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=d, bc+cr

We have b: c::b+x: =ho, and hk - ko wil. be f-betes.

To obtain ik, we have the proportiona: b::f-bec: ik.

Whence ik-bf-bc-cz; so that the pressure p x ik is represented by p ( dr2 must equilibrate.

the resistance expressed by 2

C), to which

Thus the equation becomes d-pbf-be-cr), or x2 + 2pcr-2p(bf—bc).

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For easier solution, make 2pbf-2pbc-2m, and 2pc = 2n, and we have 2+2nx=2m, an

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equation of the second degree, which makes x = √2m+n-n, which is a general formula for problems of this sort.

Returning to the values of the known quantities, in which

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From the above, then, the formula r=2m+n2 -n becomes = √25 4+ 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.), which will be found 1954; multiplying this by 13, to reduce the retaining wall and the material to the same specific gravity, we have 169. This mass being supposed to slide upon the inclined plane ED, its effort parallel to that

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113

plane will be 195 × 0 95-76-p. Having found in the last formula that qs is represented by b=6.93, sr by e=476, qr by a=8.40, f= 113, d: 17-5; the thickness of the retaining wall becomes =sh-x; m=1 m=pb x=c will be

come, substituting the values

95.76 x 6'93 ×

11-3-4-76

pc

Fig. 612.

8-40x 17-50-29-52 and 2m=59 04. n= becomes
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95.76 x 4.76

8:40 x 17-50-31, and n

9-61. Substituting these values in the formula x = √2m+n2 -n, we have x =
x=
-31-5-2, a result very confirmatory of the theory.

P

✓59.04 +9.61

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 him. self 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.

PRESSURE OR FORCE OF WIND AGAINST WALLS, &C.

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1592a. Air rushes into a void with the velocity a heavy body would acquire by falling in a homogeneous atmosphere. Air is 840 times lighter than water. The atmosphere supports water at 33 ft.; homogeneous atmosphere, therefore, is 33 × 840=27,720 ft. A heavy body falling one foot acquires a velocity of eight feet per second. Velocities are as the square roots of their heights. Therefore to find the velocity corresponding to any given height, expressed in feet per second, multiply the square root of the height in feet by 8. For air we have V√27,720 166,493 × 8=1332 feet per second: this, therefore, is the velocity with which common air would rush into a void: or 79,920 feet per minute; some say 80,880 feet. (Telford's Memorandum Book). Some authors say that the weight or pressure of the atmosphere is equal to the weight of a volume of water 34 ft. in height; or 14-7 lbs. per square inch at a mean temperature; for air and all (?) kinds of gases are rendered lighter by the application of heat, because the particles of the mass are repelled from each other, or rarefied, and occupy a greater space.

15926. The force with which air strikes against a moving surface, or with which the wind strikes against a quiescent surface, is nearly as the square of the velocity. If B be the angle of incidence; & the surface struck in square feet; and v the velocity of the wind in feet per second; then if ƒ equals the force in pounds avoirdupois, either of the two following approximations may be used, viz., ƒ= vad sin2 ß; or, ƒ= '002288 v2dsin2ß. The first is the easiest in operation, requiring only two lines of short division, viz., by 40 and by 11. If the incidence be perpendicular, sin2 8=1, and these become f=440 ·002288v5. (Gregory). The force or pressure per square foot in lbs., is as the square of the velocity multiplied by 002288.

440

IMPULSE OF THE WIND ON A SQUARE FOOT.

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=

1592c.

Velocity in Feet

Per Second.

Impulse in lbs. Velocity in Feet

Per Second.

Impulse in lbs.

Velocity in Feet
Per Second.

Impulse in lbs.

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1592d. The resistance of a sphere is stated not to exceed one-fourth of that of its greatest circle. Tredgold, Carpentry, and Iron, has minutely examined the effect of the above forces, and the principle of forming the necessary resistance to them in the construc tion of walls and roofs. See HURRICANES. Where the roofs of buildings, as in the country are exposed to rude gusts and storms, it is necessary to increase the weight of the ridges hips and flashings.

1592e. The utmost power of the wind in England is said to be 90 miles per hour, o 40 lbs. per square foot. Tredgold takes the force at 573 lbs. per square foot. Dr. Nicho

of the Glasgow Observatory, records 55 lbs. per square foot, or 382 lbs. per square inch as the greatest pressure of wind ever observed in Britain (Rankine, Civil Eng. 538 During the extremely heavy gale of January 16, 1866, the pressure in London was recorde as 33 lbs. per square foot; at Liverpool it was 30-4 lbs. The velocity of the wind on th south coast of England, during January 11, when it uprooted old elm trees, averaged 6 miles an hour; later in the day it was 90 miles; the latter impetus is equal to the 40 lbs per square foot, above mentioned.

