PREDICTION METHODS FOR CAPSIZING UNDER DEAD CONDITION AND OBTAINED SAFETY LEVEL - FINAL REPORT OF SCAPE COMMITTEE (PART 4) - Yoshitaka OGAWA*, Naoya UMEDA**, Daeng PAROKA** Harukuni TAGUCHI*, Shigesuke ISHIDA*, Akihiko MATSUDA***, Hirotada HASHIMOTO**and Gabriele BULIAN** * National Maritime Research Institute, Tokyo, JAPAN ** Department of Naval Architecture and Ocean Engineering, Osaka University, Osaka, JAPAN *** National Research Institute of Fisheries Engineering, Kamisu, JAPAN

ABSTRACT scenarios. In case of the dead ship condition, under which the main propulsion plant, boilers and auxiliaries are not in Regarding capsizing of under dead ship operation owing to absence of power, a shipmaster cannot condition, this paper reports experimental, numerical take any operational countermeasure. This means that and analytical studies conducted by the SCAPE operational factors are not relevant to safety level evaluation. committee, together with critical review of theoretical Although some operational actions such as high-speed progress on this phenomenon. Here capsizing running in following waves can decrease safety level probability under dead ship condition is calculated with against capsizing, they can be avoided by operators if the a piece-wise linear approach. Effects of statistical safety level under dead ship condition is ensured. In other correlation between wind and waves are examined. This words, if perfect operation is taken, the safety level during a approach is extended to the cases of water on deck and ship’s life-cycle could be comparable to that under dead cargo shift. Further, experimental and theoretical ship condition. techniques for relevant coefficients are examined. Based For the dead ship condition, the weather criterion is on these outcomes, it also proposes a methodology for currently implemented in the IS Code and provides a direct assessment as a candidate for the new generation semi-empirical criterion for preventing capsizing in beam criteria at the IMO. wind and waves. The dead ship condition, however, does not always mean beam wind and waves. If a ship has a longitudinally asymmetric form, ship may drift to KEY WORDS: Dead Ship Condition; Intact Stability leeward with a certain heading angle. Therefore, it is Code; Beam Winds and Waves; Capsizing Probability, important to estimate the drifting attitude of a ship and to Piece-Wise Linear Approach, Water on Deck evaluate its effect on capsizing probability. It is also widely known that empirical estimation of effective wave slope coefficient in the weather criterion is often difficult to be INTRODUCTION applied to new ship-types such as a RoPax ferry, a large passenger ship and so forth. Therefore, the IMO (2006) Stability under dead ship condition is one of three recently allows us to use model experiments for this major capsizing scenarios in terms of performance-based purpose although it is not always feasible. A simplified criteria, which is requested to be developed for the revision prediction method is still desirable if accurate enough. of the Intact Stability Code (IS Code) by 2010 at the Based on the above situation, the authors presented a International Maritime Organization (IMO). It was expected methodology for calculating capsizing probability of a ship to be a probabilistic stability assessment based on physics under dead ship condition. In this present methodology, by utilizing first principle tools. This is owing to the fact capsizing probability under dead ship condition is that probabilistic approach can be linked with a risk analysis calculated by means of piece-wise linear approach. and dynamic-based approach enables us to deal with new Through this paper, we summarize this methodology and ship-types without experience. its application of this methodology to the new generation Therefore, it is important to establish a method for criteria at the IMO. evaluating capsizing probability under dead ship condition because its safety level could be a base for other capsizing First, a prediction method of the annual capsizing annual capsizing probability of Indonesian Ro-Ro ferry probability in beam seas based on a piece-wise linear accidents. Others (Umeda et al. 2002; Munif et al. 2004, approach was presented. Francescutto et al., 2004) also applied that to estimate the Second, the methodology is verified with the Monte capsizing probability of fishing vessels and a large Carlo simulation and is extended to the case with cargo shift, passenger ship. down-flooding and water on deck. The effect of several Although Belenky’s piece-wise linear method is widely important factors, such as drifting attitude, drifting velocity, utilized, some unsolved problems remain. Firstly, complete hydrodynamic coefficient estimation and statistical formulae for covering the wind effect have not been correlation between wind and waves are investigated. published. Secondly, numerical results of the exact Third, safety level in terms of capsizing probability was formulae have not yet been reported for evaluating the assessed by means of the present approach for passenger accuracy of the simplified formula. Thirdly, effects of and cargo ships as well as fishing vessels. several elements such as wind have not been examined. Therefore, Paroka et al. (2006) examined these HISTORICAL REVIEW problems with deriving the detailed formulae for the case of beam winds and waves and calculating with both exact and Several methods were developed for estimating simplified formulae for a car carrier, which has a large capsizing probability in stationary irregular waves. One is windage area. the numerical experiment known as the Monte Carlo simulation, in which the expected number of capsizing is CAPSIZING PROBABILITY BY MEANS OF THE estimated by repeating numerical simulation in time domain PIECE-WISE LINEAR APPROACH with different wave realizations (McTaggart and de Kat 2000). If a reliable mathematical model is available, this The methodology adopted by the SCAPE committee for method seems to be easily applicable. However, since calculating capsizing probability under dead ship condition capsizing probability of an existing ship is small, the is based on the piece-wise linear approach proposed by number of numerical runs could be prohibitively large. In Belenky(1993). This is partly because it rigorously takes addition, statistical variation of outcomes is not suitable for acount of nonlinear restoring moment and its efficient regulatory purposes. Sheinberg et al. (2006) attempted to calculation load enables us to obtain annual capsizing improve this approach by focusing on critical local waves. probability by integrating many combinations of wave Others are theoretical ones, in which capsizing probability height and wave period in long-term wave statistics. As is obtained as the product of the probability of dangerous mentioned above, Belenky’s formulation for beam waves condition and the conditional probability of capsizing. Some (Belenky, 1993) and its extension to beam wind and waves methods utilize stationary linear or non-linear theory for the case (Belenky, 1994) had not been complete enough so that former and non-linear time-domain simulation of Paroka et al. (2006) corrected them. Therefore, in this paper, deterministic process for the latter (Umeda et al. 1992; Shen the upgraded formulation is overviewed. and Huang 2000; Bulian and Francescutto, 2004). Sheng φ and Huang (2000) use the Markov process for probability kfi( ) density of roll and use the deterministic simulation in the final stage. Bulian and Francescutto (2004) use a simplified stochastic linearization technique, with a Poisson modelling Steady Wind heeling arm of the capsize event. Their main drawback is an assumption -φv-φCR -φm0 φ φ for the final deterministic process. Although some studies φm0 CR v φ (Jiang and Troesch, 2000; Falzarno et al., 2001) for extending the Melnikov approach into random seaways were carried out, no direct relationship with capsizing probability was not provided. Avoiding such a drawback,

