1575. In resolving the oblique effort NA into two forces, whereof one MA tends to overturn the pier by acting in an horizontal direction, and the other ƒA to strengthen it by acting vertically, as before observed; let us represent the horizontal effort MA by p, its arm of lever, equal to the height of the pier, by d, the vertical effort ƒA by n; the arm of lever of the last-named effort, being the thickness sought, will be r; from which we have the equation == ? +nx, or 2p=xx + =α

pd=

dxx

nn

2nx

.

2n
d'

1576. As the second member of this equation is not a perfect square, let there be added to each side the term wanting, that is, the square the half of the quantity which multiplied in the second term, whence

nn
dd

dd'

2nx 73

2p+ = x x + a + ã•

1577. The second member, by this addition, having become a square whose root is x + 7 - will be the general formula

we shall have ≈ +

dd

2p+ and lastly x=^

nn

dd

for finding the thickness .

d

Application of the Formula.

1578. Let p=103, n=8, d=12. Substituting these values in the formula, it will become x= √103 × 2+}} = √21 } + {−} = √21 + } − } = 4.

1579. If, for proof, we wish to calculate the expression of the resistance, by placing in the equation of equilibrium 2pd=dx × nx, the values of the quantities p, d, and x, above found, we shall have

103 × 12=12 × 4 x 2 + 8 = 128, as was previously found for EA.

1580. From the preceding rules, it appears that all the effects whose tendency is to destroy an edifice, arise from weight acting in an inverse ratio to the obstacles with which it meets. When heavy bodies are merely laid on one another, the result of their efforts is a simple pressure, capable of producing settlement or fracture of the parts acted upon.

1581. Foundations whose bases are spread over a much greater extent than the walls imposed upon them, are more susceptible of settlement than of crushing or fracture. But isolated points of support in the upper parts, which sometimes carry great weights on a small superficies, are susceptible both of settlement and crushing, whilst the weight they have to sustain is greater than the force of the materials whereof they are formed; which renders the knowledge of the strength of materials an object of consequence in construction. Till of late years it was not thought necessary to pay much attention to this branch of construction, because most species of stone are more than sufficiently hard for the greatest number of cases. Thus, the abundant thickness which the ancients generally gave to all the parts of their buildings, proves that with them this was not a subject of consideration; and the more remotely we go into antiquity, the more massive is the construction found to be. At last, experience taught the architect to make his buildings less heavy. Columns, which among the Egyptians were only 5 or 6 diameters high, were carried to 9 diameters by the Greeks in the Ionic and Corinthian orders. The Romans made their columns still higher, and imparted greater general lightness to their buildings. It was under the reign of Constantine, towards the end of the empire, that builders without taste carried their boldness in light construction to an extraordinary degree, as in the ancient basilica of St. Peter's at Rome and St. Paolo fuorì le murà. Later, however, churches of a different character, and of still greater lightness, were introduced by the Gothic architects.

1582. The invention and general use of domes created a very great load upon the supporting piers; and the earlier architects, fearful of the mass to be carried, gave their piers an area of base much greater than was required by the load supported, and the nature of the stone used to support it. They, moreover, in this respect, did little more than imitate one another. The piers were constructed in form and dimensions suited rather to the arrangement and decoration of the building that was designed, than to a due apportionment of the size and weight to the load to be borne; so that their difference from one another is in every respect very considerable.

The piers bearing the dome of St. Peter's at Rome are loaded with a weight of 14.964 tons for every superficial foot of their horizontal section.

The piers bearing the dome of St. Paul's at London are loaded with a weight of 17·705 tons for every superficial foot of their horizontal section.

The piers bearing the dome of the Hospital of Invalids at Paris are loaded with a weight of 13-598 tons for every superficial foot of their horizontal section.

The piers bearing the dome of the Pantheon (St. Genevieve) at Paris are loaded with a weight of 26 934 tons for every superficial foot of their horizontal section.

The columns of St. Paolo fuori le mura, near Rome, are loaded with a weight of 18.123 tons for every superficial foot of their horizontal section.

In the church of St. Méry, the piers of the tower are loaded with upwards of 27 tons to the superficial foot. With such a discrepancy, it is difficult to say, without a most perfect knowledge of the stone employed, what should be the exact weight per foot. The dome of the Hospital of the Invalids seems to exhibit a maximum of pier in relation to the weight, and that of the Pantheon at Paris a minimum. All the experiments (scanty, indeed, they are) which we can present to the reader are those given at the beginning of this section. In this country, the government has always been too much employed in considering how long it can keep itself in place, to have time to consider how the services of its members could benefit the nation by the furtherance of science. An exactly opposite conduct has always marked the French government: hence more scientific artists are always found amongst them than we can boast here, where the cost of experiments invariably comes out of the artist's pocket.

Ratio of the Points of Support in a Building to its total Superficies.

