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The Construction of Flood-gates or Locks.

By a series of locks on a canal, a boat is enabled to rise from a lower level to a higher, and fall from a higher level to a lower. A boat being thus capable of changing its level, can proceed up or down at pleasure.

The form of the locks should approach as nearly as pos sible to the figure of the boats which are to navigate the canal. The width should not exceed that of the boat by more than is sufficient to work the gates. They should consume a minimum quantity of water, and be capable of being filled and emptied in the shortest possible time.

To answer all these conditions, the locks must be rectangles, the length, breadth, and depth of which must depend on the dimensions of the boats, which are generally long and narrow. The principle on which a boat is raised from a lower level to a higher, and lowered from a higher to a lower, is very simple. In ascending, the gates are closed behind the boat, and a sluice opened, which allows the water to rush from above into the lock, which it soon fills, lifting the boat to the level of the upper reach of the canal. The operation is repeated at every lock. In descending, the water enclosed in all the locks is allowed to flow out by reversing the last process, by which means the boats descends from a higher to a lower level.

The engineer should keep one great principle in view, namely, that no farther excavation is required, than what is sufficient to hold the water at a given depth and breadth. Locks may be either of wood or stone. No country in the world affords such an extent of inland communication by water as this country. There are two kinds of works employed in forming the inland water communication.

These are called slack-water navigation and canals. Slack-water navigation, which consists in the improve ment of a river by erecting dams or mounds in the streams,

by which the depth of water is increased to a considerable distance up the river, where the fall is not very great. By this simple means, a river of inconsiderable depth may of ten be converted into a navigable canal. On this subject see Mr. D. H. Mahan's Elementary Treatise on Engineering.

SECTION XIII.

ON THE THEORY OF BRIDGES.

From the laws of motion and forces, we may observe, that all the corresponding weights, and actions, and positions, of an upright arch, is exactly the same in the same arch in a hanging or festoon position: changing only drawing and tension for pushing and thrusting. This necessarily happens, from the equality of the weights, the similarity of the positions and actions of the whole in both cases. (Fig. 30.)

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This is a most useful principle in all cases of equilibriums, especially to a practical man it enables him to resolve problems, which the best of mathematicians have found a difficulty to effect by mere computation. For thus, he obtains the shape of an equilibrated arch or bridge; also the position of an equilibrated roof; having a given width, and rafters which shall either be or in a given proportion to each other. It is evident that this principle contains the whole theory of the construction of arches. By varying the distance of the points of suspension, moving them nearer, or further off, the chain will take different. forms; then the frame A B C D E may be made to that form which has the most convenient shape, found above as a model.

Prob. 1.-Required to form a balanced festoon arch, on the principle of the last problem; having a given pitch and span, also a given height and form of wall or roadway over it.

Let AB (Fig. 30.) be the proposed span of the arch, O C its pitch or greatest height, DC the thickness at the crown, and APDQ B the given form of the wall. To determine the form of the curve A C B which will put that wall in equilibrio. Invert the whole figure, as in the opposite position m hm, or construct this latter figure, on the lower side of A B, equal and similar to the proposed upper one; the point d answering to the point C, and h to the point D, &c. Let a fine, but a strong line, or perhaps a fine and slender chain of small links, be suspended from the extreme points A and B, of such length that its middle point may hang at the point d, or a little below it. Divide the given span into a number of = parts, the more the better, from which draw vertical lines cutting the festoon chain in corresponding points as in the figure. Then take short pieces of another chain, and suspend them by these points of the festoon 1, 2, 3, &c. This will alter the form of the curve. If now the new curve should correspond with the point d, and all the bottoms of the vertical pieces of chain coincide with the given line of roadway m h m, the thing required is done. But if both those coincidences do not take place, then alterations must be made, by trials, in lengthening or shortening either the festoon A d B, or the vertical pieces of the chain, or in both, till such time as those coincidences are perfect, that is, the bottom of the arch with the point d, and the bottom of the append pieces with the boundary m h m.

Then trace out the upper curve A C B the same as the lower one A d B, and there will be obtained an arch sustaining the wall above in perfect equilibrium.

The theory laid down, will serve as a foundation on

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which to establish a method for constructing arches of equilibration, on any proposed curve. In this example, we shall suppose that the intrados curve is a circular arc. And also suppose the wedge pieces to form = parts of that arc, of the quantity of 6 degrees each, that is, each wedge subtending at the centre an angle of 6°, the key wedge at the centre or crown, ..., extending 3° on each side of the vertical line passing through the centre; and have 10 other wedges, of angles (6°) on each side of the key, making in all 21 wedges, which, at 6° each, will form an entire arch of 126 degrees.

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In this case the angle which the sides of the middle wedge forms with the middle vertical line, will be 3°; and the angles which the sides of the other wedges, on each side of the key wedge, form with the vertical direction, will be found by adding continually the breadth of each wedge (6°) to 3°; by which it will be found that the angles at the centre, formed with the vertical, by the said lower edges of the arch pieces, in order after the key, will be found as follows, viz: that of the second wedge 9o; that of thet hird, 15°; that of the fourth 21°; and so on to the tenth, or last on each side of the key, which will have its lower edge making an angle of 639 with the vertical direction.

Now, in regard to the rule for computing all the weights and measures, according to the theory, it is very simple, viz that the weight of any part of the arch, counted from the crown downward, is always proportional to the angle of inclination of the lower wedge to the vertical, while the oblique pressure, in direction of the curve, is proportional to the secant of the same angle, and the constant horizontal thrust is proportional to the radius.

So, in calculating the said several weights and oblique pushes of the arches, we have but to take out, from a trigonometrical table, the tangents and secants of the several an

gles of inclination to the vertical, and multiply all the tan gents and secants by the number expressing the constant horizontal thrust, for all the values of the several weights and pressures; the products of the tangents being the several weights of the half arches, and the product of the secants being the oblique pressures of the same in the direction of the arch.

a

2

To find the constant horizontal thrust which is the constant multiplier: It may be proved that this horizontal thrust is every where in the same proportion to the weight of half the middle key stone, as radius is to the tangent of half the angle of that wedge; that is, t: 1:::÷t=horizontal thrust, putting a for the weight of the key piece, and t for the tangent of half its angle: now put a=1, then this will become ÷t-horizontal thrust. Now, in the example, the angle subtended by the key stone is 6°, the half of which is 3o, and its tangent=m; then or 5-m= horizontal thrust. Hence then, this constant number is to be multiplied by the tangents of all the vertical angles, to give the weights of the semi-arch, and by the secants of the same angles to give their oblique pressures.

To find the value of each wedge, you must take the difference of the numbers calculated from the crown down, which will give the weights of the single wedges taken separately.

Now to examine in what manner the preceding calculation of the weights of the voussoirs may be employed to give an easy mechanical construction, that may approach a true balanced arch. To do this, we must consider, that since the bases of all the voussoirs are, and the breadth very small, which is the case in practice, it will thence happen that their weights will have nearly the same ratios as their lengths, taken perpendicularly to the under curve.

Let us suppose the under curve (Fig. 31,) a semicircle having its span H R 100 feet, and consequently its pitch

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