Statistics of local ice load peaks on ship hulls

Structural Safety 40 (2013) 1–10

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Statistics of local ice load peaks on ship hulls A. Suyuthi a,⇑, B.J. Leiraa, K. Riskab,c a

Department of Marine Technology, NTNU, Trondheim, Norway Centre of Ships and Offshore Structures (CESOS), Trondheim, Norway c ILS OY, Helsinki, Finland b

a r t i c l e

i n f o

Article history: Received 15 September 2011 Received in revised form 21 August 2012 Accepted 2 September 2012 Available online 21 September 2012 Keywords: Statistical inference Ice load Peak process Ship hull

a b s t r a c t This paper focuses on statistical inference in relation to the peak process of local ice loading on ship hulls. Such inference comprises selection of a statistical model (by means of P–P plot, Q–Q plot, and probability paper), parameter estimation (by means of LSM, MoM, and MLE), evaluation of the confidence interval of the estimators, and goodness-of-fit testing of the initial distribution. Several different approaches are considered for each of the steps and numerical experiments were performed in order to reaffirm suggestions from previous works and extend the analysis. A particular data set provided by full scale measurements on the coast guard vessel KV Svalbard during the Winter 2007 was analyzed by means of the present approach. Four probability models were tested and the best fit to the data was provided by the Weibull distribution with the shape parameter being located in the range 0.7 < k < 1.0. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction A complex interaction between a ship hull, sea-ice and sea-water occurs during the ice breaking process. The ice-induced load on ship hulls is a function of crushing, bending, and frictional forces between the ship and the ice as well as a function of static (buoyancy) and hydrodynamic support forces. Moreover, these forces are also speed dependent and precise physical relations between them are not well understood. Accordingly, these phenomena can be considered as source of ‘‘internal’’ statistics, i.e. randomness of ice-induced loads due to the ice-breaking process itself even though the ice condition is uniform. When a ship is progressing in ice covered waters, homogeneous sea ice conditions are never observed in reality. The sea ice conditions with respect to the ice physical properties (i.e. micro-structure, thickness, salinity, porosity, and density) and the ice mechanical properties (i.e. tensile, flexural, shear, uni- and multi-axial compression strength, elastic and strain modulus, Poisson’s ratio, fracture toughness, and friction) usually vary significantly. Many of these properties are not well understood [1]. These variations could be referred to as ‘‘external’’ statistics, i.e. the randomness of the iceinduced loads due to the variation of ice conditions. Therefore, because of internal and external variations during the ice-breaking process, the local ice-induced loads are associated with a high degree of randomness. Seen in this perspective, statistical analysis for the local ice load is important. ⇑ Corresponding author. E-mail address: [email protected] (A. Suyuthi). 0167-4730/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.strusafe.2012.09.003

Previous work dealing with the statistics of ice loads (also gave attention to the short term extremes) are e.g. [2–5]. The work by Suominen and Kujala [4] focused on descriptive statistics employing a ‘‘classical’’ approach, which described the short term statistics of field measurement on board MS Kemira during the winters of 1987 and 1988. The term ‘‘classical’’ here means any statistical approach which requires the random process to be represented by a distribution model. In the study [4], the Weibull distribution was found to be the most suitable to describe the measurement data, if its shape parameter equals 0.75. Another ‘‘classical’’ approach is described by Kujala and Vuorio [2,3], which was also mainly concerned with descriptive statistics, both for short term and long term conditions, related to the field measurement on board the Icebreaker Sisu during the winters of 1979–1985. Kujala and Vuorio [2,3] found that the measured peak amplitudes of the ice induced loads follow an exponential distribution. Their works [2,3] were extended to obtain the short term largest value as well as the extrapolation of Gumbel asymptotic distributions fitted to the long term daily maxima. While other works [2–4] applied the classical method, the work of Lensu and Hanninen [5] introduced another approach to the assessment of short term ice load, which was referred to as the ‘‘time window’’ approach. The first three of the referenced works above [2–4], which were applying the classical method, left some open questions. If the ‘‘exact’’ classical method [6] is applied for prediction of extremes, any differences associated with choice of the initial distribution will be amplified in relation to prediction of extremes because the initial distribution will be raised to the power of n (number of events) to obtain the extreme distribution. The term ‘‘exact’’ here

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A. Suyuthi et al. / Structural Safety 40 (2013) 1–10

means that the random process is assumed to be modeled by a well established probabilistic model. Therefore, it is clear that the choice of initial distribution of the ice load under consideration is very important before proceeding to prediction of the extremes. However, this issue was not adequately addressed in previous work. This implies that several important questions were not considered in detail. Examples are how the empirical cumulative distribution function should be defined, what methods are available to identify the initial distribution which underlies the load process, how the distribution parameters should be estimated and which methods that are available for this purpose, how will these parameters vary from one data set to the other, and what about the goodness-of-fit test of the selected distribution, etc. The present paper accordingly makes an attempt to answer such questions in a systematic manner by collecting well established sources dealing with these issues, reaffirming their conclusions or procedures by numerical experiments, and applying the results in relation to statistical inference of local ice-induced load peaks. Comparison between different alternatives is made, and in some cases numerical experiments are performed for illustration purposes. 2. Statistical inference of local ice-induced load peak A random process of the ice-induced load peaks, denoted as X, is considered. Due to the necessity of applying a lower threshold, a, we need to introduce another random variable Y, such that:

Y ¼Xa

ð1Þ

Sometimes an ice load measurement system needs to employ a lower threshold in order to distinguish the ice load from other loads. However, in the absence of such a threshold, a simply could be set equal to zero. According to the properties of linear transformations, the expected value of Y, E[Y], and its variance, Var[Y], are E[Y] = E[X  a] = E[X]  a and Var[Y] = Var[X] respectively. A question being raised in the first place is always about the initial distribution underlying the random process of Y. A previous study suggested that the exponential distribution was found to fit the measured data best as compared to the Rayleigh, the lognormal and the Weibull distributions [3]. On the other hand, another study concluded that the Weibull distribution gives the best fits to the measured data and the exponential distribution fits only in rare case [7]. Supporting the latter study, Suominen and Kujala [4] mentioned that the most suitable model is the Weibull distribution with a shape parameter of k = 0.75. When evaluating pressure distribution of the ice induced loads, Jordaan et al. [8] observed that for most cases, the tail of the distribution is well represented by an exponential distribution. In summary, previous work suggested that the probability distribution of the peaks of the ice-induced loading can be well modeled by either the exponential or the Weibull distribution. This seems reasonable, since the exponential distribution is a special case of the Weibull distribution. 2.1. Empirical cumulative distribution function In order to identify the unknown distribution which is underlying a random process, a graphical method can be applied. The method requires that the empirical cumulative distribution function is estimated from the data. The empirical cumulative distribution function and the probability plotting methods play key roles in the statistical analysis of one-dimensional samples [9]. If we have a random sample of the ice load peak values Y1, Y2,   ,Yn, and their order statistic is Y(1), Y(2),   ,Y(n), where Y(1)  Y(2)      Y(n), then the empirical cumulative distribution function is generally defined as given in Eq. (2) [10,11]

Table 1 Various plotting positions. c

Proposed by

Valid for

0.000 0.300 0.333 0.375 0.400 0.440 0.500

Weibull [12] and Gumbel [13] Benard and Bos-Levenbach [14] Tukey [15] Blom [16] in Barnett [17] Cunnane [18] Gringorten [11] Hazen [19] in Gerson [20]

any distributions normal distribution any distributions normal distribution any distributions double exponential distribution any distributions

F n ðyðiÞ Þ ¼ pi ¼

ði  cÞ ; n  2c þ 1

for 0  c  1 and a < y < 1

ð2Þ

Eq. (2) is called as plotting position. Several different plotting positions exist, however the most common is given by Eq. (2). Table 1 summarizes various plotting position for different values of c. Previous studies dealing with the evaluation of various plotting positions were performed by e.g. [17,18,21,22]. Due to different point of views and the nature of the problems, the results somehow seem to contradict each other. Nevertheless, Makkonen [22] and Gumbel [13] strongly supported the utilization of Weibull’s plotting position (c = 0), which subsequently was applied in the present work. 2.2. P–P plot, Q–Q plot, and probability paper There are at least three graphical methods which are available in order to identify the unknown distribution which underlies a random process, i.e. P–P (percentile) plot, Q–Q (quantile) plot, and probability paper. Refs. [9,20] give the definitions of the first two types of plots, i.e. P–P and Q–Q plots, as graphical methods for comparison of two different distributions. In our case, the two distributions that will be compared are the sample distribution (empirical c.d.f.) and the theoretical distribution. A P–P plot of a sample and its theoretical (possibly standardized) distribution is a scatter plot of psy ðqÞ versus pty ðqÞ for various q, where psy ðqÞ is the empirical c.d.f. of y, pty ðqÞ is the theoretical c.d.f. of y, and q is a specific quantile value. Various formulations for the sample quantile are elaborated by Hyndman and Fan [23]. If the sample was actually coming from the theoretical distribution, then a P–P plot for psy ðqÞ and pty ðqÞ will be a straight line configuration oriented from (0, 0) to (1, 1). The P–P plot is usually sensitive to discrepancies in the middle of a distribution rather than in the tail area [9]. A Q–Q plot of a sample and its theoretical (possibly standardized) distribution is a scatter plot of qsy ðpÞ versus qty ðpÞ for various p, where qsy ðpÞ is a specific sample quantile, qty ðpÞ is the quantile of the theoretical distribution of y, and p is the percentile. If the sample actually originated from the theoretical distribution, then the plot of the qsy ðpÞ-quantiles versus qty ðpÞ-quantiles will be a straight line configuration with slope 1, passing through the origin. The Q–Q plot tends to emphasize a comparison of the tail behavior. The reason for this is that the quantile is a rapidly changing function of p where the density is sparse (in the tails) and a slowly changing function of p where the density is high (in the middle), see e.g. [9]. The last graphical method for the purpose of deciding which distribution underlies the ice load process is probability paper. This could be the most straightforward approach as compared to the two earlier methods. When developing a probability paper, the distribution parameters do not have to be predetermined. Also, no quantile needs to be calculated. The idea of a probability paper is that when given the empirical c.d.f. of the ordered sample, by transforming the vertical scale (and sometimes also the horizontal scale), generally the graph of the empirical c.d.f. can be

3

A. Suyuthi et al. / Structural Safety 40 (2013) 1–10 Table 2 Formulation of p.d.f. and c.d.f. for various distributions. Distribution

Probability density function, f(y)

Cumulative density function, F(y)

Exponential Lognormal

f ðyÞ ¼ k expðkyÞ

FðyÞ ¼ 1  expðkyÞ   1 1 lny  l FðyÞ ¼ þ erf pffiffiffiffiffiffiffiffiffi 2 2 2 2 r   lny  l FðyÞ ¼ 

2

lÞ 1 f ðyÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi g expf ðIny 2r2 2 ð2pr Þ

y

r

f ðyÞ ¼ 1b expf ðyb aÞg exp½expf ðyb aÞg  k  kyk1 y f ðyÞ ¼ exp h h h

Gumbel Weibull

Table 3 Axes transformations for probability papers.

FðyÞ ¼ exp½expf ðyb aÞg  k  y FðyÞ ¼ 1  exp h

Table 6 MLE for several distributions.