1583. In the pages immediately preceding, we have, with Rondelet for our guide, explained the principles whereon depend the stabilities of walls and points of support, with their application to different sorts of buildings. Not any point relating to construction is of more importance to the architect. Without a knowledge of it, and the mode of even generating new styles from it, he is nothing more than a pleasing draughtsman at the best, whose elevations and sections may be very captivating, but who must be content to take rank in about the same degree as the portrait painter does in comparison with him who paints history. Hereafter will be given the method of properly covering the walls, one which has occupied so much of our space; namely, when we treat of the subject of Roors, and the method of framing them. It is equally important, and of as high value to the architect, as that which we are now quitting, to which we regret our limits do not allow us to add more: but previous to leaving it, we must subjoin a table of great instruction, showing the ratio of the points of support to the total superficies covered in some of the principal buildings of Europe.

TABLE SHOWING THE RATIO OF THE WALLS AND POINTS OF SUPPORT OF THE PRINCIPAL EDIFICES OF EUROPE TO THE TOTAL AREA WHICH THEY OCCUPY.

The above table exhibits also the comparative sizes of the different buildings named in it.

Pressure of Earth against Walls.

1584. It is not our intention to pursue this branch of the practice of walling to any extent, the determination of the thickness of walls in this predicament being more useful, perhaps, to the engineer than to the architect. We shall therefore be contented with but a concise mention of it. Rondelet has (with, as we consider, great judgment) adopted the theory of Belidor, in his Science des Ingenieurs, and we shall follow him. Without the slightest disrespect to later authors, we know from our own practice that walls of Revêtement may be built, with security, of much less thickness than either the theories of Belidor, or, latterly, of modern writers require. We entirely leave out of the question the rules of Dr. Hutton in his Mathematics, as absurd and incomprehensible. The fact is, that in carrying up walls to sustain a bank of earth, nobody, in the present day, would dream of constructing them without carefully ramming down the earth, layer by layer, as the wall is carried up, so as to prevent the weight of the earth, in a triangular section, pressing upon the wall, which is the foundation of all the theory on the subject. With this qualification, therefore, we shall proceed; premising, that if the caution whereof we speak be taken, the thickness resulting from the following investigations will be much more than the outside of enough.

1585. Earth left to itself takes a slope proportionate to its consistence; but for our purpose it will sufficiently exhibit the nature of the investigation, to consider the substance pressing against the wall as dry sand or pounded freestone, which will arrange itself in a slope of about 55 with the vertical plane, and therefore of 34° with an horizontal plane, as Rondelet found to be the case when experimenting on the above materials in a box, one of whose sides was removable. Ordinarily, 45° is taken as the mean slope into which earths recently thrown up will arrange themselves.

1586. Belidor, in order to form an estimate for the thrust or pressure into which we are inquiring, divides the triangle EDF (fig. 611.) representing the mass of earth which creates the thrust, by parallels to its base ED, forming slices or sections of equal thickness and similar form; whence it follows, that, taking the first triangle a Fb as unity, the second slice will be 3, the third 5, the fourth 7, and so on in a progression whose difference is 2.

1587. Each of these sections being supposed to slide upon an inclined plane parallel to ED, so as to act upon the face FD, if we multiply them by the mean height at which they collectively act, the sum of the products will give the total effort tending to overturn the wall; but as this sum is equal to the product of the whole triangle by the height determined by a line drawn from its centre of gravity parallel to the base, this last will be the method followed, as much less complicated than that which Belidor adopts, independent of some of that author's suppositions not being rigorously

correct.

Fig. 611.

1588. The box in which the experiment was tried by Rondelet was 16 in. (French) long, 12 in. wide, and 17 in. high in the clear. As the slope which the pounded freestone took when unsupported in front formed an angle with the horizon of 3410, the height AE is 11, so that the part acting against the front, or that side of the box where would be the wall, is represented by the triangle EDF.

2

1589. To find by calculation the value of the force, and the thickness which should be given to the opposed side, we must first find the area of the triangle EDF 16x11- =93; but as the specific gravity (or equal mass) of the pounded stone is only of that of the stone or other species of wall which is to resist the effort, it will be reduced to 73 x =81. This mass being supposed to slide upon the plane ED, its effort to its weight will be as 113 AE is to ED::11: 20, or 81 x =459, which must be considered as the oblique 20 power qr passing through the centre of gravity of the mass, and acting at the extremity of the lever ik. To ascertain the length of the lever, upon whose length depends the thickness of the side which is unknown, we have the similar triangles qsr, qho, and kio, whose sides are proportional: whence qs: sr::gh: ho; and as ko-hk-ho, we have qr: rs::hkho: if.

Whence, if=

(hk-ho)xqs,

gr

The three sides of the triangle qsr are known from the position of the angle q at the centre of gravity of the great triangle EFD, whence each of the sides of the small triangle is equal to one third of those of the larger one, to which it is correspondent.

Thus the equation becomes

For easier solution, make 2pbf-2pbc

ad

ad

2pc =2m, and =2n, and we have x2+2nx=2m, an equation of the second degree, which makes x= √2m+nn-n, which is a general formula for problems of this sort.

