## 3 Local Ice Load
### 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 [PERSON] [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). \[F_{n}=P_{av}A \tag{1}\]  
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 [PERSON] [12]. \[P_{av}=P_{0}A^{ex} \tag{2}\]  
Figure 3: Finite element analysis of the sloping structure with the level ice. (**a**) Basic colliding model; (**b**) Rubble deletion model. 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 [PERSON] (1965 [14], 1972 [15]) equation proposed by [PERSON] et al. [16] was used to calculate the critical load, as shown in Equation (3). \[F_{Z}=\frac{\theta}{180}\frac{2}{3}\,\sigma_{f}t^{2}\left[1.05+2\left(\frac{r}{ l_{c}}\right)+0.5\left(\frac{r}{l_{c}}\right)^{3}\right] \tag{3}\]  
where \(\theta\) is wedge angle, \(\sigma_{f}\) 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. [PERSON] [17] proposed a constant breaking length based on model experiments and actual ice test results, while [PERSON] [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.