Belenky (1993) proposed a piece-wise linear approach Fig. 1 Piece-wise linear approximation of righting arm. where both probabilities are analytically calculated by approximating the restoring arm curve with piece-wise The capsizing probability here is calculated with the linear ones. Because of its fully analytical scheme, the following nonlinear and uncoupled equation of relative or piece-wise linear approach has been utilized for practical absolute rolling angle of a ship under a stochastic wave purpose as the most promising methodology. An approach excitation and steady wind moment. Usually the ship that has a series of similarities with the piece-wise linear motion in beam seas is modeled with equations of coupled method, but that is based on a simplified statistical motions in sway and roll with wave radiation forces and linearization technique, has been developed by Bulian and diffraction forces. Watanabe (1938), however, proposed an Francescutto (2004, 2006) one-degree of freedom equation of roll angle, φ , as follows For a ship in irregular beam seas, Belenky derived exact formulae based on a piece-wise linear approach without numerical results. Then he extended his method to the case (I xx + A44 )φ&& + B44φ& + WGZ(φ)()= M wind t + M wave (t) (1) in both beam winds and waves (Belenky 1994) and provided numerical results with his simplified method where Ixx is momentum of inertia of the ship, A44 is the (Belenky 1995). Here the formulae relating to the wind hydrodynamic coefficient of added inertia, W is the ship effect were not published in detail. Recently, Iskandar et al. displacement, B44 is the hydrodynamic coefficient of roll (2001) applied Belenky’s simplified method to estimated damping, GZ(φ ) is the righting arm. Mwind (t) is the wind induced moment consisting of the steady and fluctuating P(H1/ 3 ,T01,Ws ,T ) = wind moment and Mwave (t) is the wave exciting moment based on the Froude-Krylov assumption. This is because the Pl PT ()()φ > φm0orφ < −φm0 PA A > 0;φ > φm0 + roll diffraction moment and roll radiation moment due to P P ()()φ > φ orφ < −φ P A < 0;φ < −φ sway can cancel out when the wavelength is sufficiently w T m0 m0 A m0 longer than the ship breadth. (Tasai, 1969) (6). The uncoupled equation of the absolute roll motion (1) can be rewritten by dividing by the virtual moment of Here Pl and PW denote the probability of up-rossing toward inertia as follows: leeward and that of down–crossing toward windward. Under the assumption of Poisson process, PT indicates the 2 2 probability of at least one up-crossing or down-crossing at φ&& + 2αφ& + ω0 kf i (φ )= ω0 ()mwind ()t + mwave ()t (2). the border between the first and second range, φ or m0 where kfi( φ )=GZ( φ )/GM, α: roll damping coefficient, −φm0 . PA is the probability of diverging behavior of absolute value of the roll angle in the second range ω : the natural roll frequency and kf1( φ ) : the 0 including the angle of vanishing stability. non-dimensional righting arm, m (t) : the wind induced wind These can be determined as following formulas: moment consisting of the steady and fluctuating wind moment and mwave (t) : the wave exciting moment based on the Froude-Krylov assumption. PT (φ > φm0 orφ < −φm0 )()= 1 − exp()− ul + uw T (7) As shown in Fig. 1, the restoring moment coefficient in ul equation (2) is approximated with continuous piece-wised Pl = u + u lines in leeward and windward conditions as follows: l w (8) uw Pw = ⎧ kf φ 0 < φ < φ range1⎫ ul + uw kf φ = 0 m0 i () ⎨ ⎬ (3) ∞ kf1 ()φv −φ φm0 < φ range2 ⎩ ⎭ PA ()()A > 0;φ > φm0 = ∫ f A dA 0 (9) and 0 PA ()()A < 0;φ < −φm0 = ∫ f A dA −∞

