Estimation of the Ice-Induced Fatigue Damage to a Semi-Submersible Platform under Level Ice Conditions

: This study presents a fatigue analysis procedure for inclined structures operated in level ice ﬁelds. Three methods for calculating the local ice load causing fatigue damage, namely the direct method, simpliﬁed method, and semi-analytical method, were introduced and compared. The direct method uses ﬁnite element analysis to simulate the continuous breaking of ice, while the simpliﬁed method and the semi-analytical method estimate the probability distribution of local ice loads based on theoretical equations and empirical data. The fatigue damage ratio at the target location was calculated by applying the ice load calculated by each method to a deformable ﬁnite element model of the structure. The results obtained from each method indicate that they provide a reasonable estimation of the local ice load causing fatigue damage in level ice ﬁelds. The direct method offers high accuracy but requires signiﬁcant computational time, while the simpliﬁed method and semi-analytical method offer a faster analysis time and are more suitable for long-term time domain analysis. The semi-analytical method requires empirical data to supplement theoretical formulas due to the complex natural phenomena involving various environmental conditions that must be modeled. The ﬁndings of this study provide valuable insights into the prediction of fatigue damage in ice-going ships due to long-term ice impacts. The methods proposed in this study can aid in the design of Arctic ships exposed to various conditions and provide a more cost-effective and time-efﬁcient approach to evaluating fatigue damage compared to ﬁeld measurement. Future research in this ﬁeld could investigate the application of these methods to other types of structures and further reﬁne the methodology to improve accuracy and reduce computational costs.


Introduction
The prediction of fatigue damage in ice-going ships due to long-term ice impacts starts by modeling the local ice load on the outer shell of the ship's hull; however, the behavior of ice during impact is quite complex, depending on the type, characteristics of ice, and hull shape. As a result, previous studies have often relied on field measurements to obtain ice loads (Zhang et al. [1], Suyuthi et al. [2]). However, field measurement is time-consuming and costly, making it challenging to apply the data obtained under limited conditions to the design of Arctic ships exposed to various conditions. In recent years, numerical simulations have been used to overcome these limitations of field measurement. Han et al. [3] published a procedure for evaluating the fatigue damage of structures in pack ice fields, calculating the local ice load using the discrete element method. Kim and Kim [4] proposed a method for estimating the fatigue damage ratio of ships experiencing ice floe impacts. To make these methods easily applicable to various broken ice conditions, fatigue stress and the number of cycles are calculated using hull form and ice load information. However, research to calculate ice-induced fatigue damage of structures in level ice fields 2 of 17 based on numerical simulation is scarce. Suyuthi et al. [2] and Chai et al. [5] defined the fatigue load in ice-covered water through a probabilistic method and evaluated fatigue damage based on the probability model of fatigue stress and impact frequency; however, these methods calculated fatigue damage from results measured during ship operation. To be applied in the initial design of offshore structures operating in ice-covered seas, it is necessary to utilize efficient computational techniques such as numerical analysis or analytical methods to evaluate the fatigue life under various environmental conditions. By incorporating ice failure models developed based on fracture mechanics, it becomes possible to efficiently simulate the interaction between ships and ice, as well as the resulting ice fractures [4,6,7].
In this paper, we introduce fatigue analysis procedures for inclined structures operated in level ice fields. We consider the 'Direct Method', 'Simplified Method', and 'Semianalytical Method' to calculate the local ice load causing fatigue damage.
The 'Direct Method' uses finite element (FE) analysis to simulate the continuous breaking of ice for different conditions of ice thickness and velocity. The interaction between ice and structures is simulated using a damage-based erosion model (Jeon and Kim [8]). This method has high accuracy, but also requires a large amount of computational time due to the fine element size and the short time increments characteristic of FE analysis. The 'Simplified Method' is a method for estimating the probability distribution of local ice loads based on theoretical equations for various structure shapes and types of ice. Assuming an ice plate as an elastic beam, the peak of the ice-breaking load in the form of impact can be calculated from beam bending theory (ISO 19906 [9], Croasdale [10]). The ice load calculated from the beam theory is used as the average of a 2-parameter Weibull distribution, derived for each panel on the ship's outer hull. Finally, the 'Semianalytical Method' uses an impact simulation technique that combines theoretical and empirical formulas to calculate the ice load. This method has the advantage of fast analysis time, making it useful for long-term time domain analysis. The load calculation used by Lubbad and Løset [11] to simulate the interaction between the ship and ice is based on this method. However, since complex natural phenomena involving various environmental conditions must be modeled, relying solely on theoretical formulas is limited and must be supplemented with empirical data. In this study, the fatigue damage ratio at the target location was calculated by applying the ice load calculated by each method to a deformable FE model of the structure. The results were then compared. Figure 1 illustrates the procedure for evaluating fatigue in this study, which can be divided into two methods of calculating the ice load acting on the structure: 'Method 1' and 'Method 2'. 'Method 1' involves determining the time series of local ice loads, while 'Method 2' uses the probability distribution of the local ice load to calculate the fatigue damage. For 'Method 2', the 2-parameter Weibull distribution was used, as described by Kim and Kim [4]. The direct method based on finite element (FE) analysis and the semi-analytical method correspond to 'Method 1', while the simplified method was used to calculate the probability distribution of the ice load in 'Method 2'.

