Extreme loads determination on complex slender structures

Abstract

For the dimensioning of particular structures like stingers or generic tubular elements for offshore industry, it is usual to predict the extreme values of wave induced loads. Both for model test and calculations, the traditionally adopted methods of analysis are based on Weibull distribution. The necessity to investigate severe sea state conditions together with the increased complexity of the structure is an evident source of non-linearities in the exciting force peaks distribution. In the specific, the adoption of a standard Weibull approach is not indicated for accurately predict the extreme loads. The adoption of more accurate distributions suitable to capture peaks non-linearity will ensure to overcome or capture possible multi-modal behaviour of the considered population. These enhanced techniques can be used not only for model test results analysis, but also for results coming from preliminary hydrodynamic calculations (CFD). In the present work, two different methodologies based on Mixed Weibull and Generalised Pareto distributions will be applied to the results obtained for a stinger geometry, where Morison theory is adopted to evaluate wave loads considering shield effects between the single tubular elements.

Keywords

Extreme loads extreme value theory generalised Pareto distribution mixed Weibull distribution Morison equations

1. Introduction

For the dimensioning and design of fixed offshore structures and ship appendages, numerical calculations are a useful support to determine extreme loads values prior to perform dedicated model tests. In particular, for offshore industry, it is important to determine wave induced loads and motions, even in harsh weather conditions [14,20]. In the analysis of model tests of particularly complex structures, it has been noted that the behaviour of peaks distribution has non-linearities [15] that cannot be modelled with standard analysis techniques. This kind of behaviour can also be found during preliminary calculations, leading to a possible over/under estimation of the extreme loads in early design stage evaluations. This will consequently lead to a wrong preliminary dimensioning of the structure. In fact, in ship and offshore design, it is common practice to analyse the peaks as suggested by ITTC [11], i.e. by means of a two or three parameters Weibull distribution. However, once complex structures are investigated in rough sea, then non-linear behaviours can be identified in peaks distribution [10]. In such a case, the Weibull distribution in its standard form cannot estimate extreme values satisfactory [15].

Both in case of a model test or a simulation, time series of the measured/calculated loads have to be analysed by extracting the peaks in an appropriate way. According to extreme value theory [2,8], the Generalised Extreme Value Distribution (GED) should be used when all the peaks are considered. Then, to capture the non-linearities, it is possible to adopt enhanced versions of the standard Weibull functions, as, for example, the Mixed Weibull case [16]. Once only the peaks above a certain threshold are selected for the analysis, then Generalised Pareto Distribution (GPD) should be adopted [4,17]. In such a case, only on the tail of the probability distribution is considered, controlling the error connected to non-linearities through a proper selection of the threshold.

Here, adopting the two possible peaks extraction techniques, GPD and Mixed Weibull distributions will be applied for the extreme loads determination. To determine the wave loads in the particular case of relatively slender tubular structures, preliminary calculations can be done by modelling wave forces according to Morison equation [21]. In particular, for complex tubular structures, the model suitable for vertical and horizontal cylinders can be extended for generally inclined ones [6], in such a way to evaluate all the possible sections [3]. Also in case of calculations considering non-linearities, the peaks distribution presents a multi-modal behaviour [13]. In the present work the specific case of a stinger structure is presented, where calculations have been executed by evaluating the wave induced forces on general slender structures [13,18], including the shield effects between adjacent piles.

The results obtained on the selected structure are here presented for an environmental condition, and the differences between traditional peaks analysis method and proposed enhanced procedures are shown, highlighting the possible errors that a wrong extreme distribution modelling can generate in the final loads predictions.

2. Extreme values theory

The extreme values theory is of utmost importance to properly select the best distribution for modelling the maxima of a certain quantity. For this reason, it is necessary to correctly apply this theory while estimating the extremes in the data analysis both of seakeeping tests and numerical simulations. The extreme values theory is widely applicable in all the fields where data analysis is required and is giving the indications for the right selection of the distribution to adopt, according to the approach adopted for the peaks extraction from the time record.

In fact, the most important distinctions about the possible distributions that can be used for the extreme values determination are given by the alternative ways to extract the peaks from a time series.

Fig. 1.

Block-Maxima (top) and POT (bottom) peaks extraction from a time series.

