Numerical simulation on the ice-induced fatigue damage of ship structural members in broken ice fields

Marine Structures 66 (2019) 83–105

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Numerical simulation on the ice-induced fatigue damage of ship structural members in broken ice fields

T

Jeong-Hwan Kim, Yooil Kim∗ Department of Naval Architecture and Ocean Engineering, INHA University, 100, Inha-Ro, Nam-Gu, Incheon, South Korea

A R T IC LE I N F O

ABS TRA CT

Keywords: Fatigue damage assessment Ice going ship Broken ice field Ice-structure interaction Periodic media analysis Convolution integral 2-Parameter weibull model

A fatigue damage assessment for an ice-going ship navigating through broken ice fields was carried out using a numerical model to simulate the interactions between ice floes and structure. The stress history was obtained by a numerical model and the corresponding fatigue analysis results are presented. The stress time series required for fatigue analysis could be extracted directly using an efficient interaction model, the application of a periodic media analysis method, and the convolution integral. A 2-parameter Weibull model was applied to describe the probability distribution of the stress amplitude, and the fatigue damage was obtained using the Palmgren-Miner rule. Finally, the fatigue damage considering the actual ice conditions of the Baltic Sea was calculated using the proposed method and LR method, and each result was compared.

1. Introduction Arctic sea ice is melting rapidly due to global warming, resulting in the potential opening of the Arctic route. If the current warming trend continues, a new transport route will open with an increase in the number of days where the Arctic Ocean will be navigable, shortening the distance between Europe and Asia by a third [1]. One of the areas of major interest to ship-owners who are preparing for opening Arctic Ocean voyages is the fatigue problem of ice-going vessels. This is because fatigue cracks caused by ice are reported as often as those due to waves [2]. Moreover, the Arctic region is sensitive to pollution, so no oil spill is allowed. The pollution in the Arctic would be more difficult to eliminate due to the lack of light, very low temperatures, drifting ice, high winds, and a variety of other factors [3]. Therefore, an assessment of iceinduced fatigue of ship hulls is required in the design stage of an ice-going ship. In particular, it is important to estimate the fatigue strength in broken ice fields because a ship normally follows a sea path covered with ice broken by an icebreaker [4]. Nevertheless, the number of related studies is very small. There is no guidance that designers can simply refer to, except for the Lloyd's Register's guidance. Lloyd's Resister [5] announced the ShipRight FDA ICE, a fatigue design evaluation procedure to evaluate fatigue damage in hull structures under ice loads. A deterministic ice-induced fatigue analysis procedure was developed based on empirical knowledge and in-situ measurement results. The target of the procedure is an ice-strengthened vessel navigating in firstyear ice conditions with icebreaker assistance and the ice channel formation [5]. Strength of LR method is its versatility so that fatigue damage can be calculated even though the required analysis data are not fully available [6]. On the other hand, this method is designed to be simple for the designer to follow the procedure; hence, it contains many assumptions, and it is difficult to predict the reasonable fatigue life. Suyuthi et al. [6] proposed a systematic procedure for a fatigue damage assessment of ships navigating in ice-

∗

Corresponding author. E-mail addresses: [email protected] (J.-H. Kim), [email protected] (Y. Kim).

https://doi.org/10.1016/j.marstruc.2019.03.002 Received 14 September 2018; Received in revised form 31 January 2019; Accepted 12 March 2019 0951-8339/ © 2019 Elsevier Ltd. All rights reserved.

