Conventional and Linear Statistical Moments Applied in ... - CiteSeerX

Leonardo Sant’Anna do Nascimento Bureau Veritas, Rua Joaquim Palhares, 40, 7th Floor, Centro Rio de Janeiro 20260-080, RJ, Brazil e-mail: [email protected]

Luis Volnei Sudati Sagrilo COPPE/UFRJ, Av. Athos da Silveira Ramos, 149, Bloco B, 1 Andar, Ilha do Fund~ao, Rio de Janeiro 21941-909, RJ, Brazil e-mail: [email protected]

Gilberto Bruno Ellwanger COPPE/UFRJ, Av. Athos da Silveira Ramos, 149, Bloco B, 1 Andar, Ilha do Fund~ao, Rio de Janeiro 21941-909, RJ, Brazil e-mail: [email protected]

Conventional and Linear Statistical Moments Applied in Extreme Value Analysis of Non-Gaussian Response of Jack-Ups This work investigates numerically two different methods of moments applied to Hermite derived probability distribution model and variations of Weibull distribution fitted to the short-term time series peaks sample of stochastic response parameters of a simplified jack-up platform model which represents a source of high non-Gaussian responses. The main focus of the work is to compare the results of short-term extreme response statistics obtained by the so-called linear method of moments (L-moments) and the conventional method of moments using either Hermite or Weibull models as the distribution model for the peaks. A simplified mass-spring system representing a three-legged jack-up platform is initially employed in order to observe directly impacts of the linear method of moments (L-moments) in extreme analysis results. Afterward, the stochastic response of the threelegged jack-up platform is analyzed by means of 3D finite element model. Bias and statistical uncertainty in the estimated extreme statistics parameters are computed considering as the “theoretical” estimates those evaluated by fitting a Gumbel to a sample of episodical extreme values obtained from distinct short-term realizations (or simulations). Results show that the variability of the extreme results, as a function of the simulation length, determined by the linear method of moments (L-moments) is smaller than their corresponding ones derived from the conventional method of moments and the biases are more or less the same. [DOI: 10.1115/1.4028899]

Introduction The non-Gaussian or nonlinear response time-series are often found in the stochastic analysis of many types of offshore structures [1] submitted to environmental loadings; this source of nonlinearity may be caused by many factors as soil–structure interaction, free surface wave effects, hydrodynamic loading nonlinear effects, etc. A good reference for readers is Cassidy et al. [2] which provides a table of the different nonlinearities that have been accounted for in different analyses. Specifically in this work, the main source of nonlinearity comes from the square fluid–structure velocity term in the drag component of Morison’s equation and from the free surface wave effects. As, in general, no analytical solutions are available for the estimation of extreme statistics of non-Gaussian responses, a large number of practical numerical approaches are nowadays available in literature for this purpose, such as Weibull-based methods [3], Hermite polynomials-based approach [4], ACER method [5]. Weibull and Hermite models employ sample statistical moments of the sample peaks and of the whole time-series, respectively, which are prone to statistical uncertainties due the limited size of the sample. Recently, Hosking [6] proposed the application of the so-called linear method of moments or L-moment in the estimation of distribution parameters as an alternative to the conventional method of moments (mean, variance, coefficients of skewness, and kurtosis). Due to the low-order estimates, it is expected that L-moments present less statistical uncertainties and, consequently, any Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received September 15, 2012; final manuscript received May 18, 2014; published online November 17, 2014. Assoc. Editor: Bernt J. Leira.

extreme estimate based on them could present lower variability when compared with estimates based on conventional statistical moments. The purpose of this work is to investigate numerically the use of the L-moments in extreme response analysis of a jack-up platform model and compare the results obtained with previous results presented in Nascimento et al. [3] where conventional method of moments has been applied for extreme value estimations using Hermite and Weibull approaches. To achieve this objective, the most probable extreme value (MPEV) of the jack-up surge is analyzed. A reference theoretical value for comparisons is obtained by fitting a Gumbel distribution fitted to a sample of episodical extremes taken from various independent realizations (nonlinear time dynamic simulations).

