## 3 Ice Load Prediction Sub Models
### Ice Crushing-Shearing Model
They are defined based on Popov [15]:  
\[l\ =\ \sin(\alpha)\cos(\beta^{\prime}),\ m\ =\ \cos(\alpha)\cos(\beta^{ \prime}),\ n\ =\ \sin(\beta^{\prime}). \tag{14}\]  
For a symmetrical collision on the stem, the relationships become:  
\[l\ =\ cos(\gamma),\ m\ =\ 0,\ n\ =\ sin(\gamma). \tag{15}\]  
The roll, pitch, and yaw lever arm can be calculated, respectively:  
Figure 5:0  
The added mass terms in surge, sway, heave, roll, pitch, and yaw can be obtained from [15]:  
\[ \begin{array}{c}AMx\ =\ 0,\\ AMy\ =\ \frac{2T}{B},\\ AMz\ =\ \frac{2 BC_{\rm sp}^{2}}{3T\left(C_{b}\left(1+C_{\rm sp}\right)\right)}, \\ AMrol\ =\ 0.25,\\ AMpit\ =\ \frac{B}{\Gamma\left(3-2C_{\rm sp}\right)\left(3-C_{\rm sp}\right)}, \\ AMyaw\ =\ 0.3+\frac{0.05L}{B}.\end{array}  \tag{17}\]  
The squared mass radii of gyration in roll, pitch and yaw can be expressed as:  
\[ \begin{array}{c}rx^{2}\ =\ \frac{C_{\rm sp}B^{2}}{11.4C_{\rm m}}+\frac{D^{2}}{1 2},\\ ry^{2}\ =\ 0.07C_{\rm sp}L^{2},\\ rz^{2}\ =\frac{L^{2}}{16}.\end{array}  \tag{18}\]The mass reduction coefficient \(C_{o}\) thus can be obtained [15]:  
\[ \begin{array}{ccccc}C_{o}&=&\frac{\rho^{2}}{1+AMx}&+\frac{m^{2}}{1+AMy}+\frac{m^ {2}}{1+AMz}+\frac{\lambda_{1}^{2}}{m^{2}(1+AMrol)}+\frac{\mu_{1}^{2}}{ry^{2}(1+AMpt )}\\ &&+\frac{\eta_{1}^{2}}{\pi^{2}(1+AMyur)}.\end{array}  \tag{19}\]  
The normal impact force \(F_{n}\) and the energy \(IE\) can be calculated as:  
\[F_{n} = p_{o}\cdot fa\zeta_{n}^{fx-1}, \tag{20}\] \[IE = p_{o}\cdot fa\cdot\frac{\zeta_{n}^{fx}}{fx}, \tag{21}\]  
where \(p_{o}\) is the pressure on \(1\)\(m^{2}\), and equivalent to the ice crushing strength, \(\sigma_{c}\), it equals to \(2000\) kPa \(\zeta_{n}\) is the ice indentation from the initial contact point along the normal direction of contact area and \(fx\) is a function of \(ex\), where \(ex\) is constant. \(fa\) is a function of the geometric parameter. Based on Equations (9) and (21), the kinetic energy can be expressed as:  
\[KE_{e} = p_{o}\cdot fa\cdot\frac{\zeta_{n}^{fx}}{fx}, \tag{22}\]  
with the normal indentation displacement \(\zeta_{n}\) as:  
\[\zeta_{n} = \left(\frac{KE_{e}\cdot fx}{p_{0}\cdot fa}\right)^{\frac{fx-1}{fx }}, \tag{23}\]  
Substitute Equation (23) into Equation (20), the normal force can be calculated:  
\[F_{n} = p_{0}\cdot fa\cdot\left(\frac{KE_{e}\cdot fx}{p_{0}\cdot fa}\right)^{ \frac{fx-1}{fx}}. \tag{24}\]  
It can be observed that the indentation displacement \(\zeta_{n}\) plays an important role. It is affected by the impact geometry configuration. Two indentation geometry scenarios are introduced. They refer to the bow stem (or head on) collision and the bow shoulder collision respectively, as seen in Figures 7 and 8. The two collision scenes are illustrated as a symmetrical- and an unsymmetrical-contact, as shown in Figures 7 and 8. The indentation functions, \(fx\) and \(fa\) are defined in [PERSON] et al. [15]. The ice sheet dimension and bow geometry at the collision area shown in Figure 9, and the indentation areas for the two collisions and two contact area cases can be derived based on the indentation functions [15]. Figure 7: Case I-bow front collision: symmetric V wedge indentation. (**a**) Indentation does not go through the ice thickness; (**b**) Indentation penetrates through the ice thickness.