Statistical analysis of wave-induced extreme nonlinear load effects using time-domain simulations

Applied Ocean Research 28 (2006) 386–397 www.elsevier.com/locate/apor

Statistical analysis of wave-induced extreme nonlinear load effects using time-domain simulations MingKang Wu a,b,∗ , Torgeir Moan b a Norwegian Marine Technology Research Institute (MARINTEK), N-7450 Trondheim, Norway b Centre for Ships and Ocean Structures (CeSOS), Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway

Received 10 September 2006; received in revised form 27 February 2007; accepted 7 March 2007 Available online 23 April 2007

Abstract A new hybrid method for the time-domain nonlinear simulation of the hydroelastic load effects and the peak-over-threshold (POT) method for the calculation of the short-term extreme responses are briefly described and applied to a flexible containership of the latest design. Statistical analysis has been carried out to study the sensitivity of the predicted extreme vertical bending moments and vertical shear forces to the changes in the threshold of the POT method, as well as the statistical uncertainty in the prediction due to the limited duration of the nonlinear simulation. It is recommended that 90%–95% quantile should be used as the threshold in the POT method and more than 100 h of time-domain simulation should be carried out in order to obtain satisfactory predictions of the short-term extreme nonlinear load effects. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Statistical analysis; Extreme load effects; Peak-over-threshold method; Nonlinear time-domain simulation; Hydroelasticity; Computational method; Marine engineering

1. Introduction Ship motions and wave-induced load effects in a shortterm sea state depend on many different parameters, such as loading condition, vessel speed U , significant wave height Hs , average zero-crossing period Tz , relative wave heading β, wave spreading θ, etc. It is well established in linear theory that the short-term ship response is a stationary and ergodic Gaussian narrow band stochastic process with zero mean. Therefore, the peaks and troughs, which are regarded as negative peaks, have a probability structure described by the Rayleigh distribution. Applying order statistics, the probability of the short-term extreme response Re exceeding level r per unit time can be expressed as Pshort-term (Re > r ) = ne

−

r2 2σ 2

(1)

where σ is the standard deviation of the stochastic process. n is the average number of peaks per unit time. ∗ Corresponding author at: Norwegian Marine Technology Research Institute

(MARINTEK), N-7450 Trondheim, Norway. Tel.: +47 73595500; fax: +47 73595870. E-mail address: [email protected] (M.K. Wu). c 2007 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter doi:10.1016/j.apor.2007.03.001

In linear theory, it is assumed that the wave environment a ship encounters in her life time consists of many short-term sea states and relative wave headings. In each combination of short-term sea state and relative wave heading, the probability structure of linear ship response does not vary with time and position in the wave field. That is, the statistical parameters that describe the response are constant. Therefore, the longterm probability of exceedance is a superposition of all shortterm probability of exceedance weighted by the probability of occurrence of different combinations of short-term sea states and relative wave headings. When the relative wave heading β is statistically independent of the short-term sea state characterized by its significant wave height Hs and average zero-crossing period Tz , the long-term probability of exceedance per unit time can be written as XXX Plong-term (Re > r ) = P(β)P(Hs , Tz ) Tz

Hs

β

× Pshort-term (Re > r |β, Hs , Tz ).

(2)

It is often assumed, and justified by statistical data for most ships, that β is uniformly distributed between −π and +π . The joint probability P(Hs , Tz ) is given in the form of a scatter

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diagram for any ocean area. The ship’s speed is absent in Eq. (2) because it is not an independent variable. The design speed, or more likely a reduced speed, is usually specified in any short-term sea state. The synthesis of response to collection of short-term sea states and relative wave headings as a strategy to overcome the statistical nonstationarity of real seaways over the long term is not limited to linear ship responses. The same rationale can be applied to nonlinear ship responses with the exception that Pshort-term (Re > r |β, Hs , Tz ) in Eq. (2) can no longer be substituted by Eq. (1), since nonlinear response is not a Gaussian process in most cases even though it is still reasonable to assume that it is stationary and ergodic in a short-term sea state. There are several different ways to calculate the short-term probability of exceedance Pshort-term (Re > r ) for nonlinear ship responses. One is to use a more general form of probability function to describe the distribution of the nonlinear peaks, such as the generalized gamma probability function introduced by Ochi [1] in his study of ship roll motion, which is dominated by viscous and other nonlinear hydrodynamic forces. Wu and Moan [2–4], Wu and Hermundstad [5] applied the same distribution to the wave-induced nonlinear responses in ships with structural dynamic effects. The probability of the shortterm extreme response Re exceeding level r per unit time is expressed as (λr )c(m−1) e−(λr ) Pshort-term (Re > r ) = n Γ (m)

c

(3)

where m and c are two shape parameters, and λ is its scale parameter. n represents the average number of peaks per unit time. Eq. (3) reduces to Pshort-term (Re > r ) = ne−(λr )

c

(4)

