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Area of Midship Section (X A).

The area of this, or any of the other sections on the displacement table, is calculated by taking the half-breadths of the water lines and integrating them as explained for water-line area. The sum of the products thus obtained is multiplied by the distance of water lines apart, and that result by 2 for both sides. Where the vessel has little rise of floor a half water line should be introduced, and the bottom half-breadth proportioned to the rise line, as pointed out in the displacement calculation. In the example with which we are dealing, however, the vessel has considerable rise, so that this subdivision has been omitted.

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The coefficient of this area, or ẞ, is a very important element of the design as explained elsewhere, and is obtained by dividing the midship area by the area of the rectangle formed by the molded breadth and the draught, or

Mid. area

44.62

Breadth x draught 60

.743 coefficient of mid. area.

Its relation to the midship-section cylinder or prismatic co

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and "p" is equal to the volume of dis

β

placement divided by the length × mid. area, thus:

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Centre of Buoyancy (C.R.).

The centre of buoyancy of the displaced water is simply its centre of gravity, and its location below the load-water line is greater or less in accordance with the form of the immersed body. This distance may be found by dividing the under-water part into a number of planes parallel to the load line, and multiplying the volumes, lying between these water planes, by their depth below oad-water line. These moments divided by the displacement volume will give the location of centre of buoyancy below loadwater plane. So that by taking the functions of the products at each water plane on the sheet we have been working and multiplying them by the number of the water line they represent below L.W.L., and dividing the sum of those products by the sum of the functions referred to, we shall have the number of water-line intervals (or fraction of an interval), which the C.B. is below load-water line. This result, multiplied by the common interval between water lines, will give the required distance in feet.

Functions of products

KEEL. W.L. 1. W.L. 2. W.L. 3. W.L. 4.

} .15 +50.32 + 43.34 +109.36 +31.20 = 234.37

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The centre of buoyancy may be determined from the displacement curve by calculating the area enclosed within the figure formed by the vertical line representing the draught of 5 ft., the horizontal line equal to the tons displacement at this draught and the curve itself. This area divided by the length of the horizontal line referred to, will give the depth of C.B. below L.W.L. In the present example we have: area = 138.6 sq. feet, and length of horizontal line (displacement in tons) = 73.82, and

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A like result may also be obtained by taking the sum of the products of each water line, and dividing them by the sum of Simpson's multipliers. The mean half-breadths of water lines so obtained may be then used to draw a mean section of the

vessel on stout paper, which on being cut out with a knife and swung in two positions, the points being intersected afterwards, will give the centre of gravity (buoyancy) very accurately.

Various approximate methods are in vogue for finding this centre, some of which are fairly accurate.

(1) Approx. C.B. above base=

(5a-28).

1/d V

(2) Approx. C.B. below L. W.L. = +
32 A

where A is the area of load-water plane.

This centre, as will be explained, has an important bearing on the stability of the ship.

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Common Interval = 10 ft. x .06 0.6 ft. C.B. abaft No. 5.

The locus of the centre of buoyancy in a fore-and-aft direction is of course the centre of gravity of the displacement, and is the

ivotal point or fulcrum for the moments of all weights placed orward or aft of this position. It will be obvious, therefore, that is location is of great value in determining the trim of the vessel, nd the various alterations thereof due to rearrangements of weights on board. Its position is calculated by taking the areas of he sections and putting them through the multipliers; these unctions of areas are in turn multiplied by the number of interals, (each one is forward or aft of the mid-ordinate,) and the ifference between these forward and after moments divided by he sum of the area functions. The quotient resulting is the umber (or fraction) of intervals that the centre of buoyancy is orward or aft of the length according as the moment preponlerates forward or aft respectively.

This centre should be calculated for various draughts, as of course it changes with different draughts and alterations of trim, wing to the changing relationship between the fineness of fore und after bodies at different immersions and trims.

Transverse Metacentre (M.C.)

The position of this element is, in conjunction with the centre of gravity, the most vital in the design of the ship. As its name implies, it is the centre or point beyond which the centre of gravity of the ship may not be raised without producing unstable equilibrium in the upright position, or, otherwise stated, if the ship be inclined transversely to a small angle of heel, the centre of buoyancy which originally was on the centre line will move outboard to a new position; but, as it acts vertically upward, it must somewhere intersect the centre line. This point of intersection is known as the metacentre. One of the factors in the determination of its location above the centre of buoyancy has already been calculated, viz: the volume of displacement V; the other, the moment of inertia of the water plane about the centre line of ship, we shall proceed to compute. The height M above the C.B. or

B.M. is found by : —

Moment of Inertia of Water Plane

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Volume of Displacement

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The moment of inertia of the water plane is a geometrical measure of the resistance of that plane to "upsetting," or when taken about the centre line, as in the case of calculating for transverse metacentre, to "careening." So that the greater the waterline breadth the higher will be its value; for we must imagine the water plane as being divided into a great number of small areas, and each of these multiplied by the square of its distance from the

centre line of ship, when the sum of these products will equal the moment of inertia of half the water plane, about the middle line of vessel as an axis. As both sides of the water plane are sym metrical, the total I will be this result multiplied by 2. Applying this principle to W.L. 4 in the example with which we are con cerned, we get the following tabular arrangement:—

Moment of Inertia of Water Plane (I).

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The calculation for Moment of Inertia and Transverse Metacentre above C. B. may be more easily remembered if we treat the cubes of water line half-breadths as the ordinates of a curve twothirds the area of which will equal I, and this, in turn, divided by V will give B. M.

However, when we know a, or the coefficient of water line, we may arrive very accurately at the moment of inertia of the water

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