1592f. Wind exercises a tendency to overthrow a building upon the external edge of posite to the line of its advance, equivalent to the surface of the face receiving the impu sion multiplied by the force of the wind, and by a lever which on the average may be take to be equal to half the height of the building. To secure the stability of the latter, i

weight multiplied by a lever equal to half the base must exceed the sum of the elements of the wind's action.

1592g. To determine the pressure of wind in pounds per square foot, equal to the stability of a square stalk, multiply the weight of the stalk in pounds, by twice its width in feet at the base, and divide the product by the square of its height in feet, and by the sum of its top and bottom breadths in feet.

Let w=

Then

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p=pressure of wind in pounds per square foot equal to stability of stalk

h=height of stalk in feet = 50 feet

b= breadth of stalk at base in feet = 4 feet

e=breadth of stalk at top in feet = 2 feet

90,000 x 4 x 2

2,40 (4 x 2) =p=48 pounds per square foot.

If the stalk be circular, then, to

determine the pressure of wind, proceed as before, but replace the breadths by the diameter, and multiply the result by 2. Campin, Engineers' Pocket Remembrancer, 1863.

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SECT. X.

BEAMS AND PILLARS.

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 to expedients for another material, and it has become greatly superseded by iron.

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 generally 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 fullfrown 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 Timber may be said

For

a English cube foot of French oak is about 58 English pounds. 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. stance, 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 cr 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 ad thickness, 16 ft. long, placed in the same way. In respect of the absolute force, the diference 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 361 specific gravity, and a cube foot, therefore, weighing 49 lbs.

10

Cohesive Force of Pieces drawn in the Direction of their Length.
First experiment.

A small rod of oak 0-0888 in. (= 1 French line) square, and 2·14
in. in length, broke with a weight of
Another specimen of the same wood, and of similar dimensions,
broke with

Another specimen

115 lbs. avoirdupois

105

110%

The mean weight, therefore, was, in round numbers, 110 lbs.
A rod of the same wood as the former, 0·177 inch (=2 French lines)
square, and 2.14 inches long, broke with a weight of
Another specimen

Another specimen

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459 lbs. avoirdupois

418

4511

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1599. Without a recital of all the experiments, we will only add, that after increasing the thickness and length of the rods in the several trials, the absolute strength of oak was

found to be 110 lbs. for every 10000
888 of an inch area (=1 French line superficial).

The Strength of Wood in an upright Position.

1600. If timber were not flexible, a piece of wood placed upright as a post, should bear the weights last found, whatever its height; but experience shows that when a post i higher than six or seven times the width of its base, it bends under a similar weight before crushing or compressing, and that a piece of the height of 100 diameters of its base is incapable of bearing the smallest weight. The proportion in which the strength decreases as the height increases, is difficult to determine, on account of the different results of the experiments. Rondelet, however, found, after a great number, that when a piece of oak was too short to bend, the force necessary to crush or compress it was about 49-72 lbs. fo every 10000 1888 of a square inch of its base, and that for fir the weight was about 56-16 lbs Cubes of each of these woods, on trial, lost height by compression, without disunion o the fibres; those of oak more than a third, and those of fir one half.

10000

1601. A piece of fir or oak diminishes in strength the moment it begins to bend, so tha the mean strength of oak, which is 47:52 lbs. for a cube 868 of an inch, is reduced to 2.16 lbs. for a piece of the same wood, whose height is 72 times the width of its base From many experiments, Rondelet deduced the following progression :

For a cube, whose height is 1, the strength =1

12,

24,

36,

48,

60,

72,

Thus, for a cube of oak, whose base is 1066 in. area (=1 square in. French) placed upright, that is, with its fibres in a vertical direction, its mean strength is expressed by 144* × 47·52=6842 lbs. From a mean of these experiments, the result was (by experiment) in lbs. avoirdupois

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47.52 × 5
6

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For a rod of the same oak, whose section was of the same area by 12.792 in. high
(=12 French in.), the weight borne or mean strength is 144 x =5702 lbs.
From a mean of three experiments, the result was
for a rod 25-584 (=24 French) in. high, the strength is 144 x
For a rod 38.376 (=36 French) in. high, the strength is 144 ×
For a rod 51.160 (= 48 French) in. high, the strength is 144 ×
For a rod 63.960 (=60 French) in. high, the strength is 144 x
For a rod 76.752 (=72 French) in. high, the strength is 144 x
For a cube of fir, whose sides are 1·066 in. area (=1 square in. French), placed as
before, with the fibres in a vertical direction, we have 144 x 56.168087 lbs.

