## 3 Local Ice Load
### 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 (\(\alpha\)). Figure 5: Visualization of the semi-analytical method. Figure 6: Ice breaking load by the semi-analytical method. Vertical:  
\[\mathrm{V_{B}}=\mathrm{Ncos}(\alpha)-\upmu\mathrm{Nsin}(\alpha) \tag{4}\]  
Horizontal:  
\[\mathrm{H_{B}}=\mathrm{Nsin}(\alpha)-\upmu\mathrm{Ncos}(\alpha) \tag{5}\]  
where \(\mathrm{V_{B}}\) is the vertical component of the load, N is the load component in the direction normal to the slope, \(\alpha\) is the slope angle, and \(\upmu\) 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]. \[V_{B}=0.68\sigma_{f}w_{B}\bigg{(}\frac{\rho_{w}gl^{5}}{E}\bigg{)}^{0.25} \tag{6}\]  
\[w_{B}=\bigg{(}\frac{\pi^{2}}{4}\bigg{)}L_{c}\,\ if\ 2R>D \tag{7}\]  
\[w_{B}=D+\bigg{(}\frac{\pi^{2}}{4}\bigg{)}L_{c}\,\ if\ 2R<D \tag{8}\]  
\[L_{c}=\bigg{(}\frac{El^{3}}{12\rho_{w}g(1-v^{2})}\bigg{)}^{0.25} \tag{9}\]  
\[R=\bigg{(}\frac{\pi}{4}\bigg{)}L_{c} \tag{10}\]  
where \(\sigma_{f}\) is the flexural strength of ice, \(\rho_{w}\) 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). \[N_{load}=\frac{0.68\sigma_{f}w_{B}}{\cos(\alpha)-\upmu\sin(\alpha)}\bigg{(} \frac{\rho_{w}gl^{5}}{E}\bigg{)}^{0.25} \tag{11}\]  
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 [PERSON] 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). Figure 6:0  
where \(q\) and \(\zeta\) are the Weibull scale parameter and shape parameter, respectively. Research aimed at estimating the shape parameters of ice loads has been conducted by [PERSON] et al. [1] and [PERSON] 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 [PERSON] et al. [1]. Figure 6:1  
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). Figure 6:2  
Figure 6:3