Distribution

Absica transf.

Ordinat transf.

Distribution

MLE

Exponential Lognormal Gumbel Weibull

x(i) lnðxðiÞ Þ x(i) lnðxðiÞ Þ

lnð1  pi Þ U1[pi]

Exponential

^  k¼y Pn Pn ^ Þ2 ðlnxi Þ ^2 ðlnxi  l l^ ¼ i¼1 r ¼ i¼1 n n Pn ^ Þ2 ðlnxi  l r^2 ¼ i¼1 Pn n x xi expð ^i Þ b ^¼ b x  Pi¼1j ; n x

Lognormal

lnðlnðpi ÞÞ lnðlnð1  pi ÞÞ Gumbel

i¼1

Table 4 Distribution estimators based on probability paper.

Weibull

Distribution

Estimators

Exponential Lognormal Gumbel

^ k¼m r^ ¼ 1=m ^ ¼ 1=m b

l^ ¼ r^ c ^ a^ ¼ bc

Weibull

^¼m k

^ ^ h ¼ expðc=kÞ

h

Distribution

l1

l2

Exponential Lognormal

h

–

Weibull

2

e0:5r l + 0.5772b kþ1

C

k

2

2

er ðer  1Þ  2 b2 p6 kþ1 2  C k C kþ2 k

transformed into a straight line [14]. Vertical and horizontal scale transformations for several distributions, i.e. the exponential, the lognormal, the Gumbel’s, and the Weibull’s probabilistic models, are given in Table 3. These four distributions will be employed throughout this work and their p.d.f. and c.d.f. formulations are given in Table 2. Thus, the only pre-requisite for constructing a probability paper is a set of transformations and the empirical c.d.f. For more details about this plot, reference is made to e.g. [24]. 2.3. Methods for estimation of the distribution parameters There are at least three methods for estimation of the distribution parameters, i.e. the graphical method by means of probability paper, the method of moments (MoM), and the maximum likelihood estimate (MLE). The least square method was applied to the probability paper for which the slope (m) and the ordinate intercept (c) of a given probability paper can be obtained. The distribution parameters are then estimated by simple algebra. The estimators based on probability paper are given in Table 4. More detail about this method is referred to e.g. [25]. Another method for estimation of the distribution parameters is the method of moments (MoM), which is generally fast and simple to apply with no graphical representation needed. The method was

expð ^i Þ

Pn

b

xi i¼1 expð b ^ Þ ^ 1=k

P ^ ^ h ¼ ½ð1nÞ ni¼1 xki  ; ^ k ¼ 1 Pn k^ n Pn ð ^Þ

Table 5 Theoretical i-th moment.

Gumbel

a^ ¼ b log½1n

x log xi 

i¼1 i

i¼1

log xi

proposed for the first time by Pearson [26,27]. More detail about this method is referred to e.g. [24,25,28]. The MoM estimators for several distributions are given in Table 5. The last method for estimation of the distribution parameters is the maximum likelihood estimate (MLE). The method was proposed for the first time by Fisher [29]. More detail about this method is referred to e.g. [24,25,28]. The MLEs for several types of distributions are given in Table 6. 2.4. Confidence intervals of distribution parameter estimates It is to be expected that for different data sets drawn from the same parent distribution, the estimated parameters will be different. It means that the estimates themselves are random variable. Fisher [29] defined the criterion of efficiency to be satisfied by estimators which, when derived from large samples, tend to a normal distribution with the least possible standard deviation. It has been shown by Newby [30] that the moment estimators for the two^ are asymptotically norparameter Weibull distribution, ^ h and k, mally distributed. Since the exponential distribution is a special case of the Weibull distribution with shape parameter k = 1, then the same assumption will be valid for this case and the variability of the estimator can be treated in a similar fashion. The 95% confidence interval of an estimator is as follows

 ¼ l  1:96r X X X

ð3Þ

 , l and r are the mean of the estimators obtained from where X X X the sample to be random variables X, the unbiased mean and standard deviation of the estimators, respectively. An expression for the standard deviation of the estimators of the Weibull distribution (rk^ and r^h ) can be obtained by assuming the estimates to be normally distributed, see e.g. [30]. He developed a table of the standardized measures for the Weibull parameters, including the standardized asymptotic variance–covariance matrix, which is a function of the shape parameter (k) only.

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A. Suyuthi et al. / Structural Safety 40 (2013) 1–10

Fig. 1. Strain sensor arrangement.

Fig. 2. Typical time history of ice induced load on ship hull.

2.5. Goodness-of-fit by means of graphical method and Kolmogorov–Smirnov test 2

Two goodness-of-fit tests referred to as the k2 and k0 tests [31] (which do not seem to have gotten the proper attention) are applied. The tests by default are incorporated into one of the graphical methods, i.e. P–P plot. Another test which is applied is Kolmogorov–Smirnov test [32–36], which is chosen due to the fact that this method can be implemented directly in combination with a probability paper. This is contrary to other tests, such as the v2 test, Anderson–Darling test [37,38], and the Cramér–von Mises test [33,35,39,40]. The two tests introduced by Gan and Koehler [31], i.e. the k2 and 2 k0 tests, based on assessing the linearity of the points on a standardized percentage–percentage (P–P) probability plot. A standardized P–P plot according to Gan and Koehler [31] has to be ^ =r ^ Þ against pi, where l ^ and constructed by plotting Z i ¼ Fð½X i  l r^ are estimators of the location and scale parameters respectively and pi is an appropriate plotting position. Since for the case of iceinduced load we do not know exactly the distribution that applies ^ and r ^ would need to to the relevant population, the parameters l be estimated, e.g. by means of the MLE or the MoM estimators as described earlier. The first test, k2, is simply the squared correlation coefficient

P 2   ½ n ðZ i  ZÞðp 2 i  pÞ k ¼ Pn i¼1 2 Pn   2 ½ i¼1 ðZ i  ZÞ i¼1 ðpi  pÞ 

ð4Þ Fig. 4. Ice thickness (m), propulsion power (MW), and speed over ground (m/s) for the selected time interval.

