or CENTRE, in a general sense, signifies a point equally distant from the extremities of a line, figure, or body. The word is formed from the Greek *κέντρον*, a point.
CENTRE of an Arch. Under the article Bridge, the different forms of arches have been particularly considered.
Under this article, it comes very properly to be ascertained in what manner the arch-stones are supported till the arch is completed, and the most commodious and least expensive manner in which this can be accomplished. When the span is small, and upon a limited scale, as cellars, and vaults below ground, the foundation of the side walls is dug out, the earth rounded off betwixt, the arch thrown over upon it, and the earth is afterwards dug out and carried away. This must have been done on any account. By this method the wood and workmanship are saved; but it is only in particular instances that this can be done. When the arch to be cast is on land, and at no great height above the surface of the earth, a frame for supporting the arch-stones can be raised from the earth, and bound together, frequently, with a great profusion of wood, which on account of the smallness of the arch is not taken into account; but, when the span is great, or at a great height above the surface of the earth, the expense of a frame formed in the same manner, would be enormous, and in many cases impracticable; but whether the arch be great or small, high or low, a proper economy ought to be observed; and the less the expense in wood and workmanship incurred, so much the more advantage to those concerned, and the purpose being obtained, so much more credit is due to the engineer.
It is again to be considered, on the other hand, that in order to save some expense, either in wood or workmanship, the frame or center, as we shall call it, is made too slight, and so unconnected in its parts, that the pressure of the arch-stones is greater than it can support. The whole work is brought down, and the saving on the one part produces a more serious loss on the other; so that both the workmen and proprietors agree, that it is better that the centre be too strong than too weak; better have too much wood in it than too little. To assist the mechanic in this important affair, is the design of treating this article with particular attention; for which purpose we shall be at pains to acquire every affiance that can be collected, from the most experienced engineers, and from the researches and experiments of the most distinguished philosophers who have treated of such arts as may enable us to elucidate the subject, and make it worth the attention of engineers and mechanics who may have occasion to exert their genius in that line.
In the first place, it will be necessary to consider the weight to be supported: 2dly, The quantity of the materials to be used, that shall be of strength sufficient to support such a weight: 3dly, The most effective method to apply these materials, as supported by the most approved authorities, or practised by the ablest engineers. The weight to be supported is the arch-stones. Suppose an arch of 20 feet span, (see figures for the arches, a new figure being unnecessary.) It has been shown under the article Bridge, that the arch can be raised to 30 degrees and upwards, without the support of the center; after which it begins to rely upon the frame of which the center is composed, if the arch is a semicircle, or semicircular; if a segment of a circle, it will press sooner upon the center, and the more so the flatter the arch is. 1st, Suppose a semicircle; then there is 120 degrees of the arch to be supported by the center, the diameter supposed is 20 feet. One hundred and twenty degrees will measure 20.94393 feet; but as it is advisable to give the advantage to the center, we call it 21 feet in an arch of 20 feet span. If the stone is of a durable and hard quality, perhaps an arch-stone of 12 or 14 inches might be of sufficient strength; yet it is not probable that any one would think of less than 18 inches for the thickness of the arch, for it will not have too heavy an appearance if it should be two feet thick. We shall calculate the weight at 18 inches square; the thickness of the stone is not here to be considered, as the weight of the whole is to be supported till the key-stone is driven: the specific gravity of good freestone is 2.532, the solid feet in an arch of 120 degrees; the span 20 feet is 21 feet, nearly as above. The stone 18 inches square by 21 feet gives 47.25 solid feet; the weight by the above specific gravity is 7477.3276 lb. avoirdupois, about 66.753 cwt. being the weight that one rib of the center frame must sustain, without warping, or by the pressure on its haunches make it rise in the crown; neither must it sink under the pressure: in either case the consequences would be fatal, either in causing the arch to give way, upon striking out the center, or in weakening it in such a manner as to shorten its durability; being twisted in its shape, the equilibrium would be destroyed, and the consequence would be either to spring the key-stone, or, if that was prevented by the weight above it, the same weight would cause it to yield at about, or a little above, 30 degrees from the spring of the arch. From all which the necessity of the strength and firmness of the center frame is evident.
If the arch exceeds 20 feet, suppose 50, the weight will evidently become greater, and an additional strength necessary on that account; and likewise on account of its greater extent, the frame that would be sufficiently firm at 20 feet would be supple at 50. To prevent any error on this account, another calculation for 50 feet will become necessary. In the span of 50 the arch of 120 degrees measures 52.36 feet; suppose the arch-stone, 25 feet deep by 2, is five superficial feet, multiplied by 52.36 is 261.8 solid feet, and at the above specific gravity gives 41429.7154748 lb., avoirdupois, equal to 369.928 cwt. Here the weight is increased upon the center frame, in the proportion of 66.5 to 369.9, that is, more than five times, besides what allowance it will be necessary to make for the difference of the stiffness of the center frame; both which will be considered in their proper places.
Let us now consider what will be the increase of weight upon a span of 100 feet. The rise of the arch, before it presses on the center frame in a semicircle, being in the same proportion, the arch of 120 degrees in 100 feet span measures 104.719 feet; the arch-stone may be supposed abundantly strong of 4 feet length, for the depth of the arch, and 3 feet broad, which makes a superficies of 12 feet, and multiplied by 104.719 gives 1256.628 solid feet, the specific gravity, that is, the stone is supposed of the same durability gives 198,861.381 lb. avoirdupois, equal to 1775.548 cwt. about five times more weight than upon the arch of 50 feet span. If the arch is 130 feet span, 120 degrees measure 1361.13556 feet. Suppose the arch-stone 5 feet, as in the arch-stones of the bridge over the Dee at Aberdeen, at least they are between 4½ and 5 feet. The Aberdeen granite is a very hard stone, and perhaps exceeds the specific gravity above. The arch-stone is here supposed to be 5 feet by 3, equal to 15 square feet, multiplied by 1361.13556, gives 2342.0334 solid feet. According to the above specific gravity, the weight to be supported till the key-stone is drove, is 2885.2838 cwt. The weight of the key-stone in the whole of the above may be deduced.
As center-frames must likewise be used for iron bridges, we shall consider them, and take the span 236 feet, still supporting a semicircle.
It may be proper to take the weight that it would be if the arch were the segment of a circle, the span of the arch 236, the height above the spring of the arch, or the vertex line of the arch, 34 feet, in which case the diameter of the circle would be 44.4 feet nearly; the arch-stones in this segment would press upon the center-frame, at about 18 feet from the spring of the arch. Suppose the arch-stone 5 feet by 4, equal to 20 superficial feet, the whole measure of the arch is 444.154 linear feet, the solid content is 4131.84 feet, and weight 318.689 tons; but the weight of the iron was only 265 tons. It may not be improper here to observe, that in a stone bridge of that span, 5 feet of arch-stone would be too small to sustain the arch. It may perhaps be admitted, that it would be sufficient to support its own weight; and if so, the arch being smoothed above, a second arch of a five-feet stone may be thrown over above it. These two together may form a stronger arch than a stone of ten feet depth would do. And thus a stone arch may be extended to any span, and made of abundant strength; and experience has shown its durability to withstand the weather. Thus the old London bridge has performed its faithful services to the public for 600 years: that it was an incumbrance in passing up and down the river, and clumsy in its construction, were owing to the taste of the times. Perhaps few will be found that would be willing to infuse an iron bridge against the ruin occasioned by the weather for the same time, or perhaps much above one-half that time. But this is not a fit place to enter into the full discussion of this subject. To return to the weight prefiguring upon a center-frame. Having now taken a view of the weight to be supported, it comes next to be considered what strength of wood is necessary to resist this force, and the most proper and commodious manner of combining the parts. To determine this, we must have recourse to such experiments as have been made for trying the strength of different species of stone and wood.