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covered waters. Closed form expressions for fatigue damage based on the Palmgren-Miner rule, which reflects the variation under ice conditions, vessel's speed, and operational modes, were derived for several different statistical models of the stress amplitude, i.e. exponential, Weibull's, and three-parameter exponential distributions. As a result of examining the applicability of each probability model using the KV Svalvard full scale measurement results, it was concluded that the best fit probability model is a three-parameter exponential model. However, since the main parameters for fatigue analysis are obtained from the field measurement data, it is difficult to apply to different conditions. Chai et al. [7] examined the short-term extreme value statistics of ice loads acting on a ship hull and the fatigue damage due to ice loads. They found correlations between the ice-induced load statistics and prevailing ice conditions using probabilistic methods. For the extreme value prediction, a novel method, namely the average conditional exceedance rate (ACER) method, was applied, and the short-term fatigue damage was estimated based on the S-N curve approach. Han and Sawamura [8] calculated the fatigue damage of a ship passing through broken ice fields. They applied the discrete element method and described the ice load peaks using the Weibull model. Finally, the fatigue damage was determined based on the stress amplitude derived from structural beam theory. However, this method has a disadvantage that each ice floe is represented by a two dimensional circle. Three main methods can be used to obtain data, such as the stress amplitude and number of cycles required for the ice-induced fatigue analysis: field measurement, model test, and numerical analysis. Thus far, most studies have been based on field measurement data [6,7,34]. Among these methods, the numerical analysis can be a good candidate to get the required data. Numerous simulations are possible with limited computation efforts, and various ice conditions can be easily implemented [6]. Nevertheless, attempts to calculate the ice-induced fatigue damage using numerical analysis are still lacking. The reasons why the attempt is difficult can be summarized in two major ways: 1. Analyses over a long period of time are required to obtain probabilistic characteristics of calculated data. This depends on the speed of the ship, but considering the impact frequency, it is considered that at least a few 10 min should be simulated to acquire sufficient data [6]. To do this, there must be a fast and accurate simulation method. 2. It is difficult to derive stress time series for ice-induced fatigue analysis, due to the computational burdens that existing numerical scheme inherently holds. Some simulation tools are available for long term analysis, such as GEM [9], but most of these tools are specialized in calculating the ice load so it is difficult to calculate the stress of the target structural element. Hence, additional ways are needed to convert the ice load to stress if someone wants to use these methods to calculate the stress of the target point. Many studies have applied the simplified method using beam theory in this case, and many assumptions are included in the process. Several numerical models have been developed to implement the interaction between broken ice floes and structures. Millan and Wang [10]; Wang and Derradji-Aouat [11] and Kim et al. [4] applied the finite element method to calculate the global ice load acting on structures by broken ice floes. They employed the arbitrary Lagrangian-Eulerian method to implement the ice-water-structure interaction problem. Since the discrete element method (DEM) was developed by Cundall and Strack [12]; many studies using DEM have been introduced to the simulation of broken ice. Hansen and Løset [13] developed a DEM model to investigate the behavior of the turret moored FPSO in broken ice fields. Sun and Shen [14] applied DEM to implement pancake ice floes, which were modeled as 3-D dilated particles. In addition, Alawneh [9] developed GPU-based event mechanics, which enables hyper-real-time simulations of ice-structure interactions using the graphics processing unit (GPU), for the simulation of pack ice condition. In this study, a fatigue damage assessment for an ice-going ship navigating through broken ice fields was carried out. A numerical model to simulate the interaction between ice floes and structure developed using the finite element method was introduced, and time series of stresses calculated by the model and the corresponding fatigue analysis results are presented. This model enables long term analysis through an efficient interaction model, the application of periodic media analysis method and the convolution integral, and it allows the stress to be extracted directly using the finite element method. A 2-parameter Weibull model was applied to the calculated stress history to describe the probability distribution of stress amplitudes, and the fatigue damage was obtained using the Palmgren-Miner rule. Finally, the fatigue damage considering the actual ice conditions of the Baltic Sea was calculated using the proposed method and LR method, and each result was compared. The comparison with LR method needs to be understood as the validation of the appropriateness of the proposed method, rather than accuracy check. 2. Analysis methodology Fig. 1 presents the calculation procedure for the ice-induced fatigue analysis introduced in this study. The entire analysis methodology can be divided into four stages. First, the contact pressure time series on the outer-shell of a ship under broken ice field conditions was calculated. The next step was to convert the contact pressures into stresses using the convolution integral. Here, it is important to calculate the impulse response function (IRF) of the target position with respect to the pressures for each element of the outer-shell. To calculate the contact pressure and IRF, a numerical model was developed to simulate the interaction between ice floes and structure based on the finite element method using ABAQUS/Explicit software. With the aid of this mature and proven commercial software, it is easy to generate user-defined external loads, and also a robust contact algorithm for multi-body interactions can be employed usefully. In addition, it has flexible contact interaction models, and multi-bodies can be modeled easily via script language. Once the stress time series are calculated, the short-term distribution of the stress amplitude can be derived by applying peak analysis and an appropriate probability model, and the corresponding short-term fatigue damage can be calculated using the proper S-N curve. Finally, the total fatigue damage can be calculated using the Palmgren-Miner rule. 84

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Fig. 1. Calculation procedure for ice-induced fatigue analysis.

2.1. Calculation of the contact pressure time series The finite element models required at this stage are an outer-shell model for calculating the contact pressure and ice floe models for broken ice fields. The outer-shell model of the ship is implemented as rigid elements. Ice floes of a random size, thickness, shape, and distribution were generated using an automatic modeling scheme. For rapid calculations, ice-fluid and ice-structure interactions were implemented in simplified manners. Important parameters required for the simplified equations, such as the drag force coefficient or contact interaction parameter, were determined through detailed analyses. 2.1.1. Modeling of ice floes Each floe is made by rigid solid elements, and the process of generating ice elements can be explained simply as follows: First, the thickness distribution of the ice floes in the target field is determined. The thickness of broken ice is dependent on the thickness of the original ice sheet [15]. Collection of ice charts can be used as the data to determine the ice thickness distribution of the field [6]. Then circles as the basis of ice element generation are generated for each thickness. The diameters of the circles are set randomly according to criterion for determining an unbreakable ice floe developed by Lindseth [16]. The target ice was managed broken ice, so it was assumed not to be broken. 2D circles were then generated and placed one after another on a space under the predefined concentration. The placement continued until the concentration reached the specified level within the area. Later, points were generated randomly along the perimeter of the generated circles to make polygon-shaped ice. For example, assuming that a pentagonal ice floe is produced, five points are created at random intervals on the circle's perimeter, and the points are connected to produce a polygon. The shape of broken ice fragments is normally perceived to be random (Aboulazm, (1989). However, according to Aboulazm [15]; the shape is mainly triangular or rectangular based on aerial photographs of Arctic areas, and broken ice generated by icebreakers is usually crescent shaped close to the triangular shape. A 3D solid was then generated by extruding the polygon with a depth corresponding to the ice thickness, and the finite element mesh was finally generated on the solid. The finite element models for ice floes are shown in Fig. 2. 2.1.2. Modeling of the ice-sea water interaction To simulate the motion of floating ice floes, which are subject to several hydrodynamic loads, such as drag, inertia, and buoyancy ⎯→ ⎯ forces in seawater, six degrees of freedom motions were taken into account. As expressed in Eqns. (1) and (2), the drag force ( Fd ) and

Fig. 2. Finite element models for ice floes. 85

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Fig. 3. Ice floes used for the ice movement comparison.

⎯⎯⎯→ drag moment ( Md ) on an ice floe were applied for the planar motions, such as surge, sway, and yaw [17].

