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One process consists in first computing, by Simpson's rules, the areas of the several vertical sections; and then treating those areas as the ordinates of a new curve upon the base, AB, in order to compute the volume of the displacement, by the method of Chapter IX. Sometimes, though not always, that curve is drawn to a scale, and is called "Peak's Curve," or the Curve of Sectional Areas. The other process consists in computing the areas of the several water-sections, and then treating those areas as the ordinates of a new curve upon a base equal to the distance between the load-water-line and the lowest water-line, in order to compute the displacement, not only up to the load-water-section, but up to each water-section of the series.
The area of a given water-section represents also the displacement in cubic feet per vertical foot of immersion at that watersection; which, being divided by 35, gives the displacement in tons per foot of immersion. This again, being divided by 12, gives the displacement in tons per inch of immersion.
Water-section in square feet.
It is customary, in drawing the curve of water-sections on the plans of a ship, to lay down its horizontal ordinates to such a scale that they shall represent, not the areas of the water-sections themselves, in square feet, but the 420th parts of those areas, or the tons per inch immersion. The use of that quantity has already been illustrated in Chapter II. It enables us to calculate how much deeper a given ship will be immersed by a given addition to her lading.
Curve and Scale of Displacement.-(Fig. 15.) The displacements themselves, in tons of 35 cubic feet, corresponding to different draughts of water, are laid down on the drawing as the horizontal ordinates of a curve, OdD. For example, the ordinate, HD, represents the load displacement, and the ordinate, hd, the displacement at the draught, Oh. A scale of tons is marked along the longest ordinate, HD.
Computation of Cross-Sections.-As each vertical cross-section consists of two similar halves, it is customary to begin by computing the half area of each vertical section, and afterwards to multiply it by 2. The appearance of the vertical sections upon. the body-plan enables the naval architect to judge where and to
what extent subdivisions of the vertical intervals is required; and that subdivision should be made by means of intermediate watersections running the whole length of the ship, for the sake of uniformity in the calculations, the neglect of which is apt to lead to confusion and mistakes.
Computation of Water-Sections.-The water-sections, like the cross-sections, consist of two similar halves; and therefore, in general, the half areas are computed first, and afterward multiplied by 2. The process of measurement and calculation requires. no special remark, being almost always performed by means of Simpson's First Rule, with subdivided intervals where they are required (Chapter VIII.), of which the naval architect judges from the appearance of the half-breadth plan.
Computation of Displacement in Layers.—The computation of the load displacement presents no peculiarity; it is performed by treating the areas of the water-sections just as the ordinates are treated in computing areas of cross-sections, the series of multipliers being exactly the same.
In computing the series of displacements up to the other watersections, the particular rule employed must be varied according to the circumstances of the particular calculation.
If the displacement up to the 7th W.L. was required, the waterline next below the load-water-line, it could be computed by the following rule, and subtracted from the load displacement, viz. : To five times the area of the L.W.L. add eight times the area of the 6th W.L. and substract the area of the 5th W.L.; multiply the remainder by one-twelfth of the vertical interval or depth of the ayer: the product will be the volume of the layer.
The volume of any even number of equally deep layers is to be computed by Simpson's First Rule, and that of three equally deep layers by Simpson's Second Rule.
The name of Appendages is given to small portions of the ship which project beyond the net-work of water-sections and crosssections, and whose volume must therefore be found by special calculations, and added to the main part of the displacement. They usually consist of the keel below its rabbet, the false keel (if any), part of the stem, part of the stern-post, the rudder and the rudder-post in screw steamers.
Computation of Midship-Section in Layers.-It is a common practice to compute the area of immersed midship-section for a series of different draughts of water, like the displacement; and
the process is perfectly analogous. The areas can then be represented by the horizontal ordinates of a curve, which usually in general appearance is somewhat like the curve of displacement. It was formerly supposed that the resistance of a given ship at the same speed, and at different immersions, varied proportionally to the area of the immersed midship-section; but that supposition was founded on an imperfect theory of the resistance of fluids, and has not been corroborated by experience.
Determination of Centre of Buoyancy.—The nature of the centre of buoyancy, and the use of finding its position, has been explained in Chapter III.
As the immersed part of a ship floating upright consists of two symmetrical halves, one on each side of the central vertical plane which traverses AB, Figs. 1 and 2, Plate 6, it is obvious that the centre of buoyancy of a ship floating upright must be in that plane; so that in order to find the position of that centre completely, it is only necessary to find its horizontal distance from the plane of the cross-section through A, and its vertical depth below the load-water-section.
To find the horizontal distance of the centre of buoyancy from a transverse vertical plane through A, the first step is to compute the moment of the volume of displacement relatively to that plane by the rule for moments and centres of volumes, page 30, Chapter X.; that is to say, the area of each cross-section is to be multiplied by its distance from A, and the products treated as the ordinates of a new curve. The moment thus found, being divided by the volume of the displacement, gives the distance required.
To find the depth of the centre of buoyancy below the loadwater-section, the first step is to compute the moment of the volume of the displacement relatively to the plane of that section, by the rule just referred to; that is to say, the area of each watersection is to be multiplied by its depth below the load-watersection, and the products treated as the ordinates of a new curve. The moment thus found, being divided by the volume of the displacement, gives the depth required.
In performing these calculations, time is saved by the method already explained in Chapter X., of multiplying the sectional areas in the first instance not by the leverages themselves, but by the number of intervals to which those leverages are proportional, and performing a multiplication by the common interval after the addition has been made.
HALF BREADTHS OF ORDINATES FOR COMPUTING THE DISPLACEMENT OF THE U. S. STEAMER " S. STEAMER "ANTIETAM.
0.79 1.17 1.50 1.79 2.08 2.37
0.66 Section No. 1
0.66 0.79 1.08 1.47
5.00 8.42 11.50 14.12 16.25 17.83 18.00 19.70
Length of the keel to be added to the cubical contents of each body. 156.25 x 0.83 ×
Common interval between sections in both bodies, 7.81 ft.
Common interval between water-lines, 2 ft.
Coefficients of Fineness-Tonnage- Burden.
IF two ships have figures so far similar, that every ordinate or half-breadth in one of them bears an uniform proportion to the corresponding or similarly situated ordinate in the other ship, it is evident that the displacements of those two ships will be to each other simply in the proportion of the products of their lengths, extreme breadths, and immersed depths or draughts of water; that is, of the rectangular solids circumscribed about their respective immersed bodies.
Hence, if it has been ascertained that the displacement of a given ship is a certain fraction of the circumscribed rectangular solid, the displacement of any other ship of similar figure (as above defined) may be found by multiplying the product of her length, extreme breadth, and immersed depth, by the same fraction. That fraction is called a coefficient of fineness; because, by being greater in ships with bluff ends and flat floors, and smaller in ships with fine ends and rising floors, it furnishes a sort of indication of the general character of a ship's figure.
Examples of the coefficient of fineness will be given in the sequel. Amongst its commonest values are those which range from 0.5 to 0.66; but it is occasionally as low as 0.3, and as high as 0.8.
Besides the just-mentioned coefficient of fineness of the displacement, coefficients of fineness may also be computed for crosssections and for water-sections. Thus the midship section, being divided by the rectangle of its extreme breadth and immersed depth, gives a coefficient which ranges from 0.5 to very near 1. The coefficients of fineness of water-lines, obtained by dividing the area of a water-section by the rectangle of its length and extreme breadth, range in extreme cases from 0.5 to 0.9, the more common proportions being from 0.6 to 0.75.
The mean coefficient of fineness of all the water-lines of a ship is obtained as follows: multiply the greatest immersed area of mid