A method for the probabilistic modelling of ice pressure

Cold Regions Science and Technology 118 (2015) 112–119

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A method for the probabilistic modelling of ice pressure Petr Zvyagin ⁎ St. Petersburg State Polytechnic University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia Krylov State Research Center, Moskovskoe Shosse 44, St. Petersburg, 196158 Russia

a r t i c l e

i n f o

Article history: Received 2 October 2013 Received in revised form 13 May 2015 Accepted 29 June 2015 Available online 4 July 2015 Keywords: Ice pressure Probabilistic modelling ISO Standard Design formula Probabilistic properties

a b s t r a c t This paper presents a probabilistic model for global ice pressure based on empirical formulas. In recent studies, the Monte Carlo numerical simulation has become an important instrument for analysing information about ice pressure distributions in ice/structure interactions. The aim of this paper was to develop a method for obtaining theoretical probabilities of critical ice pressure values as well as other probabilistic characteristics of random ice pressure. The method presented in this paper provides an inexpensive and fast way to make estimations for ice pressure probabilistic parameters that can be used when the equation for pressure involves the product of parameters taken to fixed powers and the distributions for the parameters are either lognormal or uniform. An overview is presented of papers that show example cases where the use of lognormal or uniform distributions is appropriate. An example of the application of the developed model is provided. The relation of powers in the ice pressure formula recommended by ISO and a mathematical estimation of the resulting pressure are investigated. The results are discussed. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The following is a typical approach for determining design ice pressures: researchers simulated discrete ice feature parameters and then recalculated the ice loads on the structure that are caused by interacting with these ice features (Bekker et al., 2013). Then, the simulated data were investigated statistically. However, there is a lack of recent papers on probabilistic modelling. In this paper, a method is proposed that allows researchers to estimate probability that ice pressure will exceed some critical level, and do it without using a Monte Carlo simulation. Let us consider the interaction process of the structure and plain ice floe without hummocks. Based on the overview of different effective pressure design formulas made by Loset et al. (1999), it is possible to say that effective pressure is usually estimated as being dependent on ice thickness, h; contact area width, d; and unconfined compressive ice strength, R. The simple class of empirical formulas for pressure, p, can be described by relation α γ

p ¼ λh d

ð1Þ

where α, γ, and λ are constants.

⁎ ul. Politekhnicheskaya 29, St. Petersburg State Polytechnic University, IPMM, Higher Mathematics Department, St. Petersburg, 195251, Russia. Tel.: +7 9313385890 (mobile). E-mail address: [email protected].

http://dx.doi.org/10.1016/j.coldregions.2015.06.020 0165-232X/© 2015 Elsevier B.V. All rights reserved.

National standards such as SNiP 1982, API RP-2N 1995 (Loset et al., 2010), as well as a number of authors (overview made by Loset et al., 1999) use the next general formula to estimate effective pressure: α

γ

p ¼ λh Rβ d ;

ð2Þ

where α, β, γ, and λ are constants. An early formula, offered by Korzhavin in 1962, is of type (2) and contains factors h, d, and R all to the power 1. Parameter λ is determined on the conditions of experiment. The ISO 19906 code (International Standards Organization, 2010, Eq. (A.8.21)) also uses formula (2) to estimate global ice pressure. All of the formulas that derive pressure as being dependent only on the aspect ratio, d/h, can also be represented as (1) or (2). Sometimes formulas (1) or (2) are supplemented (Loset et al., 2010) by constraints like μ ν

h d bA

ð3Þ

where μ and υ are real numbers and A is a constant. Formulas (1)–(2) are empirical design formulas constructed to describe the general relation between pressure on a structure caused by a plain ice sheet and the characteristics of this ice sheet. In all mentioned formulas, constant parameters are estimated by the authors to provide the best fit to the available experimental data. Usually, the parameters found by the authors meet the specific type of experimental conditions, and this information accompanies the derived formula. The problems regarding the reliability of these formulas and their relevance to the