⎧ kf0φ −φm0 < φ < 0 range1⎫ kfi ()φ = ⎨ ⎬ (4). kf ()−φ −φ φ < −φ range2 where ul and uW denote the expected number of up-crossing ⎩ 1 v m0 ⎭ and down-crossing, respectively. f(A) denotes the Here, kf0 and kf1 are the slope of the range 1 and range 2, probability density function of the coefficient A,, which is a respectively. φm0 and −φm0 are the border between the function of three random variables, namely the initial range 1 and the range 2 in leeward and windward, angular velocity of roll motion,φ&1 , the initial forced roll, p1, respectively. Effects of dividing methods of the righting and the initial forced angular velocity, p , in the second arm curve into piece-wise lines and the case using the &1 relative roll angle can be found in Paroka et al. (2006). range. Since Equation (2) is linear within each range, it can be The probability density function of the A coefficient can analytically solved without any problems. The border be determined from a joint probability density function of condition, however, should be satisfied. Therefore, these three random variables, which can be calculated by using the three-dimensional Gaussian probability density capsizing occurs when the roll angle up-croses, φm0, or (Price and Bishop, 1974).Then, the probability density down-crosses, -φm0, and then the absolute value of roll angle increases further. Therefore, capsizing probability can be function of the coefficient A in leeward can be described as calculated as the product of the outcrosing probability in the follows: range 1 and the conditional probability of divergence of the absolute value of roll angle in the range 2. Belenky (1993) 1 ∞ ∞ f A = f φ& , p A,φ& , p , p dφ& dp (10) presented the following formula for the probability that a () ∫∫()1 &1()1 1 1 1 1 λ2 − λ1 −∞ 0 ship capsizes when the duration, T, passes in stationary waves. That in windward can be described as follows: 1 ∞ 0 f ()A = ∫∫f ()φ&1 , p&1()A,φ&1 , p1 , p1 dφ&1dp1 (11) λ − λ P(H1/3 ,T01,Ws ,T ) = 2PT (φ > φm0 )PA (A > 0;φ > φm0 ) 2 1 −∞∞ − (5) where Where H1/3 : the significant wave height, T01: the mean λ = −α ± ω 2kf +α 2 (12). wave period, WS: the mean wind velocity and T: duration. 1,2 0 1 Since this formula could exceed 1, Umeda et al. (2004) Furthermore, Belenky (1993) simplified Equations (10) proposed the following formula. and (11) by using the fact that roll resonance in the second range does not exist because of negative restoring slope. As a result, f(A) can be evaluated with a single integral in place of a double integral. For evaluating risk of capsizing, it is necessary to calculate annual capsizing probability with sea state statistics of operational water area. The formula of the In case of a ship has large angle of vanishing stability, the annual capsizing probability, Pan, can be found in Iskandar obtained capsizing probability could be very small but the et al. (2000) as follows: ship could capsize owing to cargo shift or down flooding when the roll angle is larger than her critical angle. In this * ()365×24×3600/T case, the calculation method mentioned above cannot be P = 1− ()1− P ()T (13) an directly applied because angle of vanishing stability cannot be determined. where ∞∞ Therefore,. Paroka & Umeda. (2006a) extend the * P ()T = ∫∫f (H1/3 ,T01 )⋅ P (H1/3 ,T01,Ws ,T )dH1/3dT01 . piece-wise linear seas approach to the case having the 00 critical roll angle. Here the probability of roll angle exceeding the critical value is defined as the probability of Here Ws is assumed to be fully correlated with H1/3 roll angle of the second range exceeds the critical value when the roll motion up-crosses or down-crosses the border VERIFICATION AND EXTENSIONS OF PIECE-WISE between the first and the second ranges of the piece-wise LINEAR APPROACH linear righting arm curve. This can be written as follows:

Comparison with the Monte Carlo simulation P(H 1 / 3 ,T01 ,Ws ,T ) =

P P ()()φ > φ orφ < −φ P φ > φ ;φ > φ + (14) For verifying the piece-wise linear approach, Paroka & l T m0 m0 CR CR m0 Umeda (2006b) calculated capsizing probability of a large Pw PT ()()φ > φm0 orφ < −φm0 PCR φ < −φCR ;φ < −φm0 passenger ship in stationary beam wind and waves and compared it with the Monte Carlo simulation. First, they where PCR is the conditional probability of roll angle confirmed that stochastic scattering of the Monte Carlo exceeds the critical value, φCR . simulation results is comparable to its confidence interval The conditional probability of roll angle exceeds the estimated with the assumption of binomial distribution. critical value is calculated with assuming that effect of Second, it was that the piece-wise linear approach coincides exciting moment on the roll motion in the second range is with the Monte Carlo simulation in moderate sea states, negligibly small. This assumption is based on the fact that such as the mean wind velocity of 28 m/s, within the resonance is impossible in the second range owing to its confidence interval but underestimates it in more severe negative stiffness. Therefore the roll motion in this range wind velocity as shown in Fig.2. The results of this depends only on the initial condition, which is the same as underestimation are that the assumption of the Poisson the roll motion at the end of the first range. The solution of process for outcrossing is not appropriate when the sea state the roll motion equation in the second range can be written is extremely severe. In addition, it is impossible for the as follows: Monte Carlo simulation to accurately quantify the capsizing probability in lower wind velocity because the probability is φ (t) = Aexp(λ t)+ Bexp(λ t)(+ φ − φ ) (15) too small. In conclusion, it is not so easy to verify the 2 1 2 v D piece-wise linear approach but it seems not to be inappropriate to use the piece-wise linear approach for the where 2 mean wind velocity of 26 m/s, which the weather criterion λ1,2 = −α ± −α + ω0 kf1 (16) deals with, but there is a room for further improvement. m φ = 0 (17) D ω 2 kf Wind Velocity (m/s) 0 1 28.0 28.5 29.0 29.5 30.0 30.5 31.0 1 A = [φ&02 − λ2{}φm0 − ()φv − φD ] (18) 1.0E+00 ()λ1 − λ2 1 B = [λ {}φ − ()φ − φ − φ& ] (19) λ − λ 1 m0 v D 02 1.0E-01 ()1 2

Here φ&02 indicates the initial angular velocity of roll motion

1.0E-02 in the second range. m0 indicates the non-dimensional steady wind moment. In order to investigate whether or not the roll angle exceeds the critical value, the following Numerical Experiment

Capsizing probability 1.0E-03 +Err iteration procedure is proposed. Firstly the initial angular -Err velocity of roll motion in the second range with negative A Present Method coefficient in the equation (18) is arbitrarily choose. 1.0E-04 Secondly the time of the angular velocity of roll motion Figure 2. Comparison of capsizing probability of the becomes zero is calculated as follows: large passenger ship between the piece-wise linear approach and the error tolerance of Monte Carlo 1 ⎛ Bλ ⎞ t = ln⎜− 2 ⎟ (20) simulation. (Paroka & Umeda, 2006b) m ⎜ ⎟ ()λ1 − λ2 ⎝ Aλ1 ⎠