Experiments have been made to ascertain the strength of timber, and many of them appear to have been conducted with great care and attention. Some of these the reader will find collected and detailed under the article Strength of Materials. We shall here state the result of some of the curious experiments which were instituted by the count de Buffon to ascertain this point. According to these experiments, the batten of five inches square, whose length was 14 feet, and which supported a weight of 5300, which may be called its breaking force, should have double the strength of a batten of 28 feet long. But it has a great deal more. The latter by the experiment is equal to 1775 only; whereas the half of 5300 is 2650. But it is to be considered, that the power of the lever is in proportion to its distance from the fulcrum; this power arising from the weight of the log, is the weight of one foot of wood acting as a weight at a distance from the fulcrum. The log increases in its power to break by its length: 12 inches of this log, five inches square, weighs about 10.4 lb. somewhat more or less; and 10.4 lb. at 13 feet distance, acts with a force of 135.2 lb.; this we consider the last term; and 2, the point of fracture, is the first term: the first and last term, multiplied by half the number of terms, are equal to the sum of all the terms; that is, $135.2 \times 6\frac{1}{2}$, amount 878.8 lb. added to 1775, equal 2653.8; so near to the half, that the difference may easily be accounted for, from the real weight of the wood on which the experiment was made; and our taking the weight from tables of specific gravity, or the supposed 62 lb. To take another example, a batten of nine feet is double the strength of one of the same size of 18 feet long. The weight that breaks a batten of nine feet, five inches square, is 830 lb.; the half is 415; but by the experiment, 3700 lb. break the batten at 18 feet. N.B. The weight being laid upon the middle, $\frac{9}{2}$ is the number of terms, one-half is 4.625. Seventeen feet one-half is $\frac{17}{2}$; 10.4 lb. multiplied by 8$\frac{1}{2}$, is 102 $\times$ 4.625, half the number of terms, is 471.25 + 3700, is 1171.25, somewhat greater, but which is so near, that the smallest accident for failure, not discernible in the wood, will occasion the difference. Now to reduce the experiment of this given size to any other of greater dimensions; suppose one foot: similar solids of the same altitude are to one another as their bases; that is, 25, the base of the five inch square, is to 144, the base of the 12 inch square, as the weight that would break the batten of nine feet, to the weight that will break another of the same nine feet length, and of one foot square (5.6. El. 12.), that is, as the base 25 is to the weight 8308, so is 144 to 47854 lb.; equal 213.8125 ton, and the proportion as above, for greater or less length of logs or spars. As we have no experiments made of logs of 12 inches square; unless there is something in the texture of the fibres, in pieces of different diameters, we have every reason to conclude, the above proportion will give the proper strength of the material used. It must, however, not be forgot, that the pieces upon which the experiments were made, were nicely chosen for the purpose. It will scarcely be practicable to find a piece of 12 inches square, and even of nine feet length, equally well adapted to bear a proportionate strain; and much more difficult to find a piece of still greater length. These experiments and proportions afford a safe criterion for proper limits to be attended to in practice. In this, we do not mean to apply such a load upon the beam as will break it; we intend the beam to support the load, without giving way or yielding to it.
In the same experiments, we are told by the author, that two-thirds of the weight broke the beam in the space of two months; that one-half the weight gave a set or bend which it did not recover, but showed no farther tendency to break; that one-third of the weight, after long continuance, did not give it a set; but the weight being removed, the beam returned to the same position as before it was loaded. Betwixt one-third and the half of load or weight that would break the beam, is the strength we allot to it for permanent use. Before we proceed to put the above observations into full practice, let us examine whether the log is necessary to be square to give it the greatest strength; practice, in a great measure, determines that it is not. It is, however, necessary to inquire what breadth to a given depth is sufficient as a maximum that we ought not to exceed; or what is the minimum that we may use, so as not to lose the principal intended effect. Belidor has made a series of experiments on the transverse strength of bodies, which are detailed in his Science des Ingenieurs, but the spars are only of one inch, not exceeding two inches in breadth or thickness. Among these, we select one spar two inches breadth, one inch depth, and 18 inches length; which at the medium of three trials was broken, lying loose at both ends, by 80 lb. Another one inch board, two inches deep, and 18 inches long, broke with the force of 1580 lb.; nearly in the proportion of the square of the depth, being only a diminution of 20 lb. weight. In the present case, the quantity of matter is the same in both.
It may therefore be concluded from this experiment, that a batten of any depth, and one-half breadth, is equally strong in that position as if it had been square timber; and that the strength is according to the depth, if the breadth is only such as that it does not yield in that direction. And hence the advantage in point of economy; for if the piece is set upon its edge, suppose nine inches deep and one broad, provided that by straining the piece in depth, it shall not yield in the lateral direction, it will bear as much strain as if nine inches square. The experiment may be performed upon a small scale. Suppose five inches, and one inch broad, the thin section may be enclosed at different distances with pieces five inches square. Suppose at the distances of 1, 2, 3, &c., fig. 1. Plate CXXXVIII. and the weight applied that broke the five inch square of the length of 14 feet, viz. 5300 lb. All All the experiments which have been alluded to above were made upon scantling of sound oak. But it has already been observed, that in practice, such pieces cannot always, if at all, be selected. But the practical mechanic, confining himself to between one-third and one-half of the absolute strength, according as his judgement directs him, respecting the soundness of the piece he uses; there can be no doubt, that, upon occasions, he will be convinced, that he cannot, with safety, allow even one-third of the absolute strength, but must take it considerably below that proportion.
As to other species of wood, trials have also been made; and the result from different experiments has occasioned some deviation. We are told that Buffon makes fir about \( \frac{5}{6} \)ths of the strength of oak, Parent \( \frac{1}{2} \)ths, and Emerson \( \frac{3}{4} \)ds; all of them different. The difference between Buffon and Parent is \( \frac{1}{12} \)th; between Parent and Emerson is \( \frac{1}{8} \)th; and between Buffon and Emerson is \( \frac{1}{4} \)th. It is easy to conceive that the different states of the wood, and different circumstances in the same species of fir and oak, will make a considerable difference; although the same persons were employed on the same materials, the experiments would probably vary; much more, may it be allowed that at different times different states of the wood must make the results different.
The experiments made by different persons vary in their amount. Belidor's experiments agree one part with another, and so do Buffon's, but differ in their results from Belidor's. Belidor's slips of oak are only of one inch square, and Buffon's are from four to eight inches square, and from 7 to 28 feet in length. When the one is reduced to the standard of the other, they do not agree: the difference may arise from various causes. We know that there is a difference in the strength of oak of different growths, and from different soils, as well as in other species of wood; there is likewise a difference in the degree of seasoning of the wood. Buffon gives the weight of his wood, Belidor does not. If Buffon's log or batten, four inches square, weighs about 60lb. that is, about 77lb. the solid foot; whereas a solid foot of dry oak will not weigh above 60lb.; but Buffon acknowledges that his wood was in the sap, as vapours issued at both ends in the bending. These differences may make all the odds in the breaking, unless the proportion was established to be, as the squares of the diameter of the battens; but this is not the case, for in Buffon's experiments, the square of four, to the square of five of the seven feet batten, the breaking force is 830clb.; but the experiment gives it 11525; that of six inches square 16:36::5312.11952; exp. 18950. In the seven inch square 16:49:5312.16268; exp. 32200. In the eight inch square 16:64:5312:21248; exp. 47619, the difference between the four and five inch square is one-third part of the experiment weight; the difference between the four and six, is somewhat more than one-third the experiment weight; and in the seventh, the difference is a little less than half the experiment weight; between the seventh and the eighth the difference is greater than half the experiment weight.