Procedure Description
Each type of ice load is transformed into stress at the target fatigue point using a coefficient factor C i . In 'Method 1', which uses the time series of stress, the number of stress amplitudes is calculated through peak counting, and the fatigue damage is determined based on the Palmgren-Miner rule. In 'Method 2', the Weibull parameters of the load calculated from each panel of the hull shell are converted into stress amplitude parameters using the influence coefficient method. Assuming the Palmgren-Miner rule is applied, the fatigue damage at the target fatigue point is calculated for each individual panel, and the final fatigue damage is obtained by summing the linear fatigue damages. The detailed calculation method is explained in Section 3. Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 18

Direct Method
The direct method calculates the time series of ice loads through dynamic collision analysis using FE models of both ice and structures. The magnitude of ice load is significantly impacted by the failure mode of the ice; therefore, it is essential to accurately simulate the ice breaking and crack propagation that occur during the collision. In order to achieve this, Jeon and Kim [8] proposed a damage-based erosion model that models the interaction between ice and a structure and estimates the ice load by considering the erosion of ice elements. The direct method calculates ice load by using the ABAQUS commercial FE analysis program with a damage-based erosion model. As shown in Figure 2, the direct method can simulate the fracture patterns of level ice. As the structure encounters the edge of level ice, the initial occurrence is the formation of radial cracks. As the structure progresses, circumferential cracks begin to emerge, leading to the generation of broken ice flakes. In the case of an inclined structure that experiences ice collision, the resulting ice load can be divided into two parts: the breaking force and the rubble force. The breaking force is caused by bending and is represented as an impact load with a relatively constant period, which is dependent on the physical properties of the ice and the angle of inclination.

Direct Method
The direct method calculates the time series of ice loads through dynamic collision analysis using FE models of both ice and structures. The magnitude of ice load is significantly impacted by the failure mode of the ice; therefore, it is essential to accurately simulate the ice breaking and crack propagation that occur during the collision. In order to achieve this, Jeon and Kim [8] proposed a damage-based erosion model that models the interaction between ice and a structure and estimates the ice load by considering the erosion of ice elements. The direct method calculates ice load by using the ABAQUS commercial FE analysis program with a damage-based erosion model. As shown in Figure 2, the direct method can simulate the fracture patterns of level ice. As the structure encounters the edge of level ice, the initial occurrence is the formation of radial cracks. As the structure progresses, circumferential cracks begin to emerge, leading to the generation of broken ice flakes.

