# Analysis of a Collision-Energy-Based Method for the Prediction of Ice Loading on Ships
## 2 Background
### Identification of Assumptions Behind the Popov Method and the PC Rules  
Based on [PERSON] et al. [9], [PERSON] [14], [PERSON] [10], [PERSON] et al. [37] and IACS [38] we identify the following assumptions behind the Popov Method and the related PC rules:  
* Assumptions related to the definition of the ship-ice collision scenario. The Popov Method assumes two types of ship-ice collision scenarios defined in Table 1: (a) Ship-ice floe collision, and (b) ship-ice field collision. The PC rules, on the other hand, assume a single type of ship-ice collision scenario, namely a glancing impact with thick level ice. * Assumptions related to the definition of contact pressure. For typical speeds of ships operating in ice, [PERSON] et al. [9] suggested that the contact pressure \(p\big{(}\dot{\alpha}_{tr},\dot{\zeta}_{in}\big{)}\) can be assumed equal to the ice crushing strength \(\sigma_{cr},[MPa]\) over the whole contact area. Accordingly, the maximum ice load corresponding to the maximum indentation depth \(\zeta_{max},[m]\) can be defined as per \[F=\sigma_{cr}\int_{0}^{\zeta_{max}}dA,\,[MN]\] (4)  
Figure 1: Considered contact geometry models: (**a**) Round; (**b**) Angular; (**c**) Symmetric \(\mathbf{v}\) wedge; (**d**) Right-apex oblique; (**e**) Spherical; (**f**) Pyramid (Adapted with permission from [PERSON] [34], International Conference on Port and Ocean Engineering under Arctic Conditions, 1999, and from [PERSON] & [PERSON] [35], The American Society of Mechanical Engineers, 2010). However, this is a simplification, as the crushing strength of ice depends on multiple factors and varies between 1.25 - 10 _MPa_[9]. In the PC rules, on the other hand, the contact pressure is assumed to correspond to the pressure-area relationship following: \[P=P_{0}A^{ex},[MPa],\] (5) where \(P_{0},[MPa]\) is average pressure acting on 1 \(m^{2}\) (class dependent), \(A,\left[m^{2}\right]\) is the contact area, and \(ex\) is a coefficient empirically determined as \(-0.1\). * Assumptions related to the definition of the contact geometry case. The application of specific contact geometries requires making assumptions regarding individual contact geometry parameters. When applying the round contact geometry model, [PERSON] et al. [9] recommend the use of R-values in the range of 10-40 m. For angular contact geometry, [PERSON] et al. [9] recommend the use of ice opening angle values (\(\phi\)) in the range of 45\({}^{\circ}\)-145\({}^{\circ}\). In the PC rules the assumed contact geometry case is always angular (also referred to as a wedge shape) with a constant \(\phi\) value of 150\({}^{\circ}\)[37]. * Assumptions related to the definition of the size of the nominal ship-ice contact area. As per the Popov Method, the nominal ship-ice contact area is assumed to be dependent on the form of the hull and ice at the point of impact as per Equation (2). The PC rules, on the other hand, assuming a wedge-shaped contact geometry, assume a single nominal contact area defined per \[A=\frac{\zeta_{m}{}^{2}\cdot\tan\left(\frac{\phi}{2}\right)}{\left(\cos\beta^ {\prime}\right)^{2}\cdot\sin\beta^{\prime}}\left[m^{2}\right]\] (6) Naturally, this is a simplification as the actual contact area depends strongly on the assumed contact geometry, and thus on the prevailing ice conditions and the hull form. * Assumptions related to the definition of load length and load height. The load length and load height, both of which affect the structural requirements in accordance with the PC rules, are derived as a function of the indentation depth [34; 39]. In the PC rules, the design load length \(w\) is assumed to be defined per \[w=W_{nom}{}^{wex},[m]\] (7) where \(W_{nom}\) is the nominal contact length of the assumed wedge-shaped contact geometry and _wex_ is an ice spalling parameter with an empirically determined value of 0.7. The ice spalling parameter accounts for ice edge spalling effects by reducing the size of the contact patch.