2  with 0.5 The second test, k0 , is obtained by replacing Z

P Þ2 ½ n ðZ i  0:5Þðpi  p 2 k0 ¼ Pn i¼1 P  Þ2  ½ i¼1 ðZ i  0:5Þ2 ni¼1 ðpi  p

ð5Þ

 tends to be close to 0.5. Both statistics have similar properties when Z Gan and Koehler [31] conducted Monte Carlo simulation to 2 obtain the percentage points of k2 and k0 in order to test the fit for the normal, Gumbel, and exponential distributions. A number of 15,000, 10,000, and 5000 replications were generated for the small, intermediate, and large sample sizes. A simple formula was given to 2 approximate the lower pth percentiles for k2 and k0 as follows 2

1  kp ¼ ðap þ bp nÞ1

Fig. 3. KV Svalbard at an offshore position to the east of Edgeøya Island.

ð6Þ

The above formula was obtained from the least square regres2 sion with respect to n for the simulated values of (1  kp )1. Values for ap and bp are given in Table 7. Values for the Gumbel and

exponential distributions were taken from [31], while values for the lognormal and Weibull distributions were obtained by similar Monte Carlo simulations which were performed by the authors. The simulation involved random number generation, construction 2 of the P–P plot, and calculation of k2 and k0 for each sample set with a particular sample size n. Sample sizes of n = {5, 6, 7,   , 50, 55, 60,   , 100}; n = {100, 110, 120,   , 200}; and n = {200, 300, 400,   , 1000}; were simulated with sample sizes of 15,000; 10,000; and 5000, respectively. Therefore, for each sample size, there will be either 15,000, 2 10,000, or 5000 values of k2 and k0 . Based on these values, we could 2 establish the empirical c.d.f. for k2 and k0 which are referred to as 2 ^ ^ F k2 and F k2 . Now, we are concerned with the values of k2 and k0 0 ^ which correspond to F k2 ¼ F k2 ¼ f0:001; 0:005; 0:01; 0:05; 0:1g and 0

5

A. Suyuthi et al. / Structural Safety 40 (2013) 1–10 Table 7 Values of ap and bp to evaluate the formula in Eq. (6) for several distributions. Distributions Level

Statistics

Exponential

Lognormal

Gumbel

Weibull

ap

bp

ap

bp

ap

bp

ap

bp

2

0.5574 0.3699

0.3207 0.1912

.5323 .4547

0.3846 0.3415

0.3990 0.1822

0.3845 0.3699

0.8222 0.8401

0.3861 0.3553

2

0.6464 0.6382

0.4145 0.2565

0.5592 0.4395

0.4740 0.4295

.1716 0.3600

0.5104 0.4663

1.3613 1.4028

0.4844 0.4505

2

0.4366 0.6794

0.4784 0.2959

0.9696 0.8890

0.5368 0.4858

0.6345 0.5044

0.5624 0.5262

1.1222 1.3101

0.5535 0.5127

2

0.8781 0.8797

0.6837 0.4540

0.5608 0.2383

0.7779 0.7177

1.7074 0.8366

0.7847 0.7443

1.3649 1.2739

0.7910 0.7443

2

0.8334 1.0049

0.8380 0.5780

1.1644 1.0970

0.9380 0.8657

2.9917 1.0999

0.9321 0.9055

1.3120 1.1036

0.9609 0.9116

0.001

k2

0.005

k0 k2

0.01

k0 k2

0.05

k0 k2

0.1

k0 k2 k0

Fig. 5. Ice induced loads for the selected time interval represented as stem plots. 2

2

which are designated as ka and k0;a . A straight line could be 2 2 2 expected if the ka and k0;a are rearranged into 1/(1  ka ) and 2 1/(1  k0;a ) respectively and these values are next plotted in a graph as function of sample size n. Subsequently, the constants ap and bp can be estimated by means of least square regression based on the given percentage points previously, see Table 7. Another goodness-of-fit test that can be incorporated into probability plots is the Kolmogorov–Smirnov test. Darling [33] attempted to give a fairly complete exposition of the history, development, status, and problems related to the Kolmogorov– Smirnov test. If F0(x) is the estimated population c.d.f. and SN(x) is the sample c.d.f. obtained from SN(x) = k /N, where k is the number of events less than or equal to x, then the sampling distribution of D = max |F0(x)  SN(x)| is known and is independent of F0(x), if F0(x) is continuous [32]. In his work, a table of critical points, e, for the distribution of D for various sample sizes was given. Other tables of critical points, e, are given by Birnbaum [41], Miller [42] and Maag and Dicaire [43]. If the value of D exceeds the critical value, e, in the table, the hypothesis that the observations are from the F0(x) population is rejected [34]. Lilliefors [34] also gave a special table for testing of the exponential distribution. However, in this work the authors selected the critical points, e, proposed by Miller [42].