Effect of critical angle for down-flooding or cargo shift Then the roll angle with the time, t , can be calculated m by means of the equation (15). If this roll angle is smaller than the critical one, the initial angular velocity is increased trapped on deck have been published (e.g. Belenky, 2003) and we repeat these processes. When the maximum roll however its effect to capsizing probability has not been angle has been larger than the critical value, the iteration is investigated. stopped and the initial angular velocity is set as the critical Paroka & Umeda (2006c) extended the piece-wise linear angular velocity. The probability of roll angle exceeds the approach to the case with trapped water on deck. Here the critical value then can be calculated as the same as the amount of trapped water was estimated with model probability of the initial angular velocity of roll motion experiments (Matsuda et al., 2005) by measuring heel and higher than the critical angular velocity. The probability of sinkage of the model in irregular beam waves. Water the initial angular velocity larger than the critical angular ingresses and egresses through the bulwark top or freeing velocity can be calculated as follows: ports and the balance of these can be determined duringb the experiment. As a result, it was found out that the trapped ∞ ⎧ ⎫ water on deck could increase the capsizing probability for 2 ⎪ φ&02 ⎪ P()φ& > φ& = exp⎨ ⎬dφ& (21) this ship type as shown in Fig. 4. This is because the trapped 02 CR ∫ 2D 02 2πDq& ⎩⎪ q& ⎭⎪ water can significantly reduce the mean restoring arm up to φ&CR the angle of immersion of the bulwark top, which is

approximated with piece-wise linear curves as a function of where φ&CR indicate the critical value of the angular stationary sea state.. velocity in the second range. The probability of roll angle As a next step, Paroka & Umeda (2007) incorporated a exceeding the critical value in a certain exposure time can numerical model for describing water ingress and egress be calculated by means of the equation (14) with the with a hydraulic model into the piece-wise linear approach. conditional probability of roll angle exceeds the critical As a result, capsizing probability with water on deck can be value is replaced by the equation (21). calculated without a model experiment. Here the dynamic Numerical results of the car carrier for several critical effect of trapped water on deck such as change in added angles ranging from 55 degrees to 75 degrees are shown in mass, damping coefficient and exciting moment are ignored Fig. 3 (Paroka et al., 2006c). The probability of dangerous for simplicity’s means. condition increases when the critical angle decreases. This is because the roll angle can exceed the critical value even if the roll motion in the second range does not diverge. The GM without Water on Deck (m) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 difference disappears when the unstable heel angle owing to 1 steady wind moment is smaller than the critical value. When the unstable heel angle is smaller than the critical value, the 1E-5 roll angle can exceed the critical value only if the roll 1E-10 motion in the second range diverges. 1E-15

1E-20 Without Water on Deck With Water on Deck 1E-25 Long-term Capsizing Probability Capsizing Long-term

1E-30

Fig. 4 Long-term capsizing probability with and without water on deck for the purse seiner operating off Kyusyu. (Paroka & Umeda, 2006c)

Effect of drifting attitude and velocity

The dead ship condition does not always mean beam wind and waves. If a ship has a longitudinally asymmetric hull form, ship may drift to leeward with a certain heading Fig. 3 Dangerous probabilities of the car carrier for different angle. Therefore, it is important to estimate the drifting critical roll angles (Paroka & Umeda., 2006a). attitude of a ship and to evaluate its effect on capsizing

probability. Effect of water on deck X

For smaller ships such as fishing vessels, the effect of Wind U trapped water on deck should be taken into account. This is x because these ships have smaller freeboard but higher χ β Y bulwark comparing with their ship size. As a result, when y 0 shipping water occurs, water can be easily trapped by Wave bulwark. Although a certain amount of water can flow out y ψ φ through freeing ports, the balance between ingress and Y z egress can be trapped on deck. The righting arm can 0' Z decrease because of the water trapped on deck. Many experimental and numerical works about effect of water Fig. 5 Co-ordinate systems the computed drift speed is close to the measured drift speed, For estimating drifting velocity and attitude of a ship in although the drift speed in beam waves only (wind speed = regular waves and steady wind, Umeda et al. (2006b, 0 in Fig. 7) was underestimated. It is confirmed that present 2007a) proposed a method for identifying equilibria of a method is useful for the quantitative computation of the surge-sway-yaw-roll model and for examining its local drift speed under dead ship condition. stability. Equilibria can be estimated by solving 4-DOF non-linear equations in the following co-ordinate systems 360 shown in Fig. 5; 330 300 ()m + m u − m + m vr = X + X + X (22) 270 x & ( y ) H A W 240 210 m + m v + m + m ur − m l φ&& = Y + Y + Y (23) ()y & ()x y y H A W 180 150 ()I z + J z ψ = N H + N A + NW (24) && 120 forward(stable) && & 90 ()I x + J x φ − mylyv& − mxlxur + 2μφ + W ⋅ GMφ heading angle (degrees) forward(unstable) (25) backward(stable) = K + K 60 H A 30 backward(unstable) where 0 u,v ; ship speed in x-axis and y-axis, 0 30 60 90 120 150 180 ψ , φ ; yaw and roll angle, angle between wind and waves (degrees) m ; ship mass, W ; displacement, Fig. 6 Heading angle of fixed points as a function of relative GM ; , angle between wind and waves (wind speed is 26 m/s, wave mx, my; added mass in x-axis and y-axis, amplitude is 2.85m and wave length ratio is 2.94). (Umeda Ix, Iz; moment of inertia around x-axis and z-axis, et al., 2006b) Jx, Jz ; added moment of inertia around x-axis and z-axis, lx, ly; vertical position of centre of mx,and my Drift speed due to wind & wave X, Y; external forces in x-axis and y-axis Cal.( wind only) 3.5 Cal. (wave only(H1/3=9.5m, T01=12.4sec)) N, K; yaw and roll moments Cal.(wind&wave(H1/3=9.5m, T01=12.4sec)) 3 Exp.(wind only) Exp.(wave only(H1/3=9.5m, T01=12.4sec)) 2.5 Exp.(wind&wave(H1/3=9.5m, T01=12.4sec)) Here the subscript H, A and W means hydrodynamic Cal.(Beaufort) force/moment, wind force/moment and wave-induced 2 1.5