Direct Method
The direct method calculates the time series of ice loads through dynamic collision analysis using FE models of both ice and structures. The magnitude of ice load is significantly impacted by the failure mode of the ice; therefore, it is essential to accurately simulate the ice breaking and crack propagation that occur during the collision. In order to achieve this, Jeon and Kim [8] proposed a damage-based erosion model that models the interaction between ice and a structure and estimates the ice load by considering the erosion of ice elements. The direct method calculates ice load by using the ABAQUS commercial FE analysis program with a damage-based erosion model. As shown in Figure 2, the direct method can simulate the fracture patterns of level ice. As the structure encounters the edge of level ice, the initial occurrence is the formation of radial cracks. As the structure progresses, circumferential cracks begin to emerge, leading to the generation of broken ice flakes. In the case of an inclined structure that experiences ice collision, the resulting ice load can be divided into two parts: the breaking force and the rubble force. The breaking force is caused by bending and is represented as an impact load with a relatively constant period, which is dependent on the physical properties of the ice and the angle of inclination. In the case of an inclined structure that experiences ice collision, the resulting ice load can be divided into two parts: the breaking force and the rubble force. The breaking force is caused by bending and is represented as an impact load with a relatively constant period, which is dependent on the physical properties of the ice and the angle of inclination. However, the rubble force, which arises from the accumulation of broken ice fragments, takes a considerable amount of time to build up and results in a continuous load on the structure. Although the average load may increase due to the massive weight of the rubble, it is expected to reduce the amplitude of the impact generated by the breaking force. To achieve accurate fatigue calculations, the implementation of rubble accumulation is crucial. However, in this study, the 'Direct method' focused on ice breaking, making it challenging to accurately represent rubble accumulation. As it is the amplitude of the load that is used for fatigue evaluation, a conservative fatigue evaluation was performed by ignoring the effect of rubble accumulation in this study, as shown in Figure 3. Further study is needed to incorporate rubble accumulation into the calculation of ice impact force. However, the rubble force, which arises from the accumulation of broken ice fragments, takes a considerable amount of time to build up and results in a continuous load on the structure. Although the average load may increase due to the massive weight of the rubble, it is expected to reduce the amplitude of the impact generated by the breaking force. To achieve accurate fatigue calculations, the implementation of rubble accumulation is crucial. However, in this study, the 'Direct method' focused on ice breaking, making it challenging to accurately represent rubble accumulation. As it is the amplitude of the load that is used for fatigue evaluation, a conservative fatigue evaluation was performed by ignoring the effect of rubble accumulation in this study, as shown in Figure 3. Further study is needed to incorporate rubble accumulation into the calculation of ice impact force.

Semi-Analytical Method
The semi-analytical method is a simulation technique that utilizes existing closedform equations to calculate the load time series. Unlike numerical analysis using finite element models, it is ideal for long-time domain analysis as it does not require repeated calculations or a dense time interval for calculation convergence. However, the accuracy of the results is dependent on the validity of the applied theoretical formula, requiring strict verification of the theory. The flowchart of the semi-analytical simulation process is shown in Figure 4. The simulation begins with the ice moving a small distance until it makes contact with other objects. Upon contact, the contact area is calculated based on the shapes of the ice and the structure, followed by the calculation of the local crushing load. If the crushing load exceeds the predetermined ice failure criterion, the ice breaks, and the simulation moves on to the next step with the broken ice.

Semi-Analytical Method
The semi-analytical method is a simulation technique that utilizes existing closed-form equations to calculate the load time series. Unlike numerical analysis using finite element models, it is ideal for long-time domain analysis as it does not require repeated calculations or a dense time interval for calculation convergence. However, the accuracy of the results is dependent on the validity of the applied theoretical formula, requiring strict verification of the theory. The flowchart of the semi-analytical simulation process is shown in Figure 4. The simulation begins with the ice moving a small distance until it makes contact with other objects. Upon contact, the contact area is calculated based on the shapes of the ice and the structure, followed by the calculation of the local crushing load. If the crushing load exceeds the predetermined ice failure criterion, the ice breaks, and the simulation moves on to the next step with the broken ice.
From the moment of impact until the ice breaks, the local ice load is the result of the local crushing of the ice. In this study, the method presented by Daley [12], which defined the crushing load as the product of the contact area (A) and the average pressure (P av ), was used to calculate the local crushing force (F n ) in Equation (1).
where the average pressure is determined by the product of the ice pressure constant (P av ), which is determined according to the PC (polar class) of the ice, and the area (A ex ) in which the pressure-area relationship index (ex) is considered. Polar class has seven notations from PC 1 to PC 7 according to the regulations of IACS, and the ice pressure constant (P 0 ) corresponding to each is shown in Table 1. In this study, −0.1 was applied for the area-pressure relationship index based on the findings of Daley [12].  From the moment of impact until the ice breaks, the local ice load is the result of the local crushing of the ice. In this study, the method presented by Daley [12], which defined the crushing load as the product of the contact area ( ) and the average pressure ( ), was used to calculate the local crushing force ( ) in Equation (1).

