# High-Expansion Foam: A Risk Control Option to Increase Passenger Ship Safety during Flooding
## 2 Damage Stability Assessment of a Passenger Ship
### Static Assessment  
The assessment of s-factor through static calculations is based on empirical formulations derived from the static residual GZ curve for all the intermediate and final stages of flooding associated with a damage case. According to SOLAS [7], the \(s_{ijk}\) contributing to \(A_{jk}\) calculation should be determined from:  
\[s_{ijk}=\min_{1<h<N_{stage}}\left(s_{int_{ijk}}s_{final_{ijk}}s_{M_{ijk}}\right) \tag{3}\]  
where \(s_{int}\) are the survivability factors for all the \(N_{stage}\) intermediate flooding stages until the final equilibrium stage, \(s_{final}\) is the survivability in the final equilibrium stage, and \(s_{M}\) is the survivability to external healing moments in the final equilibrium stage. These quantities can be calculated as follows for passenger ships:\[s_{int_{ijk}}= \begin{cases}0&\text{if $\varphi_{e}>15^{\circ}$}\\ \left(\frac{\min(GZ_{max},0.05)}{0.05}\frac{\min(Range\tau)}{7}\right)^{\frac{1} {4}}&\text{if $\varphi_{e}\leq 15^{\circ}$}\end{cases}  \tag{4}\]  
\[s_{final_{ijk}}=K\bigg{(}\frac{\min(GZ_{max},TGZ_{max})}{TGZ_{max}}\frac{\min( Range,TRange)}{TRange}\bigg{)}^{\frac{1}{4}}\text{ with} \begin{cases}K=1&\text{if $\varphi_{e}\leq 7^{\circ}$}\\ K=0&\text{if $\varphi_{e}\geq 15^{\circ}$}\\ K=\sqrt{\frac{15-\varphi_{e}}{8}}&\text{otherwise}\end{cases}  \tag{5}\]  
\[s_{Mijk}=\min\bigg{(}\frac{(GZ_{max}-0.04)\Delta}{M_{hel}},1\bigg{)} \tag{6}\]  
where \(GZ_{max}\) is the maximum of the residual righting lever curve for each damage case and loading condition. \(GZ_{max}\) must fall within the positive interval _Range_ between the equilibrium angle \(\varphi_{e}\) and \(\varphi_{v}\), the minimum between the capsize and the unprotected opening immersion angle. \(TRange\) and \(TGZ_{max}\) assume different values for cases involving flooding of car deck in Ro-pax ships (0.20 m for \(TGZ_{max}\), 20\({}^{\circ}\) for \(TRange\)) and not (0.12 m for \(TGZ_{max}\), 16\({}^{\circ}\) for \(TRange\)). \(\Delta\) is the ship displacement and \(M_{hel}\) is the external moment, which is the maximum between moments due to wind, movement of passengers and launch of survival crafts. The s-factor formulations in Equations (4)-(6) are all with values in \([0,1]\), thus the s-factors represent a probability to survive a specific event in a specific flooding condition. Figure 1 shows an example of a residual GZ curve with the associated parameters relevant for stability assessment. From these calculations, three case categories can be identified, according to the s-factor value:  
1. \(s_{ijk}=0\): cases where the vessel can be considered statically capsized or with insufficient residual stability margin. 2. \(0<s_{ijk}<1\): cases where there could be a reduced reserve of stability that may lead to capsizing in case a wave environment is faced. 3. \(s_{ijk}=1\): cases where the vessel can be considered safe and potentially have a sufficient reserve of stability to face waves. The resulting \(s\) values combined with the associated \(p\) lead to the determination of the attained survivability index \(A\), which is expected to be compared with the required subdivision index \(R\) defined by SOLAS [7], ensuring that \(A_{jk}\geq 0.9R\). \(A\) index is not the only metric suitable to measure survivability, and alternative parameters can be preferred,  
Figure 1: Residual GZ curve for static damage stability calculations. as probability of loss of lives (PLL) and GM margin. From an operator's perspective, the former is the most important and familiar measurement with direct impact on the operability of their vessels. However, from a static assessment, other information can be derived, concerning the vulnerability of the ship. Such indications are relevant to identifying critical zones, requiring attention by the designer. The static survivability assessment is performed on the damage cases derived from the sampling of marginal distributions, grouped in unique damage scenarios with associated p-factors, and determining the associated survivability through Equations (4)-(6).