On Performance-Based Criteria for Intact Stability

Transcription

1 On Performance-Based Criteria for Intact Stability Vadim Belenky 1), Jan Otto de Kat 2), Naoya Umeda 3) 1) Corporate Technology, American Bureau of Shipping Houston, Texas, USA 2) Technical Organization A.P. Møller - Mærsk A/S Copenhagen, Denmark 3) Department of Naval Architecture and Ocean Engineering, Osaka University Suita, Osaka, Japan Presented at PRADS 2007, Houston, TX, USA, 1-5 October 2007 and reprinted with the kind permission of the 10th International Symposium on Practical Design of Ships and Other Floating Structures (PRADS) Abstract The paper addresses the principal issues related to development of performance-based criteria for intact stability of ships. The motivation for the development is examined and terminology for stability failures and types of criteria is considered. Physics of three modes of stability failures (dead ship conditions, broaching and variation of righting arm, including pure loss of stability and parametric roll) is reviewed with numerical examples. Special attention is paid to problems that need to be addressed during the development of probabilistic performance-based criteria, including time dependence, choice of time interval, and the problem of rarity. Different hypotheses of capsizing are examined. The paper also contains a review of methods and techniques of solutions for all three modes of stability failures. Keywords Intact Stability, Performance-Based Criteria Introduction The International Maritime Organization (IMO) started to revise its Intact Stability Code (IS Code) in 2002, triggered by serious accidents due to head-sea parametric roll, as well as design difficulties for large passenger ships and RoPax ferries (Francescutto, 2004). IMO s sub-committee on stability, load lines and on fishing vessel safety (SLF) has used its five sessions for this task. A review of this work is available from Francescutto (2007). As a result, a door was opened for the development of performancebased, or dynamics-based, stability criteria. As the development of performance-based stability criteria in the IS Code is at a very early stage, this paper s intention is to review the state of the art, making an inventory of what is available for the development of a new generation of standards. Sevastianov (1994) described development of any technology standard as a four-step process: 1. Formulation of objectives for the standards 2. Development of criteria 3. Justification of boundary for the criteria 1 4. Assessment of the quality of the newly developed standard and consequence of its implementation. It could be expressed in the form of warranty, the likelihood that the declared objectives will be achieved if the standard is fully implemented. While the main focus of this paper is on the formulation of objectives and criteria development (as in SLF 50/4/4), relevant issues will be considered from a wider perspective. Objectives Motivation for the Development Existing criteria for intact stability originated from works of Rahola (1939) based on statistics of 14 vessels that capsized between 1870 and Despite the rather small amount of statistical data, Rahola s criteria were used in many countries for a number of years, and the method of drawing the line between acceptable and not acceptable GZ curves were used in the development of current IMO criteria. Details of this method as well as the history of IMO work in this area can be found in (Kobylinski and Kastner, 2003). 1 Sometimes, justification of the boundary for criteria is included in the previous step On Performance-Based Criteria for Intact Stability 295

2 Apart from statistics-based criteria, the weather criterion is based on physical considerations. It assumes that a vessel with complete loss of power being turned by the wind into a beam seas position experiences heavy resonant roll due to wave and wind gust action. Still, a significant amount of empirical information was used to formulate this criterion (Kobylinski and Kastner 2003). How good are these criteria now? New types of vessels may have unexpected modes of stability failure the case of the Post-Panamax C-11 container carrier may serve as a good example (France et al 2003). The phenomenon of parametric resonance, well-known to naval architects (Paulling and Rosenberg 1959), occurred unexpectedly in head seas with a large containership (this was expected to be more of a potential problem for small and medium-size vessels with insufficient stability in following seas). Head sea parametric rolling had been observed experimentally earlier with a cruise ship at low forward speed (Dallinga, 1993). Extensive research on the dynamic stability of naval ships and development of performance-based criteria started in the 1990s (de Kat et al., 1994) and is still ongoing. Also, for naval ships, the current criteria are based on old ship casualties involving pre-world War II destroyers. The Naval Stability Standards Working Group 2 aims to develop rational, physics-based criteria for intact and damaged vessels based on numerical simulations (FREDYN program developed by Cooperative Research Navies ) backed up by model tests. Significant progress was achieved in ship dynamics research during the past two decades. This is mostly related to the development of nonlinear dynamics, advanced model testing techniques and computational methods. New tools provide hope that some of these adverse phenomena can be predicted and prevented, either at the design stage or during the operational stage of the life-cycle of new vessels. Applicability It is not completely clear at this moment, which type of vessel should be addressed during this development. The IMO document SLF 50/4/4 refers to unconventional vessel as having unconventional hull particulars or geometry or modes of operation for which there is insufficient operational experience. It is clear, however, that the ways of more formal distinction should be developed as a part of the research aimed at the performance-based criteria. One of the possible ways to answer this question is the development of simplified vulnerability criteria. These criteria could only be capable of answering yes or no in response to a question of whether this particular vessel could be, in principal, vulnerable 2 NSSWG comprises navies from Australia, Canada, France, Netherlands, United Kingdom and United States, United