drifting force/moment. (m/s) speed Drift In this model, a mathematical model for hydrodynamic 1 forces under low speed manoeuvring motion (Yoshimura, 0.5 0 1988), an empirical method for wind forces (e.g. Fujiwara, 0 5 10 15 20 25 30 35 2001), and a potential theory for wave-induced drifting Wind speed (m/s) forces (Kashiwagi, 2003) are used. Fig. 7 Relation between the wind speed and the drift speed The Newton method was used to identify equilibria of the in wind and waves. (Ogawa et al., 2006b) equations. As an initial input, the equilibria in the beam wind and waves from starboard and those from port side are 0 20 40 60 80 100 calculated. And then, equilibria for different angle between 1.00E+00 the wind and waves, χ , are traced. The calculation are 1.00E-04 done in two ways: the first one is that χ changes from 0 1.00E-08 degrees to 180 degrees (forward) with the wind direction 1.00E-12 fixed, and the second is that χ changes from 180 degrees 1.00E-16 to 0 degrees (backward). Stability of equilibrium is 1.00E-20 evaluated by calculating the eigenvalues of the Jacobean 1.00E-24 matrix in the locally linearised system around equilibrium. 1.00E-28 The example of results is shown in Fig 6. 1.00E-32 These figures indicate that when the relative angle capsizing probability (case1) between wind and waves is 0 degrees, the ship is drifting 1.00E-36 angle between wind and waves (degree) leeward in beam wind and wave. Having the relative angle between wind and waves larger from 0 degrees, the heading Fig.8 Capsizing probabilities under two coexisting drifting angle gradually increases from the beam wind. conditions as functions of relative angle between wind and Ogawa et al.(2006b) examined the present method waves. (Umeda et al., 2006b) through the comparison with measured drifting motions. Fig. 7 shows the sway velocity in wind and waves as a function An example of capsizing probabilities under two of wind speed. In this calculation, the significant wave coexisting stable steady drifting states are shown in Fig 8. height and mean wave period were of the same as in the Umeda et al. (2006b, 2007a) Here the mean wind velocity is experiments (H1/3 = 9.5 m, T01 = 10.4 sec). Only the wind 26 m/s and the duration is one hour. The fully developed speed was varied in the present calculation. It is found that wind waves under this wind velocity are assumed. In this calculation, the effect of relative angle to wind is modelled simultaneous equation for determining two dimensional as the change in lateral windage area, the effect of relative flows, however, a further simplified method is expected to angle to waves is done as the change in effective wave slope be developed. For this purpose, Sato et al. (2007) carried coefficient and the effect of drifting velocity is done as the out a series of model tests by means of two dimensional change in the exciting frequency. In addition, the heeling models covering large breadth to draft ratios. They lever due to drift are estimated with the model tests by confirmed that measured effective wave slope coefficient Taguchi et al. (2005). It is confirmed that the most could be much smaller than that from a strip theory in case dangerous condition for this subject ship under dead ship of very large breadth to draft ratios. This fact coincides with condition is one in beam wind and waves, i.e. χ is 0 the experiments by Ikeda et al. (1992) for a small degrees. It is clarified that drifting motion obviously high-speed craft. depends on underwater and above-water ship geometry.

Effect of hydrodynamic coefficient estimation Guide Rope s=1/60 Looped Wire System It is also widely known that empirical estimation of M echanical Guide 25 NMRI (Guide Rope) effective wave slope coefficient in the weather criterion is often difficult to be applied to modern ship-types such as a 20 large passenger ship, a RoPax ferry and so forth. Based on this background, the IMO allows us to estimate the effective wave slope and damping coefficients as well as heeling 15 lever owing to beam wind by model experiments with the interim guidelines (2006). 10 Taguchi et al. (2005) and Ishida et al. (2006) carried out almost full set of model tests for a RoPax ferry and verified roll amplitude (deg) 5 the interim guidelines as the alternative assessment of the weather criterion. It was clarified that the evaluation of 0 weather criterion considerably differs that with the 0.7 0.8 0.9 1 1.1 1.2 1.3 combination of tests. Umeda et al. (2006a) applied this model test approach to the Ro-Pax ferry for further tuning factor investigating its feasibility. Based on the interim guidelines, the model experiments were executed with three different constraints: a guide rope method, a looped wire system and Fig. 9 Roll amplitude of the RoPax ferry in regular beam a mechanical guide method. The guide rope method means waves measured with different constraining methods that the model was controlled by guide ropes. The looped (Umeda et al., 2006a). wire system means that, by utilizing a looped wire attached 30 to a towing carriage, the yaw motion of the model is fixed 26.56 25.22 26.18 but the sway, heave and roll motions are allowed. The 25 mechanical guide is the combination of a sub-carriage, a 20 heaving rod and gimbals. Among them, as shown in Fig.9, 15 the looped wire system is less accurate because natural roll 10 period and damping was slightly changed owing to a looped (deg) amplitude roll wire. In case the yaw angle increases with larger wave 5 steepness, the guide rope method has inherent difficulty for 0 keeping a beam wave condition. The guideline allows the direct method three step method parameter identificatin three different methods for determining the roll angle from Fig. 10 Comparison of estimated roll amplitude by means the experiment: the direct method, the three step method of three different estimation methods. (Umeda et al., 2006a) and the parameter identification. As shown in Fig 10, the difference between the three methods is negligibly small. In Umeda & Tsukamoto (2007c) investigated several conclusion, the interim guidelines can guarantee reasonable possible simplified versions of theoretical methods and accuracy if appropriate constraint is used. finally proposed a formula to calculate the effective wave When we calculate capsizing probability with the slope coefficient of the equivalent rectangular sections with piece-wise linear approach, the same difficulty exists. This the Froude-Krylov component on its own. Here the local is because fluid-dynamic coefficients such as effective wave breath and area of the rectangular section should be the slope coefficient should be separately estimated in advance. same as those of original transverse ship sections. As shown Therefore it is desirable to execute model experiments in Table 1, this simplified formula agrees well with the following the IMO interim guidelines. Utilizing model tests, experiment and a strip theory for existing ships having large however, could be not always practical owing to cost and windage areas probably because the diffraction component time. Therefore, it is desirable to develop an empirical or a can almost cancel out the radiation owing to sway and the theoretical method to accurately predict obtained results local breadth and area are responsible for restoring moment. from the model test. Furthermore, they indicate that difference in the effective It is well known that a strip theory can explain the wave slope coefficient between the present formula and effective wave slope coefficient in general. (Mizuno, 1973) experiments results in only small effects on the estimation Since a strip theory requires numerical solution of of the one-hour capsizing probability in stationary beam wind and waves as shown in Fig.11. Therefore, it is * ∞∞∞ recommended to use this simplified formula for a regulatory P ()T = ∫∫∫f (H 1 / 3 ,T01 ,Ws ) 000 (27) purpose. ⋅ P()H ,T ,W ,T dH dT dW 1 / 3 01 s 1 / 3 01 s Table.1 Comparison of effective wave slope coefficients under resonant condition. Here IMO means the formula in Ogawa et al. (2006a) calculated the annual capsizing the IMO weather criterion and FK indicates the probability of the RO-PAX ferry. Fig. 12 shows her righting Froude-Krylov prediction. (Umeda & Tsukamoto, 2007c) arm (GZ) curves. The limiting KG of this ship is governed by the weather criterion so that it was used as a loading condition for the safety assessment. The approximated line of the righting arm curve, of which GM, the area under the GZ curve and the angle of vanishing stability are not changed, is also shown in Fig. 12. Fig. 13 shows the capsizing probabilities within one year as a function of wind speed. In addition to the capsizing probability by means of the present method, two kinds of capsizing probability are calculated by means of two other wave scatter diagrams. One is the wave scatter diagram normalized for each mean wind velocity (Normalized) and the other is the wave scatter diagram without correlation of wind (No correlation). The wave scatter diagram without correlation with wind was conducted by summing up the present scatter diagram of wave height, wave period and