Returning to the values of the known quantities, in which

From the above, then, the formula = √2m+nn-n becomes = √254 +5.20 −2·28 = 3.22, a result which was confirmed by the experiment, inasmuch as a facing of the thickness of 34 inches was found necessary to resist the pressure of pounded freestone. By Belidor's method, the thickness comes out 4 inches; but it has been observed that its application is not strictly correct. In the foregoing experiment, the triangular part only of the material in the box was filled with the pounded stone, the lower part being supposed of material which could not communicate pressure. But if the whole of the box had been filled with the same material, the requisite thickness would have been found to be 5 inches to bear the pressure.

1590. In applying the preceding formula to this case, we must first find the area of the trapezium BEDF (fig. 612.),

95 76 p. Having found in
the last formula that qs is re-
presented by b=6'93, sr by
c=4.76, qr by a=8·40, f=
11.3, d=175; the thickness
of the retaining wall becomes
f-c
=sh-x; m=pb x* will be-
ad
come, substituting the values

11.3-476

95.76 x 6'93 x

=

Fig. 612.

95.76 x 4.76
8:40 x 17.50

=31, and nn =

8:40 x 17:50 29-52 and 2m-59.04. 9.61. Substituting these values in the formula x=/2m+nn-n, we have 59'04 +9.61 -3.15.2, a result very confirmatory of the theory.

1591. In an experiment made on common dry earth, reduced to a powder, which took a slope of 46° 50', its specific gravity being only of that of the retaining side, it was found that the thickness necessary was 3 inches f

1592. It is common, in practice, to strengthen walls for the retention of earth with piers at certain intervals, which are called counterforts, by which the wall acquires additional

strength; but after what we have said in the beginning of this article, on the dependence that is to be placed rather on well ramming down each layer of earth at the back of the wall, supposing it to be of ordinary thickness, we do not think it necessary to enter upon any calculation relative to their employment. It is clear their use tends to diminish the requisite thickness of the wall, and we would rather recommend the student to apply himself to the knowledge of what has been done, than to trust to calculation for stability, though we think the theory ought to be known by him.

SECT. XI.

MECHANICAL CARPENTRY.

1593. The woods used for the purposes of carpentry merit our attention from their importance for the purpose of constructing solid and durable edifices. They are often employed to carry great weights, and to resist great strains. Under these circumstances, their strength and dimensions should be proportioned to the strains they have to resist. For building purposes, oak and fir are the two sorts of timber in most common use. Stone has, doubtless, the advantage over wood: it resists the changes of moisture and dryness, and is less susceptible of alteration in the mass; hence it ensures a stability which belongs not to timber. The fragility of timber is, however, less than that of stone, and its facility of transport is far greater. The greatest inconvenience attending the use of timber, is its great susceptibility of ignition. This has led, in this as in every age, to expedients for another material, and in public buildings the object may be attained. In private buildings, the cost of the substitute will not permit the employment of other than the material which is the subject of our section.

1594. Oak is one of the best woods that can be employed in carpentry. It has all the requisite properties; such as size, strength, and stiffness. Oaks are to be found capable of furnishing pieces 60 to 80 ft. long, and 2 ft. square. In common practice, beams rarely exceed 36 to 40 ft. in length, by 2 ft. square.

1595. In regard to its durability, oak is preferable to all other trees that furnish equal lengths and scantlings: it is heavier, better resists the action of the air upon it, as well as that of moisture and immersion in the earth. It is a saying relating to the oak, that it grows for a century, lasts perfect for a century, and takes a century to perish. When cut at a proper season, used dry, and protected from the weather, it lasts from 500 to 600 years. Oak, like other trees, varies in weight, durability, strength, and density, according to the soil in which it grows. The last is always in an inverse proportion to the slowness of its growth; trees which grow slowest being invariably the hardest and the heaviest.

1596. From the experiments made upon oak and other sorts of wood, it is found that their strength is proportional to their density and weight; that of two pieces of the same species of wood, of the same dimensions, the heavier is usually the stronger.

1597. The weight of wood will vary in the same tree; usually the heaviest portions are the lower ones, from which upwards a diminution of weight is found to occur. In fullgrown trees, however, this difference does not occur. The oak of France is heavier than that of England; the specific gravity of the former varying from 1000 to 1054, whilst the latter, in the experiments of Barlow, varies from 770 to 920. The weight, therefore, of an English cube foot of French oak is about 58 English pounds. Timber may be said to be well seasoned when it has lost about a sixth part of its weight.

1598. In carpentry, timber acts with an absolute and with a relative strength. For instance, that called the absolute strength is measured by the effort that must be exerted to break a piece of wood by pulling it in the direction of the fibres. The relative strength of a piece of wood depends upon its position. Thus a piece of wood placed horizontally on two points of support at its extremities, is easier broken, and with a less effort, than if it was inclined or upright. It is found that a smaller effort is necessary to break the piece as it increases in length, and that this effort does not decrease strictly in the inverse ratio of the length, when the thicknesses are equal. For instance, a piece 8 ft. long, and 6 in. square, placed horizontally, bears a little more than double of another, of the same depth and thickness, 16 ft. long, placed in the same way. In respect of the absolute force, the difference does not vary in the same way with respect to the length. The following are experiments by Rondelet, to ascertain the absolute force, the specimen of oak being of 861 specific gravity, and a cube foot, therefore, weighing 49 lbs.