States Coast Guard and MARIN to a stability loss phenomenon. It would be especially efficient if such criteria would be based on a physical consideration and therefore applicable to any type of vessel. This approach was implemented in ABS (2004) in a form of susceptibility criteria based on Mathieu equation. The susceptibility criteria being applied to a Post-Panamax containership (which is known to be vulnerable) warns about parametric roll, while an attempt to use it for a VLCC (for which parametric roll is not a problem) indicated that parametric roll is impossible (Shin et al 2004). Definition of Stability Failures The authors propose to use the term stability failure, to be consistent with the current practice of risk and reliability analysis where the term failure is widely used. A capsizing inevitably leads to a complete or total loss of ship operability forever or at least for some time (there have been cases when capsized vessels were salvaged and were put back to service), it is proposed to identify the event of capsizing as a total stability failure. A partial stability failure is defined as an event that includes the occurrence of very large roll angles and/or excessive roll accelerations, which will not result in loss of the ship, but which would impair normal operation of the ship and could be dangerous to crew, passengers, cargo or ship equipment. Modes of Stability Failures Based on the extended discussion, the 48 th session of the SLF Sub-Committee of the IMO has decided to define three distinct physical phenomena responsible for stability failure. Physical mechanisms of these phenomena are meant to be modeled within the performance-based criteria. A rather complete overview of capsize modes has been provided by ITTC (1999). The first mode is related to restoring arm variations in waves; the restoring moment becomes larger on the wave trough and smaller on the wave crest. It is mainly the result of the changing underwater hull geometry. This effect has been known to naval architects for a long time; see for example, (Paulling 1961). Fig. 1 illustrates these changes. Most of the vessels have relatively slender lines at the aft and fore ends to decrease wave resistance and facilitate proper flow around a propeller. Further up, the lines become wider; bow flares are needed to protect the deck from the green water in rough weather, while stern overhang is fitted to protect the propeller and provide a place for a steering gear. Wave trough amidships Wave crest amidships Fig. 1 Change of waterline in waves (dashed line correspond to a waterline in calm water) 296 On Performance-Based Criteria for Intact Stability

3 When a wave trough is located amidships, relatively wide sections are submerged and the actual waterline becomes wider in comparison to calm water. On the other hand, when a wave crest is located near the midship section, the waterline comes through the slender parts of the hull lines (fig. 1). The wave waterline becomes narrower, which leads to a decrease in initial stability. These changes in stability go beyond initial stability. The GZ curve also changes as the wave passes (insert on Fig. 2 where the GZ curve for the wave crest and trough are shown). Physical reasons of stability changes in waves, however, are not limited just by the forces of hydrostatical nature. Pressure in waves is distributed differently than in calm water, so the Smith effect also contributes, as well as ship waves, radiated and diffracted waves. Changing of stability in waves invokes two physical mechanisms of stability failure, pure loss of stability and parametric roll resonance Pure loss of stability in waves could happen if a vessel encounters a single large wave in following or quartering seas and spends a considerable amount of time on the wave crest. If stability of this vessel on the wave crest deteriorates too much, the vessel may capsize or attain a very large heel angle, see example Fig.2 with capsizing of midsize cargo vessel on 4 m wave, speed 25 knots φ, deg Righting Heeling Arm, m Wave trough Calm water Phase of heeling moment Magnitude of heeling moment φ,deg Wave crest -0.5 Time history of roll and capsizing Phase of encounter wave Fig. 2: Example of total stability failure caused by pure loss of stability t, s The heeling moment was applied at the moment when the vessel was on the wave crest and its stability was at its lowest; the capsizing occurred, despite the stability being on its way to recovery. Decreasing the magnitude of heeling moment arm from 0.24 m till 0.2 m converts the stability failure from total to partial (see Fig. 3) φ, deg Phase of heeling moment Time history of roll t, s -20 Phase of encounter wave -40 Fig. 3: Example of partial stability failure caused by pure loss of stability Another physical mechanism of stability failure caused by restoring arm variation in waves is parametric roll resonance (the term parametric roll is commonly used). It is a result of periodic changes of stability with certain frequency ratio with natural frequency of roll. If the frequency of stability changes (that is related to encounter frequency of waves) is about twice that of the natural roll frequency, amplitude of sub-harmonic roll motions will start to increase. This is the case of principal parametric resonance. If the frequency of stability change is nearly the same as the natural frequency of roll, parametric roll may still exist. It is called fundamental parametric resonance. The frequency band where the fundamental parametric resonance is possible is quite narrow and the likelihood to encounter such a condition is small, but not impossible. Fig. 4 shows simulated capsizing caused by parametric roll resonance for the mid-size cargo ship, wave length equal to ship length, wave height 4 m for the speed of 7 knots in following seas φ, deg The second mode to be considered is stability under dead ship condition as defined by SOLAS regulation II-1/3-8. The beam seas considerations (currently used in the weather criterion) actually came from the steam era, when most vessels had the superstructure in the middle and the upper and underwater body configuration was more or less symmetrical relative to the midship section. For such a vessel, in case of power loss in a storm, aerodynamic wind moment and wave drifting moment will turn it into beam seas, so it made perfect sense to consider a vessel without power in beam seas subject to the action of gusty wind and severe waves. The beam sea assumption is not necessarily universally applicable nowadays, as the modern fleet could be characterized by a much wider variety of configurations. As a result, the dead ship conditions could be more complex than just beam seas so that it is necessary to estimate drifting attitude of ships without power by solving an equation of equilibrium of a surge-sway-roll-yaw motion see (Umeda et al., 2006a). Experiments reported by (Ogawa et al, 2006) with t, s Fig. 4: Example of total stability failure caused by parametric resonance On Performance-Based Criteria for Intact Stability 297