19 21 23 25 27 29 31 33 35 wind speed. Then the capsizing probabilities using two Wind Velocity [m/ｓ] 1.0E-01 kinds of diagrams is defined as follows: 1.0E-03 1.0E-05 1.0E-07 (Normalized) 1.0E-09 Probability 1.0E-11 * ∞∞ 1.0E-13 IMO P ()T ,Ws = ∫∫f N (H 1 / 3 ,T01 ;Ws ) 00 (28) 1.0E-15 Strip theory 1.0E-17 EXP ⋅ P()H 1 / 3 ,T01 ,Ws ,T dH 1 / 3 dT01 1.0E-19 simplified Froude-Krylov (No correlation) Fig.11 The effect of the effective wave slope coefficient ∞∞ estimation on the one-hour capsizing probability of the * P ()T ,Ws = ∫∫f NC (H 1 / 3 ,T01 ) RoPax ferry in stationary beam wind and waves. (Umeda et 00 (29) al., 2007a) ⋅ P()H 1 / 3 ,T01 ,Ws ,T dH 1 / 3 dT01

Effect of correlation between wind and waves where fN(H1/3, T01; Ws) denotes the normalized wave scatter For the provision of the criteria based on diagram and fNC(H1/3, T01) denotes the wave scatter diagram performance-based approaches, the capsizing probability without correlation with wind. For comparison sake, the should be assessed preferably with long-term sea state short-term probability in the constant wind and waves is statistics. Ogawa et al. (2006a,2007) constructed a scatter also shown in Fig. 13. diagram of wave height, wave period and wind speed by It is found that capsizing probability of the RO-PAX means of the wind and wave database developed from ferry complying with the weather criterion is adequately hindcast calculation of wind-wave interactions and small if the statistical correlation between wind and waves examined the effect of the correlation of winds with waves is taken into account. This probability does not depend on on the long-term capsizing probability. Table 2 shows the mean wind velocity because the occurrence probability examples of the scatter diagram of wave height, wave of wind is taken into account in the calculation. The period and wind speed for a water area near Minami Daito capsizing probability with the normalized scatter diagram is Islands in the Northern Pacific. It is found that the larger than the present method when the mean wind velocity probability of large wave height and long wave period is larger 16 m/s. This indicates that occurrence probability increases as the wind speed increases. of such strong wind is not so large in this water area during The long-term probability of capsizing of a ship within a year. The capsizing probability with the no correlation N years can be described as follows: scatter diagram is larger than the present method. This is probably because unrealistic combinations of wind and ()N × 365 × 24 × 3600 / T waves are included in the calculation. ⎛ * ⎞ P = 1 − ⎜1 − P ()T ⎟ (26) The capsizing probability taken from the stationary sea N ⎝ ⎠ state is larger than the present method because the occurrence probability of such sea state is not large during a where year in this water area. On the other hand, the capsizing probability from the stationary sea state of the mean wind velocity of 26 m/s is much smaller than the present method. This suggests that the occurrence probability of sea state should be taken into account when we estimate the annual Capsizing probability (T=3600sec.) risk from the short-term capsizing probability. 0 5 10 15 20 25 30 35 0.0 -5.0 Table.2 Samples of wave height –period diagram in each -10.0 wind speed (Autumn, 1994-2003, Ws=2, 20, 40(m/s)). -15.0 -20.0

(Ogawa et al., 2006a) (probability)