3. Application: KV Svalbard expedition 2007 3.1. Ice load measurements and pre-treatment of data KV Svalbard was instrumented with fiber optic strain sensors at several locations along the hull (eight frames at the bow area and one frame at the starboard side) to measure the strain responses, see Fig. 1. The shear strain measured is then converted into shear stress. By integrating the (pre-assumed) shear stress distribution over the cross section of the frame, the total shear force is obtained. Prior to the conversion, the strain time history was filtered in order to remove the high frequency noise. The filter cut-off frequency was set to 100 Hz, which was considered high enough to represent the expected ice loads [44]. For a more detailed explanation about

Fig. 6. Histogram of ice induced loads for the selected time interval.

the ice load estimation based on the strain measurements and the associated ice load validation, reference is made to [45–47]. It has been observed in previous works (e.g. [2,48,49,47]) that the time series of the measured ice induced load looks like a sequence of spikes, see Fig. 2. A large number of much smaller peaks between two consecutive spike-like loads are considered to be noise and are hence removed from the time series. When the hull gets in contact with the ice edge, a crushing failure mechanism takes place first and consequently the spike is getting higher and higher. Looking carefully at the spike-like load, there is evidence of a typical saw-tooth shape which indicates intermittent crushing when the hull is advancing into the ice (e.g. [50,51]). When the accumulated force is high enough to initiate bending failure of the ice at a certain distance in front of the contact surface, a sudden drop of the load is observed. Afterwards, there is no event in the time series record until the next contact of the hull with the next ice edge at which the same spike-like load is repeated. The same data as in [47] is also applied in the present work, i.e. based on the conversion from measured response into estimated load. Hence, the data forming the input to the present analysis consists of the magnitude and corresponding time stamp for each of the load peaks, see Fig. 5 for the ‘‘stem’’ plot representation. The ice load, or more precisely the ice force, here is to be understood as a line load (unit: kN/m), which is the total load between two consecutive frames divided by the distance corresponding to the frame spacing. By having the load peak magnitudes and the time stamps, it is sufficient in order to apply either the ‘‘exact’’ method or the ‘‘approximate’’ method as proposed by Ochi [6], see Section 1. It is also possible to apply the ‘‘asymptotic’’ method. This method requires the maxima, which can be provided by picking up the maximum peak for each time interval.

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A. Suyuthi et al. / Structural Safety 40 (2013) 1–10

Table 8 Ice condition and operation characteristics during the expedition for the selected time intervals. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Date

Time

Duration

Concentration

(dd.mm)

(hh:mm)

(min)

(%)

25.03 25.03 25.03 25.03 25.03 25.03 26.03 26.03 26.03 26.03 28.03 28.03 28.03 28.03 28.03 28.03 28.03 28.03 28.03

16:44 18:11 20:00 20:45 21:17 21:39 19:10 19:40 20:07 20:24 7:35 7:55 9:00 9:35 9:51 10:08 10:38 11:08 11:22

30.0 40.0 30.0 30.0 9.0 18.9 30.0 27.0 15.0 20.0 20.0 29.0 30.0 13.0 14.0 30.0 30.0 13.5 12.0

99.90 57.56 65.66 86.81 84.45 77.92 100.00 84.00 100.00 99.79 78.72 93.11 95.86 85.67 98.28 97.48 97.77 99.55 85.19

Ice thickness

Propulsion power

Speed over ground

l(m)

r(m)

max

l(MW)

r(MW)

RSD(%)

l(m/s)

r(m/s)

RSD(%)

1.14 0.18 0.31 0.78 0.65 0.45 0.83 0.41 0.97 1.00 0.74 0.52 0.52 0.41 0.85 0.71 0.64 0.44 0.32

0.57 0.38 0.46 0.74 0.50 0.43 0.42 0.33 0.40 0.51 0.65 0.39 0.21 0.33 0.53 0.43 0.49 0.17 0.22

4.63 2.29 2.47 5.11 3.02 2.90 2.80 2.35 3.02 4.10 2.77 2.95 1.40 2.19 2.96 3.12 3.85 1.34 1.39

3.63 3.71 3.65 1.45 0.52 0.85 2.34 2.48 2.30 3.71 1.36 1.41 3.19 2.38 3.31 3.47 3.63 3.60 1.62

0.11 0.03 0.03 0.06 0.03 0.04 0.08 0.04 0.07 0.11 0.05 0.05 0.09 0.13 0.15 0.30 0.05 0.14 0.05

2.91 0.90 0.83 4.41 4.82 4.50 3.61 1.69 3.24 2.88 3.62 3.73 2.88 5.64 4.49 8.70 1.43 3.93 2.94

4.90 7.19 7.05 3.98 3.11 4.18 5.05 6.05 4.90 5.30 3.39 3.79 2.64 4.41 3.70 5.58 6.29 6.92 5.91

0.80 0.43 0.30 0.60 0.34 0.66 0.54 0.25 0.51 0.50 0.55 0.51 0.64 1.21 1.02 0.50 0.62 0.07 0.35

16.39 5.97 4.24 15.10 10.78 15.75 10.67 4.11 10.36 9.45 16.24 13.43 24.35 27.46 27.43 8.99 9.91 0.97 5.84

Ice load (kN/m) N

E[FX]

max(FX)

143 20 32 71 6 15 190 135 85 154 121 276 152 121 153 349 191 86 49

64.13 70.51 46.52 60.30 40.48 65.38 55.52 51.15 58.97 62.27 53.36 51.29 51.49 54.15 60.06 58.77 54.66 52.25 53.18

393.73 265.03 95.36 299.53 57.40 291.69 248.19 151.07 255.89 208.54 166.30 170.40 167.01 147.46 224.13 240.59 229.56 214.27 110.07

3.2. Selected time series for statistical analysis

Fig. 7. P–P plots to identify the initial distribution of ice induced loads. The coefficient of determination, R2, i.e. the squared values of the correlation coefficient between the estimated distribution and the empirical one are also given.