4 a modern cruise vessel freely drifting without power while subjected to wind and waves was observed to drift at 10 to 20 degrees away from a beam sea heading angle. The freeboard height can significantly change the dynamics of stability failure. If the freeboard is relatively high, greenwater is not a factor; once the deck is in the water, it is mostly a roll damping influence. If the freeboard is low, the water will be trapped into the deck well relatively early in the process. Once the greenwater has been trapped, the dynamical properties of the system change dramatically. The vessel becomes sluggish, angle of heel increases slowly and it takes several waves before capsizing occurs Heave motion may invoke another mechanism of stability failure in dead ship conditions. As the underwater geometry changes with heave motion, the GZ curve also changes. Periodic changes of the GZ curve may lead to parametric roll even if a vessel is in beam sea; as the dead ship condition is not limited to beam sea. Parametric roll may be a concern in dead ship condition as well (Munif, et al 2006). The third mode includes maneuvering-related problems in waves such as broaching-to. Broaching-to (the term broaching is also widely used) is a violent uncontrollable turn of a vessel in following or quartering waves. During the turn, a very large heel angle may be developed that may eventually lead to capsizing or represent a partial stability failure. Broaching is often associated with the surf-riding, another phenomenon of following /quartering seas. Essentially, surf-riding is the catching and dragging of a ship by a wave. Physically, surf-riding is caused by the equilibrium created by the longitudinal wave force, thrust and resistance. During surf-riding, the vessel sails with the speed equal to the wave celerity. Such mode of motion may become directionally unstable if a ship is captured at wave down slope; as a result, a ship may experience the uncontrollable turn. Dynamics of surf-riding in quartering seas (as well as consequent broaching) is considered by Spyrou (1996). To model broaching, it is essential to take into account coupling with maneuvering and steering. For this purpose, a surge-sway-yaw model (Umeda and Renilson, 1992) and a surge-sway-yaw-roll model (Umeda and Renilson, 1994) were utilized to identify the surf-riding threshold in stern quartering waves as a heteroclinic bifurcation of a vector field. (Umeda, 1999). Here, the methodology used for the uncoupled surge motion was extended from a two-dimensional case to an eight-dimensional case. A procedure for identifying this threshold with an example is available from Umeda et al., (2006c). Once surf-riding occurs in stern quartering waves, the ship is captured by a wave near a wave trough where she is directionally unstable. Directional instability does not always result in broaching. Nonlinear timedomain simulation, however, indicates that most ships suffer broaching if their rudder checking ability is insufficient. An example of such a time series is shown in Fig. 5. Firstly, the ship is captured near a wave trough as constant and negative pitch angle indicates. Then, the absolute yaw angle exponentially increases despite the maximum opposite rudder application. Finally, the ship attains large roll angle toward the direction of centrifugal force due to large yaw angular velocity (Umeda et al, 1999). In addition, if the roll restoring is not sufficient, broaching may result in capsizing. (Umeda et al., 2006b) If so, the threshold of capsizing due to broaching can be estimated as a heteroclinic bifurcation of the 8-dimensional vector field (Umeda, 1999). If the ship stability is sufficient, the ship can undergo repeating broaching in regular waves (Umeda and Matsuda 2000). 20 Pitch, deg Roll, deg Yaw, deg Rudder, deg t, s t, s Experiment Calculation Implementation Design of any complex structure always involves compromise, including sometimes finding a right balance between safety and performance. In a sense, parametric roll of modern container carriers is a result of good hull design from a traditional hydrodynamic point of view. Cargo in containers is relatively light; therefore, a containership has to be designed to maximize volumetric capacity. On the other hand, a modern container carrier has to be fast. As a result, there is no other way, but to design a fine underwater hull with full shape above the water plane. It means significant bow flare, wide and flat underwater stern sections and stern overhang exactly the elements of the hull geometry that drive stability change in waves and parametric t, s t, s Fig. 5: Typical time series of capsizing due to broaching. A purse seiner model marginally complying with the current IS Code runs in stern quartering waves with the wave steepness of 1/10, the wave length to ship length ratio is 1.637, the autopilot course of 10 deg from the wave direction and the nominal Froude number of (Umeda et al., 1999) 298 On Performance-Based Criteria for Intact Stability

