Proceedings of OMAE 02 21 st International Conference on Offshore Mechanics and Artic Engineering June 23-28, 2002,Oslo, Norway OMAE2002-28477 RANDOM WAVES AND CAPSIZE PROBABILITY BASED ON LARGE AMPLITUDE MOTION ANALYSIS Jan O. de Kat MARIN Wageningen, the Netherlands Dirk-Jan Pinkster MARIN Wageningen, the Netherlands Kevin A. McTaggart DREA Dartmouth, NS, Canada ABSTRACT The objective of this paper is to apply a methodology aimed at the probabilistic capsize assessment of two naval ships: a frigate and a corvette. Use is made of combined knowledge of the wave and wind climate a ship will be exposed to during its lifetime and of the physical behavior of that ship in the various sea states it is likely to encounter. This includes the behavior in extreme wave conditions that have a small probability of occurrence, but which may be critical to the safe operation of a ship. Time domain simulations provide the basis for deriving short-term and long-term statistics for extreme roll angles. The numerical model is capable of predicting the 6 DOF behavior of a steered vessel in wind and waves, including conditions that may lead to broaching and capsizing. INTRODUCTION The ability to predict the large amplitude motions of a ship in a seaway accurately is a necessary but not sufficient condition for quantifying capsize risk levels. It is also necessary to consider realistic operational scenarios over a given part of the ship's lifetime, including operational areas, encountered wave and wind climate, ship loading and ship handling aspects. This can be done within a probabilistic framework that makes use of efficient numerical modeling techniques and reliable data. Extreme rolling and - ultimately - capsizing in critical wave and operational conditions may occur due to the following mechanisms [1]: Static loss of stability (e.g. in wave crest) Dynamic loss of stability (e.g. parametric resonance) Broaching Combined modes with additional factors such as water on deck or wind To simulate such dynamic behavior, a numerical model is used that captures important parts of the physics involved. The numerical model and some properties of irregular waves have been discussed in [2], and the methodology of the probabilistic capsize analysis outlined above has been presented in [1]. This paper will focus on the following issues: - determination of short-term distribution of extreme roll angles based on time domain simulations - determination of long-term (yearly) distribution of extreme roll angles - influence of steering system (autopilot) on short-term probability of extreme rolling - relationship between high probability of capsize conditions to sea state and extreme wave properties The above analysis is applied to a corvette and a larger frigate ship in a normal loading condition for a range of heading angles (head to following seas) per sea state, for a large number of different sea states. Critical heading conditions in terms of extreme rolling tend to lie between following to beam seas. In these conditions the way in which a ship is steered may have a significant influence on the roll motions. To study this aspect, a typical "PID" controller is used (with fixed Proportional and 1 Copyright 2002 by ASME
Differential control coefficients) in the simulations and compared with an adaptive autopilot in terms of resulting roll probability distributions for some selected conditions. SHIP CAPSIZE PROBABILITY IN RANDOM WAVES General methodology The method used for obtaining capsize probabilities in random waves is in principle the same as described in [1]. In the present study it is applied to two ships of different length. Each ship is subjected to the same operational area and wave statistics: annual North Atlantic wave climate, where use is made of a wave scatter diagram that is based on a slightly modified Bales wave scattergram [1]. For a ship in a seaway of duration D (e.g., one hour), the probability of capsize P(C D ) is: P( CD ) = p( V ) p( β) p( Hs, Tp) P( CD V, β, Hs, Tp) (1) where p(x) is the probability density function for discretized variable X, V is ship speed, β is the relative wave heading, Hs is significant wave height, and Tp is peak wave period, and p(hs,tp) is their joint probability density. The last term of Equation 1 denotes a conditional capsize probability given a set of operational and seaway conditions, which is derived from numerical simulations using the model described in the next section. Similarly, the exceedence probability for maximum roll angle is given by: Q( φ max, D ) = p( V ) p( β ) p( Hs, Tp) Q( φ V, β, Hs, Tp) max, D where Q(X) is the exceedence probability for variable X. When predicting capsize probability, one must chose a