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ABS TECHNICAL PAPERS 2007

Range of Stability applied to the Internal Subdivision of Jack-up Lessons Learned 325

Eleventh International Conference The Jack-Up Platform

Design, Construction & Operation

Range of Stability applied to the Internal Subdivision of Jack-up Lessons Learned

J. Andrew Breuer - Chief Engineer -Offshore Engineering Dep. ABS Joseph H. Rousseau – Manager Offshore Engineering Dep. ABS

Karl-Gustav Sjölund - Consultant Researcher - SeaSafe AB, Sweden

Abstract

More than two years have passed since the adoption of a new subdivision standard for Jack-up. The actual application of the Rules has been successful but it also resurfaced an old offshore stability question. The resolution of this challenge became a key to the correct application of the Rule and has derived in the development of a new approach to stability analysis. The findings question the validity of methodologies that have been applied since the dawn of offshore drilling and the deeper understanding of the quasi-static approach to the stability evaluation of floating structures.

Keywords: Offshore, Stability, mobile offshore unit, damage, range of stability

Introduction

The new standard for subdivision published in the ABS MODU Rules and applicable to all Jack-up contracted after January 1st, 2005 has become common practice but what appears today a simple ex-ercise of stability analysis has met complications that demanded urgent resolution. The new criterion requires a minimum range of stability after the flooding of any single compartment. Questions with the calculation of righting arms have been documented before by van Santen et al [1] but until re-cently, the question was regarded mostly as an academic curiosity. With the investigation of the range of stability after damage, a deeper understanding became crucial to the correct application of the stability for damaged jack-up. Thus, the question set in 1986 now demands an answer.

In the process of investigating this matter, a new stability analysis methods has been developed that has open a new approach to the entire stability research and has resolved the question set by refer-ence [1]

History

The revaluation of the stability of Jack-up that followed rowan GORILLA I, the INTEROCEAN II, and the WEST GAMMA in 1988, 1989, and 1990 respectively has been the subject of much re-search, and many reports, papers and Safety Notices by various regulatory bodies. Perhaps, the most valuable finding of all the research is the root cause and the sequence of events that prevailed in some 25 jack-up losses since 1956. A consequence of this finding has been the publication of a rec-ommended operations practice that has resulted in a drastic reduction in the frequency of jack-up in-cidents during tow.

* The views expressed in this paper are those of the authors and do not necessarily reflect those of the American Bu-

reau of Shipping or Seasafe, AB


326 Range of Stability applied to the Internal Subdivision of Jack-up Lessons Learned

On the theoretical arena, no practicable solution for the problem was found, and it was determined that current subdivision practice was generally good despite the lack of standards. Research has also found out that some designs did not follow the practice of conventional subdivision and the safety of such units was questioned. In 2001, ABS carried out a research of all the ABS classed units operat-ing in the UK sector and found and reported that there where no concerning issues with the Jack-up in the ABS record. That research was followed with the development an improved subdivision stan-dard that would be a reflection of the best practice in Industry.

The result is an additional damage stability criterion that is a higher standard, mostly because it ap-plies to all compartments of a jack-up, regardless of exposure to collision or other sources of flood-ing. The new standard is also comprehensive in that it consolidates a number of flooding scenarios, and resolves a many questions raised in what appeared to be a weakness in certain practices.

THE NEW SUBDIVISION STANDARD

The assumption that flooding by way of collision by attending vessels or leakage through the bottom shell is the main approach to subdivision of a jack-up is adequate but not sufficient. Further, flood-ing through the deck is the most frequent cause of loss of jack-up in transit. This led development of adequate standards of internal subdivision of Jack-up. The work done under the umbrella of an-ad-hoc committee convened by ABS was reported in two papers presented at the 9th and 10th Interna-tional Conference in the jack-up platform [2] [3].