2.1. Peaks extraction techniques

The first operation that must be carried out in the analysis of extreme values is to extract the peaks from the data set record. According to standard procedures used in null-mean data analysis, mainly two different kinds of extraction can be made: the Block-Maxima and the Threshold Value Maxima. The first method is considering the maxima on several intervals of length τ inside the records or, once the interval is coincident with the sample time, extracting all the maxima of the record. On the other hand, it is possible to consider all the peaks above a certain threshold value ( $| u |$ ), so the method is also known as Peaks Over Threshold (POT) technique. In the common applications regarding seakeeping and offshore tests and simulations the Block-Maxima method is widely used, while POT technique is more related to research fields where the tail modelling of a population is mostly considered instead of the entire one. In Fig. 1 the qualitative differences between the two methods are shown on a generic time record. The selected method for the peaks extraction will have an impact on the way the entire analysis has to be carried out. In fact, according to the extraction method, different distributions should be used according to extreme value theory.

2.2. Generalised extreme value distribution (GED)

In the case that the Block-Maxima method is selected for peaks extraction, the limit law to identify the distribution of maxima is given by the Fisher–Tippet–Gnedenko theorem [20], stating that the GED should be used to describe the phenomenon. Briefly, GED can be expressed with the following cumulative density function (CDF): $\begin{matrix} (1) & F (x) = e^{- t (x)} \end{matrix}$ where: $\begin{matrix} (2) & t (x) = \{\begin{matrix} {(1 + z β)}^{- \frac{1}{β}} & if β \neq 0 \\ e^{- z} & if β = 0 \end{matrix} \end{matrix}$ and: $\begin{matrix} (3) & z = \frac{x - γ}{η} \end{matrix}$ The three real constants, appearing in equations (2) and (3), are the shape parameter β, defined in $(0, + \infty)$ , the scale parameter η, defined in $(0, + \infty)$ , and the location parameter γ, defined in $(- \infty, + \infty)$ . According to the value of the shape parameter, inside GED distribution, three particular sub-cases are obtained: the Weibull, the Gumbel and the Frechet distribution.

The Gumbel distribution, having the shape parameter equal to zero ( $β = 0$ ), is used once data are supposed to follow an exponential distribution. Frechet distribution, obtained by positive shape parameter ( $β > 0$ ), but with a reversed formulation of the distribution (changed sign of x axis), is used for particular populations presenting a significant amount of data in the tail (so-called fat-tail distributions); in fact, this distribution is exhibiting a slower decay when increasing the x values, compared to other distributions. The Weibull distribution, with a positive shape parameter ( $β > 0$ ), is representative of all the cases not covered by the previous two alternatives and is, therefore, widely used for several engineering problems such as defect data analysis, weather forecasting and for the prediction of extreme loads in seakeeping and offshore experiments [11].

2.3. Generalised Pareto distribution (GPD)

By changing the peaks extraction technique by applying the POT method, the aim of the analysis is to estimate the distribution function $F_{u}$ of x values above a certain threshold $| u |$ . Therefore, $F_{u}$ is called conditioned excess distribution function, and, applying the Pickands–Balkema–de Haan theorem [1,19], a suitable form can be identified in the GPD distribution.

Then, according to the above theorem, it is possible to represent the GPD with the following CDF: $\begin{matrix} (4) & F (x) = \{\begin{matrix} 1 - {(1 + β z)}^{- \frac{1}{β}} & if β \neq 0 \\ 1 - e^{- z} & if β = 0 \end{matrix} \end{matrix}$ Shape, scale, location parameter and z are defined as per equations (2) and (3), but with the following limitations: β is varying in $(- \infty, + \infty)$ , η in $(0, + \infty)$ and γ in $(- \infty, + \infty)$ . It should be noted that the shape parameter γ in equation (3) can be identified by the threshold $| u |$ .

Also GPD distribution is presenting typically three different behaviours, related to the value of the shape parameter.

3. Weibull distribution

Weibull distribution is widely used in engineering to perform a defect data analysis to predict the time to failure of a certain component. However in seakeeping problems it is usual to perform a similar analysis on the peaks of a sample record with the aim of evaluating the occurrence of those peaks and extrapolating the extreme values of the selected quantity. This is the typical case of extreme loads, forces or wave heights determination for long-term predictions.