⎯→ ⎯ → ⎯→ ⎯ → ⎯→ ⎯ 1 Fd = − CdF ρw Asub ( V − Vw ) V − Vw 2

(1)

⎯⎯⎯→ 1 Md = − CdF r 2ρw Asub → ω→ ω 2

(2)

⎯→ ⎯ → where CdF is the drag force coefficient. ρw is the water density and Asub is submerged area. V is the velocity of the ice floe and Vw is → the water velocity. r is the mean distance between the centroid and each vertex of floe, and ω is the angular velocity of the floe. The drag force coefficient CdF was obtained by detailed fluid-structure interaction analysis using the Coupled Eulerian-Lagrangian method. Fig. 3 presents the detailed analysis model for obtaining the drag force coefficient, and the proposed model. Ice was modeled in the Lagrangian domain, and sea water was implemented in the Eulerian domain. With the ice floes floating in the sea, the ship model, which was modeled as a rigid body, was advanced to collide with the ice floes. The analysis using the proposed method was performed under the same conditions and the results were compared to derive the proper drag force coefficient. Four ice floes near the ship were selected for the comparison, as shown in Fig. 3. A single representative value that can cover all situations is required because the shape of the projected frontal area of each ice floe is different and it changes continuously as the floes collide with each other. Therefore, the drag force coefficient showing results closest to the detailed analysis was selected through trial and error [18]. The determined drag force coefficient was 5.25. Fig. 4 shows the change in the position of the ice floes in 2D space during the simulation. It was found that the positions of ‘2’ and ‘4’ relatively match each other well, whereas in cases ‘1’ and ‘3’ some deviations are observed, especially when time passes. Deviations at later stages are what have been observed when the load transmission from ice to structure is over. Fatigue damage is more relevant to this load transmission stage, rather that the later stage when there is no contact between ice and structure. The buoyancy pressure, shown in Eqn. (3), acts on the floe surface so that the heave, roll, and pitch have hydrodynamic restoring forces and moments. For the application of buoyancy in a heeled situation due to roll or pitch, pressure is applied against the local x-y coordinate system of the corresponding surface of the floe. Pressure is also applied to the sides of the floe to consider the large angular movement, as shown in Fig. 5.

P (x ′, y′) = ρw gh (x ′, y′)

(3)

where x ′ and y' are the local coordinates for the surface. g is the gravitational acceleration, and h is the vertical displacement of a given position with respect to its initial position. The added mass was included in the form of a constant value and considered by multiplying the mass by 1+Cm in the motion equations. The added mass coefficient used in the simulation was 0.41 with reference to Newman [19]. 2.1.3. Modeling of the ice-structure interaction In this study, instead of realizing the local failure occurring during the collision of ice with detailed analysis considering the elasto-plastic material properties, a global approach was taken, which allows the penetration of colliding objects caused by the local failure of ice near the contacting area. The penetration can be determined using a predetermined pressure-penetration relation, as 86

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Fig. 4. Comparison of the positions between the detailed analysis model and proposed model in 2D space.

Fig. 5. Pressure profiles acting on the floe surface.

illustrated in Fig. 6, which was obtained by separate detailed numerical analyses. According to Kärnä, [20]; when ice and a structure are in contact, penetration occurs according to the contact pressure. This pressure-penetration relation approach is based on the hypothesis that the local failure of ice during a collision can be expressed by a single pressure-penetration relationship. The reason why a pressure-penetration relation is applied instead of a force-penetration relation is that it is possible to consider the shape of the

Fig. 6. Contact pressure-penetration relation. 87

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Table 1 Analysis conditions for the pressure-penetration curve. Number

Thickness (m)

Size (m)

Collision angle (°)

1 2 3 4 5 6 7 8 9 10

0.3 0.6 0.6 0.9 0.9 0.3 0.6 0.6 0.9 0.9

3×3 3×3 6×6 4.5 × 4.5 9×9 3×3 3×3 6×6 4.5 × 4.5 9×9

0 0 0 0 0 45 45 45 45 45

contact surface in the event of a collision so the contact phenomenon between objects with various shapes can be implemented realistically. A series of detailed ice-structure collision analyses were performed using a Coupled Eulerian-Lagrangian (CEL) method to derive the pressure-penetration curve. Kim and Kim [18] presented a detailed description of the analysis. The failure of ice was realized using a crushable foam material model, whose detailed parameters were tuned using the experimental results reported by Kim et al. [21]. The pressure-penetration relation can also be affected by a range of parameters, such as thickness, size, and collision angle. Therefore, in this study, a pressure-penetration curve was derived under different combinations of parameters, as summarized in Table 1. The target ice of this study was first-year ice. Because the majority of the ice thickness is less than or around 1 m, the thicknesses used in the case study were set to be 0.3, 0.6, and 0.9 m [22]; [5,23,24]. Ice was modeled as a square. The sizes of the ice floes were determined according to Lindseth's standard [16], which indicates the maximum size of the unbreakable ice floe based on its thickness. Two different collision directions were also considered: one in the direction of the parallel side and the other one in the direction of the corner side with a 45° angle deviation. Fig. 7 shows the examples of the detailed analysis. Under the assumption that there is a single pressure-penetration curve, linear regression analysis was performed for the data derived from detailed numerical analyses. As shown in Fig. 8, the derived slope of the pressure-penetration curve that passes through the origin is 35,367 Pa/m. 2.1.4. Periodic media analysis Statistically converged ice load histories are essential for a fatigue life calculation, so that a long duration simulation in the time domain is inevitable, which is time consuming. Considering the high frequency of the impact load, a short-term distribution can be obtained by performing analysis for at least 5–10 min. For the time domain ice-structure interaction analysis during that period of time, a sufficient number of ice elements are required considering the analysis time and speed. For example, to analyze for 10 min with a ship speed of 4 knots, at least 1200 m of ice-covered water needs to be prepared. In this case, modeling, calculating, and postprocessing are almost impossible because it takes considerable time. In this study, the periodic media analysis method was applied to solve this problem. This technique is a functionality of ABAQUS/EXPLICIT, which is a technique designed to simulate a system that is geometrically repetitive in nature, such as a manufacturing process involving conveyor belts. Fig. 9 gives an example of moving boxes on conveyor belts to illustrate the schematic representation of the periodic media analysis method. The overall model was decomposed into several blocks that are topologically identical, and their edges were connected. A block consisted of a box and a belt of unit length, i.e., the distance between I and A, and it was repeated to model the entire periodic media. The connection points of each block are marked as I, A, B, C, D, and O. Here, I and O denote the inlet and outlet, respectively. When the entire model moves in the moving belt direction and meets the trigger plane, the outlet block is shuffled back to the inlet. The trigger plane controls the timing that the shuffling process occurs. As a result, infinite repetition is possible with a limited number of elements. When this method is applied to a simulation of ice-structure interaction analysis, each box will be replaced with a group of ice floes. Virtual belt, connected to the ice floes, will be modeled to drive the modeled ice floes, but it has no influence on the physics

