust. The roofs over the Great Northern Railway station at London are of about the same span, and are of similar construction, except that the ribs are of 1½ deal in 16 thicknesses, and the spandrels are filled in with ornamental cast-iron work in form of circles, guilloches, &c., which have a very pretty effect. Like the Libourne roof, the thrust was compensated by massive brickwork.

To prevent this last defect, as well as to stiffen the rib at the springing, M. Émy designed the roof for the cavalry school at Saumur. This is intended for a span of upwards of 130 feet. Each truss is composed of two sets of ribs, similar to those before described, kept apart at the foot by a series of trellis, and joining together as one rib about halfway up the curve. Instead of one thick wall, it was intended to carry the ends of the trusses partly on an outer wall, which has strong double pilasters, and partly on one large square pilaster, intended to stand under the inner part of the end of the truss, as shown in the figure given in his work; so that, though the space is available for spectators, and little room is lost, the truss itself has a level bearing 18 feet in width from whence to spring.

But of all roofs ever projected, either in wood or iron, the most gigantic, the most original, and the boldest, is one for another riding-house, also by Colonel Émy. This has a span of 328 feet, or at least half as much again as the largest existing roof in the world—that at New Street, Birmingham. It is intended to be composed of two ribs similar to those before described, with another intermediate rib carried up about two-thirds of the span, and braced, as shown in the figure. The building was to have been surrounded by an ordinary wall, at right angles to which, under the foot of every truss, was a return wall 50 feet long and 4 feet thick, perforated below with arches to form passages, and to serve for spectators, as in the former instance.

Iron Roofs.

Like most important discoveries, the use of iron in roofs has grown up from the smallest beginnings to the largest and most extraordinary results. From the simple substitution of an iron rod for a king or queen post in a wooden truss, we have attained the art of covering spaces so vast as to throw all other modes of construction into insignificance, so light as to appear the work of fairies rather than of human beings, and yet so strong as to bear their weight with ease, and to resist unshaken and unhurt the action of the roughest wind. The theory of these roofs, however, is just the same as those of timber, allowing only for the difference of weight and power of resistance of the materials. The tie keeps the walls together, the king and queen rods prevent the tie from sagging; all these are in a state of tension; while the struts prevent the rafters from bending, and are in a state of compression.

Fig. 48 shows the earliest use of iron as the king-post of roofs suited to those of from about 20 to 30 feet span. In Plate CLXII., Carpenter, figs. 38, 40, is shown the method of employing iron as king or queen posts of still larger dimensions. Fig. 49 shows a very convenient method of using iron rods as king-post and tie-beam in a collar-beam roof where height or head-room is of consequence; but this was shortly after superseded by the form fig. 50, the adjustment of the screws, &c., at A being more convenient than at fig. 49. With large spans, however, the wooden tie-beam continued in use for a long time. One of the best instances for the time it was executed may be found in the roof over the passenger-station at London Bridge of the Croydon Railway. This roof, which is about 54 feet in span, is constructed with an iron king-post and ten sets of iron queens, with wooden struts. (See fig. 51.) It is, however, somewhat twisted on the face by the winding of the timber. The great fire at the Houses of Parliament, and subsequently at the Royal Exchange, drew attention strongly to the importance of fire-proof roofs; and the first of these structures of any size or importance was designed by Sir Charles Barry for the new palace at Westminster. Fig. 52 shows the section of that over the committee-rooms in the front next to the river. This is entirely of wrought iron, excepting the shoes, by which the ends of the various pieces are connected. The iron is flat bar, simply cut to lengths, and punched to receive the bolts that pass through them and the shoes. They vary in width from 2 to 3½ inches, and in thickness from 3½ths to 4ths. These light principals are not quite 3 feet apart, and are covered with a species of galvanized iron tile reaching from one to another, hung on a sort of connecting-rod without purlins. It is stated that the sulphurous acid in the smoke of London is already causing serious oxidation, and several methods have already been tried to prevent premature decay. This is to be regretted, as the material is very picturesque in character, as well as light, and not expensive. Some fault at the time was found with the construction of the shoes or connecting-plates, but they stand perfectly well, and there has been no failure. The suspension-rods at each side of the king are a peculiar feature; they not only keep up the tie, but bind the roof together at each intersection.

Fig. 53 shows the roof over the House of Lords. This is of much larger span, being 45 feet, while the former is 28 feet 3 inches. It is of exactly similar construction, except that the struts, wall-plates, purlins, and bearers are of cast-iron, the section of the struts being the form of a cross. The suspension-bars are double. The principals are 7 feet 6 inches apart—more than twice as much as the other roof, which of course necessitates the use of purlins.