5 roll. A rational choice of lines (especially around the stern) allows a decreasing likelihood of parametric roll inception, but does not eliminate it completely (Levadou and van t Veer 2006). Two ways remain to solve the problem: motion control devices and operational measure. Levadou and van t Veer (2006])show the positive influence of bilge keels and fins, while, according to Shin et al (2004) and Umeda et al (2005), small anti-rolling tanks may be very effective against parametric roll. The other way is to develop a ship-specific onboard guidance that allows a decreasing likelihood of the situations when the parametric roll is possible, such as recommended by ABS (2004). A similar situation could be observed with surf-riding and broaching. Both these phenomena concern relatively high-speed vessels capable of reaching a speed comparable with wave celerity. No hull geometry design could eliminate the longitudinal wave force that creates surf-riding equilibrium at certain speed ranges. So, surf-riding is unlikely to be avoided merely by hull geometry design. However, fitting the vessel with purpose-designed rudder (Umeda and Matsuda 2000) or optimized rudder size with tuning autopilot (Umeda et al 2006b) can avoid broaching. Alternatively, a ship-specific onboard guidance for keeping propeller revolution below the surf-riding threshold can be more cost-effective (Umeda et al 2004). This is why the performance-based criteria for parametric roll and broaching should be developed also for an operational use in the form of on-board shipspecific guidance. Development of the Criteria Definitions for the Criteria The word criterion has a Greek origin and literally means instrument for judgment. In the context of technology regulations, a criterion consists of a value (or values) that represent a measure of the regulated quantity. The second part of a criterion is a requirement expressed in the form of the boundary for the measure. A measure could be derived directly by using mathematical and physical modeling of the phenomena. An adequate mathematical model of stability failure can predict how and if the phenomenon is going to happen. In this case, the value of criterion is a measure of response of a vessel to a given environmental excitation. Therefore, it can be said that such a criterion measures a performance of a vessel concerning the judged quality. That is why it is logical to define a criterion with a mathematical model of stability failure as a performance-based criterion. On the other hand, the measure could be based on some parameters of a vessel. The relationship between these parameters and actual phenomenon is known from experience of operation, model tests, numerical simulations or engineering considerations. Because it is not necessary to evaluate response of a vessel to calculate this type of measure, such a criterion could be defined as parametric. Use of mathematical models to evaluate a measure for the criterion inevitably will invoke a question about modeling the environment. Information about wind and waves comes in the form of statistical averages; therefore, it would be logical to use a stochastic mathematical model. If the mathematical model cannot use probabilistic input (say it is only formulated for regular waves), there should be a way to find a regular wave, equivalent in a certain sense to the given sea state. As a result, depending on the kind of mathematical model used, the performance-based criterion could be either probabilistic or deterministic. Summarizing the above considerations, four types could be defined for stability criteria: probabilistic performance-based criterion, deterministic performance-based criterion, probabilistic parametric criterion, deterministic parametric criterion. On Probabilistic Criteria The principal advantage of probabilistic criteria is that information on the statistical description of waves and wind could be fully utilized. Another advantage is related to reliability of criteria themselves. Analyzing reliability of intact stability criteria, Sevastianov (1994) introduces a concept of warranty of a criterion that is essentially a probability that a stability failure will not happen if this criterion is satisfied (see also Belenky and Sevastianov (2007)). Such a warranty could be a really useful tool for assessing different proposals for the criteria. As a probabilistic criterion is expressed in a form of probability of failure, by itself it represents a warranty or a level of safety. Sevastianov (1994) shows that using probability of stability failure (equally applicable both to total and partial stability failures) may be difficult from a practical point of view, as this value may have problems with sensitivity to change of design parameters. It is intuitively clear that different weather conditions entail different levels of probability of stability failure. Moreover, even if the weather does not change, damage and failures are more likely if a vessel is exposed to these conditions for six hours rather than for 30 minutes. Considering stability failures within frames of general reliability approach, a random event of stability failure could be treated as a Poisson flow event. Three conditions need to be satisfied in order to use Poisson flow: random events must be ordinary (happens once at a time), events must have infinitely small probability of occurring in a particular time instant and be independent. These conditions are met for total stability failure, but not always for partial stability failure, especially for parametric roll (Belenky and Breuer 2007). Once the three conditions of Poisson flow are satisfied, the Poisson distribution can be used to obtain probability of n F stability failures during time T On Performance-Based Criteria for Intact Stability 299

6 ( p T ) F ( p T ) PT ( nf ) = exp F (1) n! F nf Here, p F is a parameter of Poisson distribution. It has a meaning of average number of stability failures per unit of time. As the intention of regulator, designer and operator is to avoid all the stability failures, the probability that the operation will be failure-free for a time T, can be expressed as: ( p T ) P ( n = 0) = exp (2) T F F Probability of complimentary even, namely that at least one stability failure will occur is expressed as: PT ( nf 0) = 1 exp( pft ) (3) Formula (3) can also be considered as cumulative probability distribution of random time before the first stability failure occurs. Its distribution density is known as exponential distribution and is expressed as: f ( T ) = pf exp( pft ) (4) The parameter p F may be considered as a good candidate for a form of probabilistic criterion. In this case, treating stability failure will be very similar to the general reliability approach. Such a form of criterion for ship stability failure was first proposed by Sevastianov in 1963, details in English are available from (Sevastianov 1994), (Belenky and Sevastianov 2007). On Performance-Based Criteria Development of performance-based criteria means the development of methods for prediction of stability failure. It may be especially challenging for the total stability failure due to the