suitable duration D for evaluating Equation 1. This duration should correspond to the observation interval for the wave statistics described by the joint distribution p(hs,tp). Wave statistics typically have observation intervals between one and three hours, during which time the wave conditions are assumed to be stationary. Once the probability of capsize for duration D has been computed using Equation 1, the associated annual probability of capsize can be computed as follows: P(C α 1year/D [ 1 - P(C ] annual ) 1 - D ) = (3) where α is the fraction of time spent at sea. (2) Capsize Probability in Given Conditions Using Fitted Gumbel Distribution The probability of capsize under given conditions is evaluated using the approach described in [3]. This approach considers the variation of maximum roll angle with seaway realization by considering maximum roll angles from a moderate number of time domain simulations (e.g., between 10 and 50). Ship speed, wave height, significant wave height, and peak wave period remain constant among the simulations for given conditions; however, each simulation uses a different seed number for generating random wave phases of sinusoidal components that are summed to model a long-crested random seaway. Between 20 and 100 wave components are typically used to model a random seaway. Using results from simulations for given conditions, distribution-free estimates of the cumulative distribution function F(X) = 1-Q(X) are as follows: ˆ i F( X i ) = (4) Ns + 1 where in this case X is used to denote φ max,d V,β,Hs,Tp, and i denotes the rank (increasing order) among simulation results, and N s is the number of simulations for the conditions V,β,Hs,Tp. Using the simulation results, a Gumbel distribution can be fitted to model maximum roll angle for given conditions: b X F( X ) = exp exp (5) a where a and b are scale and location parameters, with b being the 36.8 th percentile of X. The probability of capsize for given conditions can then be determined by: b φ P( C = C D V, β, Hs, Tp) 1 exp exp (6) a where φ C is the capsize roll angle (e.g., the angle of downflooding or zero static stability). Fitted Gumbel parameters are determined by using a least squares approach applied to ln[-ln(f(x))] in the following: b X ln [ ln( F( X ))] = (7) a A significant advantage of the least squares method is that it can be applied to a limited variable range of greatest interest. For predicting ship capsize risk, it appears appropriate to fit the Gumbel distribution to the upper 30 degree range of simulation 2 Copyright 2002 by ASME
roll angles for given conditions because the upper range is most important for ship capsize. Figure 1 shows an example of a Gumbel distribution fitted to maximum roll angles from simulations. Exceedence probability Q 1 0.1 0.01 Simulations Gumbel fit 0 30 60 90 Maximum hourly roll (deg) Figure 1: Roll Exceedence Probability for Frigate in One- Hour Seaway, V = 10 knots, β = 75 degrees, Tp = 12.4 s, Hs = 9.5 m, Wind speed = 40 knots Alternatives to Fitted Gumbel Distribution The theoretical basis of the Gumbel distribution is the maximum of a Gaussian process over a relatively large number of cycles [4]. Small amplitude ship motions in a random seaway can be considered to be a Gaussian process, and a Gumbel distribution provides an excellent model of maximum response. For ship roll motions approaching capsize, the response is no longer Gaussian and the Gumbel distribution is no longer ideal for modeling maximum response. As an alternative to using the fitted Gumbel distribution approach described above, capsize probabilities have been analyzed using a distribution free method, whereby capsize probabilities under given conditions are estimated using linear interpolation between sample values from Equation (4). The advantage of the distribution free method is that no assumption is required regarding the distribution of maximum roll angle. The main disadvantage of the distribution free method is that it does not facilitate extrapolation beyond the range of observed values. For example, the probability of exceeding the largest observed maximum roll angle for given conditions is taken as being zero. given very similar capsize probabilities when summing probabilities over a large number of conditions using Equation 1. The good agreement of summed capsize probabilities from the fitted Gumbel and distribution free approaches arises because distribution free estimates have no bias and fitted Gumbel estimates have only small bias. The generalized extreme value distribution (GEV) has also been considered for modelling maximum roll angle under given conditions. The GEV distribution is discussed in [5] and can be written as follows: 