In this work, the ice load peaks were selected by applying a lower threshold of 31.25 kN/m (i.e. corresponding to a total load acting on one frame of 25 kN and a frame spacing of 0.8 m) in order to identify the start and the end of the force peak [44]. This lower threshold is somewhat relatively high. Kujala et al. [7] suggested that 10 kN/m could be set as a lower threshold since the lower values are considered to be noise or loading corresponding to open water conditions. Due to the saw-tooth shape of the ice load, just specifying a lower threshold to the time series is not sufficient. Many consecutive peaks will occur at very short time intervals, which still represent only one single event of ice-induced load. Therefore, we need to implement a so-called Rayleigh separation, where a specified value of the separator is chosen in the beginning [7].

The statistical inference methods relevant for ice-induced loading, that were described in Section 2, were applied to a selected subset of the total measurements. A total of 19 selected subsets were analyzed. The result is tabulated in Table 8. As a first selection, a time series with a duration of 30 min that was recorded on March, 28, 2007 at 07:55–08:25 was analyzed. It was measured at sensor location number 7 (in fact, all ice-induced loads treated in this work correspond to the particular sensor location number 7). At this particular time and date, KV Svalbard was operating in the Barent Sea, approximately 60 km to the east of Edgeøya Island. See Fig. 3. This selection (including the duration) was mainly due to quite stationary conditions, in terms of the ship propulsion power of 1.41 ± 0.05 MW, see Fig. 4. The relative standard deviation of the propulsion power (i.e. coefficient of variation) was 3.7%, which implies that the assumption of a constant power is reasonable. However, it must be noted that the ship’s speed itself has a somewhat larger variation with a relative standard deviation of 13.4%. It is suspected that this amplified variation is mainly due to the considerable variation of the prevailing ice thickness. According to the log-book [52], the ice concentration during the selected time interval was generally 90–100%, with a mixture of big and small first year level ice floes and consolidated ridged floes. There were small openings, but these were mostly refrozen. The average ice thickness was 52.1 ± 39.4 cm. Observing the ice thickness variation in Fig. 4 in detail, there is an evidence of a broken ice field, which is characterized by half of the ice floes being of small size (20–100 m), and another half being a mixture of medium size floes (100–500 m) and big size floes (500–2000 m). The smallest floe was 28 m, the largest floe was 1280 m and the average floe was 329 m. These floe sizes are not necessarily the actual size of the floes in the horizontal plane, as they were estimated based on the ice thickness measurements only, which actually represent only the ship traveling distance across the floe. The openings were ranging from 3 m at the narrowest up to 28 m at the widest and its average was 12 m. The ice coverage was 94%, which is in agreement with the visual observation. Some ridging and rafting causing thicker ice at the edges of the floes were also detected. The maximum ridge thickness (keel to sail) was 2.95 m, see Fig. 4. The ice load time series during the selected time interval is presented in Fig. 5. There are 276 ice load events. The mean and the standard deviation are 51.29 kN/m and 20.27 kN/m, respectively.

7

A. Suyuthi et al. / Structural Safety 40 (2013) 1–10 Table 9 Distribution parameter estimates obtained by LSM, MoM and MLE for given sample. Distribution

LSM Estimates

MoM Estimates

ML Estimates

95% Confidence Interval ^; ^ ^; a for ^ k; l h

Exponential

^ k =21.0872

Lognormal Gumbel

l^ = 2.3973 a^ =10.5628

Weibull

^ h =19.9515

^ k =20.0405

r^ = 1.3580 ^ =16.6445 b ^ = 0.9554 k

l^ = 2.6558 a^ =10.9376 ^ h =19.9610

17.8715 - 22.6321

^ k =20.0405

r^ = 0.8270 ^ =15.7709 b ^ = 0.9908 k

Fig. 8. Q–Q plots to identify the initial distribution of ice induced loads. The coefficient of determination, R2, i.e. the squares of the correlation coefficient between the estimated quantile and the data quantile are also given.

The lowest and the highest peak values are 31.39 kN/m and 170.40 kN/m, respectively. The first, second and third quartile are [Q1; Q2; Q3] = [36.27; 44.94; 59.63] kN/m. The zero level of the ice load event at the start of the time series is an evidence that there was open water and small intermittent ice floes to start with as shown in Fig. 4. Zero levels of the ice load are similarly observed at a relative distance of 860–960 m and 4850–5300 m. A histogram of the ice load is presented in Fig. 6. 3.3. Initial distribution of ice-induced load peaks The distribution of the ice-induced load peaks is assessed by application of the three available graphical methods as described in Section 2.2. The empirical cumulative distributions, which are applied by all the three different graphical methods, are based on the Weibull plotting position due to the reasons outlined in Section 2.1. The results are presented in Figs. 7, 8, and 9 for the P–P plot, the Q–Q plot and the probability paper respectively. When constructing the P–P and Q–Q plots, the MLE method was employed in order to estimate the distribution parameters. For the probability paper, fitting was made by means of the least square method for estimation of the parameters. For comparison, the estimates obtained by the different methods are given in Table 9. The confidence intervals, which were based on the maximum likelihood estimates, are also given in Table 9. These are based on the asymptotic behavior of the MLE (for large samples), i.e. that

l^ = 2.3973 a^ =11.8622 ^ h =19.8757

^ ^ k ^ ; b; for r

r^ = 1.3002 ^ =12.5614 b ^ = 0.9809 k

2.2432 - 2.5514 10.3171 - 13.4082

1.2000 - 1.4188 11.3680 - 13.8832

17.5121 - 22.5583

0.8942 - 1.0761

Fig. 9. Probability papers to identify the initial distribution of ice induced loads. The solid lines are the fitted distribution, while the dashed and dotted lines are the Kolmogorov–Smirnov’s critical lines of a = 0.05 and a = 0.005 respectively.