It is common practice to perform the peak extraction according to the Block-Maxima procedure, considering the time interval equal to the sample time. In such a way all the peaks of the record are selected for the analysis. So applying the Fisher–Tippet–Gnedenko theorem, it is possible to represent the peaks distribution with a Weibull law.

Fig. 2.

Example of 2 parameters Weibull analysis.

In Fig. 2 a typical example of the complete process of analysis is given; starting from the time signal up to the representation in the Weibull plane.

As Fig. 2 shows, Weibull law is the most suitable solution between the possible GED cases, because no fat tail is normally present in this kind of record.

For this reason it is selected to perform a detailed study on the representation of variables with Weibull distribution.

3.1. Two-parameters Weibull distribution

The most simple case of Weibull distribution is the two parameters Weibull distribution. In equations (1), (2) and (3) the Weibull cumulative density function F was presented as a three parameters distribution, however to simplify the model the location parameter γ can be omitted ( $γ = 0$ ).

When this occurs the probability density function (PDF) can be written as: $\begin{matrix} (5) & p (x) = \frac{β}{η} {(\frac{x}{η})}^{β - 1} e^{- {(x / η)}^{β}} \end{matrix}$ and the CDF as follows: $\begin{matrix} (6) & F (x) = \int_{0}^{\infty} p (x) d x = 1 - e^{- {(x / η)}^{β}} \end{matrix}$ where β and η have been defined previously.

The probability distribution function is defined for $x > 0$ and non-negative values of β and η. As the two main parameters change, the shape of the function change too.

3.2. Three-parameters Weibull distribution

In certain particular situations, the representation of the peaks sample on Weibull plot seems not properly follow a straight line. In such cases it is possible to re-conduct data to the case of a two-parameters Weibull distribution, adopting a simple change of variable.

To rebuild the situation shown with the two-parameters Weibull distribution, it is necessary to introduce the new variable γ, the location parameter of equation (3).

Using the location parameter the PDF of the distribution becomes: $\begin{matrix} (7) & p (x) = \frac{β}{η} {(\frac{x - γ}{η})}^{β - 1} e^{- {(\frac{x - γ}{η})}^{β}} \end{matrix}$ The CDF of the three-parameters Weibull distribution is given by equation. The PDF and the CDF of the three-parameters Weibull distribution are exactly equal to the two-parameters case, except for the shift of γ in x direction. Since a two-parameters Weibull distribution is defined for $x > 0$ , it follows that a three-parameters Weibull distribution is defined for $x > γ$ .

There are no limitations in sign for γ, it can be negative such as positive. The sign of γ is changing the convexity of the distribution in the Weibull plot.

To avoid that, it is possible to represent the three-parameters Weibull distribution on a special Weibull plot, considering in the abscissae $ln (x - γ)$ instead of $ln (x)$ .

3.3. Mixed Weibull distribution

Analysing some records on the Weibull plot, it can happen that the extracted peaks are not only far from a straight line fit but also can show more than one convexity.

In such kind of cases, also a three-parameters Weibull distribution is not sufficient to describe correctly the population. This is the particular case of multi-modal distributions, means sample data that could contain inside more than one significant population. To describe this model use can be made of the mixed Weibull distribution. This particular distribution is the combination of two or more two or three-parameters Weibull distributions.

The general PDF of a mixed Weibull distribution is: $\begin{matrix} (8) & p (x) = \sum_{i = 1}^{N} q_{i} \frac{β_{i}}{η_{i}} {(\frac{x - γ_{i}}{η_{i}})}^{β_{i} - 1} e^{- {(\frac{x - γ_{i}}{η_{i}})}^{β_{i}}} \end{matrix}$ the CDF can be written as: $\begin{matrix} (9) & F (x) = 1 - \sum_{i = 1}^{N} q_{i} e^{- {(\frac{x - γ_{i}}{η_{i}})}^{β_{i}}} \end{matrix}$ where N is the number of sub-populations considered and $q_{i}$ is the percentage of one sub-population on the total sample; in such a way $\sum q_{i} = 1$ .

All the other parameters are the same described in equation (2) for the generalised extreme value distribution. There are no limits to the number of sub-populations that can be used, just the number of parameter to estimate will consequently increase.

For example, when two sub-populations are considered, the number of parameter to estimate is equal to 7, considering three sub-populations the parameters rise up to 12 and so on. In the present study use will be made of Mixed Weibull distributions with 2 sub-populations.