when the response peaks follow the Weibull distribution (m = 1). It can further reduce to Eq. (1) for the extreme response from a Gaussian narrow band stochastic process. The parameters m, c, and λ can be evaluated by equating certain moments of the observed data to the theoretical ones or by a weighted curve fitting. The method of moments assumes equal importance of all peak values. However, accurate modelling of large peaks in the upper tail is more crucial for the estimation of short-term extreme values. A distribution function for an overall fit to all peaks may fail to accurately describe the high peaks and thus induce model error in the extrapolation. On the other hand, a weighted curve fitting can force the distribution function closer to the simulated data in the highvalue region, which is of most interest, by giving large weight in that region. Unfortunately, there is no theoretical method for selecting the best weighing function. Larger weight in the highvalue region can produce better distribution tail as compared to the simulated data. However, it will also increase the statistical uncertainty of the short-term prediction due to the randomness of individual time-domain simulations of limited duration. Another alternative way of predicting short-term extreme response is to use peak-over-threshold (POT) method, which

was proposed by ISSC [6] and implemented earlier by Wang and Moan [7] for rigid ships, Wu and Moan [4] for a highspeed flexible vessel, Graczyk et al. [8] for a LNG tank. This method only uses the peak values that exceed certain threshold while leaving out those under the threshold. One advantage of the POT method is that the distribution of the excesses asymptotically approaches the generalized Pareto function (Pickands [9]) for high thresholds no matter what the parent distribution might be. Pickands’ finding plays an important role in the extremes statistics as the central limit theorem in the statistics. Applications of the method involve the selection of a suitable threshold. The threshold should be sufficiently high in order to use the asymptotic distribution. However, higher threshold implies a smaller sample and greater statistical uncertainty. In this paper, we present the POT method and some of its properties that may be used to determine an appropriate threshold. We also give a brief description of the hybrid time-domain simulation of the wave-induced hydroelastic load effects in irregular seas developed recently by Wu and Moan [3]. Then, we apply the POT method to a containership of latest design to study the sensitivity of the predicted extreme nonlinear vertical load effects to the changing threshold. Further, we investigate the statistical uncertainty of the numerical prediction derived from any individual simulation of limited duration and how this uncertainty can be reduced by increasing the simulation time. We choose containership in our case study because this type of ship usually has small block coefficient beneath the mean water line and large flare above it in order to reach higher speed and carry more cargo. As a result, wave-induced vertical bending moments and shear forces are more nonlinear in the wave frequency region and hydroelastic effects, such as whipping and springing, become more significant in the high frequency region compared to other types of ships. Consequently, direct calculations of waveinduced load effects are often necessary in the design of containerships, particularly when the ship length and capacity are increased into uncharted territory (Mewis and Klug [10]). 2. Peak-over-threshold method Since the tail of the distribution determines the extreme behaviour of a random variable, it could be convenient to use the tail of interest only to fit an appropriate function based on which the extreme order statistics are further calculated. This is the essence of the POT method. The cumulative distribution of the response peaks in its tail part can be expressed as F(r ) = F(u) + [1 − F(u)]

F(r ) − F(u) , 1 − F(u)

r >u

(5)

u denotes the threshold. Pickands [9] proved that the conditional distribution function of the peaks over a sufficiently high threshold, P[R ≤ r |R > u] =

F(r ) − F(u) , 1 − F(u)

r >u

(6)

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asymptotically follows the generalized Pareto distribution if the parent distribution F(r ) is in the domain of attraction from any of the three extreme value distributions (Weibull, Gumbel and Frechet). Most continuous distribution functions lie in the domain of attraction of a certain extreme value distribution. The generalized Pareto distribution (GPD) has the cumulative form  1 − (1 + cr/λ)−1/c , c 6= 0 G(r ) = (7) 1 − exp(−r/λ), c=0 c and λ are the shape and scale parameters, respectively. For c 6= 0, the range of r is 0 < r < ∞ if c ≥ 0, and 0 < r < −λ/c if c < 0. When c = 0, the GPD reduces to the exponential distribution with mean λ. When c = −1, the GPD reduces to the Uniform U [0, λ]. When c > 0, the GPD becomes the Pareto distribution, hence the name generalized Pareto distribution. The GPD as a limit distribution for exceedances is justified by the fact that for large enough n, (  − 1 )  c r − δ . (8) [F(r )]n ≈ exp − 1 + c η

 =1− 1+

cr cu + λ

− 1 c

(12)

which means R − u, given R > u, has a GPD(c, cu + λ) if R has a GPD(c, λ). This property implies that if a given data set and a threshold u 0 is consistent with a generalized Pareto distribution, it will be for any other threshold value u > u 0 . c and λ can be estimated by the maximum likelihood method, which maximizes the likelihood of the observed sample, or the method of moments (Hosking and Wallis [12]),   1 µ2 c= 1− 2 , (13) 2 σ   1 µ2 λ= µ 1+ 2 . (14) 2 σ µ and σ are the sample mean and standard deviation. It is noted that the mean and the higher order moments do not exist when c > 1. Eq. (5) can be rewritten as F(r ) = F(u) + [1 − F(u)]G(r − u),

r >u

(15)

The right-hand side of Eq. (8) is the von-Mises family of distributions for maxima or the maximal generalized extreme value distributions. It is the only possible parametric family for maxima as a limit; see e.g. Galambos [11]. Taking the log on both sides gives

for sufficiently high threshold u. If (R1 , R2 , . . . , Rn ) is a data sample of size n and R1 , R2 , . . . , Rn are independent and identically distributed with cumulative distribution function F(r ), then the cumulative distribution function of the extreme value Re = max(R1 , R2 , . . . , Rn ) is given by

   1 1 r − δ −c log F(r ) ≈ − 1 + c . n η

Fe (r ) = F n (r ) = P(Re ≤ r ) = 1 − α.