P. Zvyagin / Cold Regions Science and Technology 118 (2015) 112–119

physical reality are discussed by a number of authors, starting from Korzhavin (1962) through the present (Loset et al., 1999, 2010). It is possible to conclude that design formulas for ice pressure usually have the form of a weighted product of powered variables. For the aims of probabilistic modelling, these variables can be considered to be random. In the next section, we conduct an overview of papers with field experiments that show these variables can be often be modelled using a lognormal distribution. In later sections, some statements are given that allow us to determine the distribution law for the left parts of relations (1) or (2) and to solve the problem of probability estimation that pressure would exceed some critical level. 2. Lognormal distribution of the ice regime parameters Using a lognormal distribution for components that are in the right part of expressions (1) or (2) is promising because of the particular properties of this distribution. When talking about the correspondence of the available data set to the lognormal distribution, we mean that we accept the hypothesis that the “general population from which data elements are independently taken is distributed lognormally.” This approach implies two important assumptions: 1) all of the considered data elements are generated by a single, general population; 2) all of the data elements are generated independently. The hypothesis mentioned above can be proven by a number of tests, such as the Pearson chi-squared test or Kolmogorov-Smirnov test. Sometimes, the lognormal probability density function does not provide the best fit to the data, compared to other types of distribution, but at the same time, there are no strong statistical arguments against the mentioned hypothesis. Then, for the aims of probabilistic modelling, the hypothesis about lognormal distribution can be adopted. This can be done if no sensitive analysis regarding the parts of distribution which provide relatively large discrepancy with the field data is planned. The lognormally distributed random variable ξ1 can take on any positive real number. Variable ξ1 is a function of random variable ξ and is normally distributed with the mean a and standard deviation σ: ξ1 ¼ λeξ ; ξ∈Nða; σ Þ; λ ¼ const:

ð4Þ

From this relation, we can easily find the probability density of ξ1: 8 > <

2

1 ð ln x−a− ln λÞ pffiffiffiffiffiffi exp − f ξ1 ðxÞ ¼ xσ 2π 2σ 2 > : 0;

! ;

xN0

ð5Þ

The probability of the random event of “the value of variable ξ1 will be from the segment [c, d],” c ≥ 0, d ≥ 0, can be easily calculated by the next formula:     lnd−a− ln λ ln c−a− lnλ −Φ ; P ðc ≤ξ1 ≤dÞ ¼ Φ σ σ

1 ΦðxÞ ¼ pffiffiffiffiffiffi 2π

Zx

t2

e− 2 dt:

  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ ðξ1 Þ ¼ λ exp a þ σ 2 =2  expðσ 2 Þ−1:

ð7Þ

The k-th percentile xk/100 of random variable ξ1, given by relation (4), is a solution of the next equation:   P ξ1 bxk=100 ¼ k=100; and can be calculated by the formula   xk ¼ exp σ ξ Φ−1 ðk=100−0:5Þ þ aξ þ ln λ ;

!  2 σ ðξ1 Þ þ1 ; σ ¼ ln Mξ

a ¼ ln Mξ1 −

1 ln 2

!  2 σ ðξ1 Þ þ 1 − lnλ: Mξ1

ð12Þ

where Φ−1(y) is the inverse function for the Laplace function (11). The values of Φ−1(y) for different values of y can be found in the tables of Laplace function values (Ledermann, 1984). The graph of the probability density function (5), where a = −1.9 and σ = 0.525, and hence Mξ1 = 0.17, σ(ξ1) = 0.096, is presented in Fig. 1. Let us briefly cite several papers to justify that the conditions exist where the components of the right side of expressions (1) and (2), such as the ice thickness, ice strength, and ice floe velocity, are distributed lognormally. Johnston et al. (2009), in their overview of Arctic multi-year ice thickness, proposed the lognormal distribution for the data. In their study, they collected 4987 measurements compiled from different studies. However, collecting all the data from different sources smoothed the final histogram (Fig. 2). Bourke and Garrett (1987) presented histograms of ice thickness observations that do not contradict the hypothesis of a lognormal distribution of ice thickness as being plausible. In the International workshop proceedings on ice thickness (ed. Wadhams and Amanatidis, 2006), there were several histograms of ice thickness measurements presented (pp. 139, 162) that apparently correspond to the lognormal distribution. Sinitsyna et al. (2013) used a lognormal distribution as well as the other distributions to adjust sea ice strength. Nevertheless, the lognormal distribution gave the best fit for the ice strength data obtained