Fig. 7. Example of detailed analysis. 88

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Fig. 8. Derived pressure-penetration curve.

Fig. 9. Schematic representation of the periodic media analysis method [25].

taking place during ice collisions to a structure. The potential repetition of an ice load due to repetition of the ice geometry was checked and it was confirmed that the ice load history does not repeat because of the distraction of ice floes before a collision with a structure. Further details are given in Chapter 3.2.2. 2.1.5. Verification of the numerical model Verification of the numerical model has been carried out based on the model tests conducted in an ice tank at the Korea Research Institute of Ships and Ocean engineering (KRISO) using a 1:18.667 scale mode ship of the Korean icebreaking research vessel (IBRV) (Jeong et al., 2017). Three different constant speeds (0.119, 0.357 and 0.595 m/s in model scale) and two different ice concentrations (60% and 80%) were considered in the pack ice channels. Detailed of the model test can be found in Kim et al. (2017). Numerical simulations using the proposed method were performed for the same conditions as the model test. As shown in Fig. 10, the numerical model is composed of the target ship, channel wall, and broken ice floes. The results of the model test and the simulation were compared in terms of ice resistance. It would be ideal to directly compare the amplitudes of the stresses required in the fatigue calculations, but since the data available is limited, the numerical model was verified using the data derived from the ice resistance test. Fig. 11 compares the results of the simulations with the model tests in

Fig. 10. Simulation model for the verification test. 89

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Fig. 11. Result of the verification test.

terms of the ice resistance. The results show that resistances increase with increasing ship speed and ice concentration, as expected. Despite the simplicity of the simulation, a good agreement was found between the computed resistance and the experimental measurements at the 60% ice concentration. However, in case of the 80% concentration, some deviations were found for 0.119 m/sec and 0.595 m/sec cases. The differences are believed to be due to the simplified interaction modeling. Further research is needed to derive more realistic results. In addition, a comparative study is recommended in terms of the stress amplitude for the local structure. 2.2. Calculation of stress time series By applying the method described in Chapter 2.1, a time series of contact pressure was calculated for each element of a rigid body ship model. In addition, to determine the influence of the contact pressures on the target fatigue point, the impulse response function (IRF) on the target point was applied. A time series of stress can be obtained by applying the convolution integral to the calculated contact pressure and the IRF. When a system is linear and time invariant, the response of the system can be expressed in terms of an IRF through a convolution equation. The IRF represents the response of the system to an instantaneous unit impulse applied at the origin in time. The response of continuous linear systems can be expressed in the time domain using the convolution integral as follows: t

Δσij (t ) =

∫ Fj (τ ) hij (t − τ ) dτ

(4)

0

where Δσij (t ) is the stress of target point i by the contact pressure of element j at time t . Fj (τ ) is the contact pressure measured at element j . hij (t ) is the stress of target point i that occurs when a unit impulse is applied to element j . Because the stress calculated by Eqn. (4) is a value for one element of the outer-shell, the stress time series σi (t ) at the target position i can be calculated using Eqn. (5). n

σi (t ) =

∑ Δσij (t ) (5)

j=1

where n is the number of the contacted elements on the outer-shell at time t . A verification test for the calculation methodology using a simple model was carried out. A simplified ship model was made and collided with eight rectangular ice models. In one case, the stress time series was extracted by colliding ice with the entire ship structural model composed of deformable elements. In the other case, the calculation was performed using the convolution integral used in this study, where the calculation processes for the contact pressure and the IRF were separated. In Fig. 12, the method using the convolution integral was divided into step (a) to calculate the contact pressure and step (b) to calculate the IRF. In Fig. 12 (a), the time series of the contact pressure generated by the collision between the ship and ice floes is shown by each element of the outershell. Fig. 12 (b) presents the response of a unit impulse applied to the outer-shell in the form of pressure. Here, the structure model was modeled as a deformable body, and the IRF was obtained through the transient dynamic FE analysis. The applied critical damping ratio was 0.5%. Fig. 13 presents the results of the verification test. The results of the analysis using the entire ship structure model ('Simulation' in Fig. 13) and the results applying the convolution integral (‘Convolution' in Fig. 13) were almost identical. 2.3. Short-term distribution of the stress amplitude Unlike the wave induced fatigue analysis using the stress range, ice induced fatigue analysis usually employs the stress amplitude 90

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Fig. 12. Verification of the calculation methodology using the convolution integral.