(9)

For large r , F(r ) → 1, the left-hand side of Eq. (9) can be replaced by F(r ) − 1 and then    1 r − δ −c 1 1+c 1 − F(r ) ≈ . n η

(10)

It follows that FR−u|R>u (r ) = 1 − P(R > u + r |R > u) = 1 − P(R > u + r )/P(R > u) = 1 − [1 − F(u + r )]/[1 − F(u)]  1  1 + c(u + r − δ)/η − c = 1− 1 + c(u − δ)/η  − 1 c cr = 1− 1+ . c(u − δ) + η

α = P(Re > r ) is calculated as the solution to 1

F(u) + [1 − F(u)]G(r − u) = (1 − α) n α (17) ≈ 1 − + o(α 2 ) n with n being the average number of peaks per unit time. Substituting Eq. (7) into Eq. (17) yields P(Re > r ) = α   1    c(r − u) − c , = k 1+ λ  k exp[−(r − u)/λ],

(11)

This is the cumulative distribution function given in Eq. (7) with λ = c(u − δ) + η. One of the most important properties of GPD is its stability with respect to truncations from the left. It is shown by the fact that P(R − u ≤ r |R > u) = P(r + u ≥ R > u)/P(R > u) = [G(r + u) − G(u)]/[1 − G(u)] (   1 ) h h cu i− 1c cu i− 1c c(r + u) − c 1+ = 1+ − 1+ λ λ λ

(16)

c 6= 0 and r > u (18) c = 0 and r > u

for small α/n. k is the average number of peaks over the threshold u per unit time and F(u) in Eq. (17) has been replaced by (n − k)/n. The choice of the threshold u is essential but it remains an open matter. In the literature on the POT method, not much attention has been given to this aspect. Davison and Smith [13] proposed to use the mean excess plot. Indeed, the mean excess function of the GPD is given by the linear expression, E(R − u|R > u) = (cu + λ)/(1 − c).

(19)

Accordingly, if a GPD is appropriate, the scatter plot of the mean observed excess over u versus u should resemble a straight line with a slope of c/(1 − c) and an intercept of λ/(1 − c). If the points in this scatter show a linear relationship, then the GPD assumption seems reasonable.

M.K. Wu, T. Moan / Applied Ocean Research 28 (2006) 386–397

Another way of assessing whether the threshold is high enough is to use the stability property expressed in Eq. (12). The shape parameter c plotted against the threshold u should remain constant when u is sufficiently high. Further, the scale parameter λ as a function of u is λ(u) = cu + λ0

(20)

where λ0 is the value of λ associated with u 0 . Therefore, we expect a constant trend and a linear trend in the plots of the estimated c and λ versus the threshold u. 3. Time-domain simulation of hydroelastic load effects Modal superposition has been widely used to account for hydroelasticity; see e.g. Bishop and Price [14]. In this approach, any time-domain hydroelastic response y(t), such as bending moment or shear force, is expressed as an aggregate of dynamic flexible modal responses pid (t), y(t) =

∞ X

ci pid (t)

(21)

i=1

where ci are the modal contribution coefficients. In practice, this infinite series will be replaced by a partial sum. However, the number of required global flexible modes varies from ship to ship and from response to response. It can only be determined by a convergence study. Generally speaking, more modes are needed to represent shear force than to represent bending moments. Similarly, more modes are needed to calculate bending moments at cross-sections towards the ship bow and stern than to calculate the bending moment amidships. If large amount of global flexible modes are required, the modal superposition approach is less attractive. More importantly, a convergence study should be done for each and every bending moment and shear force. In some cases, it is quite difficult to know if they have converged or not. If global structural dynamic effects are insignificant in any of the flexible modes, the dynamic modal responses pid (t) reduce q to quasi-static ones, pi (t), and y(t) =

∞ X

q

ci pi (t)

(22)

i=1

is equivalent to that obtained by the conventional direct response evaluation familiar to naval architects. Since the global structural dynamic effects exist only in the few lowest flexible modes, particularly the first two-node mode, the hydroelastic response can be approximated adequately by y(t) = c1 p1d (t) +

∞ X

q

ci pi (t)

i=2 q

= c1 [ p1d (t) − p1 (t)] +

∞ X

q

ci pi (t)

i=1

=

c1 [ p1d (t) −

q p1 (t)] +

y r b (t)

(23)

y r b (t) is the response calculated by the conventional approach q where the ship hull is regarded as a rigid body. p1d (t), p1 (t)

