Rev 12/2008

I. Introduction

Floating platforms, being typically as wide as they are long, are subject to capsizing in various directions. This presents a challenge to the traditional calculation methods which work well with long, narrow ships that capsize primarily in the transverse direction. But if the direction in which capsize is most likely to take place is not fixed by the design of the vessel, the task of evaluating stability must include finding the direction of least stability.

In order to evaluate stability against the limits established by regulatory agencies, an idealized capsizing event is simulated. This event is represented by the righting-moment curve where the righting moment as a function of inclination is measured in various ways and compared to the limits set by the regulations. Different curves will typically result from inclining in different directions. The curve to which the stability regulation should be applied is the one which exhibits the least stability as measured against the prescribed limits.

This report is primarily about methods that might be used to find that righting moment curve representing the least stability. The additional steps needed to compose curves of maximum VCG are discussed only as they relate to this problem. Of particular concern is "regulatory stability", i.e. the evaluation of intact and damaged stability as required by regulatory agencies for normal operations. (In special cases a more detailed "engineering stability" evaluation may be necessary which might take into account forces that are not specifically addressed in regulatory stability.)

II. Theory

In general, the response of a floating platform to an overturning force cannot be assumed to be an inclination in the same direction as the direction of the force. If the vessel is not perfectly symmetrical on either side of the force it will tend to incline unevenly. However, the work done in producing the inclination is in the direction of the force. If stability is being measured by the energy available to resist the work done by the overturning force (it is assumed here that this is the case), then the curve of righting moments should be in terms of that component of inclination which is in the direction of the overturning force.

In reality, the direction of the overturning force could be influenced by various factors other than those normally considered for regulatory stability analysis. If the overturning force is wind, the vessel may tend to yaw and align itself in such a way as to avoid being inclined in certain directions. This would be due to the arrangement of the structures on which the wind acts. Anchors and dynamic thrust may also affect or limit the yaw. However, for the purposes of regulatory stability analysis, it will be assumed that the righting-moment curve possessing the least stability is the one to be assessed, regardless of any yaw response, control or limitation.

The righting moment is, in the absence of wind, determined by the characteristics of the vessel's buoyancy and weight distributions. The presence of wind, inducing an overturning moment, adds another variable, since the geometry of the vessel's structure above the water is what converts the wind into a force. Further, the vertical distribution of the geometry above and below the water converts the wind force into the heeling moment that opposes the righting moment. The net (or "residual") righting moment is the buoyant (or "absolute") righting moment minus the wind heeling moment.

Essentially it is the residual righting moment curve -- what is left after subtracting the heeling moment -- that must supply the energy to resist the transient overturning energy from waves. The direction of the wave energy is not necessarily in the same direction that the wind is acting, but the total overturning energy will be greatest if the wave energy is aligned with the wind. Therefore, in the generation of righting-moment curves, the direction in which all of the overturning work is being done will be assumed to be the same direction in which the wind force is applied.

As a matter of terminology, the direction in which the righting-moment curve is derived (the same direction in which the wind force acts if there is wind) will be called "heel", and the direction normal to that will be called "trim". Total inclination is a combination of heel and trim and typically will be more closely aligned with heel than with trim, since trim is a secondary response to the overturning moment in the heel direction. These "directions" are actually directions of rotation and are more precisely spoken of in terms of the axis about which rotation takes place. The axis about which heel takes place will be called the "axis of heel" or simply "the axis". The orientation of this axis is such that it is always parallel to the waterplane.

At a particular angle of heel about a particular axis (trim being adjusted for zero trimming moment since there are no forces available to support a trimming moment) there will typically be different incremental residual righting moments if the heel axis is changed. Changing the heeling axis incrementally is equivalent to the vessel yawing, assuming that the wind and wave direction is constant. Since this is a real possibility, it should not be assumed that the heel axis orientation must remain constant as the heel angle increases.

Certainly there are time-related inertial forces which are ignored in this analysis. It is assumed that the regulatory stability limits have been set to account for this discrepancy.

Given a particular operational category, the required final product of the regulatory stability analysis is a curve of maximum VCG as a function of draft or displacement. The maximum VCG at a particular displacement, when adjusted to reflect existing free surface in tanks, is such that the regulatory stability criterion is met, but would not be met if the VCG were any higher. Finding the maximum VCG for a particular displacement typically involves examining the intact vessel as well as in several damage conditions, finding the maximum VCG for each and taking the curve of least maximum VCGs as the one which governs.