extreme nonlinearity of this problem. It is proposed to call a set of assumptions allowing solution of this problem the capsizing hypothesis. Different researches used different capsizing hypothesis; it is convenient to divide them into two large groups: based on classical definition of stability and based on escape from the stability range. The classical definition of stability was first formulated by Euler. It reads as follows: A vessel is stable if, upon being inclined by external forces, it returns to the initial position when action of these forces ceases to exist. This definition could sound as if it describes dynamical application of a heeling moment in calm water and seems to be irrelevant for wave action really, why do the external forces have to disappear when ship stability needs be judged? The relevance of the classical definition of stability could be demonstrated by extending the GZ curve up to capsizing position (Belenky 1993). This allows the performing of numerical simulation of the rolling equation up to the upside down stable equilibrium, simulating capsizing in the most robust way possible. It also allows evaluation of the entire phase plane for free roll motions without damping, as is shown in Fig. 6. It can be done numerically, or just qualitatively, analyzing types of motions in vicinity of equilibria, two stable and one unstable between them. One of the simplest ways to qualitatively reproduce the phase plane is to use a piecewise linear dynamical system that has the same sequence of stiffness change as the actual GZ curve. The advantage of using the model with piecewise linear stiffness function is the possibility of using the solution of a linear differential equation on each of the ranges. Comparing partial solutions in range 0 and 1, one can see that the resonance is impossible in range 1 as the instantaneous GM is negative. As a result the response to wave excitation is much weaker in range 1 than the response in range 0 (Fig. 7). This is a very important conclusion. Whether capsizing occurs or not is no longer influenced by wave forces once the roll angle is larger than the maximum of the GZ curve. This explains how the wave forces could disappear when the vessel attains large roll angles. As it can be seen from the review of the methodology for criteria development, the methods that use the classical definition of stability as the capsizing hypothesis imply that stiffness is modeled up to the capsized position. GZ φ & φ m0 φ v Range 0 Range 1 Range 2 Fig. 6: Topology of roll phase plane and piecewise linear stiffness function RAO Range 0 φ m1 Fig. 7: Amplitude of partial solutions for positive (range 0) and negative stiffness (range 1) The hypothesis that a total stability failure will happen when a certain angle of heel is exceeded or the amplitude of excitation will exceed some critical value is both intuitive and practical. There are several techniques that use this hypothesis, including the current weather criterion. Time history of work done and energy changes for the steady state motion of a linear system is shown in Fig. 8. This figure shows how the energy balance is divided into separate balances. The work of the damping forces is compensated by the work of the active component of the excitation. A separate φ Range 1 ω φ 300 On Performance-Based Criteria for Intact Stability

7 balance is achieved by the potential and the kinetic energies together with the work done by wind heeling moment and by the synchronizing component of the wave excitation. There are two assumptions made to enable use of the energy balance method for stability criteria. Application of a sudden wind gust leading to transient motion and the nonlinearity of the GZ curve do not break the separation of the energy balances, formally achievable for the steady state motion of a linear system. These assumptions, however, may be quite reasonable, taking into account the small influence of waves on the roll motion beyond the angle of the maximum of the GZ curve. Development of nonlinear dynamics in the 1980s and significant enhancement of computational capabilities opened a new era in ship stability research, providing brand new tools for the criteria development. The Safe basin approach is one of them. Safe basin is defined as a set of initial conditions, in the phase plane, that do not lead to capsizing under a given regular excitation. Potential energy Kinetic energy Work of damping Work of active component of wave excitation Work of synchronising component of wave excitation Work of heeling moment time time time Fig. 8 Energy balance of linear system Numerical studies of the safe basin were done by Rainey, et al (1990), Rainey and Thompson (1991), Kan and Taguchi (1991), Kan, et al (1992), McMaster and Thompson (1994) and others. The safe basin can be calculated using a mapping method. The phase plane was presented as a grid of initial conditions. Numerical solution with the given excitation is computed for each of the initial conditions. If this pair of initial conditions leads to capsizing, it is marked white; if not, it is marked black. See inserts in Fig. 9. Area of SAFE BASIN 1 SMOOTH BOUNDARY a b FRACTAL BOUND- FINAL CRISIS Normalized amplitude of excitation Fig. 9 Deterioration of safe basin with increasing of wave amplitude c The safe basin for small amplitude of the excitation, shown in black in the insert on Fig. 9 has a smooth boundary almost like the separatrix (boundary of safe basin for a system without excitation). Increasing of the excitation amplitude first does not change the boundary of the safe basin (Insert b on Fig. 9) or the relative (normalized) area of the safe basin (curve in Fig. 9). Further increase starts some erosion of the safe base (insert c), however, without significant decrease of the area. Somewhere around the value of the normalized excitation of 0.7 the erosion of the safe basin becomes significant, the boundary of the safe basin becomes fractal and the area of the safe basin decreases dramatically (inserts d, e and f in Fig. 9). The boundary of the safe basin (also known as invariant manifold) can be calculated directly from the equation of motion by using the Poincarè mapping. This technique was applied by Falzarano and Troesch (1990), Falzarano et al (1992, 1995) and others. Falzarano et al (1992) proposed to use the Melnikov function to find the critical wave amplitude that leads to quick