389

and y r b (t) consist of their respective linear parts and nonlinear modification parts. The linear response is obtained from its frequency-domain counterpart through inverse fast Fourier transformation (FFT). The nonlinear modification comes from the convolution of the impulse response function and the nonlinear modification forces caused by slamming and other components that are not accounted for in the linear theory. Details of the formulation can be found in Wu and Moan [3]. This approach is hybrid in the sense that it is a combination of the conventional direct load evaluation for a rigid body and the modal superposition for a flexible body. It still accounts for the dynamic effects in the few lowest global flexible modes but eliminates the need for calculating the quasi-static responses in the higher global flexible modes when the convergence is slow or the global modes are entangled with the local ones. It will, therefore, reduce the simulation time. This time reduction is important for any simulation-based prediction of extreme nonlinear responses. The hybrid method and its implementation PC code WINSIR (Wave-INduced ShIp Responses) is a further improvement on the theory published earlier by Wu and Moan [2], Wu and Hermundstad [5]. There are three unique features in this approach when compared to other practical nonlinear time-domain simulation methods (ISSC [6]). First, it avoids convergence study and hence makes the calculation more efficient and robust. Second, it can use 2D, 2.5D or 3D velocity potentials to evaluate hydrodynamic coefficients under different circumstances, such as forward speed and slenderness of the ship hull. Third, the time-domain total nonlinear response is expressed as the summation of the linear response and the nonlinear modification. The linear and the nonlinear modification parts can be validated or calibrated separately. And other nonlinear modification forces can be easily included later on. When simulating ship responses in the short-term sea states, the irregular incident waves are often described by the ISSC spectrum, also referred to as the modified Pierson-Moskowitz spectrum, for the design of ships with worldwide operation, "     # 5 2π 4 5 2π 4 Hs2 exp − . (24) S(ω) = 16 T p 4 Tp ω ω5 This wave spectrum represents fully developed sea states. The significant wave height Hs and the spectrum peak wave period T p = 1.408Tz are two independent parameters. Eq. (24) shows the wave energy distribution with respect to wave frequency without giving any information about the wave direction or wave spreading. A general formula of wave spectrum that accounts for wave spreading (short-crested waves) would be a 2D function S(ω, θ ). In practice, this 2D wave spectrum is written as the product of a point spectrum and a spreading function D(θ ), S(ω, θ ) = S(ω)D(θ ).

(25)

This formula implies that the wave spreading is frequency independent. The most widely used spreading function for

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ocean-going ships has the form  2 cos2 θ/π, |θ| ≤ π/2 D(θ) = 0, |θ| > π/2

(26)

θ = 0 corresponds to the primary direction of wave propagation. The directional wave spectrum S(ω, θ ) in Eq. (25) is not limited to that given in Eqs. (24) and (26). Any other form of incident wave spectrum, whether it is short-crested or longcrested, narrow-banded or wide-banded, single peak or multipeak, can be employed in the time-domain simulation if the spectrum better represents the short-term sea states in which a ship might be operated. If we assume that the incident irregular waves are longcrested, which is more likely in heavy sea states, then all the wave energy is concentrated in x-direction and the irregular wave system can be generated numerically, ζ (t, x) =

N p X

2S(ωm i )∆ω sin(ωi t − ki x + εi ).

Fig. 1. Body plan of the containership.

(27)

i=1

S(ω) is described by Eq. (24) without any wave spreading. ωmi is the midpoint of the i-th frequency interval while ωi is chosen randomly inside this frequency interval. ki = ωi2 /g represents the corresponding wave number in deep water. εi denotes the random phase angle distributed uniformly between 0 and 2π . The randomness in ωi will avoid undesirable repetition in the history of any simulated response. Fig. 2. Longitudinal mass distribution.

4. Case study A 280 m long containership of the latest design has been chosen as an example to investigate the sensitivity of the short-term extreme vertical load effects to the changes in the threshold of the POT method and the statistical uncertainty associated with the time-domain simulation of limited duration. Extensive model tests for this containership have been carried out at CeSOS to study the influence of hull flexibility (hydroelasticity) on the fatigue damage (Drummen et al. [15]) as well as extreme responses. The model tests will also be used to verify the hybrid method and the computer code of the timedomain nonlinear simulation described in Section 3, and the results will be published in a separate paper. The main particulars of the containership are given in Table 1. The body plan, the longitudinal mass and stiffness distributions of the containership are depicted in Figs. 1–3, respectively. The ship hull is modelled as a flexible body and the damping ratio of the first two-node vertical flexible mode is set to 1%. The short-term extreme hydroelastic load effects are not sensitive to the damping ratio and the longitudinal stiffness distribution (Wu and Moan [16]). The calculated dry natural frequency of the first vertical flexible mode is 0.8 Hz. The short-term nonlinear simulation is a time-consuming process. However, it is not necessary to carry out simulations for all possible combinations of average zero-crossing periods, significant wave heights and wave headings described in Eq. (2). Earlier studies (Baarholm and Moan [17], Wu and Hermundstad [5]) have shown that only a few short-term sea

Fig. 3. Longitudinal stiffness distribution. Table 1 Main particulars of the containership Parameter

Value

Unit

Length overall Length between perpendiculars Breadth Draught amidships Displacement Block coefficient Centre of gravity forward of AP Centre of gravity above base line Radius of gyration in pitch Maximum service speed