=
(1) where the average pressure is determined by the product of the ice pressure constant ( ), which is determined according to the PC (polar class) of the ice, and the area ( ) in which the pressure-area relationship index ( ) is considered. Polar class has seven notations from PC 1 to PC 7 according to the regulations of IACS, and the ice pressure constant ( ) corresponding to each is shown in Table 1. In this study, −0.1 was applied for the area-pressure relationship index based on the findings of Daley [12].

=
(2) When the local crushing load reaches the critical ice breaking load, cracks form in the ice sheet, leading to its fracture. The critical load is determined based on the failure mode of the ice. In this study, only out-of-plane bending failure is considered as the failure mode of the ice, assuming that the target ice is the level ice. The modified Nevel (1965 [14], 1972 [15]) equation proposed by Lu et al. [16] was used to calculate the critical load, as shown in Equation (3).  When the local crushing load reaches the critical ice breaking load, cracks form in the ice sheet, leading to its fracture. The critical load is determined based on the failure mode of the ice. In this study, only out-of-plane bending failure is considered as the failure mode of the ice, assuming that the target ice is the level ice. The modified Nevel (1965 [14], 1972 [15]) equation proposed by Lu et al. [16] was used to calculate the critical load, as shown in Equation (3).
where θ is wedge angle, σ f is flexural strength, t is ice thickness, r is half width of the contact area and l c is characteristic length of the ice.
To simulate the out-of-plane bending failure of ice, the ice breaking length must be determined. Enkvist [17] proposed a constant breaking length based on model experiments and actual ice test results, while Wang [18] proposed a wedge radius as a function of characteristic length and relative velocity of ice. In this study, the breaking length obtained from the finite element analysis results, which was obtained using the direct method, was applied to the semi-analytical method to realize the out-of-plane bending failure of ice. Figure 5 shows the visualized result of simulation using the semi-analytical method. The red square in Figure 5 represents the fixed structure, and the blue lines indicate the edges of the broken ice. Figure 6 shows the ice breaking load time series obtained by the simulation in Figure 5. Similar to the direct method, the semi-analytical method does not take into account the effect of the broken ice, so the breaking force is only observed as having zero to peak impact. Figure 5 shows the visualized result of simulation using the semi-analytical method. The red square in Figure 5 represents the fixed structure, and the blue lines indicate the edges of the broken ice. Figure 6 shows the ice breaking load time series obtained by the simulation in Figure 5. Similar to the direct method, the semi-analytical method does not take into account the effect of the broken ice, so the breaking force is only observed as having zero to peak impact.   The red square in Figure 5 represents the fixed structure, and the blue lines indicate the edges of the broken ice. Figure 6 shows the ice breaking load time series obtained by the simulation in Figure 5. Similar to the direct method, the semi-analytical method does not take into account the effect of the broken ice, so the breaking force is only observed as having zero to peak impact.

Simplified Method
The direct and semi-analytical methods discussed in Sections 3.1 and 3.2 are capable of simulating the collision between the ship and the ice and calculating the resulting ice load over time in three dimensions. However, they have the disadvantage of being time-consuming, as a new analysis is required each time the conditions change, making it difficult to use these methods during the initial design stage where rapid results are needed.
The simplified method is an alternative approach to estimating ice load by using only information about the ship's hull form and ice conditions. This method is limited to the scenario where level ice collides with a wide, inclined structure. It has the advantage of being able to quickly reflect various conditions, as the load is calculated based on theoretical and empirical formulas.
In this study, only the ice breaking load is considered, and the normal component (N) of the load has a relationship with the vertical and horizontal components using the inclination angle (α). Vertical: Horizontal: where V B is the vertical component of the load, N is the load component in the direction normal to the slope, α is the slope angle, and µ is the friction coefficient between the ice and the structure. The bending load is calculated using beam theory in the elastic region, considering a 2-dimensional model. Although this theory can be extended to three dimensions for wide incline structures, it tends to underestimate the load for narrow structures. This issue can be addressed by selecting an appropriate w B and correcting for the 3D effect based on the size of the structure, as specified in ISO 19906 [9].
where σ f is the flexural strength of ice, ρ w is the density of water, h is the ice thickness, and E is the elastic modulus of ice. D is the breadth of the structure. By substituting Equation (6) into Equation (4), the load component in the normal direction is obtained as Equation (11).
Assuming that the peak ice load follows a certain probability distribution, the ice load calculated from Equation (11) can be considered as a mean value of the probability distribution, as noted in Suyuthi et al. [2]. In this study, the probability distribution of ice load is assumed to be a 2-parameter Weibull distribution as expressed in Equation (12).
where q and ζ are the Weibull scale parameter and shape parameter, respectively. Research aimed at estimating the shape parameters of ice loads has been conducted by Zhang et al. [1] and Suyuthi et al. [2]. In this study, the shape parameter of ice load is derived using the formula proposed by the Lloyd Classification as expressed in Equation (13) as described in Zhang et al. [1].
where t is thickness of the level ice. The scale parameter can be obtained from the definition of the expectation value in the Weibull distribution as Equation (14). Assuming that the ice load calculated in Equation (11) is the mean value of the ice load, the scale parameter can be derived using N load and shape parameters as given in Equation (15).

