Tomasz Hinz, Polish Registry of Shipping;Tomasz.Hinz@prs.pl Jerzy Matusiak, Aalto University School of Science and Technology FUZZY MONTE CARLO METHOD FOR PROBABILITY OF CAPSIZING CALCULATION USING REGULAR AND NON-REGULAR WAVE Summary Currently, research centres from all over the world (Italy, Greece, Japan, USA, Germany) are working on a new generation of stability regulations. IMO proposed to focus on three main scenarios of ship capsizing. One of these scenarios refers to Dead Ship Condition. The new regulations are to be of a probabilistic nature. To achieve this, it will be necessary to create appropriate methods for determining the probability of ship capsizing or the probability of loss of stability (LOSA - loss of stability accident). In order to calculate the risk of ship capsizing it is necessary to build a mathematical model of ship motion. One of the exiting models, which was developed in Finland at Aalto University in Espoo, is called LAIDYN. This article presents one of the methods used to assess LOSA probability. This method is based on the Monte Carlo simulation with the application of the fuzzy number method. In the ship s motion model both regular and irregular waves are considered. Key words: capsizing, dead ship condition, Monte Carlo
T. Hinz, J. Matusiak Fuzzy Monte Carlo Method for Probability of Capsizing 1. Introduction During some previous IMO - SLF sessions (from 47 th to 52 st ) [1], [2], [3], [4] discussions took place concerning the new generation of regulations. It was decided to develop the regulation basing mainly on the following four scenarios: Dead Ship Condition, i.e. ship without forward speed, exposed to action of waves and wind; Pure-loss of stability; Parametric roll; Surf-riding and broaching. In order to present a new approach to the stability issue, it is necessary to define the intact stability failure. The definition can be found in SLF document 51/WP.2 [4]: "Intact stability failure is a state of inability of a ship to remain within design limits of roll angle and combination of rigid body accelerations". SLF subcommittee presented four definitions of criteria, which deal with the assessment of the intact stability failure in different ways: "probabilistic performance-based" this criterion is based on the physical model of intact stability failure considering the probability of an event; "deterministic performance-based" this criterion resembles the previous one, except the fact that the event is determined; "probabilistic parametric criterion" this criterion is based on the value measurement connected with an occurrence but it does not encompass the physical model of the occurrence. It exploits one or more stochastic values; "deterministic parametric criterion" this criterion also does not contain the physical model but is based on one or more deterministic values, which take part in measuring the values connected with the occurrence. This criterion is applied to present regulations. Fig. 1 Process of assessment in the new generation of stability rules [4]
Fuzzy Monte Carlo Method for Probability of Capsizing T. Hinz, J. Matusiak Fig. 1 presents the multi-stage approach of the stability regulations. The concept is built on the performance based criteria and on the vulnerability criteria [4]: Performance based criteria criteria combined with attaining the performance; they can be based on the model test or numerical simulations and can have a probabilistic or deterministic nature; Vulnerability criteria criteria connected with a ship s susceptibility to intact stability failure for particular scenarios; these criteria can be divided into two or more levels: level 1 is based on the simple criterion combined with the ship geometry, level 2 is based on the simple physical model of an occurrence of a dangerous phenomenon. The discussion concerning the new generation of IS Code is widely described in literature [5], [6]. It is also possible to find some critical voices that do not agree with this new concept [6]. Individual countries, which participate in developing the new generation of rules, presented their suggestions on how to apply the multi-stage approach to particular scenarios, which are given in Table 1. Table 1 Summary of the methodologies for different stability scenarios [6] Stability failure mode Level 1 Level 2 Direct assessment Operational guidance Pure loss stability USA Germany Germany Parametric roll USA Germany Germany Surf riding/ broaching Japan Japan Japan Dead ship condition Italy, Japan Italy, Japan Italy, Japan, Germany 2. The Probability of ship capsizing in Dead Ship Condition The capsizing probability of non-damaged ship is the main part of the probability criteria. The probability model that is mostly presented is the model described by the following formula [7], [8], [9], [10]: (1) where: P(C annual ) probability of ship capsizing, V s ship speed, p vs probability that ship will be in giving speed β heading, p β probability that ship will be in giving heading H/λ wave steepness, p H/λ probability of appearance giving wave steepness H s,annual maximum significant wave height,
T. Hinz, J. Matusiak Fuzzy Monte Carlo Method for Probability of Capsizing Q Hs, annual probability of appearance of a certain significant wave height during a storm, H C minimum significant height of wave which causes the capsizing in the presence of parametrical data, By deleting from Formula (1) elements associated with speed we obtain the capsizing probability of non-damaged ship in Dead Ship Condition.. (2) It is possible to calculate the probability of ship capsizing as well as to compute the probability of exceeding a certain roll angle. McTaggart [9] proposed the usage of the method based on the probability calculation of exceeding a certain roll angle instead of calculating the capsizing probability directly. This method is based on calculating the distribution of maximal roll angle for different sea conditions. (3) where: Q X (X) probability of exceeding variable X. By deleting from Formula (3) elements associated with speed we obtain the distribution of maximal roll angle for different sea conditions for non-damaged ship in Dead Ship Condition. (4) In the case of this approach the ship capsizing probability can be calculated from Formula 4:, (5) In order to calculate the ship capsizing probability for Dead Ship Condition the Monte Carlo method was applied. The capsizing probability was determined from Equation 5. The Monte Carlo method can be described by Formula 6:, (6) where: number of simulations where the assumed angle was reached, number of simulations.