There is likewise a difference at the different lengths; for it does not appear that the different lengths bear a proportion to their parts; a batten of four inches square of seven feet length, is expected to be double the strength of one of the same dimensions of 14 feet length; that is, the one of 14 feet length is expected to break with one half of the weight that breaks the seven feet batten; but we find it much less; but when it is considered that the weight of the materials acting at a greater distance from the centre of motion, this must be taken into the account, and added to the weight of the breaking force. For example, the batten of five inches square and 12 inches length, weighs 13.368lb. at the rate of 77lb. per solid foot. This weight, acting upon the batten of 14 feet, taking the amount of the whole in an arithmetical ratio, is 13.368×52\(\frac{1}{2}\)=701.5lb. acting upon the whole, added to 5300, the breaking force 6001. The breaking force, at seven feet, is 11525; one half is 5762.25, one twenty-fourth part greater than the half. The batten of five inches square, the breaking force at 14 feet is 7475, the weight of 12 inches of this batten is 19.25lb. at 77lb. per solid foot; the acting force of this weight at 14 feet length is 19.25×52\(\frac{1}{2}\)=1010.625, added to 7475, equal to 8485625. Now the breaking force of seven feet length is 18950; one half is 9475, the difference is 89, that is, nine and a half times less than the half. In the seven inch batten of seven feet length, the breaking force is 32,200lb. and of 14 feet length, the breaking force is 13,225. The weight of 12 inches of the seven inch square is 2602lb. acting upon the 14 feet length, is 1370.5+13225=14600lb. which is one-ninth less than the half. Again, 12 inches of the eight inch batten weighs 34.2lb. at 77lb. per solid foot, acting upon the 14 feet length, is 17961lb. added to 19775, the force that broke it at 14 feet length, is 21757lb. about one-tenth part less than the half of 47,649lb. which broke it at seven feet length. From the above comparison, it may be allowed, that the difference of the force that broke the spar at seven feet, and that which broke it at 14, so far as it differs from the half, is accounted for upon philosophical principles; and when we consider that the spars or battens cannot be supposed to be mathematically exact in their measure, and that a difference in point of breaking, may be accounted for from that cause; but further, it may be observed that the weight of the materials is not equal in the solid foot. For example, the spar four inches square, and several feet in length, weighs 6clb.; that is, at the rate of 77.14lb. per solid or cubic foot, the eight feet spar at the rate of 76.1clb. do.; the nine feet spar at the rate of 77 feet; the 10 feet spar at the rate of 75.6; the 12 feet spar at the rate of 75lb. per cubic foot; which difference of weight, with the difference of exact mathematical measure, may fully account for all the difference that takes place in the manner of accounting for the above-mentioned difference of the weights of breaking at 7 and 14 feet; as also the difference that takes place between 8 and 16; 9 and 18, &c. The experiments being made upon green wood, cannot be approved of; they ought to have been made of such seasoned wood as is fitted for mechanical purposes, of which none of this kind can be used; or if experiments are made with unseasoned wood, as being of the greatest strength, they ought likewise to have been made with dry wood seasoned for use. A cubic foot of dry oak, oak will not weigh much above 60 lb. Those spars upon which the experiments were made, must have been very green, and very unfit for mechanical purposes, which gives an unfair account of the strength, when in a proper state for use. But experiments were made with wood of different weights, which may be supposed better seasoned. For example, the seven feet spar that weighs 56 lb., that is, 72 lb. per cubic foot; the nine feet spar is at 71 lb. per solid foot, and the ten feet spar at 73.8 lb. per solid foot, none of which are seasoned wood. And yet it is not mentioned which of these were used. This may be adduced as a very good reason why the variations were so great.
We shall now consider the force in bruising materials, according as we may be directed by experiments made in this way. And if, upon that of stone, which will in some measure lead to the prelude in the same direction upon other materials.
The experiments selected from M. Gauthey, engineer, in erecting the bridge of Chalons sur Saone, (tom. iv. Rozier Journal de Physique, November 1774), are now to be considered.
**Experiments Selected.**
| Length of the Stone | Breadth | Superficies | Force | Upon each square line | Proportion | Difference | |---------------------|---------|-------------|-------|----------------------|------------|------------| | Hard Stone | | | | | | | | 8 | 8 | 64 | 46 | 10½ | 12 | 2½ | | 8 | 12 | 96 | 164 | 27 | 24 | 1¾ | | 8 | 16 | 128 | 281 | 35½ | 36 | 1¼ | | Soft Stone | | | | | | | | 9 | 16 | 144 | 35 | 3½ | 4 | 1½ | | 9 | 18 | 162 | 53 | 5 | 4½ | 1¾ | | 18 | 18 | 324 | 183 | 8½ | 9 | 1¾ | | 18 | 24 | 432 | 131 | 12½ | 12 | 2¾ |
In general, the force is greater as the surfaces increase, but a regular proportion to fix upon a theory is not found; but the last line in the table, the weight that crushes the 432.3 surface, must be greater than 131; the stone being of the same quality: if in the proportion of \( \frac{8}{3} \) to \( \frac{12}{7} \), the crushing weight will be 272.7 instead of 131.
The measures here taken are cubic, and the preluding force is upon cubic lines, the thickness one line; where the prelude is upon a square foot, it is likewise to be understood one foot deep, or upon a cubic foot; the stone used, he terms Givry stone, of which he gives its absolute force to be 870,911, that it will bear 663,522 lb. In the cubic foot of soft stone the strength is 248,822 lb. The proportional force of the hard and soft is \( \frac{2}{3} \) to 1.
A cubic foot of a stone fixed in a wall, and projecting one foot, was broken by a force of 557,281 lb. And a cubic foot of soft, by 100,801 lb., the proportion \( \frac{5}{7} \) to 1.
A cubic foot of hard stone, supported upon two fulcrums at 1 foot distance, was broken by 205,632 lb. suspended from its middle; and the soft by 38,592, the proportion about \( \frac{5}{3} \) to 1.
In fine, a cubic foot of the hard stone was torn asunder by 45,500 lb.; and the soft by 15,850 lb., the proportion \( \frac{2}{3} \) to 1. Thus far Gauthey's account.
It is to be observed, that the above table does not strictly correspond with itself; for the proportion upon the square line, or \( \frac{1}{2} \) of an inch, in place of \( \frac{1}{3} \) is upwards of 11. Now the increase of force which crushes 96 square lines, and 128 one line thick, is 7.8 oz. nearly upon the square line, that is a little more than \( \frac{1}{8} \) of 35 oz. upon the square line; then as 128 square lines is to 4496 oz. so is 144 square lines to 5038, to which add one-fifth, viz. 1011\( \frac{1}{2} \), this makes 6069\( \frac{1}{2} \) upon the square inch, and this multiplied by 144, the square inches in a foot, is 874,022.4 oz. but Mr. Gauthey says, that the square foot of surface of one foot deep, is of the strength of 870,911 lb.
Again, there are 20,736 square lines in a square inch, the force upon a surface of 64 square inches, being about 11.5 upon each square line, is 238,464 oz. upon the square foot. Upon the surface of 96 lines, 27 oz. to the square line, gives 559,872 to the square foot. Upon the surface of 128 lines, 35\( \frac{1}{2} \) to the square line, is 878,806 to the square foot, the proportion of 238,464 oz. to 870,911 is about 53\( \frac{1}{2} \) nearly, and of 559,872 to 870,911 is \( \frac{5}{7} \) nearly; but by the experiment the number 870,911 is lbs. upon the square foot; the other numbers are only ozs. The variation between the first difference, and between the preluding force of 6069\( \frac{1}{2} \) oz. upon the square inch, makes in that proportion 874,022 oz. The increase of force from one square inch, to one square foot, must be \( \frac{1}{7} \) part of what the above experiment upon the square foot produces. Further experiments upon this therefore become necessary. In the mean time, we have no reason to doubt the experiment upon the square foot, or upon the smaller parts; intermediate experiments only can make them accord.
