The word stability is commonly, though somewhat loosely, used by naval architects to express not only the existence of a righting tendency in a ship inclined in still water, but also the amount of such tendency, i. e. the righting moment of the ship. The displacement of ships being always expressed in tons, their righting moments are naturally expressed in ton-feet.
I propose to describe and explain a method for calculating stability; but before taking up the method itself, shall state briefly a few elementary facts connected with the subject.
A ship floating at rest in still water, and acted upon only by her own weight and the buoyancy of the water, must—
1. Displace a weight of water equal to her own weight.
2. Have her center of gravity vertically above the center of gravity of the displaced water, usually called the center of buoyancy.
When the above conditions hold, the weight of the ship, which may be regarded as acting downward through her center of gravity, is exactly counterbalanced by the buoyancy of the water, which may be regarded as acting upward through the center of buoyancy.
This state of affairs is illustrated by Fig. 1, which may be taken to represent the transverse section of a ship through her center of gravity, G, and center of buoyancy, B.
Consider now Fig. 2, where the ship is shown inclined in smooth water, at the water-line WL, the displacement remaining- unchanged. In this condition the weight of the ship and the equal buoyancy of the water, while still acting in vertical lines, do not act in the same line. There is then a couple set up, which will tend to right the ship if the vertical through B falls outside of the vertical through G (as in Fig. 2), and will tend to still further incline the ship if the vertical through B falls inside of the vertical through G.
This couple is, of course, measured by the displacement multiplied by GZ, the horizontal distance between the verticals through G and B respectively, GZ is called the righting lever.
If, adopting some constant displacement, we determine values of GZ for a number of inclinations, and plot them as ordinates of a curve of which the inclinations are the corresponding abscissae, we can determine from the curve the value of the righting lever corresponding to any inclination at the constant displacement. Such a curve is commonly called a curve of stability. One is shown by Fig- 5.
The displacement being constant, a curve of righting levers is, on a suitable scale, a curve of righting moments also, for righting moment = displacement X righting lever.
Suppose now we give the ship a constant inclination and then gradually immerse her, determining for each of a number of parallel water-lines the displacement and its moment about an axis fixed in the ship. Plotting these moments as ordinates of a curve of which the displacements are abscissae, we have what is called the "cross curve" of stability corresponding to the constant inclination. Fig. 4 shows a number of cross curves for a ship at intervals of 15°.
If we have cross curves for a sufficient number of inclinations, we can take from the curve for each inclination the moment corresponding to a fixed displacement, and plot a new curve for the fixed displacement having moments for ordinates upon inclinations as abscissae. If the center of gravity of the ship at the fixed displacement falls on the axis about which the moments for the cross curves were found, this new curve is the same as the ordinary curve of stability explained above.
If the center of gravity of the ship does not fall on the axis, a simple correction will be necessary in order to obtain the ordinary curve of stability.
The center of gravity of a ship is always calculated approximately when she is designed, and is usually determined with considerable accuracy after the ship is completed, by an "inclining experiment."
I propose to discuss a method for determining stability, on the supposition that the position of the center of the gravity of the ship is always known.
The explanation of the method as applied to a ship is simplified by considering first a single section as shown in Fig. 3,
Let —90° P90° denote the position of a water-line when the section is upright. Call the point P the pole. Let A denote the position of the axis about which the righting moment is desired.
Let r denote a radius from the pole, a subscript being used to indicate the angle which its radius makes with the vertical. Thus r90 denotes the radius from P to 90°. In the figure, radii are drawn at intervals of 15° on each side of the vertical, as shown.
Consider now the small triangle Pa1a2, a1 and a2 denoting, in circular measure, the angular distances of the ends of the base of the little triangle from the vertical, as indicated.
In the form headed "displacement," the initial displacement is entered opposite the inclination of 0°; and the "displacement corrections" being suitably entered, as shown, and each added to the displacement above, we obtain the displacements for the successive inclinations. The "pole correction factor" is simply the distance of the pole from the axis multiplied by the sine of the angle of inclination.
Multiplying this factor by the corresponding displacement abreast it, we have the "pole correction for moments" which must be used to reduce moments about the pole to moments about the axis.
In the form headed "righting moments" the I'r^ quantities are entered and added as indicated, and the sums re-entered vertically on the left, abreast their proper inclinations. The triangular table is then filled by entering abreast each angle the product of its sine into the "sum" on the same line to the left. It should be noted, however, that abreast each 90° is entered only ½" sum " X sin 90°, for this is the end ordinate, and we are using the trapezoidal rule.