The new subdivision requirements, incorporated as a change to the 2001 ABS MODU Rules [6], and in full on the 2006 edition of the Rules [7], retained the existing expectation after damage of (a) posi-tive stability and (b) final waterline under a 50-knot wind below the boundary of weathertight protec-tion. Further, it generalized this criterion to the flooding of any one compartment regardless of expo-sure.

The new criterion, also applicable to any compartment flooded individually, requires that the unit must have a residual stability with a minimum range of stability (RoS) in accordance of the follow-ing:

SoRoS ϕ•+≥ 5.17 or

oRoS 10≥ whichever is greater.

Figure 1



Where φS is the static angle of heel after damage (Figure 1)

The new standard produced unexpected results when applied to certain damaged conditions and di-rection of stability as a peculiar calculation phenomenon occurred.

FADING STABILITY CURVES

A typical stability curve representing righting arms (GZ) as a function of heel, would look like Fig-ure 1 above. However, stability calculations extending to angles large enough to find the second zero crossing of the GZ Curve frequently fail to yield results for the full range of heels specified for the calculation run. The resulting calculations, when plotted, would look as in figure 2. The termi-nation of the curve without reaching its full extent is failure gave room for speculation and detailed analysis. It is difficult to find a good descriptive name of this phenomenon but the most frequent word used recently is “Fading Stability Curve”.

Before understanding this matter in full, we dubbed these results as “Cropped Curves”. These results occur under many circumstances; intact or damage stability, offshore or conventional hulls. How-ever, it is when trying to develop curves to the full range of stability that cropping became an ob-struction to stability analysis.

Another possible explanation of the unexpected termination of the stability curve is convergence problems. Stability software use iterative routines to achieve equilibrium of weight and buoyancy, and their respective centers of application. The sudden changes of shape typical of offshore hulls complicate the iterative routines and the process may fail to converge to a valid solution to the equi-librium equations. Most probably, such failures are related to fading stability and the difficulty to establish the definition range of the stability curve. Outside this range no valid solution exists.

A close look at the stability parameters at the final angle of the cropping curves show vanishing sta-bility in the direction perpendicular to the axis of rotation. This finding may lead to the incorrect conclusion that, while the hull is inclined in one direction, the hull capsizes transversely.

Figure 2 Fading Stability Curve



RETURN TO FUNDAMENTALS

Many of the concepts and procedures used in naval architecture are based on simplifications that are mostly applicable to conventional vessels with slender hull forms. The best known of these simplifi-cations is the acceptance that transverse stability is sufficient to establish performance. While we do not accept this simplification for the stability of offshore structures, many others ship-based simplifi-cations continue to influence the rational applied to offshore structures.

We have questioned two conventional principles to resolve the problem presented by the fading sta-bility curves; range of stability, and stability evaluation about a fixed axis – fixed or free trim.

Range of Stability

Conventional knowledge defines range of stability as the angular difference between the first and second zero crossing of the stability curve as shown in figure 1. This definition carries the sugges-tion that the motions of the hull in the process of heeling follow the path (sequence of heel-trim combinations) by the GZ curve. We have redefined RoS as an angle to which the hull can be in-clined from a position of equilibrium and from which the hull will return to the original position of equilibrium. The main innovation on this is that the “path” or sequence of hull positions from the inclined to the position of equilibrium is not significant.

Stability about a fixed axis

Stability of conventional ship hulls is calculated by rotating the hull about a longitudinal axis. The advent of the computer allowed the introduction of the free trim principle that allowed balancing the

hull longitudinally by rotating the hull about a transverse axis. The free trim concept is applied also for offshore structures. In this case the analysis needs to account for winds from any direction, not only transverse winds as for conventional ships. For this reason we introduce an evaluation axis, ξ, defined by the azimuth angle Ψ as shown in the figure below. To differentiate the rotations relative this evaluation axis from conventional heel and trim, we use the expressions “generalized heel” or G-Heel for rotation about ξ , and “generalized trim” or G-Trim for an orthogonal rotation (Figure 3).