Wherein, having mean Mξ1 and standard deviation σ(ξ1), we can express parameters a and σ2: 2

ð11Þ

−∞

where ln is the natural logarithm. We can derive the moments of ξ1,—the mean and standard deviation: ð6Þ

ð10Þ

where Φ(x) is the standardised Laplace function, which is tabulated (for example, Ledermann, 1984):

x ≤0

  Mξ1 ¼ λ exp a þ σ 2 =2

113

ð8Þ

ð9Þ Fig. 1. Probability density of the lognormal distribution law.

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P. Zvyagin / Cold Regions Science and Technology 118 (2015) 112–119

When the ice thickness h has a lognormal distribution with parameters (a, σ), then, according to Eq. (10), its probability is:   ln d− ln 20−a P ðhNd=20Þ ¼ 0:5−Φ : σ If we have two formulas for ice pressure, formula A for h N d/20 and formula B for h ≤ d/20, then, according to the law of total probability, we should use the next relation: P ðc ≤p≤dÞ ¼ P A ðc ≤p ≤dÞ  P ðhNd=20Þ þ P B ðc ≤p≤dÞ  P ðh≤d=20Þ; where PA(c ≤ p ≤ d) and PB(c ≤ p ≤ d) are the probabilities of random event c ≤ p ≤ d calculated by formula A and B, respectively. 3. Probabilistic model and its properties Let us give one well-known corollary.

Fig. 2. (Johnston et al., 2009) Histogram of individual measured multi-year ice thicknesses.

(Truskov et al., 1996) in the Northern part of the Sakhalin Shelf (Far East of Russia). Masaki et al. (1996) referred to the results of investigations conducted by Takeuchi et al. in March 1995 in Saroma Lagoon. After processing the data, the conclusion was made that the lognormal distribution law adequately describes the ice strength measurements. Masaki et al. also referred to the earlier paper of Truskov et al. (1992), who used the lognormal distribution for ice strength measurement description (Fig. 3) Nesterov et al. (2009) investigated ice drift velocity in the NorthEastern Barents Sea, and the hypothesis of lognormal ice drift velocity distribution corresponds to the histogram presented in the mentioned paper. The mean of this data, 0.17 m/s, was noted in the paper; the standard deviation was not presented in the paper. According to the information in the paper that empirical 0.9-quantile was 0.29 m/s, it is possible to estimate a standard deviation of approximately 0.096 m/s. The curve of the distribution law with these parameters is presented in Fig. 1. Sometimes empirical formula (2) is supplemented with constraints in the form of an inequality (3), such as that offered by Loset et al. (1999): 0;11 −0;37

p≈h

d

R; if d=hb20:

Assume that the contact area width d is a constant, h is random, then the inequality h N d/20 is a random event.

Corollary 1. The product ζ1 of two random variables ξ1 and η1 with lognormal distributions, raised to the constant powers α and β, α

ζ 1 ¼ ξ1 η1β ;

ð13Þ

has a lognormal distribution. The standard deviation of the natural logarithm, ζ = ln ζ1, of the left part of Eq. (13) can be found by: σζ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2ffi ασ ξ þ 2rαβσ ξ σ η þ βσ η

ð14Þ

where σξ and ση are the standard deviations of the random variables ξ and η; ξ1 = exp(ξ), η1 = exp(η); r is the correlation coefficient of variables ξ and η. If ξ and η are uncorrelated and, according to the property for variables with normal distribution, they are independent, then σζ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2 ασ ξ þ βσ η :