The moment function sum (obtained by addition) must be multiplied by the polar moment factor (from the legend) to give the moment of the wedges of immersion and emersion about the pole.
The initial displacement correction, which must now be applied, is the "constant for initial displacement correction" X sine of inclination.
Thus is obtained the righting moment about pole. Applying the pole correction (from the displacement form), we finally obtain the righting moment about axis.
The righting moments obtained so far are for inclinations up to 90°. The remainder of the form is for the purpose of obtaining displacements and corresponding righting moments for inclinations up to 180°.
Of course the "righting moment for total displacement" (corresponding to total submergence) is total displacement X distance from axis to C. B. of total displacement X sine of inclination.
I use next the well known property of floating bodies, that if we take for a given pole the displacement and righting moment corresponding to a plane at an inclination a, and deduct them from the total displacement and the righting moment for the total displacement respectively, the remainders will be the displacement and the righting moment corresponding to an inclination of 180°—a.
The steps of the process are clearly indicated on the form, and show how from displacements and righting moments up to 90° those from 90° to 180° are obtained.
Having completed Table I for three or more poles. Table II is filled in for the purpose of drawing the cross curves of righting moment.
In Table II the first three lines (D, M, C) for each inclination at which a curve is to be drawn are filled in at once from Table I. The object of the next two lines is to get the inclination of each curve at each "spot" corresponding to an ascertained displacement and righting moment.
Suppose the vertical through the C, of G. of a given water plane has a leverage of / feet about the axis.
If we immerse the ship slightly without changing the inclination, the layer of increase also has the leverage /, and one ton increase of displacement means / tons increase of righting moment. Then the inclination of the corresponding cross curve of righting moment at the corresponding spot will be the angle whose tangent is /, provided the scales for displacement and righting moment are the same.
But if, as in the case shown, the scale for righting moment is i that for displacement, the " lever of C. of G. of water plane " ( C) must be divided by 2 to obtain the tangent of the inclination.
Having the tangent, the corresponding inclination is taken from a table of natural tangents and entered in its place opposite l.
It should be said that for 90° inclination the displacement and righting moment are necessarily the same for every pole. Whatever the position of the pole, we have at 90" the same immersed volume acting—namely, the volume on one side of the central vertical longitudinal plane, called the diametral. The corresponding displacement is of course ½ of the total displacement.
It is the object of the work shown in Table III to obtain two additional "spots" for the cross curve at 90". Two sections are taken at intervals of 5' on one side of the diametral, and the area and the position of the center of gravity of each are determined. Then the tons per foot and ton-feet (about the axis) per foot are determined for the two side sections, and also for the diametral. The total displacement and moment of the slice between the diametral and the outer section are readily deduced, and by addition and subtraction to the known displacement and moment the two additional spots desired on the 90° cross curve are obtained.
Having the data in Table II, the cross curves at 15° intervals can be drawn. There are five known spots on each curve, corresponding to no displacement, to the total displacement, and to the three poles used.
For the three last spots, not only the point on the curve is known, but also the direction of the curve at the point. The inclinations at zero displacement and total displacement are also readily obtained by determining from the midship section the corresponding righting levers.
Fig, 4 shows the cross curves corresponding to Table II, being very approximately those of the U.S. S. Philadelphia.
The process by which an ordinary stability curve, corresponding to a known displacement and position of the center of gravity, is derived, is shown in Table IV.
The displacement and position of the center of gravity used are very nearly those of the Philadelphia when she was inclined for the determination of her metacentric height. The righting moments at each inclination corresponding to 5000 tons displacement are taken directly from the cross curves. The remaining steps are shown clearly in the table. The correction of the lever about the axis, on account of the position of the C. of G., is, of course : Distance from axis to C of G. X sine of inclination.
It may have been observed that much of the work in the method of calculation which I have been explaining consists in the multiplication of numbers by sines of angles. To facilitate this work and reduce the chances of error, I have calculated the appended tables of products of numbers by the sines of angles. The main table extends only to 2500, but for most purposes sufficient accuracy will be obtained (when dealing with numbers above 2500) by entering the table with the first four digits of the number being handled. For more refined work Tables A and B have been calculated. These are six-place multiplication tables of numbers by sines of angles extending to one hundred. Table A is for 15° intervals. Table B for 10° intervals. Since the sine of 30° = ½ no calculations have been made for this angle.