Derivation of Distribution Parameters by L-Moments One approach to define a theoretical probability model for a given data sample of a random variable is by means of the method of moments. The classical method of moments consists in computing the parameters of the probability model in order to match the sample n-order moments such as mean, variance, coefficients of skewness and kurtosis [7]. In extreme statistics applied to nonGaussian time series the analytical model based on Hermitepolynomials [4] also employs the four first sample moments. Both situations are well known in literature and do not need further explanations. Considering now that less uncertainty on the probability model parameters would bring higher confidence on the expected results, it has been adopted to investigate the use of linear statistical moments or L-moments instead of classical moments approach.

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For both probability models investigated in this work, i.e., Weibull and Hermite-based distribution, the relationship between L-moments and distribution parameters is available and described as follows. The Linear Method of Moments. L-moments are expectations of certain linear combinations of order statistics 0. Let Y be a realvalued random variable with cumulative distribution function F(y) and quantile function y(F), and let Y1:n Y2:n … Yn:n be the order statistics of a random sample of size n drawn from the distribution of Y. L-moments of Y are defined to be the binomial quantities in accordance to the following equation: kr ¼ r 1

r1 X

ð1Þk

k¼0

r1 E YðrkÞ:r ; k

r ¼ 1; 2; …

The theory implies that kr is a linear function of the expected order statistics E[Y(r k):r]. In particular, the first four L-moments are expressed individually from the following equations:

(4)

kr ¼

yðFÞ Wr ðFÞdF ¼

F¼0

y wr ½Fð yÞ f ð yÞdy

(5) (6)

(7)

y¼1

The terms wr[] are polynomial (weight) functions of F in the form of Eq. (8) with binomial functions given by Eq. (9). wr ½F ¼

r X

pr;k Fk

(8)

k¼0

pr;k ¼ ð1Þrk

r rþk k k

n 1X yi:n ¼ Y n i¼1

n 1X ð2Pi:n 1Þyi:n k2 ¼ n i¼1

(9)

k4 ¼

(15)

s4 ¼ k4 =k2

(16)

FðuÞ ¼ UðuÞ pffiffiffiffiffiffi 1 exp 0:5x2 /ðuÞ ¼ 2p kr ¼

ð u¼þ1

u wr ½UðuÞ:/ðuÞdu ¼ Efwr ½UðuÞg

(17) (18) (19)

To simplify this result, the cumulative distribution function U(u) can be expressed as 0.5[1 þ erf(u/(21/2)], in terms of error function erf(). By substituting this equation into Eq. (8), the weight functions in the Gaussian case become of the form of the following equations: w1 ½UðuÞ ¼ 1 u w2 ½UðuÞ ¼ erf pffiffiffi 2 u w3 ½UðuÞ ¼ 1:5 erf 2 pffiffiffi 0:5 2 u u w4 ½UðuÞ ¼ 2:5 erf 3 pffiffiffi 1:5 erf pffiffiffi 2 2

(20) (21) (22) (23)

w1[U(u)] and w3[U(u)] are even functions of u. Based on that, uw1[U(u)] and uw3[U(u)] are odd functions, then k1 and k3 are null. The nonzero L-moments, k2 and k4 are given by Eqs. (24) and (25), respectively, with corresponding L-skewness null and L-kurtosis given by Eq. (26) 1 k2 ½U ¼ pffiffiffi ¼ 0:56419 p

(24)

k4 ½U ¼ 0:06917

(25)

(10) s4;Gauss ¼ (11)

n 1X 6P2i:n 6Pi:n 1 yi:n n i¼1

(12)

n 1X 20P3i:n 30P2i:n þ 12Pi:n 1 yi:n n i¼1

(13)

k3 ¼

s3 ¼ k3 =k2

The L-Moments for Gaussian Variables. Consider the special case of a standard normal variable, denoted as U, with cumulative distribution function given by Eq. (17) and probability density function by Eq. (18). From Eq. (7), the L-moments can be obtained by means of Eq. (19)

The estimation of the L-moments from a sample of size n from the random variable Y is discussed in Ref. [8]. Commonly used estimates are presented in the following equations: k1 ¼

(14)

u¼1

From the distribution theory of the order statistics Y(rk):r it is possible to rewrite kr Y(rk):r in terms of either F(y) and y(F), see the following equation: ð y¼þ1

i 0:4 n þ 0:2

Herein, k1 and k2 are measures of the central trend and dispersion. Nonzero k3, k4,… reflect deviations of the distribution of Y from a uniform density, where k3 and k4 reflect asymmetric and symmetric deviations, respectively. Similarly to the classical statistical moments, the “linear skewness” (L-skewness) and “linear kurtosis” (L-kurtosis) are given by the following equations:

(3)

1 k2 ¼ E½Y2:2 Y1:2 2 1 k3 ¼ E½Y3:3 2Y2:3 þ Y1:3 3 1 k4 ¼ E½Y4:4 3Y3:4 þ 3Y2:4 Y1:4 4

ð F¼1

Pi:n ¼

(1)

The expectation of an order statistic may be written by the following equation: ð r! y½Fð yÞj1 :½1 Fð yÞrj dFð yÞ (2) E Yj:r ¼ ðj 1Þ!ðr jÞ!

k 1 ¼ E ½Y

where Pi:n is a distribution-free estimator of the cumulative distribution function for the random variable Y at yi:n; that is, Pi:n is an estimate of (yi:n). In this study Pi:n was taken to be evaluated by means of the following equation:

0:06917 ¼ 0:1226 0:56419

(26)

The L-Moments for Weibull-Distributed Variables. One statistical model largely used in offshore engineering to model the peaks (see Fig. 1) of a sampled time series is the Weibull distribution. The cumulative distribution function of the Weibull distribution is given by the expression represented on the following equation:

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Fig. 1 Distributions of the process, peaks, and extreme values of a time-series y(t)

! yn k FY ð yÞ ¼ 1 exp a

(27)

where n, a, and j are, respectively, the location, scale, and shape distribution parameters. Using the traditional method of moments these parameters can be directly related to the mean, standard deviation, and skewness coefficient of the sample data [9]. On the other hand, the relationship between the L-moments and these distributions parameters is given in Greenwood [10] and are expressed by the following equations: n ¼ k1 þ

5k2 s4 5s3 1

(28)

k1 n 3 a¼ 2 10 6log s4 5s3 1 7 7 C6 4 5 logð2Þ logð2Þ k¼ 5 log s4 5s3 1

(29)

1

(31)

where þ is the average rate of peaks of the response parameter and T is short-term period considered (usually, 3 h). The L-Moments for Hermite-Based Model. The Hermite polynomials-based model developed by Winterstein [4] is largely employed in the prediction of extremes and fatigue associated to non-Gaussian sampled time series. The main idea is to transform a non-Gaussian time series into a Gaussian time series through a memoryless nonlinear mapping based on Hermite polynomials. Considering a non-Gaussian sampled time series Yi, its normalized counterpart yi is defined by the following equation: y i mY sY

(32)

where mY and sY are, respectively, the sample mean and standard deviation of Y. The statistical equivalence between y(t) and a standard Gaussian time series u(t) is expressed by the following equation: N X

cn Hen1 ðuðtÞÞ

(33)

n¼1

(35) (36)

In order to compute the coefficients c3 and c4, the skewness and kurtosis coefficients of the sampled time series Yi can be employed [4]. In this work the coefficients c3 and c4 are fit numerically so skewness and kurtosis coefficients are preserved. Extreme statistics estimates for y(t) are obtained by standard analytical solutions for the extreme statistics of u(t) and then using Eq. (34). As described above, the Hermite-based model uses the conventional statistical moments of the sample time series Y(t). However, as shown by Winterstein and McKenzie [11] the coefficients b and c can also be derived using the sampled time series L-moments with values given at Eqs. (37) and (38), with parameter c expressed on Eq. (39) b¼

9:21s3 11:68 2:5c

(37)

c¼

c1 11:68 2:5c

(38)

s4 s4;Gauss

(39)

c¼

MPEV ¼ n þ a½lnðvþ T Þk

yðtÞ ¼

1 j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ 2b þ 6c þ 15c2 c3 c4 ; c¼ b¼ ð1 3c4 Þ ð1 3c4 Þ

(30)

The estimation of the extreme short-term statistics of a response parameter whose peaks sample has been modeled by a Weibull model is made through Statistics of Extremes Asymptotic Theory [9]. In this case, it has been shown that the most probable shortterm extreme peak value of a sampled time series of a response parameter is given by the following equation:

yi ¼

where Hen1() is the n-order Hermite polynomial, cn the associated coefficient and n stands for the number of polynomials considered in the approximation. Considering four terms in the approximation, y can be written by means of Eq. (34), with coefficients given by Eqs. (35) and (36), in which U is standard normal, and mY and s2Y are the mean and variance of the sampled time series Y (34) y ¼ mY þ sY j u þ b u2 1 þ cu3