To say, mean value aζ of random variable ζ is aζ = αaξ + βa η irrespective to ζ and η independence or dependence; here aξ is a mean of ξ and aη is a mean of η. Corollary 2. The product of three (or any other finite number) variables with lognormal distributions has a lognormal distribution. This corollary is derived from corollary 1 using the associativity of the summation operator for powers of these random variables. According to corollaries 1 and 2, if the parameters of the right part of relations (1)–(2) have lognormal distributions, then the ice pressure in the left part also has a lognormal distribution. Eight histograms of the pressures on panels of the Norströmsgrund lighthouse base presented by Fransson and Olofsson (2005) demonstrate a distribution that looks like a lognormal distribution (Fig. 4). Theorem 1. (Johnson and Kotz, 1972) If two variables, ξ1 = exp(ξ) and η1 = exp(η), with lognormal distributions are correlated with correlation coefficient ρ, then correlation coefficient r of the variables ξ and η can be found using:

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r        ln 1 þ ρ exp σ 2ξ −1 exp σ 2η −1 σ ξσ η

ð15Þ

Then, with the help of relations (14), (6), and (7), the mean and standard deviation of the product ξ1η1 can be found. Fig. 3. Frequency distribution of the unconfined compressive strength of ice in the Northeastern part of Sakhalin (Masaki et al., 1996).

Remark. If ρ = 1, then σξ = ση because of Eq. (8). The value of ρ should not be equal to or very close to −1.

P. Zvyagin / Cold Regions Science and Technology 118 (2015) 112–119

115

Fig. 5. Ice thickness distribution of HEM (Pfaffling et al., 2006).

parameters (a η, ση), and they are both independent, then for the random variable: α

ζ 1 ¼ λξ1 ηβ1 ; where λ = const, λ N 0, α ≠ 0, and β ≠ 0. The probability of the event “c ≤ ζ1 ≤ d” can be calculated using the expression Fig. 4. Probability density functions for segment pressures (0–414 kNm−1, Fransson and Olofsson, 2005).

 exp P ðc≤ζ 1 ≤dÞ ¼

  ln 1 þ ρσ ξ σ η : σ ξσ η If the product σξση is also very small, then

αb2 −b1 0

Corollary 3. Let us use equivalents for infinitesimal functions to simplify formula (15). If σ2ξ and σ2η are very small (for example, each of them is less than 0.05), then we can use the next relation: r≈

! 2 βσ η βaη − 2 α 2α

2 0 ln d− ln λ

∫

ln c− ln λ



Bz−βaη −α lnb2 þ B −ΦB @ jβ jσ η

 z 6 Bz−βaη −α lnb1 þ 6 B exp 6ΦB jβjσ η α 4 @



2 1 βσ η C α C C− A

2 13 βσ η C7 7 α C C7dz A5

ð16Þ where Φ(x) is a Laplace function (11) and aη and ση are the mean and standard deviation of the normally distributed variable η.

r ≈ρ:

Remark. Formula (16) can be applied using numerical integration, but it is also possible to integrate by parts and use the next relation:

Remark. In practical use of formula (15), we can substitute in Eq. (8) unknown values of σξ and ση by their estimations:

ZB

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u s2ξ1 t þ1 σ ξ ≈ ln m2ξ1 where mξ1 is the sample mean and sξ1 is the sample standard deviation of the variable ξ1 observations sample. We have a similar estimation for ση. In some cases of ice pressure probabilistic modelling, random components with nearly uniform distributions can be used. If in formulas (1) or (2) one of ice regime parameters can be uniformly distributed, for example, the diameter of the ice floe, then a direct calculation of the probability of pressures can still be determined as calculated below. For simulating unusually distributed random variables, a number of techniques are used, and a method of composition (Devroye, 1986) is among them. That method implies that the entire range of possible values of some random variable is separated to a number of intervals. On each of those intervals, the probability density function is defined in a simple way. Let us consider part of the histogram of helicopter electromagnetic measurements (HEM) of sea ice thickness h (Pfaffling et al., 2006) presented in Fig. 5. This histogram has a “heavy tail” from the right side. We can see that on the interval [b1,b2] the distribution law can be taken as uniform. Now we offer the formula for the probability density of a powered product of two independent variables, one of which is uniformly distributed and the other with a lognormal distribution. Theorem 2. If variable ξ1 is distributed uniformly on the line segment [b1, b2], variable η1 = exp(η) has a lognormal distribution with