3.4. Parameter determination

According to the adopted, two or three parameters distribution, an appropriate way to determine the regression coefficients must be selected. There are several methods available for data regression analysis, but the difficulties of parameters estimation increase by increasing the number of coefficients to be determined.

The easiest case is represented by the two-parameters Weibull distribution where, as the name of the distribution suggests, only two-parameters need to be estimated. However, once the number of unknowns increases, the parameter determination is no more an easy task [16]. In [15] the most common techniques for parameters estimation have been compared with enhanced techniques based on genetic algorithms, showing that the genetic algorithm approach is comparable with the standard procedures like Maximum Likelihood Estimation, Method of Moments and Least Square Fit methods.

For this reason the genetic approach has been adopted in this study for all the distributions, i.e., the standard Weibull and the GPD analysis.

4. GDP analysis using POT

A different approach for the analysis of extremes consists in the extraction of the peaks having a magnitude higher than a pre-determined threshold value $| u |$ . Adopting this kind of procedure (POT), the validity of the Pickands–Balkema–de Haan theorem implies that a GPD distribution should be considered instead of standard Weibull distribution with two or three parameters.

Considering the cumulative density function of the GPD given in equation (4), it can be observed that it is defined for $x > γ$ when $β > 0$ , otherwise for $γ < x < γ - η / β$ . It is not possible to state a-priori whether β will be positive or negative. Analysing the shape of the function it seems reasonable that the trend will follow a slope like the Weibull one, with a positive β.

As for the Weibull distribution, several methods for parameters estimation can be found in literature, and basically are the same mentioned in the above paragraph for Weibull distribution. In fact methods like maximum likelihood, moments or least square fitting are commonly used also for the GPD parameters estimation. Therefore, due consistence with the Weibull analysis, the same procedure based on genetic algorithm has been here used. Another important issue for the GPD is the selection of the threshold u, which is the starting point of the whole procedure.

4.1. Threshold selection

Adopting GPD to fit the peaks distribution from a sample record, implies that a suitable threshold value should be found, ensuring that the approximation given by Pickands–Balkema–de Haan theorem is applicable. The threshold selection must also take into account the fact that a sufficient number of events must lie above the selected value, in order to ensure a sufficiently accurate estimation of the unknown distribution parameters.

A suitable method to do that is based on the adoption of the “sample mean excess function”. Even though this is a simple procedure, it is currently considered [9] one of the most appropriate ones for the threshold selection. The sample mean excess function is defined by: $\begin{matrix} (10) & e_{n} (u) = \frac{\sum_{i = 1}^{n} {(X_{i} - u)}^{+}}{\sum_{i = 1}^{n} 1 |_{X_{i} > u}} \end{matrix}$ which implies that the function (10) is obtained by the sum of the excesses above the threshold, $X_{i} - u$ , divided by the number of data exceeding the threshold itself. The sample mean excess function is an empirical estimate of the mean excess function, which is defined as: $\begin{matrix} (11) & e (u) = E [X - u |_{X > u}] \end{matrix}$ From this definition, the mean excess function is representative of the expected occurrence of threshold exceedance. In any case, for signal analysis, the sample mean exceedance is adopted according to equation (10) and presented as function of the threshold value as in the example of Fig. 3.

Fig. 3.

Example of sample mean excess function plot with different threshold values.

Several authors [2,7] give the interpretation of the sample mean excess function plot, stating that once the excess function is assuming a reasonably straight line than the distribution will follow a law like the GPD. Since the signal is coming from a set of measured records, it is not possible to observe really a straight line in the plot, especially when they are representative of non-linear phenomena. For this reason, it is common to assume as indicative thresholds the points where the excess function is changing slope [12], namely $u_{1}$ , $u_{2}$ and $u_{3}$ in Fig. 3.

By considering Fig. 3 as an example, it is possible to observe more than one change in slope of the function, means that the above mentioned rule is not able to determine a single threshold value. For this reason, it has been selected to choose the last change in slope of the sample mean excess function and $u_{3}$ has been assumed as the threshold value u to adopt for the present data analysis. In this way, the chance to face multi-modality in the tail of the distribution is minimised.

5. Extreme values determination

In the previous sections, various methods have been presented and proposed to find the parameters of a suitable distribution to fit the sample data. Once the distribution law is known and the parameters determined, the extreme values of the population can be calculated from the fitted distribution.