III. Methods

A. Fixed Axis

In theory, as it was pointed out, the least-stability righting-moment curve might be produced when the axis about which heel takes place is varied as heel progresses. However, there may be reasons for not allowing the axis to vary with heel, such as: a) the geometry is such that the axis of least stability clearly does not vary with heel; b) the results obtained with fixed axis are acceptably close to those obtained with varying axis; c) the accepted method of calculation does not require varying axis; d) the available software does not provide for varying the heel axis.

If the heel axis remains fixed as heel progresses, there must be a method of finding that axis angle. The method would involve searching the range of azimuth angles and applying a criterion to evaluate the relative suitability of setting the axis at that azimuth. Because of the idealized approach to calculating stability, the criterion which works best (most reliably finds the axis of least stability) is not necessarily identical with the regulatory criterion applied to the final righting-moment curve. Some candidates for this axis-selecting criterion are: a) the same as the regulatory criterion for evaluating the stability; b) energy over the range of stability; c) energy to the angle of maximum residual righting moment; d) maximum residual righting moment; e) initial GM; f) initial slope of the residual righting moment; g) difference between total angle of inclination and heel angle (i.e. an axis angle where the free trim angle is zero).

The complexity of the stability response to VCG changes is in general such that closed-form solutions are not feasible. The solution is found using optimization techniques which are essentially trial-and error processes. As the VCG is raised, the stability, as measured by the appropriate criterion, is expected to decrease.

Two approaches are possible in applying the above-mentioned methods of selecting the axis.

1. The selection could be imbedded in the maximum-VCG-finding process. At a particular VCG the axis is selected. But there is no guarantee, in general, that the same method would yield the same axis when the VCG is changed. Therefore, some method of deciding when to reselect the axis must be used. Obviously, before the maximum VCG is reported, the axis selection should be checked and the process continued if a new axis indicated.

2. The other approach is to simply run a series of maximum VCG calculations at various axis angles, and select that which yields the lowest VCG.

B. Variable Axis

If the axis is allowed to vary as heel increases, a method is needed for selecting a new axis at each heel increment. This method might be: 1) minimum ascent (least slope of the residual righting moment vs. heel); 2) minimum GM (least slope of the absolute righting moment curve); 3) zero trim (the axis where the heel is the same as the total inclination).

C. How to Select a Method

If the task were to simulate nature, the method that best adhered to physical laws would be the best method, at least from an accuracy standpoint. But rigorous simulation is not contemplated in regulatory stability criteria. Rather, a certain level of abstraction is required, primarily that the water is flat, and that time-related forces are not directly involved. Another abstraction is that external yawing forces are ignored with the implication that the wind direction yielding least stability is to govern. So the essence of the task is to find the residual righting-moment curve having the least stability according to the given criterion.

Again, the best method of doing this cannot be judged by its adherence to physical principles because gross abstractions are inherent in the underlying methods. In the final analysis the winning method is simply whatever method finds the least stability under the given criterion.

Although a strict physical simulation is not appropriate, blindly forcing the generation of a righting moment curve at some arbitrary axis setting may encounter difficulties because some physical principle is being ignored. The well-known example of this is when the heel axis is such that the vessel becomes unstable in the trim direction; i.e. there is difficulty in maintaining zero trimming moment because of this off-axis instability. To merely discount or avoid such axis angles on that basis may be acceptable, but the justification for doing so is not obvious. It would be more satisfying to have another method of selecting the axis which a) does not excite trim instability, b) has a basis in relevant physical principles, and c) in practice finds equal or lesser stability than other methods find.

A good candidate for this axis-selecting method is the method which looks at the maximum residual righting moment (and selects the axis exhibiting the least maximum). It tends to avoid exciting trim instability because it does not need to consider heel angles beyond the angle of maximum righting moment. It avoids the problems with ill-defined angles of maximum righting moment (sometimes there are multiple angles) because it depends on the value of the maximum moment, not the angle. However it still tends to track energy (the most important measure of stability) because similar righting-moment curves will have energy proportional to the value of the maximum moment.

The answer to the question of whether to use variable axis may depend on the known characteristics of the vessel and the capabilities of the available software. However, with flooded compartments the buoyancy characteristics can change so much that it would be difficult to prove without some experimentation it that variable axis would not find lesser stability.