One example adduced is of consequence. A pillar in the church of All Saints, in Angers, of 24 feet height, and 11 inches square, supports a weight of 60,000 lb. that is \( \frac{1}{2} \) being added 85,685.9 upon the square foot, which is said not to be \( \frac{1}{4} \) part of the load that would crush it. From this it is evident, that the load it supports exceeds the weight of an arch of 50 feet span, of a semicircular form; the arch-stones being 2\( \frac{1}{2} \) feet long, or depth of the arch, and 2 feet in breadth. It is asserted under the article BRIDGE, that instead of an arch \( \frac{1}{2} \) of the opening or 10 feet thick, that a pier of 2 feet thick would be sufficient, but that it is given twice the length of the arch-stone, that is 5 feet thick in place of 103; but from this example, it is five times thicker than necessary, and has therefore superabundant strength, allowing even for the force of a current. How superfluous then will these clumsy piers be reckoned, whose sole effect is a useless obstruction to the water! But as our principal design at present is upon the strength of wood, in prosecution of this inquiry, we have paid particular attention to the strength of this material, in the transverse direction, in so far as it can be supported by experiment. Before we proceed to make particular application to its use, it will be necessary to consider its strength or power of resistance in its breadth and thickness. In this it may be with safety averred that such force will be able to crush its fibres, although only of $\frac{1}{10}$ of an inch; the same weight continued will produce the same effect upon the next stratum, till the whole piece is bruised, and its cohesive power overcome. This is supported by the experiments of celebrated mechanicians, as those of Buffon, Mutchtenbroek, Bouguer. Mutchtenbroek, in his Essai de Physique, says, that a piece of found oak, $\frac{1}{4}$ of an inch is torn asunder by 150 lb.; and that a plank 12 inches broad, and 1 thick, will just bear 189,168 lbs. These give for the cohesion of an inch 157,555, and 157,763 lbs. Bouguer in his Traité de Naüre says, that it is very well known that a rod of found oak, of $\frac{1}{4}$ inch square, can be torn asunder by 1000 lb.; this gives 16,000 for the square inch. Bouguer speaks with certainty, that $\frac{1}{4}$ inch square of found oak can be torn asunder by 1000 lb. If we reduce the above proportions of the experiment, it will appear, that the force will be much greater than 16,000, to tear asunder a piece of found oak of one inch square. It must in the mean time be allowed, that Buffon's experiments being upon a larger scale, can be followed with more security than those upon a smaller scale.
But, after all, we have not yet got sufficient data to form a criterion for an arch; nor can this be expected till we have more precisely ascertained the strength of an arch above a right line, parallel to the horizon.
In the first place, as an arch is in form, one part of it towards the perpendicular, and the other towards a horizontal line; the force that it will sustain, is between that force that a body will carry in the perpendicular, and that which produces a fracture upon any material in the horizontal direction. If the perpendicular is greater than the horizontal line, it will have more of the strength of the bruising force, than of the transverse fracture; and the force may be expressed by the ratio compounded of the bruising or crushing force, and that of the transverse fracture; or not improperly expressed, as it has been denominated by others, the absolute and relative force.
Unfortunately we have not yet a sufficient variety of experiments to ascertain the absolute force, as those made are only upon a small scale; and the number is not adequate to form a proportion of the increase for the force that will crush a piece of wood of $\frac{1}{4}$, or, as the French philosophers have done most this way, we take their measure $\frac{1}{4}$ of an inch, or one line, and from that to an inch; but the force required is found to be greater than that of the square of the diameter, as also the force to produce a transverse fracture, or to give the relative strength. This increases in a greater ratio than that of the square of the diameters; for in the above experiments, the weight that broke a batten 4 inches square, was to that weight which broke an 8-inch square batten, each of the length of 7 feet, more than double of the square of 4 to the square of 8 as above; we are, therefore, much limited as to an exact procedure.
At the same time, by keeping the experiments in view, and the observations made upon them, we shall be able to give such a ratio, as to the necessary strength, as will furnish the ingenious artificer with a pretty sure principle to act upon, and prevent his using superfluous materials, either in their application to horizontal right lines, or inclined in the right-lined direction, or in curves.
If we attend to the weight that crushes one inch of found oak, by Mutchtenbroek's experiments, we find that it is 17,300 lb., but, if computed from the increase, being as the squares of the diameters, it is only 16,000 lb. but it has been found as above, that the power to break, or make a transverse fracture in the same wood, of the same length, of different diameters, if a considerable difference in diameters is taken, the difference of weight is twice that produced by the square of the diameter. This comparison makes the proportion between the strength of stone, and that of wood, to be as 17,300 is to 6048, or 1 to 2$\frac{1}{2}$ nearly. Thus we may with a sufficient degree of accuracy substitute the one for the other in point of strength, and form a proportion between the arch and the strength of a horizontal line. As several experimentalists agree, that a square inch of wood can be crushed or pulled asunder with a weight of between 16,000 and 17,300 lb. and that a piece of wood one inch square, 18 inches in length, can be broken by 456 lb. or at 12 inches by 609, or at 6 inches by 1218; attending to the addition as mentioned above, which has been proved by comparison of experiments, to be upon the principle of the lever. If, then, the geometrical mean is taken between the elevation of the arch, as prelude or absolute strength, and the length of the horizontal line, this mean will be the strength of the arch above the horizontal line; for it is evident, that to much as the piece of wood is elevated towards the perpendicular, so much the nearer it approaches to its absolute strength, and by so much as the arch is flatter or the piece of wood less inclined, the nearer it is to a straight line, and so much the more reduced to its relative strength; the position of the arch, therefore, must be in the ratio compounded of these two.
Having now established the principles, let us endeavour to apply them to practice, in forming a center or supporting an arch, to produce the intended curvature or mould for an arch of any intended span, and at the same time, have strength to support the same. Several ingenious artificers have not only formed, but have written and laid down principles for forming these moulds, both with regard to strength and economy; at the same time we have not found any that have treated the subject upon principles that are fully established. We have, therefore, been the more particular, according to the principles laid down. 1st, We have assigned the weight to be supported, as established by uncontroverted principles; And, 2dly, established the strength of wood as to its thickness or diameter, that is sufficient to sustain such weight; which we have supported by the most approved experiments, comparing one with the other; and in the third place, we have considered the effects when the materials are applied in the horizontal direction, or elevated in any degree toward the perpendicular.
In a work of this kind, it is not only necessary to lay before our readers well grounded principles, and a well supported theory, but along with these, the different opinions, and various modes used by the most distinguished artists, who have exhibited their plans to the public, together with the principles on which they were founded, and the success they have met with, in answering the purposes proposed.
Among the most distinguished who have treated this subject, we may consider Pitot, a member of the Academy of Sciences, who wrote about the beginning of the last century. His method undoubtedly shows considerable ingenuity; but, at the same time, we must observe that he has been rather too profuse in the quantity of materials which he has employed.
To lay his plan of operation before our readers, we shall give a figure showing the constructions. The arch of the circle or ellipse being formed; as little or no weight lies upon the center, till between 30 and 35 degrees of the arch, a stretcher is extended at this height, to the same height on the opposite side; two struts support this stretcher from the spring of the arch; upon the upper part of the stretcher, immediately above, or a little within the upper end of the truss on each side, two spars joining upon the king-post, spring from about the middle of the arch, the stretcher being divided into four parts. Another strut springs from the rise of the arch, meeting the stretcher at this fourth part, from each side of the arch; these last struts are joined by a tie-beam, which gives additional strength to the first stretcher; upon these, on the upper side of the stretcher, two spars join the king-post, a little below the other; these spars are joined by bridles or cross spars, from the circular arch, to the lower strut; ribs of the same formation being placed at proper distances, according to the width of the bridge, and joined by bridging joints, which may be of greater or lesser strength, according to the span of the arch, and of consequence the weight it has to support. Pitot is the first writer who has given us any account of the method of forming frames, according to the above general description. If no rests are left at the spring of the arch, as a base for the center to rest upon; let AB, fig. 1, Plate CXXXVIII. be the ends of two planks raised from the foundation, upon which the center may rest; let CD be the stretcher, extended about 35 or 40 degrees from the spring of the arch; or, as little weight rests upon the center till that height, the stretcher may be as high as 45 degrees; let AE, AG, BD, BG be the two struts on each side; from each extremity of the center, let BE, AE, be fixed to the stretcher near C and D, and AG, BG, at ¼ of CD; their stretcher or tie-beam GG, equal to one-half of CD, the bridles, 1, 2, 3, &c. from A to C, and from B to D, are intended to prevent the arch from yielding from A to C, and from B to D. The struts EF, EF, meeting the king-post K in F, and the interior struts GH, GH, meeting the king-post in H, support the bridles 4, 5, 6, on each side of the king-post; their use is to stiffen the frame of the center, which supports the upper and more weighty part of the arch.