due to the characteristics of the ice load, which tends to resemble an impulse. Therefore, the stress due to the ice load starts from zero and increases until it reaches the peak, and then decreases to zero again in a triangular pattern [6,26]. If only one ice floe at a time influences the target position, the stress time series will form a triangle with one peak, however, in practice several ice floes can affect the target position simultaneously, so it can show a time-series pattern with several peaks at one time [27]. Therefore, in this study, only the largest peak on one triangle pattern was extracted and used in probabilistic analysis. Fig. 14 shows the peak separation to distinguish ice loads. The peaks were identified within the predefined minimum peak distance, which was estimated from the calculated data. Generally, a fatigue damage evaluation is based on the S-N data in association with the Palmgren-Miner rule. When the time series of ice-induced stress amplitudes are available, the total fatigue damage can be calculated directly by summing the fatigue damage of all stress cycles according to the linear Palmgren-Miner cumulative rule. On the other hand, fatigue damage can also be calculated in probabilistic ways when the distributions of the stress amplitudes are known. This method is used widely to predict the fatigue life of ships and offshore structures at the design stage [2,28]. The probabilistic models used to predict the probability distribution of peaks in ice loads are the Weibull distribution [6,29], exponential distribution, and lognormal distribution [30]. In Kujala et al. [31]; the exponential, gamma, and Weibull distributions were applied to the field measurement data, and the Weibull distribution with a shape parameter of 0.75 was found to fit best with the measurements. Su et al. [32] reported that the ice loads derived from numerical analysis agree well with the Weibull distribution with the shape parameter of 0.75. The Weibull model can be divided into a 2-parameter Weibull model and 3-parameter Weibull model depending on the presence of the location parameter. The probability density functions of each model are given by Eqns. (6) and (7). 91

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Fig. 13. Results of the verification test for the calculation methodology of the stress history.

Fig. 14. Peak separation to distinguish the ice loads. h−1

fs (S ) =

h⎛s⎞ ⎜ ⎟ q ⎝q⎠

fs (S ) =

h ⎛s − γ⎞ ⎜ ⎟ q⎝ q ⎠

h

⎧ s ⎫ exp −⎛⎜ ⎞⎟ ⎨ ⎝q⎠ ⎬ ⎩ ⎭ h−1

(6) h

⎧ s − γ⎞ ⎫ exp −⎛⎜ ⎟ ⎨ ⎝ q ⎠ ⎬ ⎩ ⎭

(7)

where fs (S ) is the probability density function of the stress amplitude S . h is the Weibull shape parameter, and q and γ are the Weibull scale parameter and Weibull location parameter, respectively. The location parameter is also referred to as the threshold parameter because no failure occurs before the variable S exceeds the location parameter γ [33]. The parameters of the models were estimated by the maximum likelihood estimators (MLE). 2.4. Calculation of fatigue damage Fatigue is a cumulative damage process due to cyclic loading. Operating in ice-covered waters creates loadings which are cyclic nature on ship hulls. When navigating independently in level ice, the ship is in contact with ice and breaks it into small pieces. Then 92

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the ship comes into contact with more solid ice during the process and breaks that into small pieces. When a ship is navigating in a channel made by an icebreaker or behind an icebreaker, the broken ice pieces repeatedly impact the ship hull. Consequently, the ice load causes stress cycles that can give rise to fatigue damage of ship structural members [34]. As a ship may encounter a variety of stationary conditions, the total fatigue damage (D ) can be estimated by accumulating a number of fatigue damage contributions (Dj ). t

D=

∑ Dj (8)

j=1

where t is the number of stationary conditions. According to the Palmgen-Miner rule, the fatigue damage during a particular stationary condition (Dj ) can be estimated as Eq. (9). w

Dj =

n

∑ Ni

(9)

i

i=1

where w is the number of stress blocks. ni is the number of stress cycles in stress block i . Ni is the number of cycles to failure at a constant stress range Si , which is given by the S-N curve as Eqn. (10).

Ni = K ·Si−m

(10)

where K and m are parameters defining the S-N curve. Fatigue damage for each condition can be obtained by inserting Eqn. (10) into (9), and expressed in a probabilistic manner as Eqns. (11), and (12). The mean stress effect was not considered.

D=

D=

1 K

w

∑ ni Sim =

NT K

i=1

NT K

w

∑ Sim fs (Si)ΔSi

(11)

i=1

∞

∫ Smfs (S ) dS

(12)

0

where NT is the total number of stress cycles and can be expressed as the total number of ice impacts. Therefore, it can be expressed as Eqn. (13) as the product of the impact frequency vd and the travel distance (d ). (13)

NT = vd d

A particular stationary condition refers to constant ship speed and also constant ice condition. The ice conditions can be expressed by several factors such as ice thickness, concentration, ridge thickness and ridge occurrence rate [6]. In this study, the ice condition was described by the equivalent ice thickness which considers all of these factors for simplicity. According to Zhang et al. [2]; the impact frequency can be expressed as a function of the ice thickness as Eqn. (14).

vd = Cregion

1 0.75 10.4heq − 2.0heq + 1.18

(14)

where heq is the equivalent ice thickness [35], and Cregion is a hull region factor as defined in Lloyd's Resister [5]. Eqn. (12) can be expressed using 2-parameter and 3-parameter Weibull models as Eqns. (15) and (16), respectively.

D=

D=

∞

h−1

h

NT K

∫ Sm hq ⎛ Sq ⎞

NT K

∫ Sm hq ⎛ S −q γ ⎞

⎜

0

⎟

⎝ ⎠

⎧ S ⎫ exp −⎜⎛ ⎟⎞ dS ⎨ ⎝q⎠ ⎬ ⎭ ⎩

∞

0

⎜

⎟

⎝

⎠

h−1

(15) h

⎧ S − γ⎞ ⎫ dS exp −⎜⎛ ⎟ ⎨ ⎝ q ⎠ ⎬ ⎩ ⎭

(16)

Assuming m is an integer, the Binomial theorem and gamma function can then be applied to the given analytical values of Eqns. (15) and (16). Finally, the equations for fatigue damage can be derived based on the two-parameter and three-parameter Weibull models as Eqns. (17) and (18).