Any one familiar with stability work will have observed that the method I have been describing is essentially a modification of Barnes' method, the trapezoidal rule being used instead of Simpson's, and such other changes made as are necessary in determining cross curves. These changes have been chiefly suggested by the work of Daymard, Elgar, and Jenkins.
The trapezoidal rule was adopted because for hollow curves it is quite as reliable as Simpson's. For the somewhat lumpy curves used in obtaining displacement and righting moment, the trapezoidal rule is preferable to Simpson's.
These curves differ radically from the parabolic curves for which Simpson's rules give exact results. Consequently it might reasonably be concluded that Simpson's rules, when applied to them, give unsatisfactory results. Comparative tests which I have made by applying Simpson's and the trapezoidal rule to somewhat irregular curves fully justify this conclusion, I may mention, in this connection, that it is the practice of French naval architects to employ the trapezoidal rule in all calculations, and I am informed that this is also the practice among our own civil engineers.
There is no doubt, however, that Simpson's rules, when applied to curves to which they are suited, give appreciably more accurate results than the trapezoidal, a fact quite sufficient to justify their use in such cases when extreme accuracy is desired.
In the example of work given for the purpose of illustrating the method I have taken no account of appendages, considering only the main body of the ship to the upper deck. The only appendages which would appreciably affect the result are the forecastle and the poop. With the high freeboard of the Philadelphia, the forecastle and poop afford buoyancy and righting moment only at large angles of roll. When such angles are reached, the loose water which would come on board would largely, if not entirely, neutralize any buoyancy or stability afforded by poop or forecastle.
In low freeboard ships, deck erections and poops and forecastles are of more relative importance and should be considered. The necessary changes in the forms are simple and obvious.
In considering the accuracy to which we wish to work, it should be remembered that a "curve of statical stability" is an entirely imaginary thing. We cannot, by any practical appliances, exert a twisting moment great enough to heel a large ship in still water beyond a very small angle. When ships roll it is always in disturbed water, and the actual righting moment at a given inclination depends largely upon the position of the ship at the instant with respect to the waves.
While the statical righting moment at a given inclination is a kind of mean of all possible righting moments at that inclination in disturbed water, the very fact that the actual moment is liable to material oscillations—impossible to calculate exactly—on either side of the mean, renders it unnecessary, for any practical purpose, to aim at minute accuracy in the determination of the mean. The method I have been describing is most accurate for the smaller inclinations, for which the results are practically exact. At the large inclinations, 60° and over, the results may be so much out that the righting levers determined will differ by as much as an inch, in extreme cases, from the exact righting levers. In the case of the Newark it was found by careful check calculations by other methods that the righting levers by the method just described were never in excess of the exact levers, and that the greatest defect was about ½ inch.
Such an amount of inaccuracy at large angles of heel (which are never reached in practice) is entirely negligible as regards any practical use to which curves of stability are put. I have not considered it of sufficient importance to justify the additional work involved in closer spacing of the radial planes, by which it could be reduced to almost any desired extent.
The principal source of inaccuracy is the fact that the righting moment is determined as the difference between the moment of the wedges and the moment of the initial displacement—both much larger quantities than their difference at large inclinations.
It would somewhat shorten the work and give smoother cross curves for the large inclinations if a method were adopted treating the parallel planes through equidistant poles, as the vertical sections were treated in determining additional spots for the 90° cross curves. I have, however, considered it preferable to obtain independent results for each pole, leaving the method above referred to available for fairing the cross curves if necessary. While the forms of calculation given extend beyond 90° of inclination, I have found it preferable in practice to use a graphic method to obtain the cross curves beyond this point.
Thus in Fig. 6 let OPM be a cross curve for 60° inclination. Let OD represent the displacement when fully submerged, and DM the corresponding righting moment. Lay off OH = ½OD, HC= ½DM. Draw PCQ and lay off CQ^ equal and opposite to CP. Then Q is a point on the cross curve for 120°, the angle supplementary to 60°. Any number of points on the supplementary cross curve may be thus determined and the curve for 120° drawn as shown.
Evidently the curve for 120° is simply the curve for 60° rotated 180° about an axis through C perpendicular to the plane of the paper.
It is desirable, when intending to make use of this method, to choose the axis at the center of buoyancy of the total displacement.
If this be done, the point C in Fig. 6 is the same for every curve—always coinciding with H.
(To see the complete tables and graphs, please view the PDF.)