it is normally distributed and satisfies the efficiency criterion for estimators [29]. Hence, it is also possible to check the applicability of the parameters which are estimated by other methods. It can be observed that the parameters estimated for the exponential and Weibull distributions fell within the range of the 95% confidence level both for the LSM and the MoM. The other distributions failed except for the estimated parameters of the lognormal distribution based on the LSM. The coefficient of determination denoted as R2 is given for each P–P plots (see Fig. 7) as well as the Q–Q plots (see Fig. 8). Values of R2 for all distributions are quite high (R2 ? 1), but this is not necessarily an indication of an adequate goodness-of-fit. These R2 values have to be compared with the lower pth percentiles for k2 as given in Eq. (6). If an R2 value is greater than the value of the lower pth percentiles for k2, then we may say there is no reason to reject the fitted distribution as the distribution representing the population of the ice-induced load under consideration. However, if it is smaller, than the fitted distribution has to be rejected. A comparison of the R2 values and the values of the lower pth percentiles for k2 is given in Table 10. It is observed that there is agreement between the P–P plot and Q–Q plot at the p = 0.001 level by rejecting the lognormal and the Gumbel distribution. However, for a higher p level, the Q–Q plot rejects all distributions that are being tested. This is probably due to the feature of the Q–Q plot which emphasizes the behavior in the tails [9]. As we can observe from

8

A. Suyuthi et al. / Structural Safety 40 (2013) 1–10

Table 10 Result from the k2 test versus the P–P and Q–Q plots. k2 Statistics level P–P plot 0.001 0.010 0.100 Q–Q plot 0.001 0.010 0.100

Exponential k

2

2

Lognormal k

R

Reject?

k

R2

Reject?

0.9905 0.9933 0.9962

0.9778 0.9778 0.9778

Yes Yes Yes

0.9906 0.9936 0.9962

0.9768 0.9768 0.9768

Yes Yes Yes

0.9907 0.9935 0.9962

0.9984 0.9984 0.9984

No No No

0.9905 0.9933 0.9962

0.8876 0.8876 0.8876

Yes Yes Yes

0.9906 0.9936 0.9962

0.9483 0.9483 0.9483

Yes Yes Yes

0.9907 0.9935 0.9962

0.9918 0.9918 0.9918

No Yes Yes

k

0.9888 0.9925 0.9957

0.9985 0.9985 0.9985

No No No

0.9888 0.9925 0.9957

0.9912 0.9912 0.9912

No Yes Yes

2

2

Weibull

Reject?

Reject?

2

Gumbel

R

R

2

2

Table 11 Maximum absolute difference between the empirical and estimated c.d.f., Dmax. Method

Exponential

Da1

Da2

Lognormal

Da1

Da2

Gumbel

Da1

Da2

Weibull

Da1

Da2

LSM MoM MLE

0.0426 0.0328 0.0328

No No No

No No No

0.0959 0.1503 0.0932

Reject Reject Reject

No Reject No

0.1503 0.1339 0.0962

Reject Reject Reject

Reject Reject No

0.0279 0.0295 0.0260

No No No

No No No

Note: Da1 = D[a

= 0.05; N = 276]

= 0.0731 and Da2 = D[a

= 0.005; N = 276]

= 0.0973.

Table 12 Rejection or no reason to reject (acceptance) for the estimated distributions (by MLE) for Da No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Goodness-of-fit against Da=

= 0.05

and Da

No.

0.05

Exponential

Lognormal

Gumbel

Weibull

Reject Accept Accept Accept Accept Reject Accept Accept Reject Accept Accept Accept Accept Accept Accept Reject Reject Accept Accept

Accept Accept Accept Accept Accept Accept Reject Reject Accept Accept Accept Reject Accept Accept Reject Reject Accept Accept Accept

Reject Accept Accept Accept Accept Reject Reject Accept Accept Accept Reject Reject Reject Reject Reject Reject Reject Accept Accept

Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept

= 0.005

for all selected ice load time series.

Goodness-of-fit against Da

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

= 0.005

Exponential

Lognormal

Gumbel

Weibull

Reject Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept

Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Reject Accept Accept Accept

Reject Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Reject Reject Accept Accept

Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept

Table 13 Maximum likelihood estimates for Weibull distribution. No

1 2 3 4 5 6 7 8 9 10

ML Estimates ^ h

^ k

No

25.25 37.04 14.09 28.50 7.43 19.90 22.33 19.87 23.96 31.13

0.701 0.902 0.851 0.960 0.677 0.575 0.854 0.997 0.785 1.008

Fig. 8, the upper tail of all distributions in question deviates significantly from the null hypotheses. Therefore, the k2 test result implies rejection except for a very low percentile level (p = 0.001). In contrast to the P–P plots in Fig. 7, which are usually sensitive to discrepancies in the middle part of a distribution rather than in

11 12 13 14 15 16 17 18 19

ML Estimates ^ h

^ k

20.66 19.88 19.55 22.39 27.22 25.24 21.12 20.60 22.10

0.877 0.981 0.930 0.951 0.890 0.848 0.831 0.960 1.021

the tails [9], we could observe that there is a perfect fit for the exponential and the Weibull distribution. However, for the lognormal and the Gumbel distributions, a clear deviation can be observed. Therefore, it is reasonable that the k2 test for the P–P plots shows rejection for the lognormal and the Gumbel