The arch for which Pitot allots this center, is of 60 feet span; and the arch stones seven feet in length, the weight of a solid cubic foot he makes 160lb. The Portland stone is admitted to weigh 160lb; but we do not find any other freestone of such weight. It is however to be considered, that the Paris foot is 12,783 of our inches; that is, a little more than 12½ths of our measure, which will make a difference of the weight upon the foot; as also their lb. is lighter than ours about 1.2 oz. by which the stone here mentioned is not better than ours. In a matter of this kind, such exactness is not necessary. As was proposed, we first consider the weight to be supported by the frame; and here it is evident from the figure that no strain lies upon the frame below C; the arch is raised, or can be raised to this height, before the frame is set; therefore the perpendicular CC determines the limits of the absolute pressure upon the frame. The triangle Ccc presses on the frame, and the triangle Cfg adds to the lateral pressure; the weight of the arch, that actually presses upon the frame, is contained between the perpendicular lines Cc, Dd; no more can press upon the center frame. The part of the arch below C will rest upon the abutment raised upon the pier; but if it is insisted that there is a pressure upon the lower part of the center frame, what can only possibly rest, or press upon it, must be contained between the parallels Cc and fg; although it will be admitted, that the arch can be raised to the height C, without the center frame; but to indulge such as say it is not advisable to do it, we will admit what lies between these parallels to press upon the frame. Now to determine the weight of these parts of the arch, the distance between the perpendiculars Cc, Dd is 53 feet; the arch-stone is 7 feet, and admit it to be three feet broad, 53 x 7 x 3 x 160lb = 178,080lb.
To determine the area between the two parallels Cc, fg, the line fg perpendicular to the diameter AB, is 13½, the base is 9½, and Cf perpendicular to it is 7 feet, the area is 33½ feet; Cc the base of the triangle Cfc is 7½, and fc is 7; the area is 25, the difference is 8½. If this difference had been the excess of the triangle Cfc above the triangle Cfg, it would have been a pressure upon the frame; but as it is the reverse, the pressure is upon the abutment. This distinction is requisite to be taken notice of, that an unnecessary expense of wood and workmanship be not expended where it is unnecessary; as well as its being unworkman-like, or having an appearance of ignorance in the engineer.
Let us now inquire, what strength of materials is sufficient to support this weight. It has been laid down as a principle, that the parts of wood in an arch act upon one another by their absolute strength; but are liable to the transverse fracture; in proportion to the length of the piece, in a span of 60 feet, the length of the piece may be 7 feet without sensibly impairing its strength, in reducing it to the round; and experiment gives the relative strength of 7 feet to be 47649lb. by 8 inches square. It has been formerly illustrated from experiments, that the strength is according to the depth, with this precaution, that the breadth or thickness be such, that it is prevented from warping, the absolute strength being nearly, by last experiment mentioned, as the squares of the depth. The absolute strength to the relative force has been found nearly 60 to 1, although by some it is said to be only 42 to 1; the absolute strength of the plank 12 inches broad by one thick, is 189163 lb.; if two inches, it would be no more than 189163 lb. If it had been 8 inches square, then every 7 feet of the arch might be broken with the weight 189163 lb.; but the whole weight of the arch is only 178280 lb., that is, 1108 less weight than what that part of the frame would bear; but 7 feet is only about one-seventh part of 53; the frame is therefore of sufficient strength to support the whole weight of the arch when equally divided along its whole length. This is not the case with the center frame of an arch, as it is loaded at one place, and not at another; it is therefore apt to yield between the parts where the load is laid; that is, it may rise in the middle, and thus change the form of the arch; for the center frame is not only intended to support the arch, but likewise to preserve its true form; for this cause some struts may be necessary to prevent its putting the arch out of shape. To remedy this, where the arch begins to press upon the frame at C, draw the chord line C c, fig. 2, which acts as a tie-beam to the arch, from C at 35 degrees to c at 51 degrees, as, beyond this, if the arch frame had been permitted to alter its shape, it would begin to be reformed to it, at least the force would tend that way. At that part of the arch, where its weight begins to flatten the frame, as at z, draw the stretcher z, z, which likewise acts as a tie-beam, and gives support to the bridle i, on one side, and to 3 the bridle upon the other side, from D d; and thus the arch c d is prevented from sinking by the tie-beams e d. This will effectually prevent any warping or yielding of the frame, notwithstanding the enormous load from the size of the arch-stones.
But it is necessary to attend to the relative strength of different kinds of timber of which frames may be constructed. The relative proportion of the strength of oak and fir has been ascertained by different experiments; and although the results do not exactly agree, yet the mean or least proportion may be taken. Let us take 3/5, that of Buffon. Now to reduce a frame of oak to one of fir of equal strength, divide 8 inches, the diameter of the oak, by 3/5, the relative strength of fir; this gives 13 inches. Allow 13 inches. The depth of the frame will then be 93 inches by 7 or 3 inches in breadth; that is, 93 by 23 inches. In this way the strength of the fir arch is rendered equal, and by the additional allowance superior, to the oak in strength, and of less expense in wood and workmanship.
We have here taken the most simple method of investigation and computation, that every mechanic, whether scientific or not, can easily follow it in every step, and judge of the propriety or impropriety of what is advanced.
It will now be necessary to follow Mr Pitot in estimating the quantity of materials which he allows. The ring of his arches consists of pieces of oak 12 inches broad and six thick. The stretcher CD is 12 inches square, the straining piece GG is likewise 12 inches square, the lower struts 10 inches by 8; the king-post 12 inches square, the upper struts 10 by 6, the ridges 20 by 8, French measure. Pitot allows the square inch to carry 8650 lb., that is, one half of the absolute strength, which is ascertained by experiment to be 17300 lb. nearly, and not by the square of the diameter, which would be only 16000 lb. But on account of knots he reduces it to 7200 lb. per inch. He then computes the whole load upon the frame to be 707520 lb., which is the weight of the whole arch-stones, supposing each to be 3 feet broad, and the whole to press upon the frame. This comes so very near, that it would be needless to dispute about the difference. We have shown that no more than 178280 lb. presses upon the frame; but we are not so fully satisfied as to the weight that rests upon the center. Pitot supposes it to be 3/4ths of the whole weight; but he has assigned no reason for this conjecture. Mr Couplet assumes that it presses by 3/4ths. Another writer, who makes some comment upon the whole, says that 3/4ths is nearest the truth than 3/4ths, but gives no reason for his opinion, which seems to be equally vague as the other. The pressure here allowed, and the reason of assigning such a pressure, have been already explained. Our readers, therefore, have it in their power to examine the principles, and decide for themselves.
It has been asserted by some, that the arch does not press upon the center frame below C. At the same time, were we inclined to dispute this opinion, we might state our objection in the following manner: Suppose the area of the triangle C cf was equal to the area of the triangle C fg, so that the friction above would make the triangle C cf rest upon the side cf; and as the triangle C fg is greater than C cf in the proportion of 33 1/3 to 25 1/3, the cohesion of the parts will determine the intermediate space between C c and g f, to rest upon the abutment as has been said, and not on the perpendicular, unless a fissure is made in the direction g f, in which case it would be detached from the lateral pressure, and so rest upon the center. As this is not the case, any plea for a pressure below C is entirely removed; and a method to determine with precision the actual pressure upon the center frame is thrown. If the arch is the center of a circle or an ellipse, a frame so much stronger is necessary, as more of the arch presses upon the frame; but the method of determining the strength is the same as here laid down. A second figure of the ellipse and another calculation are required. It is here to be understood, that the frame calculated for is only one rib; and the weight it supports is that of the arch-stones, between the parallels C c, D d, to three feet in breadth. If, therefore, the bridge is 42 feet broad, it requires 14 ribs of the above strength. These are joined over with planks, suppose of two inches thick, and upon these the arch-stones are laid, equally carried on from C and D, and rising equally on each side, till the key-stone is set, in which state they remain, till the engineer judges it proper to slacken the frame, by striking out the wedges at the ribs, A and B, (or as the French use logs between the frame and arch), so far as to allow the arch-stones to press upon one another, by the equilibrated curvature. curvature of the arch; after which, it being found, that the arch is perfectly just and secure, the frame is entirely removed. In the frame, fig. 2, the tie-beams are not taken into the account for strength, the arch being abundantly strong without them. Their use is merely to stiffen the frame, on account of the manner in which the weight is laid on. In an elliptic arch, it has been mentioned that it is somewhat different, requiring more strength and the binding likewise different. In what are termed elliptic arches, few or none are strictly so, the true elliptic curve being difficult to form on so large a scale. It may therefore be acceptable to our readers, and also to the ingenious mechanic, if we give the form of an ellipse that will answer nearly to the elliptic equation, and upon an universal plan, easy of construction. The greater and lesser axes of the ellipse being given, divide the excess of the greater axis above the lesser into three equal parts: set off two of these from the center of the greater axis each way; upon this distance describe an equilateral triangle on each side of the greater axis, and produce the sides of the triangle both ways from the vertex of these triangles, to the extremity of the lesser axis; describe two arches till terminated by the sides of the triangle produced gives the flat part of the ellipse. At the intersection of the produced sides of the triangle as a center, with the distance of the extremity of the greater axis, describe an arch which will meet the other arch, and complete the ellipse.
