GHS BULLETIN
Misleading, Bewildering and Unreliable Stability Criteria
1/02
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
or
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
or
LIMIT AREA FROM EQU TO 10 > 0.0053 `in m-rad
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Copyright (C) 2011
Creative Systems, Inc.