1/ c X b F( X ) = exp 1+ c (8) a where a is a scale parameter, b is a location parameter, and c is a shape parameter. The terms a and 1+c(X-b)/a in the above equation must be greater than 0. If c = 0, then the GEV distribution is a Gumbel distribution, Equation (5). For c < 0 the GEV distribution is a 3-parameter reverse Weibull distribution. In the case of c > 0, the GEV distribution represents a 3-parameter type II distribution. In comparison with the Gumbel distribution, the GEV distribution has the additional flexibility provided by the shape parameter c. Figure 2 gives an example of fitted Gumbel and GEV distributions for maximum roll angles from 500 time domain simulations. The GEV distribution parameters were determined using the maximum likelihood method presented in [6]. For small exceedence probabilities (Q(X) < 0.01), the Gumbel distribution always appears as a straight line when plotted on a log-log scale. In contrast, the shape parameter c allows the GEV distribution to appear as concave-up or concave-down (or straight) for small exceedence probabilities when plotted on a log-log scale. Experience with the GEV distribution has shown that it leads to great over prediction of capsize probabilities. This experience is based on using 10-50 maximum roll angles for given conditions when determining GEV distribution parameters. Figure 2 shows a case using 500 simulations in which the GEV distribution clearly leads to over prediction of capsize probability. Experience using fitted Gumbel distributions and the distribution free method has indicated that the two approaches 3 Copyright 2002 by ASME
Exceedence probability Q 1 0.1 0.01 Simulations Gumbel limited range fit GEV fit 0.001 0 30 60 90 Maximum hourly roll (deg) Figure 2: Fitted Gumbel and GEV Distributions of Hourly Maximum Roll, No Wind, V = 10 knots, β = 15 degrees, Tp = 13.9 s, Hs = 9.5 m MODELING OF STEERED SHIP IN WAVES Extreme motions of intact vessels To determine the distribution of extreme roll angles, on which the conditional capsize probability Eq. (6) is based, use is made of a time domain simulation model. This model is applied to a large variety of sea states, wave realizations, heading angles and operational conditions to obtain the desired statistics. Work on time domain capsize prediction started in the mid 1970s, notably at the University of California, Berkeley. Time domain simulations offer the advantage of the ability to account for nonlinearities in the ship system and external forces in a comprehensive fashion. This section presents a brief description of the time domain model (FREDYN) that has been applied in this work. The model consists of a non-linear strip theory approach, where linear and non-linear potential flow forces are combined with maneuvering and viscous drag forces. The nonpotential force contributions are of a nonlinear nature and based on (semi)empirical models. The derivation of the equations of motions is based on the conservation of linear and angular momentum. These are given in principle in the inertial (earth-fixed) reference system. Euler s method is applied for deriving the equations of motion in terms of a rotating, ship-fixed coordinate system. The equations of motion are given by: ([ M] + [ a( )] ). F x F y F z x = M x M y M z m.(wq - vr) m.(ur - wp) m.(vp - uq) ( I zz,0 - I yy,0 )qr ( I xx,0 - I zz,0 )pr ( I yy,0 - I xx,0 )pq [M] is the generalized (6x6) mass matrix of the intact ship, [a( )] is the added mass matrix, and x is the acceleration vector at the center of gravity; u, v and w are the translational velocities; p, q and r represent the rotational velocities for roll, pitch and yaw, respectively. The summation signs in the right hand side represent the sum of all force and moment contributions, which result from: Froude-Krylov force (nonlinear) Wave radiation (linear) Diffraction (linear) Viscous and maneuvering forces (nonlinear) Propeller thrust and hull resistance (nonlinear) Appendages -- rudders, skeg, active fins (nonlinear) Wind (nonlinear) Internal fluid (nonlinear) Large angles must be retained in the matrices for transformation between the ship-fixed and the earth-fixed coordinate system in relation to acceleration vectors and rotational velocity vectors. One of the most important force contributions that should be treated "exactly" is associated with the hydrostatic and dynamic wave pressure. This represents the Froude-Krylov force, which is obtained by pressure integration over the instantaneous wetted surface of the hull at each time step. This will account for a large part of the nonlinearities that affect the ship response. Linear wave theory is used to describe the sea surface and wave kinematics. In the case of irregular waves, the model makes use of linear superposition of sinusoidal components with random phasing. Linear transfer functions (from strip theory) are used in the determination of the diffraction forces. The wave radiation forces are based on linear retardation functions and convolution integrals (including forward speed terms): t F ( t) = A x ( t) B x ( t) C x ( t) K ( t τ ) x ( τ ) dτ jk jk k jk k jk k jk k 0 (10) (9) 4 Copyright 2002 by ASME