Fatigue Stress Calculation
As previously described in Section 3, the calculation of the local ice load time series is performed using both the direct analysis method and the semi-analytical method, while the Weibull parameters of local ice load amplitude are derived using the simplified method. This section explains how to obtain the probability distribution of stress amplitude using Method 1 and Method 2 as illustrated in Figure 1.

Method 1: Local Ice Load Time Series
In general, the ice belt zone is constructed using strong materials and reinforcement to withstand ice impact, resulting in a higher natural frequency compared to the frequency of the local ice load. This leads to the ice load and resulting stress being considered quasi-static. Under quasi-static conditions, the relationship between load and resulting stress can be assumed to be linear. As demonstrated in Figure 7, the stress σ i at the target point i can be calculated by multiplying the load on the outer shell by its respective influence coefficient C. The influence coefficient can be determined through a static analysis where a unit load is applied to the deformable structure. The stress time series of the target point for the load acting on each panel is expressed as follows.
where is determined as the linear summation of stress per panel as Equation (17). is the number of panels that influence the stress at the target point.
The stress time series at the target point, obtained through Equation (17), has a zeroto-peak pattern similar to the ice load, and was therefore fitted to a two-parameter Weibull distribution using peak counting. The stress time series of the target point i for the load acting on each panel is expressed as follows.

Method 2: Probability Distribution of Local Ice Load
where σ i is determined as the linear summation of stress per panel as Equation (17). N is the number of panels that influence the stress at the target point. The stress time series at the target point, obtained through Equation (17), has a zeroto-peak pattern similar to the ice load, and was therefore fitted to a two-parameter Weibull distribution using peak counting.

Method 2: Probability Distribution of Local Ice Load
When the simplified method is applied, the mean value and the scale and shape parameters for the local ice load peak are calculated for each panel of the hull outer-shell as Equations (11) and (15), respectively. The relationship between the average ice load peak (A load ) and the average stress peak (A stress ) is defined by the influence coefficient (C j ) in Equation (18).
Therefore, the Weibull distribution f (X) for the stress amplitude (X) in each panel j is determined using the corresponding scale (Q j ) and shape (ξ j ) parameters in Equations (19)-(21).

Fatigue Damage
This section describes the process of calculating the fatigue damage based on the probability distribution of the stress amplitude obtained from Method 1 and Method 2. Fatigue damage is calculated according to Palmgren-Miner's linear cumulative damage law as expressed in Equation (22).
where k is number of stress amplitudes, N j is the number of cycles until failure at a specific stress amplitude S j on the S-N diagram, and n j is the number of actual stress cycles of the amplitude S j .
To determine n j , the number of stress cycles, the stress range is discretized into appropriate interval ∆S to obtain probability P j using the probability density function f i S j . The calculation of P j is as follows: From a statistical point of view, where N 0,i is the number of total stress peaks that occur under condition i. From Equations (23) and (24), the number of actual stress cycles of the amplitude S j is arranged as follows: The S-N diagram is given in Equation (26), and the allowable number of stress cycles N j is expressed as Equation (27) log N = log K − m log ∆S (26) where N is the predicted number of cycles to failure for stress range ∆S, m is the negative inverse slope of S-N curve, and log K is the intercept of the log N axis on the S-N curve. Finally, we can calculate the fatigue damage for each condition using Equation (28).

