# Analysis of a Collision-Energy-Based Method for the Prediction of Ice Loading on Ships
## 2 Background
### Collision Scenario  
In an ice floe-structure interaction, there are three limit mechanisms for ice loads: Limit ice strength, limit driving force and limit momentum [24]. The Popov Method is based on the mechanisms of limit momentum, also referred to limit energy mechanism, and it considers impacts between two bodies where one body is initially moving and the other is at rest. The ice load is calculated by equating the available kinetic energy with the energy expended in crushing and the potential energy in collisions.  
Due to the non-homogeneous nature of sea ice and the complexity of the ship-ice interaction process, the number of possible ship-ice collision scenarios is large. Thus, when predicting ice loading, assumptions must be made regarding the location of the ship-ice impact (e.g., bow, side, stern), the type of ice involved (e.g., ice floe, ice field), and the resulting contact geometries [11]. [PERSON] et al. [9] consider the following ship-ice collision scenarios:  
* An impact between a ship and an ice floe. According to [PERSON], et al. [25] this is the most common collision scenario for merchant ships. As per Table 1, in this scenario, the kinetic energy of the ship \(T_{1},[J]\) is partially converted into the kinetic energy of the ice floe \(T_{2},[J]\) and partially consumed in crushing the ice edge \(U,[J]\). \(T_{1}\), \(T_{2}\), and \(U\) are calculated in accordance with the formulas presented in Table 1 (Case 1), where \(M_{red}^{ship},[t]\) and \(M_{red}^{ic},[t]\) are the reduced masses of the ship and the ice floe, and the parameters \(v_{1},[m/sec]\) and \(v_{2},[m/sec]\) correspond to the reduced speeds of the ship and the ice floe.
* An impact between a ship and the edge of an ice field. In this scenario, defined as case two in Table 1, the edge of the ice field is crushed, and the ice field bends due to the vertical component of the contact force \(P_{\sigma},[N]\). In this scenario, the ship's kinetic energy \(T_{1},[J]\) is consumed by the crushing \(U,[J]\) and by the bending of the ice field \(V,[J]\). \(T_{1}\), \(V\), and \(U\) are calculated in accordance with the formulas presented in Table 1 (Case 2), where \(f=\frac{P_{\sigma}}{2\sqrt{V_{1}\cdot P}}[m]\), \(\gamma_{i},\big{[}N/m^{3}\big{]}\) is the specific weight of ice, \(D=\frac{E\cdot H_{i}^{3}}{12\cdot(1-\mu)},\big{[}Pa\cdot m^{3}\big{]}\) is flexural stiffness of an ice plate, \(E,[Pa]\) is the elastic modulus of ice, \(h_{i},[m]\) is the ice thickness, \(\mu\) is Poisson ratio for ice.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Case 1: Ship–Ice floe Collision** & **Case 2: Ship–Ice Field Collision** \\ \hline \(T_{1}+T_{2}=U\) & \(T_{1}=U+V\) \\ \(T_{1}=1/2\cdot M_{red}^{ship}\cdot v_{1}{}^{2}\) & \(T_{1}=1/2\cdot M_{red}^{ship}\cdot v_{1}{}^{2}\) \\ \(T_{2}=1/2\cdot M_{red}^{ic}\cdot v_{2}{}^{2}\) & \(V=1/2\cdot f\cdot P_{v}\) \\ \(U=\int_{0}^{\zeta_{n}}Fd\zeta_{n}\) & \(U=\int_{0}^{\zeta_{n}}Fd\zeta_{n}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Energy components of the Popov Method as determined based on [PERSON] et al. [9]).