A. Suyuthi et al. / Structural Safety 40 (2013) 1–10

distribution and acceptance (or let us say: no reason to reject) for the exponential and the Weibull distributions for all levels of the 2 pth percentiles for k2 up to p = 0.1th. It is noted that the kp or 2 pth level for k here means that for a fraction equal to p for a random sample of a particular size, the lower value of the coefficient 2 of determination will be at least kp . The last graphical method to be discussed is the probability paper, for which the results are presented in Fig. 9. The fitting lines were produced based on the LSM principle, and the corresponding Kolmogorov–Smirnov critical lines (for a = 0.05 and a = 0.005) were drawn relative to the fitted lines. Visually, only the exponential distribution forms almost a straight line with a minor discrepancy at the upper tail. The fitted Weibull distribution accommodates the sample well at the middle and at the upper tail but have significant deviations at the lower tail, where most of the data points are located. It is clear that the lognormal and the Gumbel distributions failed to form a straight line and give significant deviations both at the lower and upper tails. There is tendency that we should reject the lognormal and the Gumbel distributions. However, it is noted that Fig. 9 also contains Kolmogorov–Smirnov critical lines for a = 0.05 and a = 0.005. If we observe carefully (by zooming to the area where the plotted sample and the critical lines seem to intersect), it is found that only the Gumbel distribution exceeds the critical D-value. Complete Dmax values for the various distributions and the different parameter estimation methods (i.e. LSM, MoM, and MLE) are given in Table 11. If the critical value is exceeded by the value of the Dmax parameter, then the relevant distribution must be rejected. From Table 11, it is clear that the critical value is always less than the Dmax value for the exponential and the Weibull distribution for the LSM, MoM, and the MLE (for the 0.005 and 0.05 significance levels). Therefore, we have no reason to reject either the exponential or the Weibull distribution at 0.05 significance levels. This implies that for 5% of the random samples of size n = 276 (number of ice load peaks during the time interval in question), the maximum absolute deviation between the sample cumulative distribution and the population cumulative distribution will be at least 0.0731. Thus for this particular case, we may conclude that the iceinduced load process could be well represented by either an exponential or a Weibull distribution. It is emphasized that this conclusion is only valid for the particular data set that was analyzed, and hence cannot be assumed to hold in general. For some data sets, the exponential distribution will also need to be rejected. We will return to this issue latter. A similar statistical analysis as demonstrated above was performed for all the selected time intervals presented in Table 8 and the results regarding rejection versus non-rejection are given in Table 12. Table 12 is based on parameters that were estimated by the MLE. The main reason for this is that the MLE is an asymptotic efficient estimator for large samples and tends to give unbiased estimates. Moreover, it was found that if the goodness-of-fit test as performed by means of the Kolmogorov–Smirnov test for the parameters which are estimated by the MLE shows rejection, then all other estimation methods (i.e. LSM and MoM) imply that the estimates are rejected. It is accordingly sufficient that only the results for the MLE are presented. Based on the results in Table 12 it is immediately observed that both for significance levels a = 0.005 and a = 0.05, there is no reason to reject the Weibull distribution. This applies to all the 19 data sets that were examined. The estimates of the Weibull distribution parameters based on the maximum likelihood approach are presented in Table 13. A particular feature that is observed pertains to the values of the shape parameter which are less then or in the vicinity of unity (k [ 1). This implies that the upper tails of these distributions are relatively slowly decaying. It is noted that

9

the very low shape parameter values (k < 0.7) belong to cases 5 and 6 could be considered as unreliable. It is due to the fact that the sample sizes are very small, i.e. 6 and 15, respectively. Focusing on the applicability of the exponential distribution for a significance level of a = 0.05, it is rejected in more than 25% of the selected cases. For the much lower significance level of a = 0.005, which is of course less restrictive in relation to accepting the distribution (less rejection), there is one case (5.26%) for which the exponential distribution is rejected. Therefore, one should be somewhat cautious before selecting the exponential distribution for representation of the ice induced load magnitudes. Somewhat surprisingly, according to Table 12, the lognormal distribution actually exhibits a very similar performance as the exponential distribution, i.e. 25% of rejection for a significance level of a = 0.05 and 5.26% of rejection for a significance level of a = 0.005. Obviously, the worst performance for these cases is observed for the Gumbel distribution, which scored 50% of rejection for a significance level of a = 0.05 and 15.79% of rejection for a significance level of a = 0.005. 4. Conclusion This paper reviewed some useful sources of classical statistical inference procedures that could be applied for modeling the local ice-induced loads on ship hulls. In this present work, only the initial distribution was considered. The goodness-of-fit tests showed that the distribution of the peaks of the ice-induced loads on ship hulls is well modeled by the exponential and/or the lognormal distribution based on some (not all) of the available data sets. However, the Weibull distribution seems to be the best model for all data sets. The Weibull shape parameter varies, such that values in the range 0.7 < k < 1 are observed. Therefore, it is natural that some data sets are well modeled by the exponential distribution, because these correspond to k  1, which represent to a special case of the Weibull distribution. By applying a systematic procedure for statistical inference as described above, a well established distribution as probabilistic model of the peaks of the ice-induced loads can be selected with a higher confidence. Such a model can be a basis for extreme load prediction and fatigue damage calculation, which will be addressed as part of future work. The procedure outlined in the present paper are based on well established sources dealing with these issues. These procedures were also verified by numerical experiments and the gap were completed. This implies that several important questions which were not considered in detail (see Section 1) by previous work in ice statistics, e.g. [2–4], have been thoroughly investigated. Acknowledgement The authors would like to acknowledge the support of Research Council of Norway (NFR) through the Institutional Strategic Program (ISP): ‘‘Energy from the North’’, at Department of Marine Technology, Faculty of Engineering Science and Technology, NTNU. The authors express their gratitude to Det Norsk Veritas (DNV) for making the ice loads measurement data from KV Svalbard available, which were obtained during the winter of 2007. References [1] Timco G, Weeks W. A review of the engineering properties of sea ice. Cold Regions Science and Technology 2010;60:107–129. [2] Kujala P, Vuorio J. On the statistical nature of the ice-induced pressures measured on board I.B. Sisu. In: Proceedings of the 8th International Conference on Port and Ocean Engineering under Arctic Conditions, Greenland, 1985.

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