Although the methodology presented by Winterstein and McKenzie [11] for L-Hermite coefficients, feasible results for this specific application were not obtained and among the possibilities, the methodology applied in this work proposes that the coefficients c3 and c4 be computed through standard numerical procedures preserving original skewness coefficient and applying L-kurtosis instead of kurtosis. As the main source of nonlinearity comes from Morison’s equation, the parameter c2, the kurtosis coefficient of the response sampled time history, has been computed by an approximation proposed by Najafian [8]. As can be seen in Ref. [8], Eq. (40) showed to be much more efficient than conventional method of moments for high-kurtosis distributions c2 ¼ 365:906 s34 þ 211:333 s24 þ 2:385 s4 þ 0:190

(40)

Numerical Applications: Jack-Up Models In this section the most probable extreme short-term responses of two jack-up models are evaluated by means of a Weibull approach and by the Hermite-based model using both conventional and linear statistical moments of the simulated response time-series. The first example is a single degree-of-freedom (SDOF) model, described by Jensen and Capul [12], idealized to represent the sway motion of a jack-up, with properties taken from Cassidy et al. [13]. The other one is a 3D simplified model of a jack-up operating in a water depth of 92.0 m. The Weibull model employed consists in fitting a three-parameter Weibull distribution to a subset of the sampled time series peaks which are above a given threshold. In this work, this threshold is taken as the value corresponding 75% fractile of the empirical cumulative distribution of the sample containing all time series peaks.

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Hereafter, this approach is identified simply as Weibull-3P-PoT. As shown in Nascimento et al. [3], among various Weibull-based approaches this is one with better performance in extreme predictions for the response offshore structures whose main source of nonlinearity comes from Morison’s hydrodynamic loading.

Table 2 Normalized MPEVs estimated for SDOF jack-up model for K 5 1.00 Conventional moments Simulation length (s)

SDOF Model of a Jack-Up. The first analyzed jack-up model corresponds to a very simplified SDOF system presented by Jensen and Capul [12], representing the lateral behavior of a 120 m high three-legged jack-up platform located in a water depth of 92.0 m. All the parameters from Ref. [12] have been applied in this section; it is important to notice, however, that Hs modeled in this section does not aim to represent an extreme seastate and the SDOF model results are evaluated for basic evaluation purposes. The model is described by the differential equilibrium equation shown in the following equation:

21,600

m€ xðtÞ þ cx_ ðtÞ þ kxðtÞ ¼ FðtÞ

43,200

(41)

_ where x(t), x(t), and x€(t) are, respectively, the jack-up lateral motion, velocity, and acceleration, mm is the total model mass (18,995 kN s2/m), c is the model damping (1193 103 kN s/m), k is the lateral stiffness (4682 104 kN/m), and F(t) is an idealized Morison-type force acting on the structure. Using this data one can see that the natural frequency x0 of the model is 1.57 rad/s and the damping ratio considered as 0.02. The shortterm environmental condition considered is represented by a sea state having a significant wave height (Hs) of 6.3 m and zerocrossing period (Tz) of 8.4 s. The sea state spectrum is modeled by the modified Pierson–Moskovitz spectrum. The short-term duration is considered to be 3 h (10,800 s). F(t) has been generated artificially in order cover inertia and drag dominated cases of the hydrodynamic loading through a K factor define by the following equation: K¼

kd s2u km s2u_

(42)

where su and su_ are the standard deviations of fluid velocity and acceleration, respectively; kd and km are the drag and inertia parameters. It is observed that in the case of the inertia dominated loading K ! 0 and K ! infinite when it is drag dominated. In this work three different values of K have been simulated, i.e., K ¼ 3.00, K ¼ 1.00, and K ¼ 0.25. Initially, for each K value, 100 distinct 10,800 s long sampled time series of lateral motion x(t) have been simulated through Table 1 Normalized MPEVs estimated for SDOF jack-up model for K 5 3.00 Conventional moments Simulation length (s) 1200 2400 4800 7200 10,800 21,600 43,200