Target Structure and Ice Conditions
This study focused on a semi-submersible drilling rig installed in an ice-covered area as shown in Figure 8. To prepare for long-term ice loads, a specially designed structure was applied in the ice belt zone to mitigate the impact of the ice loading. The detailed dimensions of the structure are presented in Table 2.  The material properties of the ice applied in this study are presented in Table 3.     The material properties of the ice applied in this study are presented in Table 3. The material properties of the ice applied in this study are presented in Table 3.  Table 4 shows the ice thickness and velocity conditions. For the direct analysis, we selected 12 conditions (three thicknesses and four velocities) denoted in bold in Table 4, in order to minimize computational cost. The remaining conditions were covered through linear interpolation of calculated results. In the direct analysis and the semi-analytical methods applied in this study, the probability model was made based on the result of time series analysis. Assuming that the continuous ice load action by level ice breaking is a stationary process, the parameters derived from the short-term time series analysis can be equally applied to the long-term situation. However, valid statistical properties require time series calculations for a sufficient amount of time. On the other hand, time series analysis for too long a period of a time requires excessive analysis time. Therefore, in this study, the required analysis time to show statistical convergence was determined by calculating the parameters according to each analysis time. Using the analysis time as the criterion for convergence may lead to a large variation in the number of collisions depending on the applied speed of the ice. As a result, in this study, the moving distance of ice was used as the evaluation criterion instead of the analysis time.
The subsequent paragraphs present an example of finding the effective length of level ice to obtain load data from statistical variables. For example, at velocity 9.18 cm/s, the parameters were calculated for the level ice moving as much as 25 m, 50 m, 100 m, 200 m, 400 m, and 1000 m. Figure 9a,b show the scale and shape parameters calculated by varying the moving distance of ice. The stability of statistical convergence was confirmed for ice movement conditions of 200 m or more, and the results of ice load calculation for 200 m movement were applied to the fatigue analysis. Similarly, the effective ice length was determined for the ice thickness-velocity combination indicated in bold in Table 4. parameters were calculated for the level ice moving as much as 25 m, 50 m, 100 m, 200 m, 400 m, and 1000 m. Figure 9a,b show the scale and shape parameters calculated by varying the moving distance of ice. The stability of statistical convergence was confirmed for ice movement conditions of 200 m or more, and the results of ice load calculation for 200 m movement were applied to the fatigue analysis. Similarly, the effective ice length was determined for the ice thickness-velocity combination indicated in bold in Table 4. The weld of the internal stiffener connected to the outer-shell of the hull, as shown in Figure 10, was selected as the target point for the fatigue analysis, as it is a typical fatigue prone point in the ice-strengthened area (Kim and Kim [11]). Through screening analysis using the finite element method, two panels that mainly affect the fatigue stress at the target location were selected, and the ice load and influence coefficient were calculated. The weld of the internal stiffener connected to the outer-shell of the hull, as shown in Figure 10, was selected as the target point for the fatigue analysis, as it is a typical fatigue prone point in the ice-strengthened area (Kim and Kim [11]). Through screening analysis using the finite element method, two panels that mainly affect the fatigue stress at the target location were selected, and the ice load and influence coefficient were calculated. Fixed boundary conditions were applied to the top and bottom parts of the ice-strengthened area to obtain the influence coefficient for each panel. The influence coefficient was determined as the principal stress of the target point derived by applying a unit load to each panel of the finite element model. Table 5 shows the coefficient of influence for each panel. Fixed boundary conditions were applied to the top and bottom parts of the ice-strengthened area to obtain the influence coefficient for each panel. The influence coefficient was determined as the principal stress of the target point derived by applying a unit load to each panel of the finite element model. Table 5 shows the coefficient of influence for each panel.   Figure 11 compares the mean values of the local ice load peaks on Panel 1 and 2 calculated by each method. In the simplified method, where the geometry of the structure dominates, the same ice loads were calculated in Panels 1 and 2, and the values were larger than the other methods. It can be seen that the direct method gave larger ice loads, but the magnitude and tendency of the results by the direct method and the semi-analytical method were similar to each other.   Figure 11 compares the mean values of the local ice load peaks on Panel 1 and 2 calculated by each method. In the simplified method, where the geometry of the structure dominates, the same ice loads were calculated in Panels 1 and 2, and the values were larger than the other methods. It can be seen that the direct method gave larger ice loads, but the magnitude and tendency of the results by the direct method and the semi-analytical method were similar to each other. Figure 11 compares the mean values of the local ice load peaks on Panel 1 and 2 calculated by each method. In the simplified method, where the geometry of the structure dominates, the same ice loads were calculated in Panels 1 and 2, and the values were larger than the other methods. It can be seen that the direct method gave larger ice loads, but the magnitude and tendency of the results by the direct method and the semi-analytical method were similar to each other. Figure 11. Mean of the local ice load peak. Figure 12 shows the cumulative Weibull distribution of ice load amplitudes calculated according to each method. The gentler the slope of the distribution, the greater the randomness included in the calculation process, which means that the variance was large. The direct analysis method based on the finite element model showed the widest load distribution because it can consider enough randomness that can occur in the real environment. The simplified method and the semi-analytical method showed a relatively narrow load distribution.

