Misleading, Bewildering and Unreliable Stability Criteria


Difficulties with Equilibrium Angle as a Measure of Stability

The ability to resist a heeling moment resulting from wind, passenger crowding, etc. or to stay somewhat upright after flooding is often measured by the equilibrium heel angle. However, requiring that this angle be less than some given angle or some characteristic angle such as margin line immersion, downflooding point immersion or half freeboard does not in itself guarantee finite stability. It says nothing about what happens if the heel is increased a bit farther. An assessment of the strength of the equilibrium angle is needed (which is different from the ultimate resistance to capsizing).

With some vessels in certain conditions it is possible to meet a simple equilibrium angle criterion and yet have virtually no stability beyond the angle of equilibrium. Therefore, if the criterion envisions an actual or possible operating condition, it must be augmented by some other measure of initial stability. Possible augmentations would be GM, minimum range of stability, minimum area, or a minimum residual righting arm.

Requiring a healthy GM at equilibrium is a good start. Unfortunately the GM requirement in common damage criteria is too low and does little to rule out curves that quickly fall back to zero or fail to rise significantly until the angle is greater than the intended angle of equilibrium. Even large GMs can fail in this respect.

A range of stability (angle from EQU to RA0) requirement may be effective in some cases, but neither does it adequately address cases where the residual righting arm fails to rise initially. These cases actually have little resistance to heeling for some significant range beyond equilibrium.

A minimum residual righting arm, if combined with a minimum range of stability, is usually effective in guaranteeing that there is some real strength in the angle of equilibrium. But this is similar to an area requirement. The area requirement over some initial range -- perhaps 10 degrees beyond equilibrium -- is most closely related to the real strength of the equilibrium angle. For this purpose the range over which the area is taken should not be bounded by RA0, deck immersion or downflooding. (If the range goes beyond RA0, the negative area at the end of the range is a penalty which must be overcome by more area in the beginning of the range.)

A GHS LIMIT command which would enforce modest initial stability is
   LIMIT AREA FROM EQU TO 10 > 1.0    `in ft-deg
   LIMIT AREA FROM EQU TO 10 > 0.0053 `in m-rad
Note: See also the "AntiLoll" bulletin.

GM at Equilibrium

When the equilibrium angle is not upright, increasing the VCG tends to increase the equilibrium angle. If it so happens that this new heel angle encounters new buoyancy of sufficient magnitude to greatly boost the transverse moment of waterplane inertia, the new GM may actually be higher than the original. Therefore, it is possible that the relationship between GM and VCG at equilibrium is the reverse of what would normally be expected. Therefore, while GM may be useful in guaranteeing initial stability it can also make it impossible to find a maximum VCG that just meets the criterion.

Difficulties with Angle of Maximum Righting Arm

The angle at which a righting arm curve reaches its maximum value can never be known precisely. The reason for this is that it is a property of the derivative of the righting arm curve, not of the curve itself. A property such as the angle of equilibrium is an angle at which the righting arm curve is zero. The angle at maximum is the angle at which the derivative of the curve is zero.

While a mathematical curve may have a precise derivative, a curve representing data obtained by experiment always contains some degree of uncertainty. Differentiation always exaggerates the uncertainty just as integration diminishes the uncertainty.

Whether measured by inclining a physical ship or inclining a computerized model of the ship, the righting arm curve contains experimental errors which are exaggerated by differentiation. Therefore the angle of maximum righting arm cannot be known with a precision comparable to the precision of the angle of equilibrium.

The differentiated curve (which plots the slope of the original curve) is typically more "bumpy" than the original; and it may go through many more slope reversals than the original. The closer you look; i.e. the more samples or angles you take, the bumpier it gets. This is because for closely-spaced samples the predominant difference is the experimental error: an attempt to examine the "fine structure" of the curve finds only the "noise" of the experimental error. Indeed, if no distinction is made between the property being measured and the noise, the results will be unrepeatable and bewildering.

Take, for example, a righting arm curve which actually has a "ledge"; i.e. for some range of angles the arm stays constant. Adding to this the experimental error, you find that it is not quite constant; in practice there will most likely be an angle of maximum perceived value within that range. While it may be insignificant that this so-called angle of maximum is not quite true (in fact there is no angle of maximum in the range) it may be significant that a later attempt to find a maximum in the same range comes up with a different angle due to a different stream of experimental errors.

This fact causes difficulty when the angle of the maximum becomes the controlling limit when attempting to find a maximum VCG which just satisfies the limit. Each time a slightly different VCG is introduced, a different stream of experimental errors, exaggerated by the differentiation, influence the angle of the maximum. For some range of VCG variations, the experimental errors (which are not proportional to the change of VCG) will be more significant than the actual change in the curve due to the change of VCG. This tends to bewilder the process of solving for the maximum VCG since the relationship between the VCG and the limit margin has become "noisy" and when taken at fine enough intervals is essentially random.

It is unfortunate that the angle of maximum righting arm has been included in some stability criteria. While the computational difficulties are not unsurmountable, the results often bear the effects of the experimental error. This must be understood by those dealing with the results and appropriate allowances made.

Jumps in RA0

In some damage cases as the VCG is raised the righting arm curve develops a dip which eventually comes down to the axis. At that point RA0 angle, and probably the angle of maximum RA also, make a sudden jump down. If all of the limit margins are positive until this point is reached, some are likely to go negative immediately as the VCG is raised a bit more. When this happens it's impossible to find a VCG where all margins are positive or zero and at least one is zero, which is what the maximum VCG process is trying to do.

The remedy is to add something to the criterion which is not sensitive to this discontinuity. A reliable way improve it without altering its results when more normal RA curves are present is to add a minimal area requirement that is not bounded by RA0 or any other property of the curve. For example, a limit that improves damage criteria is
    LIMIT AREA FROM EQU TO 10 > 1.0    `in ft-deg


    LIMIT AREA FROM EQU TO 10 > 0.0053 `in m-rad

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