Viscous effects include roll damping due to hull and bilge keels, wave-induced drag due to wave orbital velocities, and nonlinear maneuvering forces with empirically determined coefficients. The quasi-steady hull forces resulting from the motions in the horizontal plane consist of a linear and nonlinear part; for instance, the sway force is: F 2, H (t) = F H,L (u,v,r,θ;t) + F H,NL (v(x,t), T(x,t), C Dx (u(t),x), u,v,r,η t ) (11) developed an adaptive controller. The autopilot is based on Linear-Quadratic (LQ) control, which is a state-space technique for designing optimum dynamic control systems. It enables one to trade off control performance against control effort by means of a control performance criterion and weighting matrices. This adaptive controller has been implemented in FREDYN as an alternative autopilot. Figure 3 shows a schematic of the autopilot system. where v(x,t) is the local transverse velocity at (sectional) location x, T(x,t) is the local draft, C Dx is the local cross-flow drag coefficient, and η t is the local wave orbital velocity in transverse direction. The roll moment resulting from lift and hull drag forces has the following nature: M x,h (t) = F H,L.z(t) + F H,NL (t).z(x,t) + K ur.u(t).r + K up.u(t).p + K pp p p (12) Actual state (position, course, speed) Controller Task (course control) Freq Allocation Rudder Rudder angle The propeller thrust depends on the propeller characteristics and instantaneous inflow conditions. The hull resistance is a function of instantaneous speed and draft. Appendage forces are estimated by using wing theory in the case of a rudder or active fin, and by using pressure drag in the case of a skeg. Interaction between rudder and hull is accounted for. The equations of motion are solved in the time domain using a higher order Runge-Kutta scheme. Additional information (wave and windforces) Pilot setting Figure 3: Adaptive autopilot Course keeping A control system is needed for a ship to maintain course in waves and wind. Typically a simple autopilot linked to the rudder is used to provide course keeping ability in time domain simulations; in our work we have used extensively a PID-type (Proportional-Integrative-Differential) controller: ψ c () t c ( ψ () t ψ ) + c ψ ( t) = d 2 1 (13) where ψ c (t) is the commanded rudder angle, ψ(t) is the instantaneous heading angle of the ship, and ψ d is the desired heading angle. Selection of the gain coefficients c 1 and c 2 is a matter of judicious choice, based on experience and overall performance in relevant wave conditions. The values of these coefficients can have a significant influence on the roll response, as is evident from model test and simulation results. This implies also that the role of the autopilot in probabilistic capsize assessment can be significant. For instance, the extreme roll distributions derived in [2] are valid for a ship with constant gain coefficients, i.e., the autopilot settings were kept the same for all operational conditions and sea states covered in the scatter diagram. To enable more realistic course control in simulations, MARIN s Maritime Simulator Centre Netherlands (MSCN) The model is valid for the control of relatively small track and course deviations. The model is linearized around the actual forward velocity and matching propeller revolutions; by optimizing the performance criterion a linear feed back gain is derived. When a ship is moving relatively fast the autopilot reacts differently than when it is moving slowly. The autopilot settings are defined such that a fixed relation between deviation of desired course and position results in the same response, e.g. a deviation of 2 meters in x and 3 meters in y and 1 degrees course change can result in the same required effort. It is possible to change the weighting factors and make the autopilot more sensitive to a specific deviation such as heading angle. The linear feedback gain is adapted to generate continuous or discrete (at a fixed time step) rudder force orders. In the case of the FREDYN-autopilot, the feedback system generates a requested yaw moment, N req, for course control at a fixed time step of t=0.5 s, shown by Equation (14), and allocates the requested moment to the rudder(s). The input of the autopilot consists of ship position, heading and velocities and propeller revolutions, as well as on the desired parameters related to course keeping. Based on the difference in heading ( ψ) and track deviation ( y), the control settings are calculated and the required rudder angle, δ, is passed to the rudder actuator. The 5 Copyright 2002 by ASME