Let AB, fig. 3, be the greater axis 60, and DE the lesser axis 40, be drawn at right angles, bisecting one another in C. Set off AF 40, upon AB, then the excess FB is 20, which divide into three parts; set off two of these from C to G and H; upon GH describe the equilateral triangles GHK, GHL; produce KG, KH, to any indefinite length, which may be cut by the arch drawn through D and E; from the centers KL at the intersections GH, and distance AB, let the other part of the ellipse be described; thus an universal method of describing a beautiful ellipse, and so just that it answers the elliptic equation exceedingly near, at least till it becomes very flat.
A second form of a center frame described by Pitot, is adapted to an elliptical arch. The construction differs nothing from the former, only the two upper struts are parallel; the strength as in the former is superabundant, which is easily accounted for, from not knowing the real weight that lies upon the frame, or by considering the whole weight of the arch to rest upon the frame. Both this and the former, Pitot has considered as divisible into three pieces, which renders it more manageable in erecting, particularly in large spans. See fig. 4.
Fontana has given a description of a very neat frame consisting of two pieces, the upper and the lower. The struts 1 2, 1 2 taken from fig. 4, leave a representation of Fontana's frame. Different constructions being laid before our readers, the ingenious artist may improve the hints that have been thrown out; and thus form a more simple, or better construction.
We shall now select draughts of the most approved center frames that we are able to collect; and make such remarks upon them as may occur. Fig. 5, exhibits a form, which the experienced engineer will readily allow to be neat and ingenious; but there is much more wood and work expended than is necessary. It is divided into two parts, the base or stretcher LL, of the upper part, resting upon the lower part of the frame, the greatest part of which at least must appear quite superfluous. The lower rests, EF, appear only necessary to prevent the stretcher LL from yielding, and thereby allowing the arch to lose its true curvature.
The general maxim of construction adopted by Perronet, a celebrated French architect, is to make the trusses consist of several courses of separate trusses, independent, as he supposes, of each other, and thus to employ the united support of them all. Each truss spans over the whole distance of the piers. It consists of a number of struts, set end to end, so as to form a polygon. By this construction, the angles of the ultimate trusses lie in lines pointing towards the center of the curve. It is the invention of Perrault, a physician and architect, and was practised by Mandlard de Sagonne at the great bridge of Moulins.
In the centering of the bridge of Cravant, fig. 6, the arches are elliptic. The longer axis or span is 60 feet, the semi-transverse axis or rise 20 feet. The arch-stones weigh 176 lb. per foot, and are four feet in length, which is the thickness of the arch. The truss beams were from 15 to 18 feet long, and 9 inches deep by 8 broad. The whole frame was constructed of oak. The distance between the trusses, which were five in number, 5½ feet. The whole weight of the arch amounted to 13,500,000 lb., which is nearly equal to 600 tons, making 112 tons for the weight on each truss. Ninety tons of this must be allowed really to press the trusses; but a great part of the pressure is sustained by the four beams which make the feet of the trusses, joined in pairs on each side. The diagonal of the parallelogram of forces drawn for these beams is to one of the sides as 365 to 285. Then 365:285::90:17½ tons the weight on each foot. The section of each is 144 inches. Three tons may be laid with perfect safety on every inch; and the amount of this is 432 tons, which is six times more than the real pressure on the foot-beams in their longitudinal direction. The absolute strength of each foot-beam is equal to 216 tons. But being more advantageously placed, the diagonal of the parallelogram of forces which corresponds to its position is to the side as 438 to 285. This is equal to 38½ tons for the strain on each foot; which is not much above one-fourth of the pressure it is able to bear. This kind of centering, therefore, undoubtedly possesses the advantage of superabundant strength. The upper row of struts is quite sufficient; nothing is wanted but to procure trestles for it.
In his executing the bridge at Neuilly, fig. 7, of 120 feet span, and only 30 feet rise; the arch 5 feet thick; his strut-beams are 17 by 14 inches of size, and king-post 15 by nine, the strut-beams placed in three parallel polygons, each abutting upon the king-post, he uses the binders or bridges of 9 inches square. This arch is remarkable for its flatness. The account Perronet gives of his success with this frame, and the effects it produced in his work, are as follows. Notwithstanding the different improvements he had made upon his center frame, he here found that it sunk 13 inches, before the key-stones were set, and that the crown rose and sunk as the different courses were laid. At 20 courses on each side, with a load of 16 tons upon the crown, it sunk an inch; when 20 more courses were laid, it sunk half an inch more, and continued sinking as the work advanced. When the key-stone was set, it had sunk 13 inches; and, as it sunk at the crown, and in the advance to the crown, it rose at the haunches, so as to open the upper parts of the joints almost an inch; which gradually lessened towards the crown, and of consequence the joints in the lower part opened as the upper part was compressed. This no doubt showed a suppleness in the frame, and at the same time inattention in the architect, to load the crown when he perceived it sinking with having already too much weight upon it. If he had observed the crown to rise, it would then have been proper to give it additional weight.
Let us now attend to the description of the centre frame of the bridge of Orleans, fig. 8. The architect to this bridge was Hupeau; and it is universally allowed to be an elegant structure. The arch stones are six feet in length, the form is elliptical, the span 100 feet, and rise 30. Hupeau died before any of the arches were complete. The center-frame had been placed, and some rows of the arch laid. Upon his demise, Perronc succeeded as architect, and finished the bridge. As the work advanced, he found that the crown of the center rose; he then found it sink as remarkably, which showed that there was some defect: he inserted the long beam AB, on each side; he then found the frame sufficiently stiff; for this made a change in the nature of the strut.
Having taken a view of the practice of the French architects, as to their form and effects, let us direct our attention to those of our own country, which are well worthy of notice. We shall only name some that have used trusses, and among these we find the center-frame of Blackfriars bridge, fig. 9. The span here is 100 feet; the form is elliptical, the arch-stones from the haunches seven feet, near the key-stone not quite so much, as they decrease in length from the haunch to the key-stone.
A particular description of this arch is not necessary; a view of the figure will show the use of the different parts; it may be sufficient to observe, that when the arch-stone was placed, it had changed its shape only one inch, and when the frame was taken out, the arch remained firm without any sinking of consequence. The great arch did not sink above one inch, and none of them above an inch and a half; whereas those already mentioned sunk by the suppleness of the frame 13 inches, and some of them 9 inches more when the frame was removed.