Evaluation Path

The second principle questioned is the sequence of hull rotation (about an axis). However, in the return to fundamentals, we retained the quasi-static premise of conventional stability analysis. This

Figure 3



means that Dynamics are not part of the analysis and that only gravitational forces are considered.

The return to fundamentals followed the Principle of Virtual Work to establish a rational sequence of heel-trim combinations. To do so, we selected to analyze the work to heel the hull and the corre-sponding change in potential energy.

ISO-ENERGY CONTOURS

Energy to heel and trim

The conventional approach to calculating the energy to heel the vessel consists in the integration of the righting arm moment frequently referred to as the “Area under the righting arm curve”

∫ ⋅=−Θ

θϕ

00 dGZEE PP (1)

This method is so widely used that most intact stability, and sometime damage stability, stan-dards will express requirements of heeling energy in terms “areas under the stability curve.”

The work needed to incline (further referenced as energy to incline) corresponds to the increase in potential energy (EP) and can be determined by more rigorous method.

Figure 4 Conventional calculation of energy to heel



For a constant displacement, the energy to incline can also be determined by the change in potential

energy from the equilibrium position to the inclined position. The increase is proportional to the change in vertical distance between the center of gravity and the center of buoyancy.

Figure 5 illustrates the methodology and the EP equation is:

( )BGZBEP −⋅Δ= 11δ (2)

( )BGZBEP −11~ (3)

Where ∆ is the displacement

Representing of the Energy to Incline

To visualize the energy we developed a model of a fictitious jack-up. A typical void compartment

adjacent to the aft end of the hull is also flooded with direct communication with the sea.

For the selected damage condition, with a constant position of the weight and center of gravity, an azimuth of 270o, we calculated the energy to incline for a matrix of heel-trim combinations.

The calculated energy to incline represented in an orthogonal system of Heel (X-axis), Trim (Y-

Figure 5 Rigorous determination of the energy to heel

Figure 6 Model used for calculations and illustration



axis), and EP (Z-axis). The energy at the position of static equilibrium after damage (PSE) is the en-ergy baseline.

In an orthogonal 3-axis system, the matrix of results of G-heel, G-trim, and Energy, fit a surface de-picted as a topographic chart by presenting EP in terms of iso-energy contours. Having established the increase in vertical distance between the centers of Buoyancy and Gravity (BG), the analogy be-tween the energy surface and the motions of a particle on a physical surface is evident and of help to understand statics of stability afloat.

Nature of fading stability

To understand the subject event, the heel and trim calculated in the development of the GZ curve il-lustrated in Figure 2, we have plotted the sequence of G-Heel and G-Trim in Figure 8. It can be seen that as the hull is heeled, the initial trim (2o approximately) remains relatively constant up to after 3 degrees of heel beyond the position of static equilibrium. After that point, there is an increasing G-trim and culminates at a point where G-Heel does not change and the rotation is only in the “trim di-rection”.

Because motions of the hull will not increase the heel, the point of fading stability can also described as the point of refusal. There is more to this point but will be best explained after understanding the energy surface.

Observations on the Energy Surface

The general shape of the energy surface will change dramatically with the intact or damage case and the parameters of the calculation but several points and lines are notable on all surfaces (Figure 9).

Figure 7 Energy surface and Iso-Energy contours



Because the undisturbed hull will “balance” at the lowest point of energy achievable, the low point in a depression will always be the representation of a static position of equilibrium. If a surface covers the full range of heel and trim, more than one position of equilibrium is likely to be found; usually corresponding to the upright and capsized conditions. Hulls with a CG below the CB, such as spars, submarines, drydock gates, will result in a single depression consistent with the unlimited stability of these hulls.

On conventional damaged offshore hulls, the general shape will have the depression with several “peaks” around them and the ridgelines will lie between peaks.

The ridgelines are notable because they are the slope “divide” and therefore constitute a “watershed”. This means that an analogous particle will roll down one side of the hill or the other, depending on which side of the watershed line it starts its motion.