294.0 281.0 32.26 11.75 76656 0.68 136.3 12.87 69.45 23

Metres Metres Metres Metres Tonnes Metres Metres Metres Knots

M.K. Wu, T. Moan / Applied Ocean Research 28 (2006) 386–397

states combined with head seas contribute more than 95% to the long-term extreme wave-induced vertical load effects. Further, it has been known for a long time that large bending moment and shear force depend more on the wave length and wave steepness than the absolute wave height. The most critical average zero-crossing period for this particular ship is around 11.5 s. Considering the likelihood of heavy weather avoidance (Shu and Moan [18]), we have selected one short-term sea state (Hs = 10 m, Tz = 11.5 s), one ship forward speed (5 knots, the minimum steering speed), and one wave heading (head seas) for our time-domain nonlinear simulations. In order to determine the relevant level of the short-term probability of exceedance in 3 h, we first estimate the ship’s exposure time under different wave conditions based on the IACS wave diagram (IACS [19]). The results are listed in Table 2. Taking into account the relationship between the probability of exceedance and the return period shown in Table 3, we have decided that Log10 (probability of exceedance in 3 h) = −1.5 is the most appropriate level. 300 simulations, each lasting 60 h, have been carried out with time step 0.025 s. The long-crested irregular incident waves are generated by Eq. (27), with N = 200 regular wave components. The wave spectrum, Eq. (24), is truncated at the low-frequency end when the value is less than 0.1% of the spectrum peak and at the high-frequency end when the value is less than 1% of the same peak. Adding more tail of the incident wave spectrum does not necessarily improve the results since the linear springing has little contribution to the extreme load effects. A pair of time histories is obtained for each nonlinear vertical bending moment or vertical shear force. One is the direct result from the time-domain simulation in which the ship is modelled as a flexible hull. The other is the lowpass filtered response regarded as the result from a rigid ship hull without any hydroelastic effects. The cut-off frequency is determined in such a way that the high frequency vibration is eliminated from the filtered signal. Between two consecutive zero up-crossings in the low-pass filtered nonlinear response, one highest peak and one lowest trough are recorded for the unfiltered response as well as for the filtered response. This is because the slamming-induced high frequency whipping decaying peaks and troughs are not totally independent and identically distributed as required in the order statistics. The time histories of peak values from one time-domain simulation of 60 h are depicted in Fig. 4. There is no sign of any repetition in the plots. It is assumed that the simulated responses are stationary and ergodic. Therefore, the peaks extracted from the 300 simulations of 60 h can be combined to form one single large set. This single set is further divided into smaller subsets of any required size, representing the results from the time-domain simulations of the corresponding duration. For instance, if we divide the large set into 150 subsets, the results form 150 samples of 120 h simulation. Likewise, if we divide the set into 3000 subsets, then each of those 3000 subsets will represent one sample of 6 h simulation.

391

Parameters of POT method as functions of threshold for the peak (sagging) and trough (hogging) vertical bending moment (VBM) amidships are depicted in Fig. 5 based on one timedomain simulation of 120 h. The mean excess over threshold and the scale parameter λ demonstrate a linear trend between 2.5 × 106 and 4.0 × 106 kN m, which correspond to 80% and 97% quantiles, for the sagging VBM from the flexible hull shown on the left-hand side of Fig. 5. The shape parameter c remains relatively constant within this range. The same linear and constant trends can be observed in the range between 2.0 × 106 and 4.0 × 106 kN m for the sagging VBM from the rigid hull. They correspond to 70% and 99% quantiles. Small fluctuation exists inside those two ranges. However, the fluctuation becomes larger when the threshold exceeds 99% quantile. This is an indication that the number of the peaks over the threshold is too low. Similar observations can be made for the hogging VBM amidships in the plots on the right-hand side of Fig. 5, as well as the sagging and hogging VBM and vertical shear force (VSF) at the quarter length from the fore perpendicular. We have calculated the mean value µ and the standard deviation σ of the predicted extreme VBM amidships, VBM and VSF at the quarter length from the fore perpendicular, so as to study the sensitivity of the extreme load effects to the changes in the threshold and the statistical uncertainty caused by the limited duration of the time-domain simulation. The results of the VBM amidships are given in Tables 4 through 9 for different threshold values and different simulation time but the same probability of exceedance level. The results of the VBM and VSF at quarter length from the fore perpendicular are given in Tables 10 and 11 for different threshold values but the same simulation time (120 h) and the same probability of exceedance level. The mean value of the predicted extreme VBM amidships does not change too much when the simulation time increases or decreases given the same threshold. This implies that there is no significant bias related to simulation time. When the threshold changes from 90% quantile to 98.75% quantile, the fluctuations in the mean values of predicted extreme VBM amidships are no more than 2% for most cases and the highest one is about 3%–4%. If we push the threshold as low as 60% quantile, the fluctuations in the mean values can reach 10%–15%. This indicates that the extreme load effects are not sensitive to the threshold if the threshold is around 90% quantile or higher. The standard deviation of the predicted extreme VBM amidships increases when the threshold goes higher given the same simulation time, or when the simulation time decreases given the same threshold value. It is understandable since higher threshold or shorter simulation time means fewer peaks over the threshold. For the same quantile threshold, the standard deviations of the predicted extreme VBM and VSF from the flexible hull are larger than the corresponding standard deviations of the predicted extreme VBM and VSF from the rigid hull. This is because the slamming-induced whipping does not happen every incident wave cycle and hence the number of whipping events is fewer than that of the wave-frequency response peaks. It is also observed that the standard deviations

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Table 2 A ship’s exposure time under different wave conditions based on the IACS wave diagram Wave condition

A

B

C

A∗

B∗

C∗

Probability of exposure (10−3 ) Exposure time in ship’s life (h)