objective is to regulate the differences in position (or sway) and heading around zero. N req = c 1 y c 2 ψ c 3 v c 4 r δ req = f ( N req ) δ max δ req δ max CAPSIZE ASSESSMENT FOR TWO SHIPS The procedure outlined above for determining the short and long term capsize probabilities has been applied to two generic naval ships -- a frigate with a length of 120 m and a smaller corvette with a length of 70 m. To reduce the number of simulations to a certain extent, it has been assumed that both ships travel always at a speed of 10 knots in all sea states and heading angles when at sea. The speed of 10 knots is considered as a realistic speed for operation in severe sea conditions. Ship dimensions and hydrostatics (14) (6) GZ [m] 1.2 1 0.8 0.6 0.4 0.2 0-0.2 KG=3.77m KG=3.55m KG=3.37m Generic 70m frigate -0.4 0 10 20 30 40 50 60 70 80 90 Heel angle [deg] Figure 4: GZ curves for 70 m corvette The main dimensions of the two frigates are given in Table 1. For each ship simulations have been performed for three KG conditions at the same draft, T; the KG values were chosen such that the angles of vanishing stability were the same for both ships (80, 90 and 100 degrees). Figures 4 and 5 show the righting arm (GZ) curves in calm water for the three loading conditions of frigates 1 and 2, respectively. Corvette Frigate LBP (m) 70.00 120.00 B (m) 8.24 14.12 D (m) 7.10 12.17 T (m) 2.51 4.30 (tonnes) 749 3784 GM_KG1 (m) 0.56 0.86 GM_KG2 (m) 0.78 1.23 GM_KG3 (m) 0.96 1.55 T φ (KG1) (s) 7.8 10.5 T φ (KG2) (s) 6.5 8.8 T φ (KG3) (s) 6.0 7.2 AREA_KG1 (m-rad) 0.281 0.485 AREA_KG2 (m-rad) 0.482 0.823 AREA_KG3 (m-rad) 0.678 1.171 Table 1: Ship characteristics for Corvette and Frigate Table 1 shows the GM and roll period T φ (at zero speed in calm water) associated with the three KG values for each ship. In addition, the associated area underneath the positive righting arm curve is shown in Table 1 for the three KG conditions. The GZ area represents a measure of restoring energy of a ship (also referred to as total dynamic stability ) and in previous studies was found to have a definitive influence on capsizing [7]. GZ [m] 1.2 1 0.8 0.6 0.4 0.2 0-0.2 KG=6.46m KG=6.09m KG=5.77m Generic 120m frigate -0.4 0 10 20 30 40 50 60 70 80 90 Heel angle [deg] Figure 5: GZ curves for 120 m frigate Extreme roll probability distributions To determine the short-term extreme roll probability given by Equation (1), the procedure for determining the conditional capsize probability, Eq. (7), was applied for one ship speed (V = 10 knots) for both ships in each sea state. This was done for all sea states covered by the modified Bales scatter diagram for the annual North Atlantic wave statistics. Furthermore, all heading angles with respect to the seaway were evaluated (from 0, 15,, 180 degrees) for each operational case. Subsequently the long-term (annual) probability of exceeding a critical roll angle was determined using Equation (3). 6 Copyright 2002 by ASME
All results shown below apply to the ships being kept on course with the PID-type autopilot. The annual cumulative probability of exceeding a given roll angle is given in Figures 6 and 7 for frigates 1 and 2, respectively. Generic 70m frigate capsize reduces significantly. Moreover, it is interesting to note that the lowest GM condition (KG1: = 80 deg) of the 120 m frigate results in approximately the same capsize probability of the 70 m frigate at its medium GM condition (KG2: = 90 deg). In these loading conditions both ships have the same areas underneath the positive GZ curve (approximately 0.48 m-rad), as shown in Table 1. Probability of exceedance [-] 10-3 Capsize probability [-] = 80 deg = 90 deg = 100 deg 10-4 0 10 20 30 40 50 60 70 80 90 Roll angle [deg] Figure 6: Cumulative distribution function for roll of 70 m corvette (annual probability of exceedence) Probability of exceedance [-] Generic 120m frigate 10-3 Generic 70m frigate Generic 120m frigate 10-4 80 82 84 86 88 90 92 94 96 98 100 [deg] Figure 8: Annual capsize probabilities for 70 m corvette and 120 m frigate as a function of angle of vanishing stability Although the annual capsize probabilities of the two frigates are similar for the loading conditions resulting in the same GZ area, their extreme motions can differ significantly in the same sea state. 10-3 = 80 deg = 90 deg = 100 deg 10-4 0 10 20 30 40 50 60 70 80 90 Roll angle [deg] Probability of exceedance [-] Figure 7: Cumulative distribution function for roll of 120 m frigate (annual probability of exceedence) The annual probability of exceeding an angle of 90 degrees (assumed to be a capsize in the present analysis) is given in Figure 8 for frigates 1 and 2 as a function of the angle of vanishing stability,. It is clear that the capsize probability is sensitive to and that as the angle increases the probability of Generic 70m frigate Generic 120m frigate 10 20 30 40 50 60 70 80 90 Roll angle [deg] Figure 9: One-hourly roll exceedence probability for frigate and corvette (Tp = 9.7 s, Hs = 6.5 m, 45 deg heading); = 90 deg for both ships 7 Copyright 2002 by ASME