Panel1
Simplified method Semi-analytical method Direct method Figure 11. Mean of the local ice load peak. Figure 12 shows the cumulative Weibull distribution of ice load amplitudes calculated according to each method. The gentler the slope of the distribution, the greater the randomness included in the calculation process, which means that the variance was large. The direct analysis method based on the finite element model showed the widest load distribution because it can consider enough randomness that can occur in the real environment. The simplified method and the semi-analytical method showed a relatively narrow load distribution.

Fatigue Damage Calculation
This section presents the results of the fatigue damage calculations by the methods introduced in this study. The S-N curve is given by Equation (26). The S-N parameters for welded joint by DNV, which consider cathodic protection are employed [9].

Fatigue Damage Calculation
This section presents the results of the fatigue damage calculations by the methods introduced in this study. The S-N curve is given by Equation (26). The S-N parameters presented in Table 6 by DNV, which consider cathodic protection are employed. The final fatigue damage was calculated by multiplying the unit fatigue damage distribution and the moving distance distribution of ice for each stationary condition. Figures 13-15 present the results of the fatigue damage calculations using each methodology. The left figure shows the fatigue damage per unit moving distance (200 m) for 55 ice conditions, while the final result obtained by multiplying the actual moving distance distribution is shown on the right. In contrast to the direct method, the unit fatigue damage was expressed as a function of only the ice thickness because the effect of the drift ice velocity was not considered in the simplified method and the semi-analytical method. was not considered in the simplified method and the semi-analytical method. Table 6 shows the final fatigue damage for the applied ice thickness conditions. Based on the results of Sections 6.2 and 6.3, the relationship between the probability distribution of the ice load and the corresponding fatigue damage was investigated. Although the simplified method yielded the highest average ice load peak among the three methods, as shown in Figure 11, its fatigue damage was estimated to be similar to that calculated by the direct method. This is because the direct method considers a large amount of randomness, leading to a wider distribution of ice loads. On the other hand the semi-analytical method showed lower fatigue damage due to most of the ice loads being concentrated near the mean with low levels of randomness. It can be concluded that the ice load and its distribution play a critical role in the evaluation of level ice-induced fatigue damage. Further studies are recommended to investigate the impact of the randomness of ice loads on the simplified and the semi-analytical methods. Based on the results obtained from the direct method and the model test, it is expected that the accuracy of these two methodologies can be improved.    was not considered in the simplified method and the semi-analytical method. Table 6 shows the final fatigue damage for the applied ice thickness conditions. Based on the results of Sections 6.2 and 6.3, the relationship between the probability distribution of the ice load and the corresponding fatigue damage was investigated. Although the simplified method yielded the highest average ice load peak among the three methods, as shown in Figure 11, its fatigue damage was estimated to be similar to that calculated by the direct method. This is because the direct method considers a large amount of randomness, leading to a wider distribution of ice loads. On the other hand, the semi-analytical method showed lower fatigue damage due to most of the ice loads being concentrated near the mean with low levels of randomness. It can be concluded that the ice load and its distribution play a critical role in the evaluation of level ice-induced fatigue damage. Further studies are recommended to investigate the impact of the randomness of ice loads on the simplified and the semi-analytical methods. Based on the results obtained from the direct method and the model test, it is expected that the accuracy of these two methodologies can be improved.