Following the analogy of floating stability and particle motions on a surface, the watershed line is the

boundary of positive stability. A hull inclined beyond a combination of heel and trim will not return to the position of static equilibrium. Therefore, the watershed line corresponds to the second zero crossing on the righting arm curve. Mathematically, if

Figure 9 Notable points of the energy surface



∫ ⋅Θ

0~ ϕdGZEP (4)

Then

ϕ∂∂ PEGZ ~ (5)

Thus EP will have a maximum value when GZ = 0

The watershed line segment between peaks will have a “low point” or minimum. These saddle points are special points of capsizing as they are at relative low levels of energy of the ridgeline. In the particular case shown in Figure 10, the surface has three saddle points. The “distance” – meas-ured in angular difference - between the position of static equilibrium and each of the saddle points constitute a range of stability. Further, the distance from the point of static equilibrium to the saddle is the minimum range for the family of righting arm curves calculated for that general direction.

Going back to Figure 8 the GZ calculation depicting the sequence of heel-trim combinations, the point of fading stability is reached before the point of capsizing; this dismisses the notion that the point of fading stability constitutes a loss of stability. Further, because the latter point cannot be reached by inclining the hull around the established axis, the concept of range of stability is not ap-plicable to the case.

Energy surface contours have shown to require extensive calculations. However, once a surface an-other notable aspect of the energy surface is that, the surface is independent of the sequence of mo-tions of the hull. Thus for a given displacement Ep is a function n of the vertical position of the cen-ter of gravity (VCG) and the values for any other VCG can be determined by direct calculation

Insofar as range of stability is concerned, the most important conclusion from this is that GZ fading stability curves cannot be used to establish RoS.

Figure 10 RoS on iso-energy contours



Waterplane parameters in the Iso-energy contours

Assuming the hull fixed to the coordinate system, the waterplane is described by the vector normal to the plane:

0=+• drN (6)

( )( )kCosCos

jSinCos

iSinN

ˆ

ˆ

ˆ

ϕθ

ϕθ

θ

+

++

+=

(7)

Where: N is the Plane’s normal R is a vector from the origin to any point in the plane D is the distance of the origin to the plane Θ is the angle of G-Trim φ is the angle of G-Heel

The angle Ψ from the axis of azimuth to the axis of inclination can be calculated as:

000

)/(2/)(

)/(

2

2

2

21

1

21

>=<

⎪⎩

⎪⎨

⎧

−−−

=NifNifNif

NNArcTanNsign

NNArcTanπ

πψ (8)

or, for inclinations less than π/2 radians,

000

)/(2/)(

)/(

>=<

⎪⎩

⎪⎨

⎧

−−

−=

ϕϕϕ

ϕθπθ

ϕθπψ

ififif

SinTanArcTansign

SinTanArcTan (9)

The angle ψ’ between the vessel’s longitudinal axis and the axis of inclination is:

ξψψ +=′ (10)

If we accept that rotation about a fixed axis is moot, we must accept that the range of stability is the difference between the inclination of the waterplane between the position of static equilibrium and the angle of capsizing. This includes that the rotation direction of the hull in the position of equilib-rium is not the same as the one at capsizing.

To illustrate this point, Figure 10 shows the watershed lines and a line representing an inclination of 10o beyond the angle of static equilibrium. RoS in the Iso-energy contours corresponds to the incli-nation of the hull at the position of static equilibrium and the closest point on the watershed line – most likely at the nearest saddle point

ALTERNATIVE INCLINING PATH

Dynamic descent

If heeling about a fixed axis is in conflict with fundamentals of physics, the question to resolve is



which sequence is the theoretically correct path and what assumptions we must make to reach that answer.

Following the analogy of the particle moving on a surface, we could assume that a sphere in the posi-tion of static equilibrium, receives an impulse to roll uphill. Dismissing the effect of friction, the sphere will reach a certain elevation and then endlessly roll down and up the hill. With such motion, the sum of potential and kinetic energy remains constant. While special cases could result on an end-less repeat of the same path, most likely, the sphere will roll following an apparently random path.

