## 6 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 \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ice Thickness (m)** & **Direct Method** & **Simplified Method** & **Semi-Analytical Method** \\ \hline
0.296 & 3.5111 \(\times\) 10\({}^{-6}\) & 3.8232 \(\times\) 10\({}^{-7}\) & 2.6087 \(\times\) 10\({}^{-9}\) \\
0.592 & 3.7336 \(\times\) 10\({}^{-6}\) & 5.0845 \(\times\) 10\({}^{-7}\) & 1.4196 \(\times\) 10\({}^{-8}\) \\
0.888 & 0.0024 & 0.0026 & 9.0073 \(\times\) 10\({}^{-6}\) \\
1.184 & 0.3314 & 0.2278 & 0.0010 \\
1.48 & 0.3793 & 0.7506 & 0.0012 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of final fatigue damages according to different ice thickness conditions. Figure 15: Fatigue damage—semi-analytical method. 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. Methodology, Y.K. ; Validation, Y.-C.S. ; Investigation, Y.-C.S. ; Writing--original draft, J.-H.K. ; Writing--review & editing, Y.K. ; Visualization, Y.-C.S. ; Supervision, J.-H.K. ; Project administration, Y.K. All authors have read and agreed to the published version of the manuscript. This work was supported by the Dong-A University research fund. Not applicable.