! 2 z−m z−m k2 z−m−k =α z¼B z z¼B þm α 2 α e Φ dz ¼ αe Φ −αe2α Φ z¼A; k k z¼A k z α

ð17Þ

A

where α, m, and k = const, α ≠ 0, and k N 0. Using formula (17) instead of numerical integration saves computation time. Let us consider the special case of formula (2) when contact area width d is constant: α

p ¼ ch Rβ where c = λdγ = const. Introducing randomness into components of right-hand part of this formula, we should expect the randomness of the resulting ice pressure. The mean of pressure p is a constant characteristic. But when we construct design formula of the type mentioned above and have specified lognormal distribution for its components, the resulting mean pressure will be the function of two arguments: powers α and β. The choice of these arguments varies a lot in different studies (Loset et al, 1999). In order to help choose appropriate powers, the next theorem was formulated by the author (Zvyagin, 2015). Theorem 3. Let h and R be correlated lognormal random variables, h = exp(ξ) with parameters (aξ, σξ), R = exp(η) with parameters (aη, ση), and correlation coefficient of ξ and η is r, − 1 b r b 1. Then, the global minimum of the ice pressure mean Mp is provided by the next values of powers:

αm ¼

aη σ ξ σ η r−σ 2η aξ σ 2η σ 2ξ ð1−r 2 Þ

; βm ¼

aξ σ ξ σ η r−σ 2ξ aη σ 2η σ 2ξ ð1−r 2 Þ

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P. Zvyagin / Cold Regions Science and Technology 118 (2015) 112–119 a

If h and R are independent lognormal variables, then α m ¼ − σξ2 , ξ

a

βm ¼ − ση2 . The proof can be found in the paper (Zvyagin, 2015). η

Remark. If parameter β is fixed, then αm ¼

−aξ −rβσ ξ σ η σ 2ξ

:

If parameter α is fixed, then βm ¼

−aη −rασ ξ σ η σ 2η

Corollary 4. It is evident that the values of α and β that provide the global minimum of variance σ(p) are α = 0, β = 0. 4. Probabilistic modelling Monte Carlo probabilistic modelling can be applied easily if the ice pressure model (1) or (2) is considered by researcher as valid and distributions of the components are known. Probabilistic modelling becomes especially simple when components have normal, lognormal, or uniform distribution. Modelled random numbers with such distributions can be obtained by in-built procedures of software like STATISTICA or SPSS. Besides of this, one can use special numerical algorithms to get a set of random numbers, which correspond to uniform or normal distribution law. These numbers are usually denoted as “pseudorandom,” the algorithms for them are well-known and described, for example, by Knuth (1968). While using software random generators, one should take into account the pre-assumed independence of the components in formulas (1) or (2)—ice thickness, ice strength, or the pre-assumed type of their dependence. Usually, random generators provide values that can be considered as independent. There are special algorithms for getting correlated random values. Such algorithm for correlated lognormal variables is described in the paper (Zvyagin and Sazonov, 2014). Further in this paper, the Monte Carlo method will be used to check and illustrate results obtained analytically. Having a set of ice pressures, calculated according to formulas (1) or (2), we can use estimators for parameters of interest, such as ice pressure mean, variance, and percentiles. As estimators that we use are consistent, the distance between their values and values obtained analytically should not be significantly large. 5. Practical application To estimate the global pressure p on the construction, let us use the formula of type (2), which is recommended by ISO 19906 code: p≈d

−0:16 nþ0:16

h

R

ð18Þ

where n is an empirical coefficient, taken as n = −0.5 + h/5 if h b 1 m and as n = −0.3 if h ≥ 1 m; d is the width of the contact area (see also Palmer and Croasdale, 2013). Example 1. To illustrate the theory presented above, we use a structure width d = 10 m. For the ice thickness, we use observations of the fast ice thickness in the Sea of Okhotsk made at Chaivo Station (Shirasawa et al., 2005). The average value of the fast ice thickness on the 15th of March was Mh = 0.76 m, and the standard deviation was σh = 0.118 m. The distribution law of h is assumed as lognormal. The ice thickness Monte Carlo probabilistic modelling results (2000 simulations) are presented as a black coloured histogram in Fig. 6. To avoid the randomness of the exponent in formula (18), we take the coefficient n = −0.5 + 0.76/5 ≈ −0.35. Thus, the power of h in formula (18) is −0.19.

