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On the Limits of Safety of Ships as Regards Capsizing-Distribution of Weight and Buoyancy in Ships-Measure of Fighting Efficiency of Sea-Going IronClads.
ON THE LIMITS OF SAFETY OF SHIPS AS REGARDS CAPSIZING.*
If we consider a ship heeling over in still water, and have regard simply to the statical effects of the pressure of the water and the action of gravity, we observe that these effects are the same as would be produced by a pair of parallel and equal forces. The force of gravity may be replaced by a single force acting downwards at the centre of weight of the ship, and the pressure of the water by a single force acting upwards at the centre of figure of the displaced water, or centre of buoyancy. These forces therefore constitute a couple, the axis of which is in the direction of the vessel's length; the arm is the horizontal distance between the centres of weight and buoyancy, and the moment is the product of this arm into the weight of the vessel, or, what is the same, into the weight of the water which it displaces, called the displacement. The question whether this couple is an upsetting or righting one depends upon the centre of buoyancy moving out from the middle-line plane slower or faster than the centre of weight. With a ship which has both sides alike, these are in the same vertical line when she is upright.
The determination of their motion, as the ship heels, is one of pure geometry. For the present we are only concerned with its effects. Whatever may be the details, the instant after the centre of weight has overtaken the centre of buoyancy in moving out towards the direction of heeling, there is a tendency to upset, even without any extraneous force, such as that of the wind. The action of a steady wind, after all oscillations have disappeared, and steady motion has been obtained, consists partly of linear motion of the ship and partly of another couple, formed by the resistance of the water to the lateral motion of the ship, as one
* From the Annual of the Royal School of Naval Architecture. 1871. By C. W. Merrifield, F. R.S., Principal of the School.
force, and by the resolved pressure of the wind on the sails as the other. This wind-couple, if the vessel maintains a steady inclination, must be exactly equal to the righting-couple due to the stiffness of the ship; for if we have regard only to the tendency to capsize, or the reverse, we need only consider the resolved wind-couple acting in a plane parallel to that of the stiffnesscouple. The statical measure of either of these couples is its moment, expressed in foot-tons, or some equivalent unit.
The knowledge of a ship's statical stability at any particular angle is not sufficient to determine the practical question of her capsizing. Dynamically, the difference between the moments of the wind-couple and the stiffness-couple is simply an accelerating or retarding force. Even in smooth water the effects of a varying wind, or of the sudden application of a steady wind, as may happen when a vessel passes a high head-land, depend upon the equation of work, not on the vanishing of the applied couple. Let us suppose that we have calculated the moment of the rightingcouple for all possible angles of inclination, and that, setting out equal angles at equal distances along a base-line, we set off the corresponding moments as ordinates. We then obtain the curve of stability or stiffness. I will suppose it to be as in Fig. 25. The ordinate, always beginning from zero, is here supposed to reach its maximum at 23°, when the stiffness is 1,850 foot-tons ; and the stiffness vanishes at 60°. At this point there is unstable equilibrium, and if the vessel be slowly pushed beyond it, she must continue to heel until she reaches another position of stable equilibrium. If there be such a position, short of her being bottom up, she is said to be "on her beam ends.”
Now consider the vessel to be suddenly exposed to the action of a steady breeze, producing an upsetting couple of 1,000 foottons. This wind-couple will be in excess of the righting-couple until 11° 30' of heel. It will then be balanced by the rightingcouple; but the vessel will not stop at that point, because it will have accumulated a quantity of mechanical work, represented by the area of the triangle Owp; it will continue to heel, with diminishing velocity, until this work has been expended by the action of the righting-couple in excess of the wind-couple. This will take place at about 21° of heel, when the area ptr, is equal to the area Owp, or, what is the same thing, when the total work done by the wind, represented by the rectangle Owth, is equal to the total work done against it by the righting-couple,
represented by the area Orh. The vessel will then begin a return oscillation against the wind, the applied force with which it tends to return being then measured by the line tr. Suppose now that the steady pressure of the wind-couple is 1.300 foot tons, and that the wind is again suddenly applied, the applied couple will vanish when the angle of heel is 15°, but the vessel will continue to go over beyond this until the area of the rectangle OWTH is equal to the curvilinear area OPDRH. The righting force against the wind will then be represented by the line RT, and since the points R and T are here coincident, their force vanishes, and there is nothing whatever to right the vessel. Therefore, although her statical stability does not vanish until 60° of heel, a wind which would give her a steady heel beyond 15° would capsize her if it came as a sudden gust. *
Through what follows, I neglect the diminution of the effect of the wind on the sails, by the vessel's heeling. This is not sensible until very large angles are reached, especially when bellying of the sails is taken into consideration. Besides this, the reasoning involves several assumptions which are not in accordance with observation, and it omits others which ought not to be neglected.
