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

GG, p. 223. There seems to be a little confusion in the idea of the possibility of altering the nature of the action of the fibres of a beam by altering the place of the gudgeon in this manner; but the author has very properly abstained from making any practical application of the supposed modification thus introduced. With respect to the strength required for scarfing or joggling, it may be observed, that the whole of the compressed fibres of the concave side may be considered as abutting against the whole of the extended fibres on the convex side; and this abutment is equally divided throughout the length of the beam; so that if the scarfings or joggles in the whole length of the arm of a lever, taken together, are as strong as one half of the depth of the lever, exerting half its powers, from the inequality of tension, there will be no danger of the failing of these joints; and from this principle it will be easy to determine the depth to which the joints ought to extend in any particular case. Hence also we may understand how a beam may become so short as to be incapable of transverse fracture in its whole extent; for the lateral adhesion between the different fibres of wood is generally far inferior to the longitudinal strength of the fibres and if, for example, it were only one fourth as great, a beam less than twice as long as it is deep would separate, if urged in the middle by a transverse force, into two strata, from its incapacity of affording sufficient abutment, before its longitudinal fibres would give way. HH, p. 225. If the bolts were sufficiently numerous and sufficiently firm, so as to produce a great degree of adhesion or of friction between the parts, this joint might

be made almost as strong as the entire beam, since there Carpentry. is nothing to prevent the co-operation of each side with the other throughout its extent; but much of the strength would be lost if the bolts became loose, even in an inconsiderable degree.

II, p. 227. The author has reasoned upon the direction of straps, as if it were universally necessary to economize their immediate strength only, without regard to the effect produced on the tightness of the joint; but it may happen that the principal purpose of the strap will be answered by its pressing the rafter firmly upon the beam, and this effect may be produced by a certain deviation from the horizontal position, with but little diminution of the strength of the strap; a deviation which has also the advantage of allowing the strap to embrace the whole of the beam, without weakening it by driving a bolt through it. We must not, however, run the risk of crippling the end of the beam, and the straps represented in fig. 38 may be allowed to be somewhat too erect.

KK, p. 228. It does not appear to be desirable that the ends of the rafters should be supported without any pressure on the ends of the beams, since these ends would bear a small weight without any danger of bending, and would thus lessen the pressure on the king-post.

LL, p. 228. The half length being 25 feet, and the camber 6 inches, the excess of the oblique length will be 625-25 — 25, or ʊ of a foot, that is, of an inch, which is all that the beam would appear to lengthen in sinking; nor would the settling of the roof be more "considerable" than about a quarter of an inch. But there seems to be no advantage in this deviation of the tie-beam from the rectilinear direction; and the idea, which appears to be entertained by some workmen, that a bent beam partakes of the nature of an arch, is one of the many mischievous fallacies which it is the business of the mathematical theory of carpentry to dispel. (T. Y.)

Definition. ROOF, the 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 href, who thought so much of its importance, that they called the carpenter hrof-wyrhta, or the "roof-worker.'

Varieties of covering.

Pitch.

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 were 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 III.) 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 mediaval 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 XXI.), 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 XIX., 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, stilted 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 root 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.

used.

After all, although much latitude must be given to taste, Pitch deit is probable the pitch of a roof mainly depends on the pendent or 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 æsthetics. 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, inasmuch as they have more weight to bear. We therefore pay dear in more ways than one for our liking for high-pitched roofs.

due to cli

mate.

Although it happens that both Greek, Roman, and Ita- Pitch not lian roofs are flatter than ours, and the climate is warmer, 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

Roof.

Pitch required for different materials.

Qualities

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 22°. 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. Pantiles 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 of various 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

roof coverings.

Weight of different

roof cover

ings.

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 su-
perficial nearly) of zinc will weigh about...
A square of lead, according to thickness, from...5
A square of slating from.....

A square of pan-tiling..
A square of plain tiling.....

1 cwt.

to

7

.5 to 71 ..7

.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. Mediaval 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

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 prin- Principles ciples which regulate the strength of the materials, relative which reto the manner in which this strength is exerted, and the gulate the strength of manner in which the strain is laid on the piece of matter. the mateWith respect to the first, this is not the proper place for rials. considering it, and we must refer the reader to the article borrow from that article two or three propositions suited to STRENGTH OF MATERIALS IN MECHANICS. We shall just

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 gluten, 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

Roof. cylindrical pillar, which may be taken out of its mould.

Thei:

to transverse strains

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 We can easily form a notion of wrenching or twisting it. 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 weakness or other matter, may be broken transversely. This will in relation 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.

E

Fig. i.

B

H

A

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

W

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 ƒ 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: p, and p will be the weight which will just break the thread, by hanging by

CD the point A. This gives us pƒx If the weight CA be hung on at a, the force just sufficient for breaking the

same thread will be=ƒx CD. In like manner, the force,

Ca

which must be hung on at A in order to break an equally

strong or an equally resisting fibre at F, must be = ƒ x CF 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 ƒ 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 7. We may now express the strength of the whole line CD by fx 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 7, is fx CDx C g, whilst the momentum of w is wx CA. We must therefore have ƒx CD x Cg wXCA, or fd x d = wl, and fd: w = 1:d, or fd: w = 21: 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 fdb, and we still have 21: d=fdb: w; 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 frac ture. There is some point c which lies between the fibre: 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 Dc, and the en.

ergy of the whole resistances must be less than by the se cond statement. Till we know the proportion between the dilatability and compressibility of the parts, and the relation

Roof.

between the dilatations of the fibres and the resistances which they exert in this state of dilatation, we cannot positively say where the point cis 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 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.

Practical These observations are enough to give us a distinct noinferences. tion 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.

Effect when

and loaded

ille.

Fig. 2.

D L

H

We have hitherto supposed, that the beam had one of its beams are ends fixed in a wall, and that it was loaded at the other end. supported This is not an usual arrangement, and was taken merely as at the ends affording a simple application of the mechanical principles. in the mid- 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 Ffe, Hhg, going over the pulleys ƒ 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 Ff and Hh, 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

W

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,

In like manner, the strain occasioned at the point D by the weight W hung on there, is W x DE EG

X DG; which is therefore equal to 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 WX DEX DG is as DEX DG), EG and is greatest when the load is laid on the middle of the beam.

We also see, that the strain at I, 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 con

nections.

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 joists 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.

Roof.

It is easy to deduce from these premises the strain on Inferences. 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