Fuzzy Monte Carlo Method for Probability of Capsizing T. Hinz, J. Matusiak In this article the ship heeling angle equal to 60 is used as criterion of capsizing. The value of critical angle was taken from experts opinion and intact stability rules. 2.1. Fuzzy probability of capsizing (lost of stability accident) The probability of ship capsizing can also be described using the fuzzy numbers. According to definition a fuzzy number is described with the formula (7).given below [11], (7) where. There also exists a verbal description, as well, where particular numbers or sets of numbers are assigned to a given linguistic term. The latter way of description is more natural for people [11]. In order to determine the fuzzy probability of ship capsizing it is possible to use two ways separately or both of them at the same time: with the usage of operation performed on fuzzy numbers with the usage of linguistic terms In this article, the second approach was applied to calculate the probability of ship capsizing in Dead Ship Condition, with the usage of the ranges presented in Table 2. Table 2 Probability range for linguistic terms [11] Probability (linguistic term) MTBF range (days) Failure rate (ordinal scale) Very high 1 to 5 1E-1 to 2E-1 High 5 to 50 2E-1 to 2E-2 Moderate 50 to 500 2E-2 to 2E-3 Low 500 to 2000 2E-3 to 5E-4 Remote 2000 to 10000 5E-4 to 1E-5 Fig. 2 presented form of the function linguistic terms. for fuzzy probability of capsizing describe be
T. Hinz, J. Matusiak Fuzzy Monte Carlo Method for Probability of Capsizing Fig. 2 Form of the function [12] 3. Calculations and results The calculations were conducted for a container ship ITTC A1 [13]. Table 3 gives the elementary technical data of the ship. The calculations were conducted for regular and irregular waves. The wave parameters were chosen basing on the statistics for the Adriatic [14], [15]. The Tabain spectrum [14], which is appropriate for this sea, was chosen as the wave spectrum. The program LAIDYN was applied to conduct all calculations (for regular and irregular waves). The program is under continuous development in Finland at Aalto University in Espoo [16], [17], [18], [19]. The LAIDYN method is based on the assumption that the complete response of a ship equals to the sum of linear and non-linear parts [16]. The diffraction and radiation forces are well described by the linear theory. In this method, the main part of the first order load is calculated with the linear approximation, based on the current heading and location in relation to a wave. Defining the non-linear part, hydrostatics, wave force (regular and irregular waves [19]), propeller and rudder forces, were taken into consideration. It was obtained from the computations that for angle 60 the probability of ship capsizing in regular waves is equals to. The dangerous weather conditions for this ship were chosen from the calculation for regular wave. The most dangerous area is for significant wave height ranging from 2.6 m to 5.9 m and periods taken from statistics[14], [15] for this significant wave height. In the case of irregular waves, the probability of ship capsizing is smaller the for regular waves and is about 4.279E-4.
Fuzzy Monte Carlo Method for Probability of Capsizing T. Hinz, J. Matusiak Table 3 Principal particulars of the test[13] Items Ship ITTC A1 L pp Length [m] 150.00 B Breadth [m] 27.2 D Depth [m] 13.5 T f Draught at FP [m] 8.5 T Mean draught [m] 8.5 T a Draught at AP [m] 8.5 C b Block Coefficient [-] 0.667 Pitch radius of gyration [m] 0.244 Longitudinal position of centre [m] 1.01 aft of gravity from midships x CG GM Metacentric height [m] 1.0 T e Natural roll period [s] 21 Table 4 presents the results of calculations, which concern the fuzzy probability of ship capsizing. Fuzzy probability of ship capsizing was assess using theory of fuzzy number[11] and range of linguistics term[11], [12] from Fig. 2. Table 4 Fuzzy probability of ship capsizing Regular wave Irregular wave Remote 1 1 Low 0.25 0.25 4. Conclusions The probability of capsizing of the container ship during one-year operation is found to be 4.808E-4 for regular waves and 4.279E-4 for irregular waves. The safety level for the Dead Ship Condition remains open and must be evaluated by governments or insurance companies. The Dead Ship Condition scenario is considered at IMO as one of the main scenarios for stability accidents. It is assumed as especially dangerous for ships with big "superstructures" like a loaded container ship or RoPax. Monte Carlo method, as a tool for probability computations, has some disadvantages. The most serious one is its time-consuming nature in particular for irregular waves. This method gives better results when the number of computations increases. REFERENCES [1] "SLF 50/4 - Revision of the intact stability code. Report of the working group on intact stability at SLF49 (part 2). Submitted by the chairman of the working group," 50st session of IMO Sub-Committee on Stability and Load Lines and on Fishing Vessels Safety, IMO, 2006.
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