4.91 1291

1.24 324.5

0.717 188.4

0.614 161.4

0.154 40.57

0.0896 23.55

The ship’s design life is assumed to be 30 years (262 800 h). A = {Hs ≥ 10 m}, B = {Hs ≥ 10 m; 12 s > Tz ≥ 11 s}, C = {11 m > Hs ≥ 10 m; 12 s > Tz ≥ 11 s}, A∗ = {A; β = 0◦ }, B ∗ = {B; β = 0◦ }, C ∗ = {C; β = 0◦ }. Table 3 The relationship between the probability of exceedance (POE) and the return period Log10 (POE in 3 h) POE in 3 h Return period (h)

−3.0 0.001 3000

−2.5 0.00316 949

−2.0 0.01 300

−1.5 0.0316 94.9

−1.0 0.1 30.0

−0.5 0.316 9.49

0.0 1 3.00

Table 4 Mean value µ and standard deviation σ of predicted extreme vertical bending moment amidships using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN m) σ/µ (%)

6978 9.23

7557 11.1

7824 12.7

7888 14.2

7850 15.6

7741 16.0

Rigid sagging

µ (MN m) σ/µ (%)

5553 7.16

5917 8.66

6089 9.85

6140 11.0

6156 12.2

6129 12.7

Flexible hogging

µ (MN m) σ/µ (%)

4704 6.72

5125 8.77

5355 10.5

5451 11.7

5460 12.8

5424 13.1

Rigid hogging

µ (MN m) σ/µ (%)

3981 5.60

4276 7.27

4421 8.77

4489 9.83

4522 10.6

4522 11.1

Log10 (POE in 3 h) = −1.5. Simulation time = 6 h. Table 5 Mean value µ and standard deviation σ of predicted extreme vertical bending moment amidships using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN m) σ/µ (%)

6960 5.86

7542 7.20

7819 8.15

7894 9.23

7885 10.3

7817 10.9

Rigid sagging

µ (MN m) σ/µ (%)

5539 4.51

5903 5.48

6076 6.34

6131 7.16

6155 8.00

6161 8.55

Flexible hogging

µ (MN m) σ/µ (%)

4692 4.31

5111 5.54

5345 6.51

5449 7.29

5482 8.28

5470 8.95

Rigid hogging

µ (MN m) σ/µ (%)

3971 3.45

4264 4.46

4410 5.51

4484 6.20

4524 6.79

4541 7.45

Log10 (POE in 3 h) = −1.5. Simulation time = 15 h. Table 6 Mean value µ and standard deviation σ of predicted extreme vertical bending moment amidships using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN m) σ/µ (%)

6953 4.14

7536 5.14

7817 5.80

7899 6.62

7902 7.39

7852 7.85

Rigid sagging

µ (MN m) σ/µ (%)

5535 3.30

5897 3.90

6070 4.24

6128 4.93

6154 5.62

6174 6.10

Flexible hogging

µ (MN m) σ/µ (%)

4688 3.02

5106 3.91

5341 4.72

5446 5.28

5482 6.01

5480 6.61

Rigid hogging

µ (MN m) σ/µ (%)

3968 2.50

4260 3.13

4405 3.75

4481 4.34

4523 4.84

4548 5.42

Log10 (POE in 3 h) = −1.5. Simulation time = 30 h.

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M.K. Wu, T. Moan / Applied Ocean Research 28 (2006) 386–397 Table 7 Mean value µ and standard deviation σ of predicted extreme vertical bending moment amidships using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN m) σ/µ (%)

6950 2.97

7532 3.68

7816 4.15

7899 4.61

7908 5.13

7872 5.40

Rigid sagging

µ (MN m) σ/µ (%)

5532 2.29

5894 2.78

6066 3.05

6125 3.41

6157 3.98

6176 4.22

Flexible hogging

µ (MN m) σ/µ (%)

4686 2.17

5103 2.77

5339 3.37

5444 3.78

5483 4.22

5484 4.70

Rigid hogging

µ (MN m) σ/µ (%)

3967 1.77

4256 2.24

4402 2.66

4478 3.14

4520 3.45

4548 4.02

Log10 (POE in 3 h) = −1.5. Simulation time = 60 h. Table 8 Mean value µ and standard deviation σ of predicted extreme vertical bending moment amidships using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN m) σ/µ (%)

6948 2.26

7532 2.82

7811 3.00

7898 3.34

7911 3.55

7877 3.80

Rigid sagging

µ (MN m) σ/µ (%)

5530 1.66

5893 2.06

6066 2.31

6125 2.46

6154 2.80

6176 3.11

Flexible hogging

µ (MN m) σ/µ (%)

4686 1.55

5103 2.01

5337 2.45

5443 2.65

5484 2.89

5491 3.39

Rigid hogging

µ (MN m) σ/µ (%)

3966 1.24

4256 1.66

4402 1.83

4478 2.13

4521 2.31

4551 2.70

Log10 (POE in 3 h) = −1.5. Simulation time = 120 h. Table 9 Mean value µ and standard deviation σ of predicted extreme vertical bending moment amidships using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN m) σ/µ (%)

6948 1.60

7532 2.04

7814 2.23

7902 2.34

7913 2.68

7882 2.88

Rigid sagging

µ (MN m) σ/µ (%)

5531 1.16

5893 1.42

6065 1.66

6125 1.64

6153 2.06

6180 2.07

Flexible hogging

µ (MN m) σ/µ (%)