Usually the values of engineering (design) interest are the events with the following probability values of p; i.e. 3%, 1% and 0.1%. To know these values, one should use the quantiles (inverse cumulative distribution) of the distribution. For the different distributions the quantiles have the following forms: $\begin{array}{l} (12) & Q (p, η, β) = η {(- ln (1 - p))}^{\frac{1}{β}} \\ (13) & Q (p, γ, η, β) = γ + η {(- ln (1 - p))}^{\frac{1}{β}} \\ (14) & Q (p, u, η, β) = u + \frac{β}{η} (1 - {(\frac{n}{N_{u}} (1 - p))}^{β}) \end{array}$ which are representative of the 2, 3 parameters Weibull and of the Generalised Pareto distributions respectively. In particular, for the GPD, n represents the total number of samples of the record and $N_{u}$ the number of samples above the selected threshold u.

It must be noted that none equation is referring to the mixed Weibull distribution. In fact, a general expression for the quantile of the mixed Weibull is not easy to find, because of the difficulty in inverting the CDF, due to the presence of a sums of logarithms where the argument is elevated at different exponents. For this reason the estimation of the quantile for the Mixed Weibull case has been executed with an iterative process, reaching the convergence at the desired $1 - p$ value.

6. Wave forces determination

The determination of total wave forces acting on a complex tubular structure are here determined by means of a calculation method based on Morison equation. A tubular structure like a stinger is composed by a certain number of cylindrical piles with different incidence angles with respect to the incoming flow. For such a reason the standard Morison equation, referring to vertical and horizontal cylinders, should be written in a more general form, considering the effective cylinder inclination: $\begin{matrix} (15) & {\underline{f}}_{N} = C_{M} ρ \frac{π}{4} D^{2} \dot{\underline{w}} + \frac{1}{2} C_{D} ρ D \underline{w} | \underline{w} | \end{matrix}$ where ${\underline{f}}_{N}$ is the normal force per unit length, ρ is the water mass density, D the cylinder diameter and $C_{M}$ and $C_{D}$ are the inertia and drag force coefficients respectively. The equation refers to acceleration and velocity vectors $(\dot{\underline{w}}, \underline{w})$ normal to the cylindrical pile, that can be expressed in the incoming flow reference system according to the following transformations: $\begin{array}{l} (16) & \underline{w} = (\underline{u} \cdot \underline{n}) \underline{n} \\ (17) & \dot{\underline{w}} = (\dot{\underline{u}} \cdot \underline{n}) \underline{n} \end{array}$ being $\underline{u}$ and $\dot{\underline{u}}$ are the water particle velocity and acceleration while $\underline{n}$ is the unit vector normal to the pile in the plane formed by the pile axis and $\underline{u}$ . The resulting forces breakdown on a general cylindrical pile inclined of an angle α with respect to water particle velocity is presented in Fig. 4.

Fig. 4.

Forces scheme on a generally inclined cylindrical pile.

Structures as a stingers are composed by multiple cylindrical piles, so equation (15) should be applied for each pile of the structure, considering that each part is facing a different load, due to the position of the piles with respect to the incoming wave system. By considering a simple superposition between the forces evaluated for each single pile an overestimation in the total wave force will be determined, because each pile is considered invested by an uniform velocity field.

Fig. 5.

Reference system for wake determination on adjacent piles.

In a structure like a stinger the piles are close to each other, so it is reasonable to suppose that only the piles directly facing the flow will be subjected to a total force compliant with equation (15). The other piles will be in the wake of the front ones, being subjected to a shield effect [5], which is reducing the incoming speed on the piles’ portions inside the wake of the forward ones. Adopting Schlichtling formulation extended for an Airy wave potential according to the relative positions between the piles, one was: $\begin{array}{l} (18) & u_{TOT} (ξ, η, ζ, t) = u_{0} (ξ, η, ζ, t) - u_{w} (ξ, η, ζ, t) \\ (19) & u_{w} (ξ, η, ζ, t) = k_{1} u_{0} \sqrt{\frac{C_{D} D}{ξ_{S}}} e^{- 0.693 {(η / b)}^{2}} \end{array}$ where $u_{0}$ is the incoming wave velocity and $u_{w}$ is the wake velocity reduction, with $k_{1}$ a constant equal to 1.0 and b, $ξ_{S}$ defined as follows: $\begin{array}{l} (20) & b = k_{2} \sqrt{C_{D} D ξ_{S}} \\ (21) & ξ_{S} = ξ + \frac{4 D}{C_{D}} \end{array}$ being $k_{2}$ a constant value set at 0.25. The reference scheme of the wake calculation is shown in Fig. 5, it must be noted that this formulation is considering also a bi-dimensional approximation of the wake.