Conclusions
This paper introduced three different methods for calculating fatigue damage in inclined structures subjected to level ice impact. The direct, semi-analytical, and simplified methods were utilized to estimate local ice loads that induce fatigue cracking, and the corresponding fatigue damages were derived based on different types of ice loads. The following conclusions were drawn from the results obtained in this study: -Ice failure mode In the semi-analytical method, only out-of-plane bending failure was considered as the failure mode of ice. Conversely, the direct method yielded various types of crack propagation, particularly in collisions where the contact area between the structure's collision surface and the horizontal edge of the ice was wide. Cracks exhibited highly random shapes. As the ice load is heavily influenced by the crack pattern, the semi-analytical method, which considers only one failure mode, lacks randomness in the ice load calculation. Consequently, the fatigue damage calculated using the semi-analytical method was significantly lower than that obtained from the other two methods. Although it is challenging to theoretically verify the randomness caused by the hull's shape, it can be improved by establishing and applying correlations between the crack pattern and the load occurring at that specific instance, obtained from the direct method or field experiments. -

Consideration of dynamic effect
Limitations of the simplified and semi-analytical methods were identified based on the unit fatigue damages obtained from each method. These methods calculate unit fatigue damages solely based on thickness, neglecting the impact velocity in the local ice load calculation. In contrast, the results of the direct analysis reveal that the effect of impact speed on fatigue damage is as significant as that of ice thickness. Thus, incorporating the dynamic effect in estimating the ice load is expected to provide a more accurate evaluation of ice-induced fatigue damage.  Based on the results of Sections 5.2 and 5.3, the relationship between the probability distribution of the ice load and the corresponding fatigue damage was investigated. Although the simplified method yielded the highest average ice load peak among the three methods, as shown in Figure 11, its fatigue damage was estimated to be similar to that calculated by the direct method. This is because the direct method considers a large amount of randomness, leading to a wider distribution of ice loads. On the other hand, the semi-analytical method showed lower fatigue damage due to most of the ice loads being concentrated near the mean with low levels of randomness. It can be concluded that the ice load and its distribution play a critical role in the evaluation of level ice-induced fatigue damage. Further studies are recommended to investigate the impact of the randomness of ice loads on the simplified and the semi-analytical methods. Based on the results obtained from the direct method and the model test, it is expected that the accuracy of these two methodologies can be improved.

Conclusions
This paper introduced three different methods for calculating fatigue damage in inclined structures subjected to level ice impact. The direct, semi-analytical, and simplified methods were utilized to estimate local ice loads that induce fatigue cracking, and the corresponding fatigue damages were derived based on different types of ice loads. The following conclusions were drawn from the results obtained in this study: -Ice failure mode In the semi-analytical method, only out-of-plane bending failure was considered as the failure mode of ice. Conversely, the direct method yielded various types of crack propagation, particularly in collisions where the contact area between the structure's collision surface and the horizontal edge of the ice was wide. Cracks exhibited highly random shapes. As the ice load is heavily influenced by the crack pattern, the semianalytical method, which considers only one failure mode, lacks randomness in the ice load calculation. Consequently, the fatigue damage calculated using the semianalytical method was significantly lower than that obtained from the other two methods. Although it is challenging to theoretically verify the randomness caused by the hull's shape, it can be improved by establishing and applying correlations between the crack pattern and the load occurring at that specific instance, obtained from the direct method or field experiments. -

Consideration of dynamic effect
Limitations of the simplified and semi-analytical methods were identified based on the unit fatigue damages obtained from each method. These methods calculate unit fatigue damages solely based on thickness, neglecting the impact velocity in the local ice load calculation. In contrast, the results of the direct analysis reveal that the effect of impact speed on fatigue damage is as significant as that of ice thickness. Thus, incorporating the dynamic effect in estimating the ice load is expected to provide a more accurate evaluation of ice-induced fatigue damage.
The direct method offers high accuracy but requires a substantial amount of computational time. Conversely, the simplified method and the semi-analytical method offer faster analysis times and are more suitable for long-term time domain analyses. Therefore, the selection of the method should depend on specific analysis requirements, such as the desired accuracy level, available analysis time, and available resources.
Overall, the findings of this study provide valuable insights into predicting fatigue damage in ice-going ships resulting from long-term ice impacts. The methods proposed in this study can assist in designing Arctic ships exposed to diverse conditions and offer a more cost-effective and time-efficient approach for evaluating fatigue damage compared to field measurements. Future research in this field could explore the application of these methods to other types of structures and further refine the methodology to enhance accuracy and reduce computational costs.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.