## 3 Ice Load Prediction Sub Models
### Ice Crushing-Shearing Model  
The energy method proposed by [PERSON] [7] and [PERSON] et al. [15] provides an effective way to calculate the crushing force and indentation displacement for any ice-hull contact. The initial impact collision is triggered by a ship with kinetic energy and a stationary ice sheet with bulky mass. The inputs for calculation are the ice properties, e.g., ice thickness, ice crushing strength, and the hull geometry. Moreover, the method includes analytical formulas for the different geometric contact cases. Figure 4: Description of how to obtain ice breaking force in the current model. The hull geometry parameters are described and presented in Table 1 and Figure 5. The definitions of the hull angles from literature and Finnish-Swedish Ice Class Rules (FSICR) [16] are summarized and illustrated in Figure 6. The coordinate system is set with the origin at the mass center of a ship. The \(x\)-axis is along the longitudinal direction of the hull and positive forward. The \(y\)-axis is normal to the \(x\)-axis and is positive towards starboard. The \(z\)-axis is perpendicular to the \(xy\)-plane and positive downward [17]. The relationships between ship angles are given as below:  
\[\tan\beta\ =\ \tan\alpha\tan\gamma, \tag{7}\]  
\[\tan\beta^{\prime}\ =\ \tan\beta\cos\alpha. \tag{8}\]  
The energy method is applicable for an impact between two objects. It is assumed that one object is moving while another is still. During the process, the effective kinetic energy from the moving object \begin{table}
\begin{tabular}{c l c l} \hline \hline
**Parameters** & **Description** & **Parameters** & **Description** \\ \hline \(L\) & Waterline plane length & \(\beta\) & Frame angle \\ \(B\) & Waterline plane beam & \(B^{\prime}\) & Normal frame angle \\ \(T\) & Draft & \(\gamma^{\prime}\) & Sheer(buttock) angle \\ \(D\) & Depth & \(\varphi^{*}\)1  & Rake angle \\ \(x\), \(y\), \(z\) & The collision point on the hull or the ice sheet & \(C_{b}\) & Block coefficient \\ \(cg\) & Centre of gravity & \(C_{up}\) & Water plane area coefficient \\ \(\alpha\) & Waterline angle & \(C_{m}\) & Midship section coefficient \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ship parameters. Figure 5: Definitions of geometric parameters at a ship-ice collision point. Figure 6: Definitions of hull angles at a collision point. is equivalent to the energy expanded to the another as crushing, or indentation, energy. The collided object is assumed to have infinite large mass, thus the effective kinetic energy is regarded the same as the total kinetic energy. This approach is used to determine the maximum force when all motions are ceased. According to such assumption and based on [PERSON] [7], the effective kinetic energy, \(KE_{\rm e}\), and the crushing energy, \(IE\), are equal:  
\[KE_{\rm e}\ =\ IE. \tag{9}\]  
The indentation energy can be represented by the indentation force, \(F_{n}\), and the crushing indentation displacement, \(\zeta_{c}\); the kinetic energy is expressed by the ship effective mass and velocity:  
\[IE\ =\ \int_{0}^{\zeta}F_{n}\mathrm{d}\zeta_{c}, \tag{10}\]  
\[KE_{\rm e}\ =\ \frac{1}{2}M_{e}{V_{n}}^{2}, \tag{11}\]  
\[M_{\rm e}\ =\ \frac{M_{\rm ship}}{C_{o}}, \tag{12}\]  
\[V_{n}\ =\ V_{\rm ship}l, \tag{13}\]  
where \(M_{\rm e}\) is the effective mass at the impact point; \(M_{\rm ship}\) stands for the effective mass of the ship at the contact point; \(C_{o}\) presents the mass reduction factor in six degrees of freedom; \(V_{n}\) is the normal velocity at the impact point; \(V_{\rm ship}\) is the ship velocity; \(l\) is the x-direction cosine.