Steepest descent method.

The alternative to a dynamic approach is a quasi-static path. This means that as the particle returns to the position of equilibrium, the change of potential energy does not convert to kinetic energy. Un-der such a premise, the particle would follow the steepest descent path (SDP) such that the potential energy will deplete through the most efficient path. The path is a function of the starting point and the initial direction imparted to the particle.

In the quasi-static approach, the SDP is the theoretically rigorous solution and its resolves the evalua-tion of RoS. While the dismissing dynamics is significant simplification, the adoption of quasi-static is the only way to resolve stability at a regulatory level. In the context, SDP is not only more rigor-ous than the conventional fixed axis approach but opens the door for new and more accurate methods in the realm of offshore stability analysis

The SDP has the simplicity needed to resolve stability expediently and it follows acceptable princi-ples of physics. A dynamic approach, while possible, is extremely complex and impractical due to the random nature of the environmental forces and excitation by the environment.

With the SDP approach, the starting point and the initial direction of motion determine the path. The

kCosCosjSinCosiSinN hhhhhhˆ)(ˆ)(ˆ ϕθϕθθ ++−= (11)

kCosCosjSinCosiSinN ccccccˆ)(ˆ)(ˆ ϕθϕθθ +++−= (12)

)( ch NNArcCosRoS •= (13)

))(( hcchch CosCosCosSinSinArcCosRoS ϕϕθθθθ −+= (14)

Figure 11 Steepest descent Paths



path is reversible; meaning that the path does not depend on the direction of the motion, and that the path of energy buildup path will be identical to the path of energy depletion.

If the vessel moves from its equilibrium to an arbitrary point somewhere within the boundary of the “watershed”, the moment resulting from the gravitational forces will always rotate the vessel along a SDP. This is because the moment vector is always parallel to the SDP.

When the steepest descent method is used, we intrinsically allow the hull freedom to rotate relative the direction of the heeling and righting moments. The freedom to rotate is the fundamental differ-ence between the SDP method and the conventional approach. This freedom prevents the occurrence of fading stability.

Figure shows the families of paths that follow the SDP principle. We can note the following proper-ties of the SDP:

1. All paths pass through the extreme points of the EP surface - peaks and position of static equi-librium.

2. The path are perpendicular to the contour lines at the point of intersection.

3. Boundary lines between families of lines, a dividing line connecting the point of static equilibrium with the saddles, define the direction of minimum range of stability.

4. The boundary lines are a singular case of a SDP but because the line reaches the saddle, the hull capsizes without progressing on the watershed line.

5. There is no fixed relation between the moment direction and the inclination axis.

6. There is no fixed relation between the moment direction and a body-fixed axis.

Analysis of stability along a SDP

A Stability curve can be developed in association with any SDP under certain assumptions.

The GZ-curve is a function of one parameter ξ. For the steepest descent rotation path, the natural choice of ξ is the rotation along the SDP, i.e. a rotation that is parallel to the moment vector at all times. The value of ξ in a given point thus equals the length of the path measured from the point ξ = 0. This choice has two important advantages:

1. Only through this choice will the area below the GZ-curve be proportional to the buildup in energy. This is because the rotation is always parallel to the moment vector along the SDP.

2. It is always possible to present GZ as a function of ξ irrespective of how the path twists and turns, since the value of ξ always increases along the rotation path. Therefore, the GZ curve never fades. The curve allows evaluation of range of stability and other stability parameters.

A typical SDP Righting arm curve GZ = f(ξ) is illustrated in Figure 12.



As indicated before, the boundary line between families of SDP is a singular path that must be paid special attention. Because they terminate at the saddle point, a relative minimum EP, this path is likely to be a critical one.