D=

D=

NT K NT K

m

∑ k=0 m

∑ k=0

m! k qk Γ ⎛1 + ⎞ (m − k ) !k! h⎠ ⎝

(17)

m! k γ m − kqk Γ ⎛1 + ⎞ (m − k ) !k! h⎠ ⎝

(18)

where Γ() is the gamma function. 3. Short-term distribution of stress due to ice loads The short-term distribution of stress developed by an ice collision was calculated using the proposed numerical method. As a 93

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Table 2 Analysis conditions for fatigue analysis. Ice thickness (m)

Concentration (%)

Ship velocity (knots)

0.3 0.6 0.9

40 60 80

4 6 8

starting step for calculating the fatigue life of a ship navigating the Baltic Sea, which is the target of this study, a series of calculations were performed under several conditions that the target ship can experience during navigation through the Baltic Sea. The probability distribution of the stress amplitude was derived under different conditions and the tendency of the derived Weibull parameters was examined. The Weibull parameters derived here were applied to the fatigue calculations considering the actual environmental conditions in Chapter 4. 3.1. Analysis condition In this study, the ice thickness, ship velocity, and ice concentration were considered as variables because the target ice condition is the managed broken ice. Because the ice used in this study is assumed to be first-year ice, the thicknesses used for the fatigue analysis were set to 0.3, 0.6 and 0.9 m. In case of the ice concentration, the case studies for a 40%, 60% and 80% ice concentration were added because it can be an important variable due to the characteristics of managed broken ice. In the case of the ship speed, 4 knots, 6 knots and 8 knots were applied for the case study. Table 2 lists the analysis cases for fatigue analysis. 3.2. Finite element modeling 3.2.1. Target vessel The target vessel used in this study is the Korean icebreaking research vessel (IBRV), Araon. For comparison with the LR method, an ice-strengthened ship should be applied. However, icebreaker model was applied in this study due to the limitation on the available data and model. In this sense, the comparison with LR method needs to be understood as the validation of the appropriateness of the proposed method, rather than accuracy check. Fig. 15 presents the finite element model of Araon. Assuming most ice loads affect the bow region of the ship, only the fore model was used. The model range was set to be large enough to minimize the boundary effects on the results and the fully fixed boundary condition was applied to the plane of the cut. Table 3 lists the main particulars of the ship. 3.2.2. Periodic media analysis model The periodic media analysis method was applied to the finite element model to reduce the analysis time. To apply this technique to a broken ice field, it is necessary to make appropriate modifications to the method. First, all blocks in this technology must be connected, but the broken ice field in this study is comprised of a number of discrete elements. To solve this problem, a virtual belt to connect all the blocks together was made, as shown in Fig. 16. The virtual belt and ice floes were included as a single block. Therefore, the virtual belt continues to move like a conveyor belt, and the ice floes virtually linked to the belt repeat the shuffling process according to the movement of the belt. Fig. 17 shows the whole finite element analysis model using the periodic media analysis method. The width and length of the channel applied in this study was 80 m and 600 m, respectively, and the boundary was not considered on both sides of the channel. The entire model consisted of three blocks. Because each block repeatedly disappears and reappears, the length of each block was set to be long enough for the continuity of analysis. In addition, instead of moving the ship, the ice floes were moved at the speed of the

Fig. 15. Finite element model for the fore part of Araon. 94

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Table 3 Main particulars of Araon. Items

Value

Length of ship (m) Breadth of ship (m) Draft of ship (m) Stem angle of ship (deg.)

95.0 19.0 6.8 30

Fig. 16. Block for the periodic media analysis of broken ice fields (80% ice concentration).

Fig. 17. Entire finite element model considering the periodic media analysis.

ship, which was realized by the downstream current of the same speed. Fig. 18 presents a snap-shot of the shuffling moment. After shuffling, the ice floes that have already passed the ship after colliding, disappear and are arranged and attached to the end of the last block for the next turn. A question may be asked as to whether the response is also repeated at a constant cycle because the same blocks continue to repeat. On the other hand, because the ice floes in front of the ship are disturbed by the ship and other ice floes, the load that the ship experiences is not expected to repeat, as shown in Fig. 18. To confirm this numerically, an autocorrelation of the reaction forces measured in the time domain was analyzed while shuffling was repeated using the periodic media analysis technique, as shown in Fig. 19. As a result, no periodicity was observed in the autocorrelation results, confirming the randomness of the ice loads during the analysis time.

Fig. 18. Snap-shots of the shuffling moments (before & after shuffling). 95

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Fig. 19. Reaction forces measured at the ship and autocorrelation results.

3.2.3. Screening analysis and fine mesh model To select the target position to calculate the fatigue damage, the distribution of the contact pressures according to the position on the outer-shell of the ship model was checked. Generally, several fatigue points are selected based on various conditions, and also the number of cycles and the stress of the target position should be considered together in the screening process. However, the purpose of this study is to introduce the developed methodology, so only one point was selected where the greatest contact pressure is derived for one arbitrarily selected condition for simplicity. Also, the number of cycles were not considered in this screening analysis. The checked condition was 0.9 m in thickness, 60% concentration and a 4 knots speed, and the analysis lasted for 300 s. Fig. 20 shows the set of locations of the impacts in plan view. The small dots are the nodal points of the ship elements, and the red circles are the points that experienced the upper 1% contact pressure. The maximum values acting on each node during the analysis time were sorted, and only the top 1% are represented by circles. It was observed that majority of red circles are located in the vicinity of the forebody centerline. However, these points were simply excluded from the fatigue analysis candidates because it is thought that this area is structurally strong, and the risk of fatigue damage is very small. Finally, the point indicated by the arrow was selected as the target position. Fig. 21 shows a drawing of the frame selected by the screening analysis, and the selected welded point closest to the draft is presented. Fig. 22 presents the fine mesh of the fatigue point. To derive the hot spot stress, a mesh density in the order of t × t , where t is the plate thickness, was applied [36]. 3.3. Short-term distribution of the hot spot stress amplitude A series of numerical analyses were performed using the method proposed in this study, and time series of the stress amplitudes at the target position were calculated. Considering the different thicknesses, concentration, and speed summarized in Table 2, all cases were combined, and a total of 27 cases were analyzed. To find the time required for appropriate analysis to obtain the stochastic characteristics, as shown in Fig. 23, the convergence of the Weibull fitting was examined by varying the analysis time. The 2parameter Weibull model was applied based on the following conditions: 0.9 m thickness, 40% concentration, and 4 knots ship speed,