1200 2400 4800 7200 10,800

L-moments

Hermite

Weibull 3P-PoT

Hermite

Weibull 3P-PoT

1.08 (60.288) 1.08 (60.200) 1.10 (60.177) 1.10 (60140) 1.12 (60.131) 1.11 (60.082) 1.11 (60.061)

0.94 (60.322) 0.95 (60.250) 0.98 (60.238) 0.99 (60.202) 1.00 (60.183) 1.00 (60.122) 1.00 (60.075)

1.11 (60.200) 1.10 (60.138) 1.11 (60.109) 1.11 (60.079) 1.11 (60.066) 1.11 (60.041) 1.11 (60.030)

0.97 (60.320) 0.98 (60.238) 1.00 (60.183) 1.00 (60.139) 1.01 (60.115) 1.01 (60.071) 1.00 (60.059)

numerical integration of Eq. (41) by the Euler method [14]. The largest episodical values of each generated lateral motion timeseries has been taken to form a sample of 3h extreme values. Using the method of moments [15], a Gumbel extreme value distribution has been fitted to the data and the most probable 3h extreme value (MPEV) identified. This value has been taken as the reference value for the analysis of the results obtained by both Hermite and Weibull-3P-PoT methods. Hermite and Weibull-3P-PoT estimates of the 3h most probable extreme surge motion have also been estimated using traditional and linear statistical moments different simulation lengths. Tables 1–3 present the mean value of the 100 3h MPEVs estimated by these approaches divided by the 3h MPEV value obtained from the extreme sample. These tables also include (in brackets) the 95% confidence of variance of the estimators. Figures 2–4 present graphically the bias and 955 confidence interval for the results obtained with Weibull-3P-PoT model for K ¼ 3.00, K ¼ 1.00, and K ¼ 0.25, respectively. It should be noticed that the ideal method should have a bias around 1.0 and a confidence interval very small. As already shown in Ref. [3], the Hermite model predicts biased results mainly for drag-dominated situations. However, in all cases analyzed for this example it is observed that the statistical uncertainty is a little bit smaller for predictions based on L-moments. Table 3 Normalized MPEVs estimated for SDOF jack-up model for K 5 0.25

L-moments

Conventional moments

Hermite

Weibull 3P-PoT

Hermite

Weibull 3P-PoT

Simulation length (s)

1.09 (60.339) 1.11 (60.272) 1.12 (60.172) 1.11 (60.133) 1.12 (60.112) 1.12 (60.076) 1.12 (60.066)

0.95 (60.412) 0.97 (60.345) 0.98 (60.240) 0.97 (60.174) 0.99 (60.151) 0.99 (60.117) 1.00 (60.092)

1.10 (60.246) 1.12 (60.139) 1.13 (60.092) 1.12 (60.081) 1.12 (60.063) 1.12 (60.038) 1.13 (60.028)

0.97 (60.421) 0.97 (60.259) 1.00 (60.186) 1.00 (60.154) 0.99 (60.126) 0.99 (60.083) 1.00 (60.067)

1200 2400 4800 7200 10,800 21,600 43,200

L-moments

Hermite

Weibull 3P-PoT

Hermite

Weibull 3P-PoT

0.99 (60.188) 0.98 (60.140) 0.98 (60.108) 0.98 (60.090) 0.99 (60.084) 0.99 (60.048) 0.98 (60.032)

0.94 (60.242) 0.95 (60.200) 0.97 (60.146) 0.98 (60.149) 0.99 (60.136) 0.99 (60.094) 0.98 (60.056)

1.04 (60.217) 1.01 (60.163) 0.95 (60.090) 1.00 (60.110) 1.01 (60.093) 1.00 (60.056) 1.00 (60.050)

0.94 (60.211) 0.95 (60.159) 0.96 (60.104) 0.97 (60.088) 0.97 (60.072) 0.97 (60.044) 0.97 (60.039)

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Fig. 2 Jack-up model SDOF model: MPEV estimated extremes with Weibull-3P-PoT model for K 5 3.00