The described formulations are valid for regular waves. However to properly investigate extreme value theory, the irregular waves should be considered. Considering an irregular sea state modelling by means of the wave amplitude spectrum $S_{ζ}$ , then the single amplitudes of multiple regular waves can be determined per each frequency band $δ ω$ . In such a way the irregular wave system can be described as: $\begin{matrix} (22) & ζ (t) = \sum_{i = 1}^{N} ζ_{0_{i}} cos (k_{i} ξ - ω_{i} t + Φ_{i}) \end{matrix}$ where $N = 64$ and the phase $Φ_{i}$ is generated with a random based process.

Fig. 6.

Overview of stinger geometry in working position used for calculations.

7. Test case

To evaluate the differences between the standard extreme value analysis for seakeeping/offshore purposes [11] and the one proposed according to a more accurate definition of the extreme value theory, a dedicated simulations of forces acting on an existing stinger geometry (Fig. 6) have been carried out, adopting the calculation scheme described in Section 6. The simulations have been tested in a towing tank and already used to validate the force prediction procedure [18]. All the calculations have been carried out in irregular seas. Through the validation study, several conditions in terms of sea state and incoming wave directions have been tested. Here, only one specific case is reported as example, considering a Bretschneider wave spectrum with a $H_{1 / 3}$ of 3.0 m and a $T_{z}$ of 6.0 s with an incoming wave direction of 90 deg. The obtained forces records for transversal and vertical force are reported in Fig. 7. For the presented condition the longitudinal force $F_{x}$ is negligible with respect to the other two components, so it is not herein considered.

Fig. 7.

Wave forces in x (upper), y (middle) and z (lower) direction in irregular waves for $α = 90 deg$ , $H_{1 / 3} = 3.0 m$ and $T_{z} = 6.0 s$ .

Fig. 8.

Analysis on the $F_{y}$ Weibull plot according to (a) 2 par. Weibull, (b) 3 par. Weibull, (c) GPD and (d) mixed Weibull distributions.

Fig. 9.

Analysis on the $F_{z}$ Weibull plot according to (a) 2 par. Weibull, (b) 3 par. Weibull, (c) GPD and (d) mixed Weibull distributions.

Fig. 10.

Mixed Weibull PDF for $F_{y}$ and $F_{z}$ .

From the Weibull plane representation of the peaks (Figs 8, 9) a multimodal behaviour can be noticed, since the data present both convexities and concavities with respect to a straight line. This can be further visualised also in Fig. 10, where the mixed Weibull PDF of $F_{x}$ and $F_{y}$ are represented. For this purpose it has been decided to test the Mixed Weibull approach for the extreme value determination. On the other hand another way to proceed is to consider a different extraction of the peaks from the standard record, considering a predetermined threshold, thus considering a Generalised Pareto distribution to fit the population. Also this solution can be adopted to overcome the lower value of the sample data, which are apparently distributed according to another law, generating a multi-modal response.

7.1. Analysis of the results

By adopting the described procedure with the Block-Maxima extraction, the extreme values of the populations have been predicted adopting two and three-parameters Weibull distribution, with reference to equations (12) and (13) respectively and in the iterative way for the mixed Weibull distribution with 2 sub-populations. The values of $p = 3.0 %, 1.0 % and 0.1 %$ have been extracted.

On the other hand, with reference to equation (14), extreme values for the same p have been extracted from the peaks population, generated with the POT technique, according to the GPD distribution. For the specific case of GPD distribution, the threshold values $| u |$ have been selected according to the sample mean excess procedure, resulting in different thresholds for each analysed force record. In the specific, a threshold of 97 kN for $F_{y}$ and 134 kN for $F_{z}$ . Uncorrelations between peacks has been confirmed during data analysis.