Fig. 20. Screening analysis for the fatigue analysis. 96

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Fig. 21. Fatigue point selected by screening analysis.

Fig. 22. Finite element model for the fatigue point modeled by fine meshes.

where the number of impacts is the smallest. As a result, the Weibull curve converged almost immediately after 15 min so the analysis time was determined to be 20 min. In addition, the results were fitted using the 2-parameter and 3-parameter Weibull model based on a 0.9 m thickness, 80% concentration, and 4 knots ship speed, as shown in Fig. 24. As a result, the 3-parameter model approximated the tail part better than the 2-parameter model. On the other hand, the denser part of the data fitted the 2-parameter model better. This trend makes the 2-parameter Weibull model underestimate the extreme values. On the other hand, the 2-parameter model is better suited for the fatigue problem because the stresses at the intermediate level with a large number of cycles are more important. Fig. 25 presents example plots of the stress histories obtained using the proposed numerical method. The figure shows considerable spikiness, which is caused by the collisions of ice floes near the area of investigation. Fig. 26 gives some examples of 2parameter Weibull plots. The fits are sometimes poor, especially for the tail part, but their influence is expected to be quite limited because of the few cycles of those large stress amplitudes. Fig. 27 summarizes the Weibull scale parameters derived for all cases considered in this study. The x-axis is fixed to the ice thickness, which is considered to be the most important variable in ice-induced fatigue. As a result, the Weibull scale parameter increased with increasing ice thickness in all cases. Fig. 27 (a), (c), and (e) show that the Weibull scale parameter also increases with increasing ship speed. On the other hand, Fig. 27 (b), (d), and (f) shows that the relationship between ice concentration and the Weibull scale parameter is not linear in some cases, particularly in small thickness cases. In Fig. 28, the tendency of the Weibull scale parameters according to the various conditions was examined using the mean and standard deviation under each condition. Here, circles and bars represent mean and standard deviation, respectively. As a result, the Weibull scale parameter increased with increasing ice thickness, concentration, and ship speed. In addition, the standard deviation 97

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Fig. 23. Weibull plots according to various analysis time.

Fig. 24. Weibull plots according to the number of parameters.

also increases with increasing condition. Fig. 29 presents the results of the Weibull shape parameters; it was difficult to find any trend for the Weibull shape parameters under the given conditions except for the ice thickness. Instead, the Weibull shape parameters tended to decrease with increasing ice thickness. The values of the Weibull shape parameter were distributed between 0.71 and 0.90 and the average was 0.80. For reference, the shape parameters on the same ice thickness conditions derived from the measurement by Suyuthi et al. [6] were 0.85–0.97. Similarly, the parameter and ice thickness tended to be inversely proportional. 4. Fatigue damage assessment under actual environmental conditions 4.1. Analysis condition Fatigue analysis of a ship has been carried out based on the actual environmental conditions using the calculated short-term distributions. Table 4 lists the environmental conditions used in the analysis. The trading route was assumed to be the Kemi route of the Baltic Sea, and the ship visits the port an average of 3.5 times during each winter month in its 25-year service life according to the LR FDA-ICE. The ice concentration was assumed to be 100% because the route is mostly frozen during winter. The ship speed was assumed to be 8 knots, which is the median of the ship speeds operating at an ice concentration of more than 85% in the Baltic Sea [37]. Table 5 lists the monthly sailed distance [5]. The acceptance criteria for the final fatigue damage ratio was set to 0.5, taking the 98

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Fig. 25. Stress histories under different ice conditions.

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Fig. 26. Weibull plots of the calculated data under different ice conditions.

100

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Fig. 27. Weibull scale parameters simulated under different ice conditions.

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Fig. 28. Tendency of the Weibull scale parameters simulated under different ice conditions.

fatigue damage due to wave loads into account according to the LR FDA-ICE.

4.2. Comparison of the fatigue damage ratios using the proposed method and the LR method The fatigue damage ratios for the Kemi route using the proposed method and LR method were compared. First, when using the proposed method, the Weibull scale parameters and Weibull shape parameters, which are summarized in Figs. 26 and 28, can be 102

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Fig. 29. Weibull shape parameters simulated under different ice conditions.

Table 4 Environmental conditions for fatigue analysis. Items

Value

Trading route Ice concentration (%) Ship speed (knots) Acceptance criteria

Kemi route 100 8 0.5

Table 5 Ice conditions in the Kemi route. Month

Winter type

heq (m)

25 years sailed distance (km)

Nov.

Mild Average Severe Mild Average Severe Mild Average Severe Mild Average Severe Mild Average Severe Mild Average Severe Mild Average Severe

– – 0.15 0.10 0.20 0.35 0.18 0.47 0.64 0.38 0.76 0.86 0.53 0.86 1.10 0.50 0.82 1.14 0.14 0.28 0.50

– – 1,042 313 2,917 7,917 521 26,981 9,792 1,042 59,796 20,835 3,959 59,796 20,835 3,542 35,003 14,585 313 8,751 5,000

Dec.

