Elsevier

Applied Ocean Research

Volume 28, Issue 6, December 2006, Pages 386-397
Applied Ocean Research

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

https://doi.org/10.1016/j.apor.2007.03.001Get rights and content

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.

Introduction

Ship motions and wave-induced load effects in a short-term 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)=ner22σ2 where σ is the standard deviation of the stochastic process. n is the average number of peaks per unit time.

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 long-term probability of exceedance is a superposition of all short-term 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 Plong-term(Re>r)=TzHsβP(β)P(Hs,Tz)Pshort-term(Re>r|β,Hs,Tz). 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 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], [3], [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 short-term extreme response Re exceeding level r per unit time is expressed as Pshort-term(Re>r)=n(λr)c(m1)e(λr)cΓ(m) 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 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 high-value 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 high-value 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 high-speed 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 wave-induced 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]).

Section snippets

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)+[1F(u)]F(r)F(u)1F(u),r>uu denotes the threshold. Pickands [9] proved that the conditional distribution function of

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)=i=1cipid(t) 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

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

Conclusions

The hybrid time-domain nonlinear simulation and the peak-over-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 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

Acknowledgement

The authors wish to express their gratitude to the Research Council of Norway for financial support to the Centre for Ships and Ocean Structures.

References (21)

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