deterioration of the safe basin. The Melnikov function is an approximation for the distance between two boundaries, so when the boundaries intersect, the Melnikov function equals zero. It is shown in the referred source that if the GZ curve is approximated in a form of a polynomial, the critical wave amplitude could be expressed in analytical terms. This method was further extended for irregular waves, Vishnubhota et al (2000, 2001). Summarizing: the capsizing hypotheses based on the idea of escape from the stability range could be formulated in the form of exceeding a critical angle (current weather criterion method of energy balance) or in the form of exceeding critical amplitude of excitation (safe basin mapping, boundary integration and Melnikov function technique). On Performance-Based Probabilistic Criteria The problem of a vessel capsizing or attaining a very large roll angle involves consideration of nonlinear rolling under stochastic excitation. This problem cannot always be solved with acceptable accuracy using analytical methods. Therefore, numerical simulations and model experiments may be the only possible way to attain physically robust solutions with engineering accuracy. The role of analytical methods is also very important. They provide an invaluable tool for understanding and interpretation of the results of numerical simulation and d e f On Performance-Based Criteria for Intact Stability 301

8 model tests. The most recent review of these methods can be found in the IMO document SLF-50/4/12 and Francescutto (2007). Once the time becomes a part of the probabilistic criteria, a principal decision on time scale should be made. Generally speaking, waves and wind are not stationary stochastic processes. However, their rate of change could be considered as very slow in comparison with the natural roll period. Then, it makes sense to introduce a concept of quasi-stationarity a period of time during which, average characteristics of waves and wind could be considered constant and the processes of wave elevation and wind velocities are assumed stationary. The usual assumption for time of quasi-stationarity is from thirty minutes to six hours typical time between wind and wave measurements. If the seastate, course and speed of a vessel do not change, the stochastic process of roll motions could be considered as a stationary process. Obviously, if one is interested in a larger scale of time, say, voyage, month, year or lifetime, stationarity assumption is also not applicable due to the change of average characteristics of wind and waves. Therefore, there are two approaches possible: short-term and long-term. The short-term approach uses an assumption of quasistationarity of wind and waves. It is proposed to introduce a term assumed situation as a set of conditions for the short-term approach. This set has to include definitions for average characteristics for wave and wind, as well as definitions for course and speed. The problem of a rational choice of an assumed situation may represent a significant challenge for the development of probabilistic criteria. If the assumed situation is too severe, the criteria may come out overly conservative. Consideration of several assumed situations with assigned probability to each of them leads to the longterm approach. The long-term does not use stationary assumptions, but it is possible to present the longterm problem as a sequence of short-term ones. The long-term approach was actually implemented in a research project described in van Daalen, et al (2005). One of the findings from the research project referred to above was the significant sensitivity of the resulting probability to choice of speed and course, i.e., to the operator s behavior. The probability of stability failure for an individual vessel may be very low. It means that the average time before the failure may happen to be very large in comparison with natural roll period that serves as a main time scale for the roll motion process. This constitutes The Problem of Rarity, as an attempt to evaluate probability of stability failure may lead to work with two incomparable time intervals: natural period of roll and average time before stability failure. This problem may become especially challenging when numerical simulation is used as the main tool, as it will lead to the necessity of reconstruction of very long records of wave elevations. Normally, wave elevations are reconstructed with the inverse Fourier transform: ζ W N ω Wi i= 1 ( t) = r cos( ω t + ϕ ) (5) i i Here, ζ W is the instantaneous wave elevation at the time instant t, the set of initial phases ϕ i consists of random numbers distributed uniformly from 0 to 2π; this is the only stochastic figure in the model. The amplitude of the component r Wi is obtained from the spectral density. The set of frequencies ω i has to cover a significant part of the spectral density. Choice of the frequency may be very important for the length of reconstructed record. It is well known that equal spacing of frequencies leads to the so-called self-repetition effect. The uneven frequency spacing is a commonly used technique to avoid the self-repetition effect. However, this leads to spreading of the error rather than complete elimination, see also Belenky (2004, 2005). If the spread error is found to be significant for the outcome, two ways may be considered, as simply increasing number of frequencies may significantly increase computation time. One way is to use many relatively short records or consider alternative ways to present irregular waves, auto-regression model, for example (brief description available from (Belenky and Sevastianov 2007). Another problem that affects use of numerical simulation in irregular waves is the possible absence of practical ergodicity of roll response. Ergodicity is a general property of stationary stochastic processes allowing obtaining all statistical characteristics of the process from one record if it is sufficiently long. Wave elevations are known to be an ergodic stochastic process. It is also known that a linear system always produces ergodic response on ergodic excitation. However, response of a nonlinear system may be not ergodic. Non-ergodicity is stronger if nonlinearity is stronger and it is especially noticeable for parametric roll (Belenky 2004, Bulian et al. 2006). Although the process is probably ergodic, but convergence of time-based