Ws=2(m/s) T(sec.) 10 -25.0 Present method Hw(m)3-4-5-6-7-8-9-10-11-12-13-14-15-

11.25-0000000000000 log Beaufort(short term) 10.75-0000000000000 -30.0 10.25-0000000000000 Normalized 9.75-0000000000000 -35.0 No correlation 9.25-0000000000000 8.75-0000000000000 8.25-0000000000000 -40.0 7.75-0000000000000 Wind speed(m/s) 7.25-0000000000000 6.75-0000000000000 6.25-0000000000000 5.75-0000000000000 Fig. 13 Capsizing probability in one year of the Ro-PAX 5.25-0000000000000 4.75-0000000000000 4.25-0000000000000 ferry under dead ship condition. (Ogawa et al., 2006a) 3.75-0000000000000 3.25- 0 0 0 0 0 0 0 0.000297 0.000224 0 0 0 0 2.75-0000000.0004140.00040500000 2.25- 0 0 0 0 0.000176 0.000589 0.001021 0.000847 3.91E-06 0 0 0 0 1.75- 0 0 0 0 0.000887 0.001245 0.002002 0.002682 5.05E-05 0 0 0 0 ASSESSMENT OF SAFETY LEVEL IN TERMS OF THE 1.25- 0 0 0 1.1E-05 0.001218 0.004456 0.004191 0.001685 0.000424 0 0 0 0 0.75- 0 0 0 0.000378 0.001439 0.001877 0.000579 9.28E-05 1.9E-05 0 0 0 0 0.25- 0 0 0 0.000101 4.15E-05 3.73E-05 3.91E-06 1.86E-07 1.86E-07 0 0 0 0 PIECE-WISE LINEAR APPROACH 0-0000000000000 Ws=20(m/s) T(sec.) Hw(m)5-6-7-8-9-10-11-12-13-14-15-16- 15.25-000000000000 14.75-000000000000 Passenger and cargo ships 14.25-000000000000 13.75-000000000000 13.25-00000000003.73E-073.17E-06 12.75-00000000003.35E-061.86E-07 12.25-00000000001.12E-060 Umeda et al. (2007a, 2007b) attempted to evaluate the 11.75-000000000000 11.25-000000000000 10.75-000000000000 safety levels of four passenger and cargo ships by utilizing 10.25-000000000000 9.75-000000007.83E-064.84E-0600 9.25-00000006.71E-068.24E-05000 the above mentioned capsizing probability calculation. 8.75-00000001.58E-050.000117.45E-073.91E-060 8.25-0000002.61E-067.32E-056.17E-055.78E-063.5E-050 Their principal dimensions of these ships are shown in 7.75-000001.3E-061.01E-050.0001684.84E-066.15E-061.3E-050 7.25-000009.69E-064.45E-050.0003510000 6.75-00002.24E-064.49E-050.0002019.82E-051.49E-06000 Table 3. The method for calculating capsizing probability 6.25-00004.47E-067.84E-050.0002120.0001197.27E-061.45E-0500 5.75-00005.33E-050.0003010.0003020.0001284.1E-063.82E-0500 5.25- 0 0 0 5.03E-06 0.000106 0.00066 1.29E-05 6.28E-05 8.57E-06 6.15E-06 00 here is the piece-wise linear approach with down-flooding 4.75-0003.78E-059.69E-050.00024803.73E-061.86E-07000 4.25-0005.29E-057.81E-052.5E-0505.59E-074.84E-06000 3.75-002.96E-051.9E-053.63E-050000000 angle taken into account. The mean wind velocity is 3.25-001.58E-052.83E-053.06E-050000000 2.75-0002.24E-0600000000 assumed to be fully correlated with the wave height and the 2.25-000000000000 1.75-000000000000 effect of drifting attitude and velocity is ignored. The Ws=40(m/s) T(sec.) Hw(m) 8- 9- 10- 11- 12- 13- 14- 15- 16- 17- restoring arm is approximated to keep the metacentric 15.25- 0 0 0 0 0 0 1.86E-07 0 0 0 14.75- 0 0 0 0 0 0 1.49E-06 0 9.32E-07 5.59E-07 14.25-000000004.66E-060 height, the angle of vanishing stability and the maximum 13.75-000000009.32E-060 13.25-000000009.5E-060 restoring arm unchanged. 12.75-00000005.03E-063.17E-060 12.25-0000000000 The calculation results of the annual capsizing 11.75-0000000000 11.25- 0 0 0 0 7.45E-07 0 0 0 0 0 10.75-0000000000 probabilities under the designed conditions in the North 10.25-0000000000 9.75-0000000000 Atlantic are shown in Fig. 14. Here the annual capsizing 9.25-0000000000 8.75-0000000000 probabilities of a LPS (large passenger ship) and a 8.25-0000000000 7.75-0000000000 containership under their designed conditions are smaller 7.25-0000000000 than 10-8 so that the associated risks are assumed to be

negligibly small. In particular, the safety against capsizing 1.4 in beam wind and waves for the containership is sufficient Original GZ (kg=10.63m) 1.2 Present method although her designed metacentric height is small. This is 1 because the large freeboard of the containership results in

0.8 significant increase of the slope of righting arm curve at

GZ(m) 0.6 larger heel angle. On the other hand, larger probabilities are obtained for the Ro-Pax and the PCC (pure car carrier). In 0.4 particular, the Ro-Pax has large annual capsizing probability, 0.2 although her number of persons onboard is not small, 0 0 1020304050607080 because her natural roll period is small so that resonant Angle of Heel(deg.) probability increases in ocean waves. The Ro-Pax ferry investigated here is operated not in the Fig. 12 Righting arm curves of the RO-PAX ferry. (Ogawa North Atlantic but in Japan’s limited greater coastal area, et al., 2006a) which is water area within about 100 miles from the