Different methods are employed for raising the frame, or disengaging it from its weight. We shall give a short description of Mr Mylne's method of placing and disengaging his center-frame from the mason-work. Each end of the truss was mortised into a plank of oak cut in the lower part as in the figure; a similar piece of oak was placed to receive the upper part of the polls. The blocks rested upon these polls, but were not mortised into them, pieces of wood being interposed. The upper part of this block was cut similar to the lower part of the other; the wedge E, being intended to be driven betwixt them, was notched as in the figure, and filled up with small pieces of wood, to prevent the wedge from sliding back by the weight of the arch; which, it will appear from the figure, would have been the case: the event proved the fact. When the centre was to be struck, the inserted pieces of wood were taken out, and the wedge, which was prepared for driving back by being girt with a ferule round the top, was removed by a piece of iron driven in with the head so broad as to cover the whole of the wood. A plank of wood was prepared armed with iron in the same manner at the one end, and suspended so that it could freely act in driving back the wedge to any distance, however small, with certainty. Thus, by an equal gradation, the centre was eased from the arch, which appeared to have been so equally supported throughout the whole of the operation, and the arch stones so properly laid, that it did not sink above one inch; and thus it was evident that the centre might be entirely removed, having completely answered the purpose.
The above examples may be considered as sufficient to show the effects of the trussed arches, which have been employed by the French architects. We shall now take the liberty of suggesting some hints which may tend to improve the construction, and remedy the faults and failures that have occurred in practice.
Trussed arches for center frames being found expedient in navigable rivers, and almost in every river which is apt to be raised by rains, or other rise in the river, the frame is apt to be endangered or carried off, to the great risk of bringing down the arch, and ruining the work before it is finished. In arches where there is no such danger, the frame may be properly secured by polls from below, which are made to abut upon these parts of the arch where the greatest strain must fall.
In the centre used by Pitot we have only to complain of an unnecessary expenditure of wood and workmanship. We have already shown what strength of oak is necessary, and have reduced that strength of oak to an equal strength of fir-wood for the ring of his frame, which alone ought to have the strength required to be fully adequate to the load; but as this weight must be gradually applied, the frame must likewise have such a degree of firmness as to form the exact mould of the arch that is intended. And, for this purpose, it must be prevented from yielding in any part of its arch. Now, as it has been made to appear, that the frame supports no part of the arch till it rise from the spring to about 35 degrees, if a semicircle, and so in proportion for a segment of a circle; in an ellipse, to a part similar according to the nature of that curve; the supporting struts and ties can be more particularly directed to support that part of the arch which produces the greatest strain upon the center. In fig. 2, where the necessary strength for Pitot's arch is pointed out, the frame of fir requisite to stiffen the frame, is 9½ by 2½. The tie-beam C c is joined to those parts of the arch, where the strain being greatest, would tend most to raise it in the crown. The strength of this tie-beam being 9½ by 2½, and its length 25 feet, would require a weight of 30495 lb. to make the trans-
R r 2 verse fracture; one-third of this at the bridle 1, 3, is sufficient to resist the strain at that part of the arch; and the abutment, being according to the principles laid down under the article Bridge, prevents the possibility of its rising at the haunches; but if not formed according to these principles, the two tie-beams Cc Dd are joined by a third tie-beam 2, 2 with its bridle 3, 4. Fig. 4 is Pitot's centering for his elliptic arch: the strength of fig. 2 may suffice to this by giving the ring and tie-beams ½ an inch more depth.
Fig. 6 represents two centerings used by Perronet; A is that used by him in erecting the bridge at Nogent, and B that at Maxence; they differ little from one another. That at Nogent is 90 feet by 28 feet height. The span of the latter being greater, we shall here consider the weight to be supported. This is the arch from A to C, which is an arch of 47° 45'. The measurement is 42 feet; the arch-stones 4½, and supposing them 3 feet broad, they would amount to 567.9 solid feet, which, at 160lb. per foot, is equal to 90866.88lb. This is little more than one-half of the semicircular arch; and, although it is flatter, the weight is so much less, that no additional strength is necessary to be given to the frame, fig. 2, for the 62 feet span. There is likewise abundance of strength of materials for the 90 feet arch; but on the greater extent, that it may be rendered more stiff, a tie-beam 1, 4, 3, 4 may be added on each side of the arch, as represented by the dotted line.
It is scarcely necessary to make any farther calculations on the centering used by Perronet. It appears, that notwithstanding the superabundance of wood employed, they were too supple as when fitted upon an extended arch, they rose and sunk so much, that the arch was changed from its intended form by a radius of several feet. These changes took place in erecting the bridges at Nogent and Maxence, which are represented in fig. 6. Perronet, it would appear, was not satisfied with these; and, convinced of their insufficiency, changed the form of the frame of the bridge at Neuilly. But this form is far from answering the purpose; for, when the arch-stones began to press upon the centering, it yielded to the weight. He then loaded the crown to prevent its rising there, but still sunk; he added more weight to the crown, it continued sinking as the work advanced. When the key-stone was set, it had sunk more than 13 inches, and it was found to have raised the haunches; for when the centering was slackened, the arch still sunk for about 9 inches more. The arch-stones being raised at the haunches, the joints were of necessity opened; for the pressure from the crown, when the centering was removed, forced them again into contact, by which the arch flattened to such a degree, that from an arch intended to have a radius of 150 feet, it flattened till part of it was as if formed from a radius of 244 feet. It here appeared to be settled, from which a considerable deformity must appear in the structure; which deformity took its rise from two evident causes: the want of firmness in the centering, and the bridge not properly loaded at the haunches. It is evident, that if the load at the haunches is only equal to the weight of the arch-stones from the place where they begin to rest on the centering to the crown of the arch, the pressure of the arch could never overcome itself or its equal weight upon the haunches; much more, if the weight upon the haunches, before it comes to press upon the centering, was made to exceed that part of the arch that did press upon it, the load upon the crown of the arch would have restored the figure of the centering. It seems to be a strange oversight, that Perronet, when he saw that his centering was rising at the haunches, did not apply his loading to this part of the arch, by which he might have restored it to its equilibrium before his center was struck, and before his lime had lost the band; if this is once done, it is allowed that it does not again recover it.
From the whole of this it appears evident, that filling up the haunches to a proper height, so as to make a firm abutment to the pressing part of the arch, serves two good purposes. It acts as an abutment to the center frame, in preventing its sinking by the load as the work advances; and likewise prevents the arch-stones at the haunches being raised from their beds; for it is only acted upon by a force considerably less than what they have a power to resist. Having now seen the defects of this centering, and animadverted on the manner of executing the work, let us now examine the weight of this arch, and what resistance would have prevented its change in shape, and preserved its intended form.
The part of the arch that presses upon the center, is from C to C, fig. 10, an arch of 36 degrees, and measures 94½ feet nearly; the stones 5 feet in length, and breadth 3, make 1979.035 solid feet, × 160lb. The weight of a solid foot, make the whole weight 316645.88lb. Allow each beam of the truss to be 7 feet, and its absolute strength, to tear it at 12 inches deep, by one inch thick, 189163; the absolute power of transverse fracture, 95416lb. The strength of the arch is the mean of these, or ratio compounded; taking one-third of each, the geometrical mean is 44285lb.; that each 7 feet can sustain when formed into an arch; there are 13 times 7 in 94 feet, equal to a power of 382764, to sustain the weight of 316645.88 lb equally distributed. But this not being the case, a tie-beam of about 30 feet marked cc, dd, will prevent the arch yielding to the pressure. It is supported at e by the struts Ee, Hh; and these by the joint support of ef, hf tied at k. The whole center frame is supported by the upright posts CC, DD. Two wedges A and B are placed across between two blocks which are fitted for a rest to the frame. When it is required to be slackened, and the frame withdrawn from the arch, they allow it to rest by its own pressure. This, it must appear obvious, ought to be done when the key-stone is let before the lime has begun to be dry and solid.
The center frame of the bridge of Orleans is represented fig. 8. It has been already noticed in this undertaking, that Perronet succeeded Hupeau. As the work advanced, he found the arch and frame to sink, and trying his ordinary mode of loading the crown of the arch, he was now taught by experience to strengthen his center frame, and happily succeeded by continuing his strut. By forming the base of the triangle 1, 2, 3, on each side, his frame was rendered sufficiently stiff, and the inner part below AB, AB became superfluous. The weight that presses upon this frame is great both on account of the flatness of the arch, and the length of the arch-stone. The pressing arch is an arch of 57 degrees; it measures 88.87 feet, \( \times 6 \) the length of the arch-stone, and by 3 in width, makes 1599.66 solid feet \( \times 160 \) lb., the weight of a solid foot, gives 25594.56 lb.