analysis is so slow, that it has to be considered non-ergodic for all practical purposes. This is reflected in the term practical non-ergodicity. Practical implication of nonergodicity of roll response means that it is necessary to simulate several records for statistical processing. Apart from these difficulties, work with results of simulation may require use of statistics of extreme events. See (McTaggart and de Kat 2000) (Mc Taggart, 2000). One could also consider the possible analogy with second order drift forces and the response of a moored structure in irregular waves for which long duration tests or simulations are required, see (e.g. de Kat and Dercksen, 1994). Methodology: Numerical Tools Dramatic improvement of computational capabilities and their universal availability make numerical simulation important not only as a development tool, but also for actual application during the design process. A brief review for some such tools is given below. This review does not pretend to be complete (for a comprehensive review, see Beck and Reed (2001)); the purpose is to illustrate, which capabilities are needed for the development and application of performance-based probabilis- 302 On Performance-Based Criteria for Intact Stability

9 tic criteria. FREDYN is a potential time-domain code based on strip theory (de Kat and Paulling, 1989) that includes other types of forces based on systematic model test, developed by CRNAV. FREDYN has propulsion, seakeeping and maneuvering capabilities and was validated to reproduce capsizing and broaching on the basis of frigate model tests, see de Kat et al (1994), de Kat and Thomas (1999, 2000), parametric roll for postpanamax containership (France et al 2003), and loss of fishing vessel GAUL (Van Walree et al., 2006). LAMP (Large Amplitude Motion Program) is a potential time domain code based on panel method (Lin and Yue, 1990, 1993) with the capability to include forces of other physical nature. LAMP was developed by SAIC (Science Application International Corporation) under sponsorship from US Navy, US Coast Guard, ABS and SAIC. LAMP has seakeeping and maneuvering capabilities and was validated to reproduce large wave loads and motions (Shin et al 2003) using a number of single and multi-hull vessels and parametric roll for post-panamax containership (France et al 2003). OU-BROACH is a numerical model developed by Umeda and Hashimoto (2006d) for broaching and parametric rolling in astern waves. Its simplest version is based on a linear maneuvering model with linear wave forces and is capable of predicting typical time series of broaching and parametric rolling in regular waves. Its advanced version also takes into consideration second order terms, such as interaction force between maneuvering and wave actions so that critical condition of broaching can be predicted. SLF 50/INF.2 describes two time domain solvers. ROLLS solves numerically ordinary differential equations of roll and surge while information on heave and pitch are obtained from a linear frequency domain solution. GL SIMBEL is a strip theory based potential time domain code capable of nonlinear calculation of diffraction and radiation forces. As some of the forces (e.g. roll damping and maneuvering related) must be treated empirically and can not be evaluated by calculations only, model tests may be necessary to generate input data for numerical simulation. The IMO already has developed the model test guidelines for roll damping as MSC.1/Circ For synchronous rolling under dead ship condition, linear and nonlinear roll damping should be estimated as accurately as possible for calculating capsizing probability (Paroka and Umeda, 2006). For parametric rolling, linear roll damping is responsible for its threshold prediction together with amplitude of roll restoring variation (ITTC 2005). For broaching, roll damping is not as important because of the very low encounter frequency, but the linear maneuvering coefficient should be estimated accurately. Methodology: Broaching As there are no reliable analytical methods to treat broaching in irregular waves and broaching occurs with only a few wave cycles, it is believed that deterministic performancebased criteria should be developed for broaching. Bifurcation analysis, a technique based on nonlinear dynamics, is believed to be an appropriate method for developing broaching criteria. It has already been used to develop the surf-riding threshold in MSC Circ. 707 in 1995, which was amended as MSC.1/Circ in 2006 (Umeda, 1990). A numerical simulation approach using the Monte-Carlo method remains at this moment the only way of practical evaluation of risk of capsizing caused by broaching. The methodology of simulation using FREDYN is described by McTaggart and de Kat (2000). Three modes of broaching were observed and identified. Successive overtaking waves while ship is traveling at low speed, Low frequency, large amplitude yaw motions, and High-speed broaching caused by a single wave.. Recently, Umeda et al. (2007) developed an analytical method for evaluating probability of broaching in irregular waves with a given deterministic threshold in regular waves, and well validated it with the Monte Carlo simulation using OU- BROACH. Methodology: Dead Ship Conditions By the classical definition of stability, crossing of the separatrix is associated with capsizing. If roll process can be assumed stationary (short-term approach), upcrossing theory may be applicable (Belenky and Sevastianov 2007). Upcrossing theory allows calculating the parameter p F of Poisson flow (1), if the distribution of the process and its first derivative are known. The piecewise linear method briefly described in earlier so in this paper also uses the upcrossing theory. The event of capsizing is associated with the upcrossing of the level φ m0 (see Fig. 6) with the roll rate above critical (Belenky 1993), (Belenky and Sevastianov 2007). This approach was extended for wind and drift by Belenky (1994), Paroka et al (2006). There were case studies and other applications using the piecewise linear approach Belenky (1995), Iskandar and Umeda (2001), Paroka and Umeda (2006). A comparison between the piecewise linear method and the Monte Carlo simulation can be found in Paroka and Umeda (2006). The problem of generalizing the weather criterion has been approached also in Bulian and Francescutto (2004) who developed a combined analytical/numerical procedure for the evaluation of the probability of capsizing. The spatial correlation of wind gusts has been taken into account by means of an aerodynamic admittance, whereas the stochastic sea takes into account of the effective wave slope through a hydrodynamic admittance. Future development of performance-based criteria is also considered by Bulian, et al (2006a). Methodology: Righting Arm Variation The problem of pure loss of stability from a probabilistic point of view is quite close to the problem of capsizing in dead ship conditions; generally, the same approaches could be used. For example, Umeda et al. (1990), Umeda and Yamakoshi (1993) used the classical definition of stability, integrating probability distribution density in phase plane with the boundaries of the separatrix. To take into account the changing stability in On Performance-Based Criteria for Intact Stability 303