Table 1
Extreme values of $F_{Y}$ in kN

Regression type Probability p $R^{2}$ $SSE$

$3.0 %$ $1.0 %$ $0.1 %$ [–] [–]

[kN]

2 parameters Weibull 111.17 133.75 175.80 0.991 0.0212

3 parameters Weibull 110.48 127.23 149.78 0.997 0.0035

Mixed Weibull (2 pop) 108.41 125.45 154.65 0.999 0.0011

Generalised Pareto 109.94 121.51 137.81 0.998 0.0015

Regression type	Probability p	$R^{2}$	$SSE$
2 parameters Weibull	111.17	133.75	175.80	0.991	0.0212
3 parameters Weibull	110.48	127.23	149.78	0.997	0.0035
Mixed Weibull (2 pop)	108.41	125.45	154.65	0.999	0.0011
Generalised Pareto	109.94	121.51	137.81	0.998	0.0015

Table 2

Extreme values of $F_{Z}$ in kN

Regression type	Probability p			$R^{2}$	$SSE$

	$3.0 %$	$1.0 %$	$0.1 %$	[–]	[–]
	[kN]			[–]	[–]
2 parameters Weibull	179.62	215.47	282.47	0.988	0.0307
3 parameters Weibull	163.56	187.72	229.69	0.998	0.0022
Mixed Weibull (2 pop)	163.46	188.02	230.40	0.998	0.0022
Generalised Pareto	152.64	161.87	200.34	0.998	0.0020

The regression procedure, including the parameter estimation with the genetic algorithm [22], leads to the extreme values reported in Tables 1 and 2 for $F_{y}$ and $F_{z}$ respectively. In the same tables also the regression coefficient of determination $R^{2}$ and the sum of square errors ( $SSE$ ) are reported. The coefficient of determination has been calculated according to the following formulation: $\begin{matrix} (23) & R^{2} = 1 - \frac{SSE}{{SS}_{tot}} \end{matrix}$ where: $\begin{array}{l} (24) & SSE = \sum_{i} {(y_{i} - f_{i})}^{2} \\ (25) & {SS}_{tot} = \sum_{i} {(y_{i} - \overline{y})}^{2} \\ (26) & \overline{y} = \frac{1}{n} \sum_{i = 1}^{n} y_{i} \end{array}$ and $y_{i}$ are the record data points, $\overline{y}$ is the mean of the record and $f_{i}$ are the fitted values coming from the genetic regression algorithm.

It can be seen that the standard prediction methods are not able to properly fit the peaks distribution in the so called tail, resulting in an overestimation of the extremes of about 30% in case of the two-parameters Weibull and about 20% for the three-parameters case with respect to GPD ones. The Mixed Weibull approach is not performing really well for the selected example cases. In fact, the regression is fitting well the low-loads peaks, especially for $F_{y}$ case, but it fails in the reproduction of the tail, resulting in extreme values prediction in line (or somewhat higher) with three-parameters distribution. This can lead to the presence of more than 2 sub-populations in the peaks distribution. For the reported case the GPD distribution is reproducing better than the other distributions the tail of the population, giving a more reliable result for the extreme values. For such a reason it is advisable to model extremes with a correct tail modelling, as given for the reported cases by the GPD approach.

8. Conclusions

Once it is no more possible to reproduce a certain sample of data with the commonly used distributions (two or three-parameters Weibull) adopted for the estimation of extreme values in offshore and seakeeping applications, another kind of analysis should be carried out.

In this study two possible alternatives have been proposed and presented to overcome the multimodal distribution of certain data samples. The solutions have been developed in accordance to the mathematical formulation of the extreme value theory. A first approach is to use the Mixed Weibull distributions, the second is to use a GPD distribution.

Because of the complexity of the fitting process in case of a Mixed Weibull, where at least 7 parameters must be determined, a least square process based on genetic algorithm has been implemented to fit the functions. The algorithm was thereafter used also for all the other cases, to keep the same methodology for parameter determination through the entire study.

The results of the different procedures were compared with the recommended standard ones for wave loads calculation made on the geometry of a stinger. For the represented case the results highlights that the method that gives more reliable results is the GPD one, being able to reproduce with sufficient detail the peaks distribution tail. In any case, it is not possible to exclude Mixed Weibull method as suitable to model extremes, since it could be that the investigated records present more then 2 sub-populations.

In any case, for each case regarding evaluation of extremes on complex structures, a detailed non-standard analysis should be carried out in order to accurately estimate the final loads to be used for design.

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