The length of each plank of the truss being 7 feet, depth 12 inches by 2 inches thick, the strength is 189163 lb. The weight for every 7 feet in length of the arch, one-third of this 63054.3 lb. in 88 feet, there is 12 times 7, that is 63054.3 \( + 12 = 756,632 \) lb. to support 25594.56 lb. more than 3 times stronger, without taking into account the strength of the arch, being the mean of the splitting force and transverse section: the tie-beams, as in fig. 7, will be of abundant strength to stiffen the frame.
The next we take notice of is, the truss frame, fig. 9, used by Mr Mylne, at Blackfriars bridge, London. This is supported by ties and struts in such a manner, that no sinking took place during the mason work going on, although the arch-stones at the haunches were 7 feet, gradually lessening to the crown of the arch; and, when the frame was struck, which was done by a very ingenious method, by the wedges of the constructions as in the figure, in place of sinking 9 inches, it did not sink above 1, which may well be accounted for by the compression of the mortar; whether a smaller quantity of materials might not have answered the same purpose, such as fig. 7, we shall refer to the judicious reader, or to the ingenious artist who may have occasion to depend upon such frames for support of this work, or a tie-beam, between 1 and 3 on each side, represented by the dotted line. As there is a strain upon the frame at s, s, let these tie-beams be supported by the struts a, b, c on each side, and tied at 4, 4 as represented by the dotted line 4, 4. It does not appear that what lies between the dotted line a, 4, 4b bears any part in the support or stiffness of the frame, and therefore becomes unnecessary; nor does it appear, that the different beams used as kingposts are of so much advantage for strengthening the frame, as tie-beams would be. At the same time, those used by Mr Mylne are employed with so much judgment, that none of their effects are misapplied. This cannot be said of any of the frames used by the French architects, even of that used at the bridge of Orleans. They are not often employed by the British architects; they rather prefer a tie-beam at the spring of the arch from one side to the other. This, however, might be as judiciously applied at the height where the arch begins to rest upon the frame, especially if the shoulders are properly loaded or filled up, so as to be a counterpoise to the arch-stones, that rest upon the frame. In this case they effectually prevent the necessity of a tie-beam, as a diameter at the spring of the arch; and from the spring proper supports may be given at the upper tie-beam, and from it to any part of the arch, where the greatest strain lies.
Having from the examples adduced, and the observations made upon them, found center-frames of sufficient strength to support arches of very extensive spans, and even greater extent than they have yet been applied; it may be said, why not continue these frames for the bridge, without the very great additional expense of throwing a stone arch over them? The main would answer, that the stone was more durable, and had other advantages, particularly as to neatness, when once thrown, and freed from the uncoated trusses and tie-beams necessary in the wooden frame. The carpenter would reply, that if wood was not so durable as stone, it could be raised at much less expense; and, when it failed in any part, it could be replaced at a small expense, and made to last longer than a stone arch; which latter, when it fails, requires as much expense as at first, and even more, in clearing off the rubbish of its decayed and now useless materials. As to neatness, the frame of wood vies with the arch of stone in elegance, and is erected at half the expense, and even less. But now since iron materials are introduced in place of stone, there is room for experiments with regard to neatness and extent of span.
We shall here suppose the carpenter exhibits this plan. Let A.B be a span of 60 feet, (fig. 11,) the arch a semicircle, the absolute strength of oak a plank 12 inches by one is 189163 lb. Let the arch be composed of pieces 5 feet long, 12 inches deep, and 2 inches broad; a second arch joining to this, of the same depth and breadth in close contact, but the joints of the one to the middle of the other, like brick-building, or as the carpenters express it breaking-joint. The absolute strength of this arch is, before the two struts are joined, more than 8.4 ton, as may be collected from the calculations above, which is more than 3 times what can ever come upon it. The beauty of this arch would be hurt by placing struts below to stiffen it, for which there is not the smallest occasion; for it can be stiffened to better advantage above the arch. But this is not practicable in center-frames. Let the road-way be CDEF, resting upon the perpendicular support 1, 2, 3, &c. As the carriage acts upon these in the oblique direction, transects from the arch in a radial direction, give them the advantage of equal pressure upon the arch. Each of these perpendiculars is mortised into short pieces, that will form into an arch, the pieces all abutting one upon another, and forming a fillet over the arch, and projecting so far, that the faces of an architrave of any order may be formed along the face of the arch, which adds both to its strength and beauty. Thus there is formed a rib, 12 inches deep and 4 thick, with its fillet over it 4 inches deep and 6 inches broad, to cover the faces of the architrave. Suppose the arch 44 feet wide, 7 of these ribs may give a strength not inferior to the strength of stone or any metal; but it will be said, it will not be so durable. It is well known how long wood lasts in the roofs, and joists of flooring, and even when it forms a part of the wall of a house built of brick. The interstices between these perpendicular bearings of the wood may be built up with brick; even brick on edge, or brick thick, will render its preservation equal to what it is in a house, and will preserve it from the bad effects of wet and dry; and the lower part of the ribs covered with a thin lining. A door being left in the side to observe at different times any failure in the wood, it may be repaired without interrupting the passage by the bridge. It ought to be so covered above, that water may be prevented from going through to the injury of the bridge. It has been formerly mentioned, in speaking... ing of the proportional strength of oak and fir, and by the calculation it appeared, that fir plank $13\frac{1}{2}$ inches, is equal in strength to oak of 12 inches. And thus a framing of wood does not much exceed the expense of centering either a stone or iron bridge; and is not inferior even in elegance.
The span here proposed is only 60 feet. But an arch of 600 feet may be required, which must have a centering to support the weight and preserve the figure; the size of that center frame can be made of strength equal, and even to exceed the weight it has to support. It can be rendered stiff by the method proposed for the 60 feet arch. This, therefore, will be a bridge that will support any weight that can be laid upon it, and may be of any figure, elliptical, or at the pleasure of the architect, any other curve which may be required. It may be framed in a similar manner to those formed of iron, but it is natural to suppose that one arch over the other will be equally strong and more easily preserved from the weather, constructed in the way described above.
In the simple wooden bridges not curved, it is only necessary to refer to fig. 7, where the struts E, f, h, g, will be a support for planks, that will form a straight bridge, joining so many ribs as are necessary for the bridge according to its breadth.
The joints may be secured from opening by dovetail pieces being inserted across the joints on the inside of the rib; the abutments prevent the ends of the arch from flying out. The pressure above coming upon it obliquely, may be said to tend to make it rise at the crown, especially when of a great span. In the center-frame, the only manner of preventing this is by struts and tie-beams judiciously applied. Here the rise may be prevented more effectually without hurting the ornamental part of the arch. In the abutment, which must be of mason-work, let a beam be built into the wall, the ends at G and K projecting 1 foot, corresponding to each rib, the road-way formed by the beam D.E.; let a tie-beam G.D., K.E., join these in the manner the carpenter knows to be the most secure; from this tie-beam, let the radial struts be mortised into the fillets at G, K, formerly described, instead of the perpendiculars there named, and perpendiculars joining the road-way C.P.E.F., and resting on the tie-beam G.D., K.E., supported by the radial struts 4, 5, 6, as in the figure. Thus the crown of the arch cannot rise without lifting up the whole body of the abutment at each end, and it cannot sink till the weight laid upon it is sufficient to crush the materials of which the arch is composed. In this manner a neat and elegant arch is procured, that may at a small comparative expense be kept up for centuries. Here is then a choice of three species of arches, that may vie with each other in point of strength. With the last none may compare in point of elegance, and in duration perhaps not inferior to the iron bridge.
**CENTER of Gravity**, in Mechanics, that point about which all the parts of a body do in any situation exactly balance each other.
**CENTER of Motion**, that point which remains at rest, while all the other parts of a body move about it.
**CENTER of a Sphere**, a point in the middle, from which all lines drawn to the surface are equal.
Hermes Trismegistus defines God an intellectual sphere, whose center is everywhere, and circumference nowhere.