These carry two common rafters 2 feet 6 inches apart, this being the width of the iron plate or tile with which the roofs are covered.

Much animadversion has been expressed as to these roofs. The nearness of the principals has been criticised, as well as the small subdivision of detail; in fact, we shall shortly show roofs of three times the space with a less number of junctions, the principals of which are very much farther apart; so in this respect they are very much more economical. But it must be remembered these Gothic roofs are of high pitch, while the others are very flat; and therefore more struts and queens are necessary; and that the distance of the principals was regulated in great measure by the nature of the covering. If we also consider they were the first roofs of the kind, we think they may be regarded with great commendation. The progress of railways caused a great demand for light roofs of every span and length, and the facilities for obtaining rolled T and L iron, and the machines invented for punching and riveting, combined with the fact, that iron was not only the lightest material of its strength, but was at the same time incombustible, brought it day by day into more general use.

The awkwardness of fitting wooden struts to an iron tie-rod suggested the use of cast-iron for that purpose; and a very pretty light roof, first used on the Manchester and Birmingham line, was invented (fig. 54), which has since been

of almost universal adoption for moderate spans. For small spans, a very simple and cheap contrivance was used (fig. 55). It is composed of ordinary wrought-iron tubing: A

is a common tee-piece with right and left screws; the king and tie are thus brought together, the other ends first being put into the fire, and hammered into the form of straps. The considerations before named soon caused the abandonment of the use of wood as principal rafters altogether. Some important roofs were constructed with rafters of cast-iron; but this is a material which, if it gives way, breaks quite suddenly, and without any warning; and by degrees the rolled T iron superseded its use. A very large roof, 87 feet in span, was erected at Paris on the Quai Jemappes,

over the Providence magazine, as shown in fig. 56, where

between A and B, there being too long a bearing without a strut. It was, however, much improved at Paris at the terminus of the Rouen Railway, where one of a span of 88 feet 6 inches was erected, as shown in fig. 58; while fig. 59 shows the detail of the junction of the ties and struts. The introduction of more struts has much strengthened the principal rafter; but the triangle ABC is too large, and ought to have been subdivided by another strut from B to D, the point C being unfairly loaded, and there being a tendency to sink at the point D. A still larger roof, 97 feet 5 inches in span, was shortly after put up at the terminus of the Strasburg Railway at Paris (fig. 60). This is of an arched form, the principal rafter being a huge circular arc of wrought iron framed on the trellis principle. The top and bottom ribs are of rolled T iron, 3½ inches deep, with a 3½ flange, the web of the metal being 2½ inches thick. The struts are of cast-iron. The rods which tie this vast structure together are only from 1½ to 1¾ inch diameter. The walls which support it are above 40 feet high; and the effect is almost cobweb-like, so slight do the ties appear when viewed from beneath. There is perhaps a defect in making the centre part of the trussing two parallelograms, which could be easily prevented by the introduc-

Fig. 58.

tion of a strut from A to B. The extreme inconvenience of having columns or other internal supports to a railway station, and the success of these large roofs, emboldened our engineers day by day to increase their spans; and in the year 1850 Mr R. Turner of Dublin erected the roof over the great station at Liverpool. This was 374 feet in length, of the enormous span of 153 feet, and of the principle shown in fig. 61. The height of the springing is 25 feet, and to the crown 55 feet. The principals are 21 feet 6 inches apart, and are more like a trussed girder than the ordinary form of roof. They are composed of a rib or arc of rolled iron 9 inches deep and 4½ inches of an inch thick, with top flange 4½ and bottom 3 inches wide; on this is rivetted a plate 10 inches wide and 4½ inches of an inch thick.

From the springing to the haunches these ribs are strengthened by plates rivetted on both sides. The struts radiate as shown, and are constructed like the ribs, but are only 7 inches in depth. From strut to strut there are three sets of tie-rods between the two extreme radiating struts, and two between the others, varying from 2 inches to 1½ inch in diameter. The diagonal braces are 1½ inch. The ends of the principal are secured to a chair of cast-iron, resting partly on cast-iron columns, partly on the walls of the office, and partly on a box-girder. Each purlin is composed of three pieces of T iron. The centre piece runs straight

Fig. 59.

from principal to principal, the other two branch off, so as to strut them in three points; besides this, they are crossed by diagonal braces; so that the whole roof is trussed fore and aft, and forms one solid mass of framing. The attachments are made by linking-plates, much like those in fig. 59.

The roof is covered partly with corrugated galvanized iron of No. 16 gauge, and partly with glass, in sheets averaging 12 feet 4 inches by 3 feet 6 inches, and 3/8ths of an inch thick.