Fig. 6. Histograms of the ice thickness modelling results for example 1 (black) and example 2 (grey).

For the parameter of the ice strength R, we take statistics for the 1-year ice strength observations in Okhotsk Sea made by Truskov and presented also by Sinitsyna et al. (2013): MR = 2.04 MPa, σR = 0.36 MPa. As a distribution law for R, we take the lognormal law, which agrees with the data described in the mentioned paper. Thus, h = exp(ξ), ξ ∈ N(aξ, σξ). We can estimate the parameters of the random variable ξ by formulas (8) and (9): aξ ≈ − 0.282, σξ ≈ 0.154, σ2ξ ≈ 0.024. Reasoning the same way, R = exp(η), η ∈ N(aη, ση), where aη ≈ 0.7, ση ≈ 0.173, σ2η ≈ 0.03. Let us denote ζ ¼h

−0:19

R:

ð19Þ

For the Monte Carlo simulation, we use standard tools for normal and uniform random variable simulating that exist in Statsoft STATISTICA as well as algorithms of normal random value simulating (Knuth, 1968). At first we assume that random variables h and R are independent. The approach with independent ice thickness and ice strength is currently used in the probabilistic modelling of ice loads (Su et al., 2011). It is connected, first of all, with the difficulties of correlated random variables simulation. According to corollary 1, ζ would have a lognormal distribution, with parameters aζ ≈ 0.754, σζ ≈ 0.176, wherein ζ mean Mζ ≈ 2.16 and ζ standard deviation σ(ζ) ≈ 0.38. Taking into account that d−0.16 ≈ 0.692, for pressure we obtain p = d−0.16ζ and, finally: Mp ≈ 1.495 MPa and σ(p) ≈ 0.26 MPa. The result of the probabilistic modelling of pressure p by the Monte Carlo method is presented in Fig. 7, as the black histogram. Let us calculate the 95th and 5th percentiles of random variable p, defined by formula (18), using Eq. (12): x0.95 = exp(0.289 + 0.754 + ln 0.692) ≈ 1.96 MPa and x0.05 ≈ 1.1 MPa. The 99th and 1st percentiles of random variable p are: x0.99 ≈ 2.21 MPa and x0.01 ≈ 0.98 MPa. The Monte Carlo modelling result for percentiles is (Fig. 7, black histogram): ~x0:95 ≈1:953, ~x0:05 ≈1:104, ~x0:01 ≈0:978, ~x0:99 ≈2:18 MPa, the number of modelled ice pressure simulations was 2000. Now let us assume that the ice thickness h and its strength R are correlated, and all other conditions are the same. Timco and Frederking (1990) summarised the results of the plain ice sheet strength and made a conclusion that the ice strength for plain ice normally increases with increasing ice thickness. This strengthening occurs because ice salinity decreases with increasing ice thickness. Therefore, if we consider both h and R as random variables, we can speak to the usual positive correlation between them. The dependence of the mathematical estimation Mp and standard deviation σ(p) on the parameter r, which is related with the h and R

P. Zvyagin / Cold Regions Science and Technology 118 (2015) 112–119

Fig. 7. Histograms of the ice pressure modelling results for example 1 (black) and example 2 (grey).