10 waves, the Grim (1961) effective wave concept was used. Additionally surging in irregular waves was added to the model in order to have a more realistic distribution of time spent on different phases of wave. Belenky (1997, 2000) considered extensions of the piecewise linear approach for following and quartering seas. The problem of probabilistic description of parametric roll is more complex as the development of parametric resonance involves a sequence of waves. As the probabilistic theory for parametric rolling is still under development, a deterministic approach may be tentatively used for criteria development. ABS (2004) introduces susceptibility and severity criteria. See also (Shin, et al, 2004). The susceptibility criteria are based on a Mathieu equation and they are aimed at determining if this particular vessel may be vulnerable to parametric roll or not. The severity criterion is based on the numerical solution of ordinary differential equation in regular waves in order to find out how severe the problem of parametric roll could be and whether following or head seas present most of the problem. Hashimoto et al. (2006) identified head-sea parametric rolling for their post Panamax containership model in regular, long-crested irregular and short-crested irregular waves, and reached the conclusion that their numerical models can predict amplitude of parametric rolling in regular waves but could overestimate that in irregular waves. This result is in line with previous findings by Bulian et al (2004). Then, Hashimoto et al. (2006a) measured restoring variation of the model in irregular waves and found that it agrees with the theoretical prediction with radiation and diffraction taken into account. Thus, prediction of roll restoring variation is not a major reason of discrepancy in irregular waves as well as roll damping obtained from the roll decay tests. A similar discrepancy was reported for the C11 container ship in irregular waves by Levadou, and van t Veer (2006). At the same time, numerical simulation provides a way to assess a danger presented by parametric roll for a given sea state and loading condition, and therefore could be used as an alternative procedure, both for design analysis and development of operational guidance. The first attempt to summarize this experience was made by Belenky et al (2006). Examples of Available Treatments of Rarity Problem As was noted above, the rarity problem is one of the most serious challenges for the development probabilistic performance-based criteria, see also (Bulian et al. 2006a), (Mc Taggart & de Kat 2000). A time-split scale method is an attempt to combine numerical and analytical approaches. The time-split scale method is a generalization from piecewise linear presentation. It separates the problem into two parts: rare and non-rare. Solutions for each part could be obtained with any method: analytical, numerical or experimental. Upcrossing theory is meant to be used to combine both solutions in the same manner as in the piecewise linear method see Fig 10. Rare Non-Rare φ Fig. 10 Time split-scale method: separation into rare and non-rare problem This description is only meant to serve as an example as different methods may work better for different phenomena and different types of vessels. It still seems to be too premature to make a final choice of methodology at this time Conclusions Upcrossing not leading to capsizing Capsized equilibrium φ m0 φ & 1 Upcrossing leading to capsizing Developing new intact stability criteria that would fit the future fleet and would be robust and still practical is, indeed, quite a challenging task. This paper was trying to review the entire challenge and see what is available in the knowledge and methodology inventory and what particular issues need to be addressed. The reason why new criteria are needed is the appearance of unconventional vessels with new forms of hull and new modes of operations. The new criteria are intended for unconventional vessels, while the existing intact stability code is still expected to be used for conventional vessels. The difference between conventional and unconventional vessels still needs to be established. One way is to simply define an unconventional vessel as having a certain range of parameters. Another way is to define vulnerability criteria for physical phenomena responsible for stability failures. New criteria will be both for design and operation. Two types of stability failures were defined: partial (large roll angle) and total (capsizing). Three modes of stability failure were considered: dead ship conditions, righting arm variation (pure loss of stability / parametric roll) and broaching. Four types of criteria were discussed: probabilistic performancebased, probabilistic parametric, deterministic performancebased and deterministic parametric. Particular issues to be addressed for the development were discussed: of probabilistic criteria - format of criteria and time dependence, of performance-based criteria - hypothesis of capsizing including classic definition of stability and escape from the stability range, of probabilistic performance-based criteria - choice of time interval, the problem of rarity. While discussing methodology, the following methods / techniques were highlighted: numerical tools, techniques for broaching, methods for dead ship conditions pure loss of stability and parametric roll, example of treatments for the problem of rarity. t 304 On Performance-Based Criteria for Intact Stability