If a variable-axis method is used, the one of those mentioned above that is most closely aligned with physical principles is the minimum-ascent method. The inclination path which this method produces is the one of least incremental energy at each increment of heel. Therefore it is not unreasonable to expect that it will yield a righting-moment curve having as low or lower energy characteristics than the other methods.

The traditional method is to search for the "worst axis" by comparing stability evaluations at various predetermined axis angles. This method will typically encounter trim instability if a full range of axis angles is attempted. It has the advantage of allowing the wind heeling moment curve to be given anew before the stability evaluation for each axis; thus it is easy to use with heeling moments from an external source. It has an additional disadvantage of not lending itself to automatic maximum VCG curve generation since one cannot be certain that the same axis is "worst" at all displacements, especially with damage.

IV. Experimental Results

Preliminary experiments were carried out using a simple geometry consisting of a hull plan having a trapezoidal shape and a highly directional superstructure element. An advantage of simple geometry is that it can be readily designed to produce certain stability characteristics which are expected to lead to difficulties.

The complexities of a real offshore platform all contribute to an overall response which could, in theory, be represented by a much simpler "equivalent" model. It could be argued that since the equivalent model would have proportions that represent an average, there are no grounds to expect that it would present difficulties in a stability analysis. However, experience with detailed models of real offshore platforms has shown that characteristics which lead to computational difficulties during a stability analysis are not uncommon. Flooding simulations, in particular, can cause a well-behaved intact model to exhibit a variety of responses quite different from those of the intact vessel.

The main objective in the design of this particular experimental model is to investigate the effect of nonuniform wind drag with respect to azimuth and to develop software tools that adequately handle this effect. The trapezoidal hull plan is not extreme, but the superstructure shape leads to high transverse heeling moments and low longitudinal heeling moments while the hull has least stability longitudinally.

Further experiments are needed using a variety of models representing types and sizes of actual vessels. While such experiments will certainly help to validate and refine the software tools, it is unlikely that they will be as directional with respect to wind drag and therefore will be less demanding on the methods used to deal with the heel axis direction.

Two views of the model are shown below. The hull is 40 meters long, 10 meters deep and symmetrical about its longitudinal centerplane. All sides are vertical and the bottom and deck are flat. The half-breadth is 30 meters at the aft end and 14 meters at the forward end.

The superstructure portion of the model is a rectangular prism rising 10 meters above the deck. It is 40 meters long and two meters wide. An 80-knot wind was used to produce the heeling moments.

The essential report of the experiment is the graph shown below which compares maximum-VCG curves at various fixed-axis angles with a curve derived using a variable axis where incremental heel direction changes were made using the

Click here to download report pdf.

V. Conclusion

The selection of the heel axis about which the wind heeling moment is applied should be carried out at the lowest possible level in order to discover the worst stability behavior. If this is not possible, due to current software availability, an axis that is fixed with respect to heel should be chosen at each displacement and maximum VCG. If the axis is fixed, choosing it on the basis of the least maximum residual righting arm avoids some computational difficulties.

VI. Software Tools

GHS versions 10.28 and later include the following new features which are useful in this context.

Fixed Axis Selection

AXIS MINGM

AXIS MINASCENT

AXIS MINRA

AXIS MINE:RA0

AXIS MINE:MAXRA

AXIS *

Variable Axis Method Selection

VARY AXIS:MINASCENT

VARY AXIS:MINGM

VARY AXIS:ZEROTRIM

MaxVCG with Automatic Axis Selection

MAXVCG ... /AXIS:MINRA

MAXVCG ... /AXIS:MINGM

MAXVCG ... /AXIS:MINE `(to RA0)

Wind heeling moments can be generated from the geometrical model using

WIND speed

HMMT WIND /BANDS

Note that in order to use variable axis and/or MAXVCG /AXIS the heeling moments must be generated from the model since the heeling moment must be available not only as a function of heel but also of heel axis angle.

Shielding between components within parts is complete. No shielding takes place between parts.

The recommended procedure is to use the following sequence of commands.

WIND speed

HMMT WIND /BANDS

VARY AXIS

LIMIT ...

MAXVCG drafts

(repeat with flooded conditions using MAXVCG /COMPOSITE)

System variables

These variables may be used for research purposes such as developing new stability criteria or methods.

RESMOMH - Residual heeling moment

RESMOMHS - Residual heeling moment slope in heel (dM/dH)

ROSTABH - Range of stability in heel

MAXRRAH - Maximum residual righting arm in heel

BGN - Distance from CB to CG normal to the waterplane

Copyright (C) 2007 Creative Systems, Inc.