The French inch, consisting of 144 lines.

6853

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56:16 × 5
D

For a square rod, whose base was 1·066 in. area (=1 square in. French), 12-792 in. 2=6739 lbs.

high, we have 144 x

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- 6863

Fur a rod 25-584 (=24 French) in. high, 144 ×

56'16
2

=4043 lbs.

- 3703

56-16

=2696 lbs.

3

- 2881

= 1348 lbs.

= 674 lbs.

56.16

6

56.16
12

56.16
24

= 337 lbs.

For a rod 38-376 (=36 French) in. high, 144 × For a rod 51·160 (=43 French) in. high, 144 × For a rod 63.960 (=60 French) in. high, 144 × For a rod 76-752 (=72 French) in. high, 144 + The rule by Rondelet above given was that also adopted by MM. Perronet, Lamblardie, and Girard. In the analytical treatise of the last-named, some experiments are shown, which lead us to think it not very far from the truth. From the experiments, moreover, we learn, that the moment a post begins to bend, it loses strength, and that it is not prudent, in practice, to reduce its diameter or side to less than one tenth of its height. 1602. In calculating the resistance of a post after the rate of only 10-80 for every 1066 superficial line English (=1 line super. French), which is much less than one quarter of the weight under which it would be crushed, we shall find that a square post whose sides are 1066 ft.(= 1 ft. French) containing 22104-576 English lines (=20736 French), would sustain a weight of 238729 lbs. or 106 tons. Yet as there may be a great many circumstances, in practice, which may double or triple the load, it is never safe to trust to a post the width of whose base is less than a tenth part of its height, to the extent of 5 lbs. per 1-066 line; in one whose height is fifteen times the width of the base, 4 lbs. for the same proportion; and when twenty times, not more than 3 lbs.

Horizontal Pieces of Timber.

1603. In all the experiments on timber lying horizontally, as respects its length, and supported at the ends, it is found that, in pieces of equal depth, their strength diminishes in proportion to the bearing between the points of support. In pieces of equal length between the supports, the strength is as their width and the squares of their depths. We here con

tinue M. Rondelet's experiments.

1604. A rod of oak 2·132 in. (2 in. French) square, and 25·584 in. (24 in. French) long, broke under a weight of 2488 32 lbs., whilst another of the same dimensions, but 19:188 in. (18 in. French) bore 3353 40; whence it appears that the relative strength of these two rods was in the inverse ratio of their length. The proportion is 19-188: 25-584:2488-32: 3317-76, instead of 3353-40 lbs., the actual weight in the experiment.

1605. In another rod of the same wood, 2·132 in. wide and 3.198 deep, and 25.584 in. bearing, it broke with a weight of 5532 lbs. In the preceding first-mentioned experiment it was found that a rod of 2·132 in. square, with a bearing 25-584 in. bore 2488 32 lbs. Supposing the strength of the rods to be exactly as the squares of their heights, we should have 4.54 (2·132): 10-23 (3·1982):: 2488-32: 5598-7 lbs. ; which the second rod should have borne, instead of 5532 lbs. There are numberless considerations which account for the discrepancy, but it is one too small to make us dissatisfied with the theory.

1606. In a third experiment on the same sort of wood, the dimension of 3·198 in. being laid flatwise, and the 2.132 in. depthwise, the bearing or distance between the supports being the same as before, it broke with a weight of 3575 lbs.: whence it follows that the strength of pieces of wood of the same depth is proportional to their width. Thus, comparing the piece 2.132 in. square, which bore 2488 lbs., we ought to have 2132: 3-198 2488-32 3624-48, instead of 3573 lbs.

1607. From a great number of experiments and calculations made for the purpose of finding the proportion of the absolute strength of oak, to that which it has when lying horizontally between two points of support, the most simple method is to multiply the area of the piece in section by half the absolute strength, and to divide the product by the rumber of times its depth is contained in the length between the points of support.

1608. Thus, in the experiments made by Belidor on rods of oak 3 French (=3·198 English) ft. long, and i French (=1066 in. English) in. square, the mean weight under which they broke was 200-96 lbs. avoirdupois. Now, as the absolute strength of oak is from 98 to 110 lbs. for every in. (=1 French line), the mean strength will be 104 and 52 lbs. for its half, and the rule will become (144 lines, being =1 French in.) =207-30 lbs., instead of 200.96 lbs.

144 x 52 36 (F. in.)

100000

576(144x4) x 52

1609. Three other rods, 2 French in. square (2·132 Eng.), and of the same length between the supports, broke with a mean weight of 17118 lbs. By the rule 18 1658.88 lbs. avoirdupois. Without further mention of the experiments of Belidor, we

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