But this roof, vast as it is, sinks into comparative insignificance by the side of that over the joint station in New Street Birmingham (fig. 62), which was finished in 1854, and is the largest and lightest in the world. It is about 1/4th of a mile in length, and varies from 191 to 212 feet in width, the ground being irregular. It spans at once ten parallel lines of railway, four passenger platforms, and a long carriage road. The principals are 24 feet apart, and are supported on one side by the walls of the offices, and on the other by cast-iron columns 2 feet in diameter, connected together at the top by cast-iron arched girders. The top of these, on which the principals rest, is 33 feet above the rails; the tie-rod has a versed sine of 17 feet, the curved principal is 23 feet deep, and the total height to the top of the louvre is 84 feet. The rib forming the principal rafter is of rolled iron 15 inches deep, and 3/8ths of an inch in thickness. In each edge, both at top and bottom, are riveted two angle irons 6 by 3, which together form two flanges 12½ inches wide. All junctions are made to break joint, and have plates rivetted on each side, forming a species of fish. The tie is a solid rod 4 inches in diameter, and is thickened out at every screw; so that the full diameter of the rod is preserved independent of the thread of the screws, which are right and left handed, and which meet at wrought coupling-boxes. On these the struts and diagonal braces take their seating; this is a cast-iron shoe with the requisite lugs and bolt-holes, and which clips the coupling-box, and is screwed to it from beneath. The diagonals are of 3/8ths rolled iron, varying from 5 to 3 inches in width. The struts, twelve in number, are of very original construction. They are vertical, composed of four pieces of angle iron set back to back, as if upon the four corners of a square, and are kept apart by iron crosses to which they are bolted. These crosses are larger in the middle of the strut than at the ends, so as to cause the angle irons to curve out each like a bow, and form a sort of open swelling strut, which is not only very strong, but has a very pleasing effect. At one end these vast ribs are secured to stones let into the wall; at the other, which is over the columns, are sets of plates, one attached to the foot of the ribs, and the other to the girders, between which are a series of rollers 2 inches in diameter, on which the ribs have play, so as to compensate for contraction or expansion of the metal by change of temperature. The parlins are of wood 6 inches square, and are 10 feet apart, trussed with three-quarters tension-rods. Louvres for ventilation are shown in the figure. A little more than half the roof is covered with rough-rolled fluted glass 3/8ths thick, each plate being 6 feet long and 16 inches wide; the other part is covered with galvanized corrugated iron. Some idea of this vast construction may be formed when we are told it comprehends 2 acres of galvanized iron covering, and rather more than 2 acres of glass. This last weighed 115 tons, and the whole iron work 1050 tons. The cost was £32,274, or about £19 per square; but iron at that time was exceedingly low in price—more so, in fact, than had ever been known.

A very ingenious roof (fig. 63) has just been erected over the new Royal Italian Opera House at Covent Garden. It was designed by Mr Edward Barry, and is 90 feet in span, and on the ridge-and-furrow principle. The spans being supported by a series of double-trellis girders 9 feet deep, and 19 feet 6 inches apart from centre to centre, between which are the painting-rooms, carpenters' shops, &c. They are entirely of wrought iron, the ends AB (fig. 63) being, in fact, box-girders for a length of 3 feet 9 inches. A double set of iron trellis is then constructed, as


**Fig. 63.**

Trellis Girder and Roof, Royal Italian Opera.

In fig. 63, 9 feet deep, and 6 inches apart. On the top and bottom of each set, and on both sides (fig. 64, AB), a series of angle iron is rivetted, on the top and bottom of which, again, a plate of flat iron is also rivetted, forming upper and lower compression and tension flanges 1 foot 6 inches wide. The whole may be regarded as a perforated hollow beam 90 feet long, 9 feet deep, and 6 inches in thickness, having, as has before been stated, top and bottom flanges 1 foot 6 inches wide. From each of these rises a light transverse roof 19 feet 6 inches in span, the gutter of which is on the top of the girder. The tie and king are of flat iron, and the struts, rafters, and purlins of T iron, all as shown in fig. 64. At the top are skylights which open for ventilation; the rest of the roof is covered with slabs of slate. The floor of the painting-rooms is supported by iron bearers (CC, fig. 63), on which rest common joists and the ordinary boarding. The strength is enormous, and the construction has these advantages: It is not only lighter than any known roof, but there is no thrust in the walls; on the contrary, it tends to tie them together, and there is no loss of room, as the interval between each girder forms a fine room 90 feet long, 19 feet wide, and 9 feet high. (A.A.)