correlation coefficient ρ according to the Eq. (15), is presented in Fig. 8. Mp and σ(p) decrease when r increases (and ρ as well). This happens because of the negative power of factor h in relation (19), and vice versa, the negative correlation between h and R increases the mean and variance of pressure p. We should note that some empiric formulas offered for pressure p (see Loset et al., 1999; Croasdale, 2009) contain factor h in a positive power as well as factor R, and for those cases, the positive correlation of h and R will increase the mean and variance of pressure p. Let us fix in formula (2) the power β of ice strength R with value 1, as in (18), and investigate how Mp and σ(p) change when the power α of ice thickness h changes. Curves of these dependences for the conditions of example 1 with the factors of formula (18) uncorrelated are presented in Fig. 9. The absolute values of α and β that provide the global minimum of Mp, calculated according to theorem 3 with given pairs (aξ, σξ) and (aη, ση) as in example 1, are significantly greater than 1 and are not of practical interest. We only note that α ≫ 1 and β ≪ −1 because of the negative and positive signs of aξ and aη, respectively. So, the statistical properties of random ice pressure in the left part of expression (2) significantly depend not only on the statistical properties of R and h ice regime parameters but also on the value and sign of powers α and β.

117

Fig. 9. Dependence of Mp (solid line, left axis) and σ(p) (dashed line, right axis) on parameter α, when β = 1 and r = 0.

Example 2. This example is intended to illustrate the “ice thickness scale-effect” of ice pressure design formula (18). Applying the method of composition (Zvyagin and Sazonov, 2015) to simulate ice loads generated by ice with thickness h distributed with a right “tail” in Fig. 5, let us assume that h is uniformly distributed on the segment [b1, b2]. According to Fig. 5, let us take b1 = 0.8 m and b2 = 2.3 m. The histogram of the simulated ice thickness values (2000 simulations) is presented in Fig. 6 (grey). Let us assume a lognormal distribution for R, with the same parameters as in example 1, and let us assume that h and R are uncorrelated random variables. The result of the probabilistic modelling of the pressure p according to the formula (18) is presented in Fig. 7 (grey histogram). Using the numerical calculations by expression (16), it is possible to estimate the 99th, 95th, 5th, and 1st percentiles of random variable p: x0.99 ≈ 1.98, x0.95 ≈ 1.74, x0.05 ≈ 0.96, x0.01 ≈ 0.85 MPa, which corresponds to the grey histogram presented in Fig. 7. The Monte Carlo modelling result for percentiles is ~x0:95 ≈1:738, ~x0:05 ≈0:959, ~x0:01 ≈0:84 8, ~x0:99 ≈1:938 MPa, the number of modelled ice pressure simulations was 2000. One can compare these values with percentiles obtained in previous example. Because of the negative power of h in ISO formula, the thicker ice we have, the more probable smaller values of p become, obtained according to formula (18). In addition, for this simulated data, the statistical difference between the obtained distribution and lognormal can be found only after the precise study and with the large number of data points. This occurs primarily because the absolute value of the power of uniformly distributed component is small, and hence this component has small influence the resulting pressure distribution type. The results obtained by the method described above, greatly depend on the reliability of the applied ice pressure design formula.

6. Discussion

Fig. 8. Dependence of Mp (solid line, left axis) and σ(p) (dashed line, right axis) on parameter r, when α = −0.19 and β = 1 (Zvyagin, 2015).

The method proposed in this paper is a theoretical support for a wide class of ice pressure probabilistic modelling problems. The investigator should remember that the reasoning in the proposed method is based on the assumption of the specific distributions of the ice regime parameters as well as the validity of the design formula for ice pressure. Assumptions about the ice regime parameters should be considered as a statistical hypothesis. In the preliminary step of using the method described above, these hypotheses should be tested by tests, such as the