Japanese coast. (Watanabe and Ogawa, 2000) In addition, The capsizing probability calculation in this paper her navigation time ranges only from 6 to 12 hours so that except for this section assumes that wind is fully correlated she can cancel her voyage if bad weather such as a typhoon wave height based on the WMO’s reference data (Umeda et is expected. Therefore the calculation of capsizing al., 1992). It will be a future task to compare this probabilities of the case in Japan’s limited greater coastal methodology with the capsizing probability calculated area and the case in Japan’s limited greater coastal area with taking account of the statistical correlation between wind operational limitation to avoid the significant wave height and waves. Then the final conclusion whether the statistical greater than 10 meters are executed and compared with the correlation should be considered for the performance-based case in the North Atlantic, as shown in Fig. 15. Here the criteria or not will be discussed. aero- and hydrodynamic coefficients were estimated with experimental data. The comparison indicates that the effect of water areas is not so large because worst combinations of significant wave height and mean wave period exist also in Japan’s limited greater coastal area. This seems to be reasonable because Table. 4 Principal dimension of the subject ships at full typhoons are included in the wave statistics in Japan’s load departure condition. Here FBS: freeboard, TZ: natural limited greater coastal area. heave period, hB: height of bulwark and Af: area of freeing port. (Paroka & Umeda, 2007) Table.3 Principal dimensions of ships under designed conditions. Here W: displacement, Tφ: natural roll period, Items Units new old 135 80 GT 39 GT

N: number of persons onboard. (Umeda et al., 2007b). 135 GT GT

Container Lpp Meters 38.50 34.50 29.00 23.00 Items LPS PCC RO/PAX Ship BS Meters 8.10 7.60 6.80 5.90 W (t) 53,010 109,225 27,216 14,983 d Meters 2.851 2.65 2.25 1.77 FBS Meters 0.485 0.496 0.484 0.396 Lpp (m) 242.25 283.8 192.0 170.0 Δ Tons 480.98 431.07 276.61 145.70 B (m) 36.0 42.8 32.3 25.0 GM Meters 1.95 1.65 1.57 1.79 Tφ Seconds 5.80 5.87 5.20 3.886 d (m) 8.4 14.0 8.2 6.06 Tz Seconds 3.413 3.435 3.246 2.218 GM (m) 1.58 1.06 1.25 1.41 hB Meters 1.50 1.45 1.35 1.30 2 Af Meters 7.05 4.75 3.40 2.60 T (s) 23.0 30.3 22.0 17.9 N 4332 33 25 370

0.8 1.0E+00 0.7

0.6

1.0E-02 0.5

0.4

4.98E-05 1.0E-04 3.66E-05 0.3 New 135 GT Freeboard (m) Freeboard Old 135 GT 0.2 80 GT 39 GT 1.0E-06 0.1 PROBABILITY

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1.0E-08 Metacentric Height (m)

Fig. 16 Marginal lines of purse seiners for the annual 1.0E-10 -6 RO/PAX LPS CONTAINER SHIP PCC capsizing probability of 10 . (Paroka & Umeda, 2007)

Fig. 14 Annual Capsizing Probability for passenger and 0.8

cargo ships in the North Atlantic. (Umeda et al., 2007b) 0.7

0.6 1.00E+00 0.5

3.66E-05 0.4 1.23E-05 New 135 GT 1.00E-05 0.3 Old 135 GT Freeboard (m) Freeboard 80 GT 0.2 39 GT

0.1 1.00E-10 2.92E-12 0.0

PROBABILITY 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Metacentric Height (m) 1.00E-15 Fig. 17 Marginal lines of purse seiners for the IMO weather criterion for fishing vessels. (Paroka & Umeda, 1.00E-20 NORTH ATLANTIC JPN(Limited greater coastal area） JPN(Limited greater coastal 2007) area+Operational limitation）

Fig. 15 Effect of operational conditions on annual capsizing Paroka & Umeda (2007) estimated the safety levels of probability for the RO/PAX. (Umeda et al., 2007b) four Japanese purse seiners, whose principal dimensions are listed in Table 4, as one of typical fishing vessels types. By On the other hand, the significant reduction in the utilizing the piece-wise linear approach with water trapped capsizing probability is realized by the operational on deck taken into account, combinations of metacentric height and freeboard realizing the annual capsizing limitation. This explains that the Ro-Pax ferry is safely -6 operated in actual situations with the aid of the operational probability is 10 are identified as shown in Fig. 16. The limitation to avoid bad weather. water areas used here are actual fishing grounds within the Fishing vessels Japanese EEZ for these vessels so that wave statistics are obtained from the NMRI database and actual fish catch calculated probability coincides with the Mote Carlo records (Ma et al., 2004) with assumption that wind velocity simulation results within its confidence interval for the is fully correlated with the wave height. This diagram sea state relevant to the weather criterion indicates that the designed full departure condition for each 2. The calculation method was extended to cover effects vessel has safety level above 10-6. Combinations of of flooding angle or cargo shift angle, water on deck, metacentric height and freeboard for marginally complying hydrodynamic coefficient estimation, drifting attitude with IMO weather criterion for fishing vessels and the and velocity and the correlation between wind and water-on-deck criterion in the recommendation of the waves. Torremolinos Convention are shown in Figs.17-18, 3. By utilizing the above extended methods, safety levels respectively. The comparison among these results of passenger and cargo ships as well as fishing vessels demonstrates that the weather criterion for fishing vessels are calculated as annual capsizing probability, and on its own is not sufficient for ensuring safety of these show that current designs ensures reasonable levels of fishing vessels because of trapped water on deck but the safety against intact capsizing under good operation water-on-deck criterion in the Torremolinos Convention is practice. comparable to the current probability-based calculation. ACKNOWLEDGEMENTS

0.8 A part of the present study was carried out in cooperation 0.7 with the Japan Ship Technology Research Association 0.6 through the part of the Japanese project for the stability 0.5 safety (SPL project) that is supported by the Nippon 0.4 Foundation. This research was also supported by a Grant-in 0.3 Aid for Scientific Research of the Japan Society for Freeboard (m) 0.2 New 135 GT Promotion of Science (No. 18360415). The authors thank Old 135 GT 0.1 80 GT 39 GT Drs. Y. Sato and M. Ueno, Misses E. Maeda and I. 0.0 Tsukamoto, Messrs. J. Ueda, S. Koga, H. Sawada and H. 0.00.20.40.60.81.01.21.41.61.82.0 Metacentric Height (m) Takahashi for their contribution to the works described in this paper.

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