4685 1.01

5102 1.39

5338 1.82

5445 1.85

5486 2.00

5494 2.50

Rigid hogging

µ (MN m) σ/µ (%)

3966 0.769

4256 1.19

4402 1.25

4478 1.42

4520 1.74

4550 2.19

Log10 (POE in 3 h) = −1.5. Simulation time = 240 h. Table 10 Mean value µ and standard deviation σ of predicted extreme vertical bending moment at quarter length from fore perpendicular using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN m) σ/µ (%)

3279 2.58

3528 3.02

3618 3.26

3605 3.55

3541 3.52

3480 3.72

Rigid sagging

µ (MN m) σ/µ (%)

2775 1.89

2863 2.22

2849 2.54

2794 2.46

2755 2.46

2732 2.53

Flexible hogging

µ (MN m) σ/µ (%)

1678 1.30

1812 1.85

1888 2.14

1938 2.59

1964 3.09

1978 3.42

Rigid hogging

µ (MN m) σ/µ (%)

1351 1.06

1427 1.22

1467 1.36

1488 1.57

1499 1.74

1508 1.99

Log10 (POE in 3 h) = −1.5. Simulation time = 120 h.

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Table 11 Mean value µ and standard deviation σ of predicted extreme vertical shear force at quarter length from fore perpendicular using peak-over-threshold method Threshold (quantile) (%)

60

80

90

95

97.5

98.75

Flexible sagging

µ (MN) σ/µ (%)

83.38 2.11

91.30 2.64

95.04 2.96

96.24 3.32

96.18 3.45

95.75 3.83

Rigid sagging

µ (MN) σ/µ (%)

66.60 1.53

70.97 2.00

73.16 2.18

74.05 2.44

74.37 2.58

74.88 2.73

Flexible hogging

µ (MN) σ/µ (%)

55.85 1.35

60.59 1.80

63.25 2.27

64.78 2.47

65.68 2.89

66.21 3.36

Rigid hogging

µ (MN) σ/µ (%)

49.20 1.26

52.82 1.62

54.62 1.82

55.50 1.94

55.98 2.19

56.34 2.54

Log10 (POE in 3 h) = −1.5. Simulation time = 120 h.

for the sagging VBM and VSF are generally higher than those for the hogging. When the simulation time is 6 h and the threshold is set to 90% quantile, the standard deviations of the sagging VBM amidships from the flexible and rigid hulls are about 13% and 10% of the mean values, and those of the hogging VBM amidships are around 11% and 9%, respectively. Typical model tests do not last more than 6 h full scale in any specific irregular sea state. Therefore, one should be aware of the statistical uncertainty in the prediction of the extreme nonlinear load effects based on model tests. The same can be said with respect to the prediction based on short nonlinear simulations by using more complicated 3D time-domain codes, such as LAMP (Lin et al. [20]) or SWAN (Sclavounos [21]). When the threshold remains 90% quantile and the simulation time increases to 120 h, the standard deviations of the sagging VBM amidships from the flexible and rigid hulls can be reduced to 3% and 2.3% of the mean values, and those of the hogging VBM amidships to 2.5% and 1.8%, respectively. Further investigation has shown that about 70% of the predicted extreme VBM amidships are within the range of mean value ± standard deviation as revealed in Tables 12 and 13. 95% of those are within the range of mean value ±2 × standard deviation. Almost all the predicted extreme VBM amidships are within the range of mean value ±3 × standard deviation. Therefore, time-domain nonlinear simulation of 100 h or more will give satisfactory predictions of the short-term extreme load effects in a rigid hull as well as in a flexible hull. Finally, the mean values and the standard deviations of the predicted extreme VBM amidships and VBM and VSF at quarter length from the fore perpendicular as functions of the probability of exceedance are given in Tables 14 through 16 and Fig. 6. The simulation time and the threshold are 120 h and 90% quantile, respectively. As expected, both the mean values and the standard deviations increase when the probability of exceedance decreases. 5. Conclusions The hybrid time-domain nonlinear simulation and the peakover-threshold method are applied to a containership of the latest design for the calculation of the wave-induced extreme hydroelastic load effects. Statistical analysis has been carried out to study the sensitivity of the predicted extreme VBM and

Table 12 Percentage of the predicted extreme VBM amidships inside the ranges [µ−bσ , µ + bσ ] b

0.5

1.0

1.5

2.0

2.5

3.0

Flexible sag (%) Rigid sag (%) Flexible hog (%) Rigid hog (%)

43.3 43.3 35.3 36.7

68.7 75.3 67.3 70.7

85.3 86.7 87.3 86.7

94.7 92.7 94.0 94.7

98.0 98.7 100 98.7

100 99.3 100 99.3

The total number of the predicted values is 150. Simulation time = 120 h. Log10 (POE in 3 h) = −1.5. µ and σ represent the mean and the standard deviation of the sample values. Table 13 Percentage of the predicted extreme VBM amidships inside the ranges [µ−bσ , µ + bσ ] b

0.5

1.0

1.5

2.0

2.5

3.0

Flexible sag (%) Rigid sag (%) Flexible hog (%) Rigid hog (%)

39.0 38.5 38.2 39.0

68.8 68.7 67.3 67.2

86.0 88.7 87.5 87.5

95.5 94.5 95.2 95.7

98.7 97.5 98.7 99.0

99.7 99.7 100 99.7

The total number of the predicted values is 600. Simulation time = 30 h. Log10 (POE in 3 h) = −1.5. µ and σ represent the mean and the standard deviation of the sample values.