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χ2, Kolmogorov-Smirnov (K-S), or other tests. However, the assumption of a lognormal distribution of ice strength and ice thickness can be used for a wide class of problems. For the probabilistic modelling method of the composition, the uniform distribution of the ice regime parameters can be used. In general, the proposed model provides a way to obtain fast estimations before or instead of standard Monte Carlo simulation methods. Nothing prevents the use of the proposed method to estimate not only ice pressure but also ice loads, using formulas similar to (1) or (2), corrected by the contact area. In addition to the probability evaluation for different random events, the method presented in the paper allows to investigate a number of design formulas for ice pressure and force that are currently in use. Thus, the mathematical estimation and standard deviation of pressure can be calculated as well as other moments and percentiles. The contribution of each formula's component on the final pressure parameters can be evaluated and understood better. All of these problems can be subjects of future investigation. Based on the computational example presented in the paper, it is possible to say that small absolute power values of the ice thickness factor h lead to the small influence of this factor on the shape of the pressure distribution law. Correlation of the ice regime parameters is another important factor that can influence the ice pressure modelling result. The positive or negative signs of the powers of two lognormally distributed factors influence the final product distribution in the case of the correlation presence. Thus, in the case of a positive correlation, if the factors have powers of different signs, then the mean and the variance of the product will be smaller than in the case of uncorrelated factors. The values of the powers that provide the minimum of the ice pressure mathematical estimation depend on the parameters of the lognormally distributed factors, as was stated in Theorem 3. However, there is a lack of publications about experiments where the statistics for ice thickness and ice strength were measured at the same time. Therefore, it is difficult to find univocal opinions about how these parameters can be statistically related. In this way, using such a theoretical analysis, more precise requirements and conditions for empirical ice pressure formulas can be imposed in the future.

his support. I thank Prof. Karl Shkhinek for his valuable books on the subject that he gifted to me. I am grateful to the anonymous reviewers of this manuscript for helpful comments. Appendix A Proof of Theorem 2. Let us find the natural logarithm of both parts of ζ1 = λξα1 η1β: ln ζ 1 ¼ ln λ þ α lnξ1 þ β ln η1 : Let us denote ζ = ln ζ1 − ln λ and ξ = αlnξ1, η = lnη1. Because ξ1 and η1 are continuous random variables, ζ1 and ζ are also continuous variables. Then, ξ has the distribution density function: f ξ ðxÞ ¼

8 <

x 1 exp ; x∈½α ln b1 ; α ln b2  ; α ðb2 −b1 Þ α : 0; x∉½α ln b1 ; α ln b2 :

According to the conditions of the theorem, βη has Gaussian distribution density: f βη ðxÞ ¼

 2 ! x−βaη 1 pffiffiffiffiffiffi exp − ; −∞bxb þ ∞: jβjσ 2π 2β2 σ 2η

Therefore, we have ζ = ξ + βη, where random variables ξ and βη are independent, as are the initial variables ξ1 and η1. For the sum of two the independent variables ξ and βη, we have the next probability density function þ∞ Z

f βη ðxÞf ξ ðz−xÞdx

f ðzÞ ¼ −∞

Hence f ξþβη ðzÞ ¼ f ζ ðzÞ ¼

7. Conclusions The findings of the paper can be summarised as follows: - a probabilistic model for ice pressure based on a wide class of empirical formulas and on two types of distributions of ice regime parameters is proposed; - two theorems are stated and proved (Theorem 2 and Theorem 3) regarding the distribution of the product of two random ice regime parameters—with lognormal and uniform distributions—and regarding the minimum of an ice pressure mathematical estimation depending on the powers in the ice pressure design formula; and - an example of the ice pressure estimations made by the proposed method for the Okhotsk sea conditions is provided. Curves that illustrate the influence of the design formula parameters on the resulting ice pressure statistical moments are presented. The results of the theoretical calculations fit the results of the Monte Carlo simulation modelling; - the value and sign of powers of random factors in ice pressure design formulas can significantly influence the resulting ice pressure statistical properties. Acknowledgments I wish to thank Prof. Kirill Sazonov from St. Petersburg Krylov Research Center for his helpful discussions and Prof. Oleg Timofeev for

z−α ln b1  2 ! x−βaη x pffiffiffiffiffiffi exp − −  2 dx α α ðb2 −b1 Þjβjσ η 2π z−α ln b2 2 βσ η

expðz=α Þ

∫

After transformations we have exp f ζ ðzÞ ¼

 2 ! βσ η z−βaη þ α 2α 2

2 0

α ðb2 −b1 Þ

6 Bz−βaη −α lnb1 þ 6 B ×6ΦB 4 @ jβjσ η



0 2 1  2 13 βσ η βσ η C Bz−βaη −α lnb2 þ C7 B 7 α C α C C−ΦB C7 A @ A5 jβjσ η

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