To compare the extreme roll behavior of the two ships in a critical sea state, figure 9 shows the one-hourly roll exceedence probabilities for the frigates and corvette with loading conditions resulting in the same angle of vanishing stability ( = 90 deg). It suggests that the 70 m corvette has a somewhat higher roll exceedence probability than the 120 m frigate. Probability of exceedance [-] Probability of exceedance [-] Generic 70m frigate φ =90 v Generic 120m frigate φ =80 v 0 10 20 30 40 50 60 70 80 90 Roll angle [deg] Figure 10: One-hourly roll exceedence probability for frigate and corvette (Tp = 9.7 s, Hs = 6.5 m, 45 deg heading); equal area under GZ curve for both ships In contrast, Figure 10 shows the same information as Figure 9, but in this case the loading condition for the 120 m frigate is such it results in the same area of the GZ curve (0.48 m-rad) as for the corvette. Here it is clear that the capsize probability of the frigate is significantly higher than for the smaller ship. The same roll distribution characteristics apply for another critical sea state, with Tp = 12.4 s and Hs = 9.5 m. A possible explanation for the behavior may be that these sea states contain individual waves that are more critical to the larger frigate in terms of wave length and height. Influence of autopilot To investigate the influence of autopilot on capsize probability, some of the most critical conditions were repeated in the simulations using the adaptive autopilot. Analysis of the shortterm (one-hourly) capsize probabilities suggested that one of the more critical heading angles was 45 degrees, i.e., stern quartering conditions. For the case shown in Figure 11, the adaptive autopilot results in somewhat lower roll exceedence probabilities than the PID controller for extreme roll angles, but the differences are of a moderate level in this case. Further work will continue to investigate the adaptive autopilot. Adaptive PID 20 25 30 35 40 45 50 55 60 65 Roll angle [deg] Figure 11: One-hourly roll exceedence probability for 120 m frigate with PID and Adaptive autopilot control ( = 90 deg, V = 10 kn, 45 degrees heading, Tp = 12.4 s, Hs = 8.5 m) PROBABILITY OF ENCOUNTERING CRITICAL WAVE CONDITIONS The occurrence of an extreme roll event or capsize is dependent on an encountered individual wave or group of waves with critical wavelength and height, given the ship speed and heading angle. Analysis of the short-term (one-hourly) roll distributions suggests that the highest probability of capsize can be expected for the steepest sea states. The characteristic steepness of the seaway, defined by Hs/λp (where λp = gtp 2 /2π), determines the probability of occurrence of critical, steep waves. The maximum characteristic steepness observed for ocean waves lies typically around 0.05; wave statistics suggest the average steepness for worst North Atlantic storm waves is approximately 0.035. For both ships, the capsize probability is relatively high for sea states with peak periods between 9 and 13 s; the corvette is more sensitive to sea states with shorter periods than the frigate. It is possible to make an assessment of the probability structure of waves based on the joint distribution of individual wave heights and periods. Let us consider the properties of a steep storm sea state obtained from buoy measurements in the North Atlantic, taken in deep water off the East coast of Canada. The significant wave height is 10.7 m with a peak period of 12.4 s (H S /λp = 0.044); to obtain statistically reliable distributions, the 20 minute time series with measured wave elevation data were concatenated into a stationary time series of about two hours duration. This sea state lies at the outer edge of the Bales wave scatter data and would be critical in terms of extreme roll probabilities for the ships considered. 8 Copyright 2002 by ASME