The SDP stability curve allows evaluation in the same way as a conventionally calculated stability. This includes the typical critical angles such as, first (ξ1) and second intercepts (ξ2), first (ξ0) and second zero-crossing (ξc - static equilibrium and capsizing), and the angle of maximum righting arm (ξm). However, the concept of directionality, heel and trim is no longer relevant as the angle evalu-ated is the inclination of the hull and the concepts of heeling about an azimuth axis and the trimming of the axis are no longer valid.

If the analysis assumes the superposition of a wind overturning moment, a relative rotation of the hull about a vertical axis must be accepted because the directionality of wind must be consistent with the rotation along the SDP.

ISO-ENEGY CONTOURS – FUTURE POSSIBILITIES

The determination of the range of stability by direct calculation from the contours is only one of the many possible applications.

Experience on the application of the Steepest Descent Method to intact stability has proven is useful-ness even when following the conventional methodology for establishing the critical direction of sta-bility. Because the critical direction is one of the reasons for different results from designers and regulators, designers may gain in confidence that their findings will be approved.

The biggest potential is predicted if the contours of wind energy contours are developed in the same fashion as the Iso-energy contours. This should allow the determination of the 1.4 area ratio (or 1.3 for Semi-submersibles) boundary line that would determine by direct reading the most critical direc-tion of stability. A follow-up step would be the plotting of the contour of downflooding. This will lead to an unquestionable critical direction and the allowable VCG by almost direct reading from the contours.

While these predictions of the potential for the Iso-Energy contours, the first versions of the software used for this investigation have yielded excellent results at the cost of extensive computations. Im-provements in the algorithms will result in faster performance and quicker results but we can predict that the future developments will demand optimum codes and powerful processors.

Figure 12 Steepest Descent Path Stability Curves



CONCLUSIONS

Experience shows frequent problems to establish Free-Trim GZ-curves for different types of offshore units. The evaluation on the Range of Stability (RoS) in offshore structures and unconventional hull forms is complicated by the incompatibility between the conventional methods and the scientifically more rigorous motions of the hull.

Conventional free to trim stability procedures are inadequate to establish Range of Stability because it assumes an unrealistic sequence of heel-trim combination.

Conventionally obtained righting arm curves in offshore can terminate at unexpected angles of heel. The fading of these curves is not an indication of capsizing and do not constitute a second zero cross-ing of the GZ curve. If conventional methods are applied, fading GZ curves should be disregarded if evaluating range of stability.

The energy to incline a hull is a function of the initial and final position of the hull and not a function of the sequence of heel trim combinations.

Range of stability and other stability properties can be determined by analysis of the energy to in-cline surface.

Applying the steepest descent path to the evaluation of intact and damage stability is a more rational method than the conventional free to trim.

The steepest descent path allows the rotation of the azimuth axis and resolves the problem of fading stability.

REFERENCES

Joost van Santen et Al, “Stability calculations for jack-ups and semi-submersibles”, Conference on Computer Aided Design, Manufacture and Operation in the Marine and Offshore Industries, Sep-tember 1986

Stability Calculations for Jack-up and Semisubmersibles – J. A. van Santen – Conference in Com-puter Aided Design, Manufacture and Operation in the Marine and Offshore Industry

The Future of Jack-Up Stability - Learning From Our Past. Ninth International Conference-The Jack-Up Platform, Design, Construction & Operation, 2003,London, England

Breuer, J.A., Bowie, R., Bowes, J., “Internal Subdivision of Jack-ups- a new standard to resolve an old concern”, Tenth International Conference on the Jack-up Platform, City University, September 2005

Breuer, J.A., Karl-Gustav Sjölund, “Orthogonal Tipping in Conventional Offshore Stability Evalua-tions”, STAB2006, 9th International Conference on Stability of Ships and Ocean Vehicles

American Bureau of Shipping – Rules for Building and Classing Mobile Offshore Drilling Units, 2001

American Bureau of Shipping – Rules for Building and Classing Mobile Offshore Drilling Units, 2006

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