We assume that the displacement remains invariable, and we neglect all keel-resistance and friction. Evidently, if part of the work done by the wind be taken up by these obstructions, the result will be more favorable for the vessel. Again, we assume the gust to be suddenly applied,—that is, bursting suddenly from a calm into its full force, and lasting long enough to upset the ship. Now this is quite contrary to what we know of the propagation of atmospheric waves, especially away from the coast. Therefore, the work done by the wind should be represented, not by a rectangle, but by a curve beginning from 0. Thus, the curve of stability being given by the plain line in Fig. 26, the wind curve would be given by the chain line, and safety would depend upon the area OMP being less than PNR. On the other hand, we have entirely neglected the effect of waves. These will sometimes tend to right the ship, and sometimes to upset her. In considering the limit of safety we must take the worst case.
In a stormy sea, with waves, the statical stability of a ship may be supposed to oscillate about the calm-water stability or stiffness.
According to the data stated at the court martial, this would have happened at 13° with H.B. M. ship Captain.
Thus for a particular amplitude of wave-motion we may have something of this kind:-NP (Fig. 27) being the righting moment for still water, this moment will oscillate from Np, to Np, in wave water. We have no means of calculating what this oscillation may be, because it depends upon the mechanical composition of the wave as well as on the geometrical form.
Of course, if a ship lurches beyond her proper statical heel, the curve of righting moments in wave water will oscillate about the still-water curve, like Fig. 28, which, however, represents only one particular combination of phase between the wave and the lurch. But if we could draw all the curves corresponding to every variety of phase, we should obtain a belt, the inner edge of which (envelope of the different oscillatory curves) would give a limit within which none of them would pass. If we then apply
to this curve the construction first used, we shall obtain an inferior limit to the capsizing angle-that is to say, a limit of heel short of which the righting-moment will still exceed the upsettingmoment of the wind-couple. But we have no means at present known of setting off the curve Op, and the object here is simply
to call attention to the fact that it must lie within the curve OP and considerably within it in very rough water.
This includes the rolling of the ship, so far as relates to the angle which she will bear without risk of capsizing. But it has no reference at all to the dynamical stability, or stored work, inasmuch as the phase of the wave does not remain unaltered during the period of heeling. Evidently the accumulation of work depends on the individual curve, and its limiting conditions are not to be inferred from the envelope of the family of curves.
Reverting to Fig. 26, we conclude, that for the curve OPNR, we must take, not the curve of stability given in Fig. 25, but the inner curve of Fig. 29.
It follows that the angle at which the statical stability altogether disappears may very easily be three or four times the angle of safety due to the wind measured statically. That is to say, in a stormy sea, it is conceivable that a vessel might capsize with a gust equal in force to a steady wind which would heel the vessel 15°, while yet the statical stability would not vanish until an angle of 60° was attained. For small angles a sudden gust pushes a vessel to double the statical angle; but for critical angles the statical angle has to be considered at both ends of the curve of stability. Moreover, although there is no such thing practically as an absolutely sudden gust, yet the gradual increase of the wind may be much more than compensated for by the possible diminution of stability due to the waves.
The testimony of early writers on this subject puts it almost beyond doubt that in the older types of wood sailing-ships there was generally a great excess of buoyancy in the middle, and deficiencies of buoyancy at the ends only. In later sailing-ships there were portions of the amidship length (in wake of water, ballast, and other concentrated weights) of which the weight exceeded the buoyancy; and this excess, as well as that due to the heavy extremities, was counterbalanced by the surplus buoyancy of the portions of the ship intermediate between the middle and the extremities. With the introduction of steam as a propelling agent, and of very largely increased lengths and proportions for ships, a vastly different state of things has been brought
* From Naval Science for July, 1872.