The outer contours represent the waves with smallest probability of occurrence; Figure 12 shows that the highest observed wave has a height of about 19 m and a period of 10.7 s. The same information can be represented in terms of spatial (zero-crossing) wavelength and height, represented in Figure 13. It can be seen, for instance, that waves with a height of more than 15 m have lengths ranging between 170 m and 300 m. As a last example of how such data can be presented, figure 14 shows the joint distribution of wave steepness as a function of wavelength, where the individual wave steepness is taken to be s = H/λ. The figure shows that the steepest waves have a length ranging from about 50m to 180 m; their maximum steepness is around s = 0.10. Figure 12: Joint probability distribution function of wave period and height (full scale measurements, H s = 10.7 m, T p = 12.4 s) Figure 12 shows the joint distribution (probability density function) of the zero-crossing wave periods, T z, and associated crest-to-trough wave heights, H. Figure 14: Joint pdf of wavelength and steepness (full scale measurements, H s = 10.7 m, T p = 12.4 s). Figure 13: Joint pdf of spatial wavelength and height (full scale measurements, H s = 10.7 m, T p = 12.4 s). The probabilistic description of the wave surface as discussed above can provide a useful indication of the severity of a given sea state. A sea state with a realistic occurrence of high, steep waves can be considered potentially dangerous. Using the joint probability density function, it is easy to estimate the probability of occurrence of a wave with certain critical properties. Let us assume (purely for illustration purposes and disregarding conditional aspects) that the frigate with a length L = 120 m will experience extreme motions, possibly up to the point of capsize, when it meets a wave with the following properties: Critical wavelength range: λ min < λ < λ max Critical minimum wavelength: λ min = 0.85 x L 9 Copyright 2002 by ASME
Critical maximum wavelength: λ max = 1.75 x L Critical minimum steepness: s = H/λ = 0.08 (assumed for all wavelengths) Given the above properties, the probability of occurrence of such waves in the above conditions with H s = 10.7 m and T p = 12.4 s is 0.08, i.e., 8% probability in this sea state. If the critical minimum wave steepness were 0.09 instead of 0.08, the probability would be reduced to 4.7%. CONCLUSIONS This paper provides a brief overview of the background and framework for obtaining estimates of short-term and long-term capsize probability using time domain ship motion simulations. The analysis has been applied to two naval ships, a frigate of 120 m in length and a corvette with 70 m length. [4] Madsen, H.O., Krenk, S., and Lind, N.C., Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, New Jersey, 1986 [5] Naess, A. and Clausen, P.H. (1999). "Statistical Extrapolation and the Peaks Over Threshold Method", Proceedings 18th International Conference on Offshore Mechanics and Arctic Engineering - OMAE '99, St. John s, Newfoundland [6] Hosking, J.R.M. (1985). "Maximum-likelihood Estimation of the Parameters of the Generalized Extreme-value Distribution", Applied Statistics, 34, pp. 301 310 [7] De Kat, J.O., Brouwer, R., McTaggart, K. and Thomas, W.L., "Intact ship survivability in extreme waves: new criteria from a research and navy perspective", Proceedings STAB '94 Conference, Melbourne, Florida, November 1994 From the results the following conclusions are drawn: For loading conditions resulting in the same angle of vanishing stability, the long-term capsize probability of the 120 m frigate is significantly lower than for the 70 m vessel. For loading conditions resulting in the same total dynamic stability (area underneath the positive righting arm curve), the annual capsize probability is approximately the same for both ships. Short-term roll exceedance distributions for specific wave conditions can differ significantly for both ships, even when the GZ areas are similar. Application of an adaptive autopilot results in a modest decrease of the capsize probability for the 120 m frigate. ACKNOWLEDGMENTS The authors would like to acknowledge the Cooperative Research Navies Dynamic Stability group for their continued efforts in extending the boundaries of developing and applying methods for dynamic stability assessment. REFERENCES [1] McTaggart, K.A. and De Kat, J.O., "Capsize Risk of Intact Frigates in Irregular Seas", SNAME Transactions, 2000 [2] De Kat, J.O., "Irregular Waves and Their Influence on Extreme Ship Motions", Proceedings 20 th Naval Hydrodynamics Symposium, Santa Barbara, August 1994 [3] McTaggart, K.A., "Ship Capsize Risk in a Seaway Using Fitted Distributions to Roll Maxima", Transactions of the ASME, Journal of Offshore Mechanics and Arctic Engineering, Vol. 122, No. 2, 2000 10 Copyright 2002 by ASME