3D Simplified Finite Element Model of a Jack-Up. The extreme estimates using L-moment method described before are now investigated in the analysis of the surge motion and base shear force of a 3D jack-up model operating in a water depth of 92.0 m. Due to computational effort, quantity of generated data and computer code restrictions, a simplified jack-up computer finite element-based model (shown in Fig. 5) has been used. The 3D finite element model is made-up of 130 beam elements and 109 nodes, representing the platform legs and hull. A complete detailing of the jack-up model, including hydrodynamic properties and boundary conditions can be found at Ref. [16]. Also due to these computer analysis limitations, the short-term period has been considered to be equal to 6000 s. Besides the nonlinearity imposed by the drag term in Morison’s equation, the nonlinear effects of the fluid due to free surface interaction is also accounted for in the model. Other potential

source of nonlinearities for jack-ups [1], such as the hysteretic behavior of the foundations and second-order p-delta structural behavior, are neglected herein. The main assumptions used in the analysis of this case are: (a) Short-term sea state having Hs ¼ 15.5 m, Tz ¼ 16.5 s and represent by JONSWAP spectrum; (b) Simulation lengths of the sampled time-series: 1200, 2400, 4800, and 6000 s; (c) 20 independent realizations for each simulation length; (d) Responses observed: lateral displacement of the deck, static, and base shear force. As a theoretical reference for comparisons it is taken the most probable value of the response obtained from the fitting of a Gumbel distribution to the 20 maximum episodical values observed in



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Table 4 Normalized lateral displacement 6000 s MPEVs estimated for the 3D jack-up model Conventional moments Simulation length (s) 1200 2400 4800 6000

L-moments

Hermite

Weibull 3P-PoT

Hermite

Weibull 3P-PoT

0.95 (60.154) 1.22 (60.259) 1.21 (60.151) 1.17 (60.136)

0.83 (60.151) 1.05 (60.239) 1.03 (60.143) 1.01 (60.133)

1.06 (60.143) 1.29 (60.165) 1.30 (60.104) 1.25 (60.083)

0.88 (60.207) 1.08 (60.238) 1.08 (60.131) 1.05 (60.124)

Table 5 Normalized base shear force 6000 s MPEVs estimated for the 3D jack-up model Conventional moments Simulation length (s) 1200 2400 4800 Fig. 5 Simplified jack-up FE model 6000

the 20 different 6000 s long realizations (simulations) performed with the model. Tables 4 and 5 present the normalized results and 95% confidence of variance of the 6000 s extreme MPEVs for the lateral displacement and base shear forces considering Weibull and Hermite models applying both linear and conventional method of moments. Figures 6 and 7 illustrate graphically these normalized results for Weibull-3P-PoT; the results for L-moment based model is highlighted, on the other hand, the results for the conventional moments based model [3] are presented in gray for comparison purposes. It is observed that the Weibull-based predictions lead to

L-moments

Hermite

Weibull 3P-PoT

Hermite

Weibull 3P-PoT

0.92 (60.159) 1.19 (60.267) 1.19 (60.151) 1.15 (60.136)

0.81 (60.145) 1.05 (60.261) 1.03 (60.159) 1.00 (60.145)

1.04 (60.140) 1.28 (60.165) 1.29 (60.107) 1.24 (60.087)

0.83 (60.208) 1.06 (60.246) 1.06 (60.122) 1.03 (60.120)

less biased results. However, the most important aspect to be pointed out is that the reduction in the statistical uncertainty of the predicted extremes using L-moments in connection with Weibull3P-PoT approach is not expressive, i.e., this statistical uncertainty does not differ too much when using conventional or linear statistical moments in the extreme predictions. The reduction is a little bit larger in the case of the Hermite-based model but on the other hand this approach presents larger biases.

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Fig. 6 3D jack-up model: estimated lateral displacement extreme MPV using Weibull-3P-PoT model (conventional versus L-moments)

Fig. 7 3D jack-up model: estimated base shear extreme MPV using Weibull-3P-PoT model (conventional versus L-moments)

Conclusion In this study, the main focus was to investigate numerically the application of the so-called L-moments method [6] applied to Weibull and Hermite distribution models in the extreme value prediction of non-Gaussian responses of jack-up platforms. The best Weibull-based approach for this type of structure, discussed in a previous work [3], the Weibull-PoT, has been used in this study. Based on the analyses of a very simplified SDOF model of a jack-up and another 3D finite element-based model of another jack-up platform, the main conclusion that can be drawn from the present study is that application of the L-moments method leads to lower variability in MPEV results when compared those predicted using conventional statistical moments; however, this reduction is not so expressive.

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