Jan.

Feb.

Mar.

Apr.

May

applied. The Weibull parameters for specific ice thicknesses can be obtained using the linear regression equation derived for each ice thickness. Extrapolated value was obtained by extending the dot line in Fig.28(b) to 100%. For the ice impact frequency, Eqn. (14) developed by Zhang et al. [2] was applied. Finally, the fatigue damage under each condition can be calculated using Eqn. (17). Next, fatigue analysis using the LR method was performed. Table 6 lists geometric information of the vessels required for the calculation. In the LR method, the fatigue stress at a critical location due to the ice load is determined using structural beam theory [5]. For the S-N curve, the proposed method used the D-curve, which is recommended one for hot spot stress according to Det Norske Veritas [36]. On the contrary, F2-curve was used for LR method because the LR method employs the nominal stress concept. Table 7 103

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Table 6 Hull form angles and geometrical parameters. Items

Value

Hull form angles

Angle α (degs) Angle βn (degs)

21.5 39.8

Geometrical parameters

Frame spacing (mm) Frame span (mm) Section modulus (mm3)

800 4,054 576,111

Table 7 Calculation of the fatigue damage in the Kemi route. Month

heq (m)

Scale parameter

Shape parameter

DPR

DLR

Nov.

– – 0.15 0.10 0.20 0.35 0.18 0.47 0.64 0.38 0.76 0.86 0.53 0.86 1.10 0.50 0.82 1.14 0.14 0.28 0.50

– – 2.65 2.11 3.19 4.80 2.97 6.10 7.93 5.13 9.23 10.31 6.75 10.31 12.90 6.42 9.88 13.33 2.54 4.05 6.42

– – 0.94 0.96 0.93 0.88 0.93 0.85 0.79 0.87 0.76 0.73 0.83 0.73 0.65 0.84 0.74 0.64 0.95 0.90 0.84

– – 7.42E-06 1.26E-06 3.38E-05 2.94E-04 5.01E-06 2.13E-03 1.98E-03 4.72E-05 2.25E-02 1.30E-02 4.41E-04 3.73E-02 4.54E-02 3.33E-04 1.78E-02 3.96E-02 2.00E-06 1.97E-04 4.69E-04

– – 3.44E-05 5.40E-06 1.57E-04 1.22E-03 2.33E-05 7.96E-03 6.24E-03 1.91E-04 6.17E-02 3.15E-02 1.56E-03 9.04E-02 9.09E-02 1.21E-03 4.54E-02 7.91E-02 9.23E-06 8.68E-04 1.71E-03

Dec.

Jan.

Feb.

Mar.

Apr.

May

lists the Weibull parameters and fatigue damage ratios calculated under each condition, and Table 8 compares the final fatigue damage calculated using each method. Here, DPR and DLR mean the damage ratios by the proposed method and LR method, respectively. As a result, the fatigue damage calculated by the LR method is more than twice that of the proposed method. This is because the LR method is simplified one, hence the results are considered to be more inclined to be on safe side with some safety margin.

5. Conclusions This paper proposed a novel fatigue assessment method using numerical analysis for ice impacts on a ship hull in broken ice fields. A numerical model to simulate the interactions between ice floes and structure was developed. For calculation efficiency, ice-fluid and ice-structure interactions were implemented in simplified manners, and the important parameters, such as the drag force coefficient or contact interaction parameter, were determined through detailed analyses. For efficient long-term time domain analysis, the periodic median analysis method, which is a function of ABAQUS/EXPLICIT, was applied with appropriate modifications to be used in a broken ice field. As a result, it enabled time domain analysis for 20 min with a limited number of elements; it was confirmed that the ship continues to receive new ice loads over the entire analysis time. Contact pressure time series on the outershell of the ship was converted to a hot spot stress time series by the convolution integral using the IRF on the target fatigue point. The applicability of this technique was verified using a simple test, and it is possible to estimate accurate hot spot stress. A series of numerical simulations were carried out for the various analysis cases considering the ice thickness, concentration and ship speed, and the Weibull parameters were derived for each condition. For efficiency, the 2-parameter Weibull model was applied so the scale Table 8 Comparison of the total fatigue damage ratios. Methods

Fatigue damage ratio

Proposed method LR method

0.18 0.42

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parameters and shape parameters could be calculated, and the relationships between the parameters and each condition were derived. As a result, the Weibull scale parameter is linear to the ice thickness and ship speed in all cases. In some cases, however, the scale parameter is not linear with ice concentration. For the Weibull shape parameters, it was difficult to find any tendency under the given conditions except for the ice thickness. Instead, the shape parameters tended to decrease with increasing ice thickness. A fatigue damage assessment of the ship for the Kemi route in the Baltic was carried out using the calculated short-term distributions, and the results were compared with those obtained using the LR method. The fatigue damage calculated by the LR method was more than twice that of the proposed method. This is because the LR method leads to more conservative results for safety reasons. The advantage of the proposed methodology is that it can evaluate the fatigue damage of an ice-strengthened ship for various conditions with reduced time and low cost. However, the accuracy of the method can be restricted by the simplifications and assumptions. For more realistic results, further studies about the parameters used in the simplification process are required, and a comparative study with full scale data is recommend. Acknowledgement This study was supported by the Industrial Convergence Strategic technology development program (10063417, Development of basic design technology for ARC7 class Arctic offshore structures) funded by the Ministry of Trade, Industry and Energy (MI, Korea). References [1] Melia N, Haines K, Hawkins E. Future of the sea: implications from opening arctic sea routes. Foresight, Government Office for Science. 2017. [2] Zhang S, Bridges R, Tong J. 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