VSF to the changes in the threshold of the POT method, as well as the statistical uncertainty in the predictions due to the limited duration of the nonlinear simulation. The following three conclusions are reached based on the results presented in this paper: (1) When using the POT method to predict wave-induced shortterm extreme nonlinear load effects, the fluctuation in the mean values is mostly no more than 2% provided that the threshold is around 90% quantile or higher. This implies that the prediction is not sensitive to the threshold if the threshold is sufficiently high. (2) Given a threshold, the statistical uncertainty in terms of standard deviation increases as the simulation time decreases. When the simulation time is less than 10 h, the standard deviation can reach 10%–15% of the mean value. Therefore, one should be aware of the possible error in any predictions of short-term extreme nonlinear load effects based on model tests or simulations by complicated 3D programmes. This is because typical model tests do not last more than a few hours (full-scale) in any particular irregular sea state, and it takes too much time to do longer simulation

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Fig. 4. Peak value histories of vertical bending moment (VBM) amidships from one time-domain simulation of 60 h.

Table 14 Mean value µ and standard deviation σ of predicted extreme vertical bending moment amidships using peak-over-threshold method Log10 (POE in 3 h)

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

Flexible sagging

µ (MN m) σ/µ (%)

9349 4.80

8865 4.20

8354 3.60

7811 3.00

7237 2.42

6628 1.86

5982 1.36

Rigid sagging

µ (MN m) σ/µ (%)

6750 3.51

6553 3.12

6327 2.72

6066 2.31

5766 1.89

5420 1.46

5022 1.07

Flexible hogging

µ (MN m) σ/µ (%)

6002 3.79

5806 3.36

5585 2.91

5337 2.45

5059 1.98

4746 1.52

4395 1.09

Rigid hogging

µ (MN m) σ/µ (%)

4755 2.74

4658 2.45

4542 2.15

4402 1.83

4234 1.50

4034 1.17

3793 0.85

Threshold = 90% quantile. Simulation time = 120 h.

Table 15 Mean value µ and standard deviation σ of predicted extreme vertical bending moment at quarter length from fore perpendicular using peak-over-threshold method Log10 (POE in 3 h)

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

Flexible sagging

µ (MN m) σ/µ (%)

4597 5.32

4275 4.62

3949 3.93

3618 3.26

3283 2.61

2944 2.00

2600 1.45

Rigid sagging

µ (MN m) σ/µ (%)

3253 3.89

3134 3.45

3000 3.00

2849 2.54

2678 2.07

2487 1.61

2270 1.17

Flexible hogging

µ (MN m) σ/µ (%)

2064 3.23

2014 2.88

1956 2.52

1888 2.14

1809 1.74

1716 1.35

1607 0.98

Rigid hogging

µ (MN m) σ/µ (%)

1535 1.90

1518 1.74

1496 1.56

1467 1.36

1430 1.14

1382 0.91

1321 0.67

Threshold = 90% quantile. Simulation time = 120 h.

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Fig. 5. Parameters of peak-over-threshold (POT) method as functions of threshold for the vertical bending moment (VBM) amidships based on one time-domain simulation of 120 h.

using 3D time-domain programs. When the simulation time increases to 100 h or more, the standard deviation can be reduced to less than 3% of the mean value if the threshold is about 90% quantile. (3) We recommend using 90%–95% quantile as threshold in the POT method and carrying out more than 100 h of timedomain nonlinear simulation in each combination of short-

term sea state, wave heading and ship speed. In doing so we can keep the statistical uncertainty to a low level and hence obtain satisfactory predictions of the short-term extreme nonlinear load effects. As a final note, 60 h of nonlinear hydroelastic simulation by the hybrid method described in this paper takes about 3 h of computation time on today’s typical personal computer.

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M.K. Wu, T. Moan / Applied Ocean Research 28 (2006) 386–397 Table 16 Mean value µ and standard deviation σ of predicted extreme vertical shear force at quarter length from fore perpendicular using peak-over-threshold method Log10 (POE in 3 h)

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

Flexible sagging

µ (MN) σ/µ (%)

114.5 4.78

108.3 4.17

101.9 3.56

95.04 2.96

87.88 2.38

80.37 1.82

72.46 1.31

Rigid sagging

µ (MN) σ/µ (%)

80.86 3.28

78.66 2.93

76.11 2.57

73.16 2.18

69.73 1.79

65.74 1.40

61.12 1.02

Flexible hogging

µ (MN) σ/µ (%)

69.89 3.50

67.97 3.11

65.78 2.69

63.25 2.27

60.35 1.84

57.02 1.41

53.20 1.01

Rigid hogging

µ (MN) σ/µ (%)

58.74 2.68

57.62 2.42

56.26 2.13

54.62 1.82

52.65 1.50

50.26 1.17

47.37 0.86

Threshold = 90% quantile. Simulation time = 120 h.

Fig. 6. Extreme vertical bending moment (VBM) and vertical shear force (VSF) versus probability of exceedance in 3 h.

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