# Incorporating Epistemic Uncertainties in Ship Operability Study
## 2 Methods
### Reliability Problem Formulation  
As explained in the preceding section, the accuracy of the seakeeping calculations is uncertain, as well as limiting values of seakeeping criteria. Therefore, the seakeeping criteria are exceeded with a certain probability, which is calculated based on the limit state functions. In general, the limit state function \(g(X)\) is given as:  
\[g(x_{1},x_{2},\ldots x_{n})<0, \tag{15}\]  
where \(x_{1},x_{2},\ldots x_{n}\) is a vector of random variables and \(g\) takes negative value when the limiting criterion is exceeded. In the present study, the random variables are epistemic uncertainties of the seakeeping results and criteria. To describe the procedure, let us consider the simplest limit state function, which is for pitch angle. The limit state function, in this case, reads:  
\[x_{\theta}-\sqrt{\int\limits_{0}^{\infty}|\varphi_{\theta}\Phi_{\theta}( \overline{\omega})|^{2}S_{I}(\overline{\omega})\cdot d\overline{\omega}}<0 \tag{16}\]  
\(x_{\theta}\) in Equation (16) is a normally distributed random variable representing pitch criterion, with a mean value equal to 1.5\({}^{\circ}\), according to Table 2, and CoV of 0.05. \(\varphi_{\theta}\) is also normally distributed with the mean value calculated by Equation (14), depending on the heading angle and ship speed, and a standard deviation of 0.19. \(\Phi_{\theta}(\overline{\omega})\) is a transfer function of pitch, given by Equation (2), while the wave energy spectrum \(S_{I}(\overline{\omega})\) is defined by Equations (8)-(10). The limit state function for the roll is defined in similar way, only that the random variable representing criterion uncertainty \(x_{roll}\) with a mean value of 2.5\({}^{\circ}\) is used. The transfer function of roll motion is \(\Phi_{\varphi}(\overline{\omega})\), defined by Equation (7). The limit state function for vertical acceleration at FP is a bit more complicated, as the transfer function for vertical acceleration at FP (Equation (6)) depends on heave and pitch transfer functions (Equation (5)). Each of these transfer functions is multiplied by the corresponding epistemic uncertainty, calculated by Equation (14), using parameters from Table 1. The limit state function is then readily defined using an expression like Equation (16). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Criteria** & **Statistical Property** & **Mean Value** \\ \hline Pitch & RMS & 1.5\({}^{\circ}\) \\ Vertical acceleration at FP & RMS & 0.05 g \\ Roll & RMS & 2.5\({}^{\circ}\) \\ Slamming & Probability & 0.03 \\ Propeller emergence & Probability & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean values of operability criteria limits [8]. The limit state function for propeller emergence requires the calculation of the probability of the propeller emergence:  
\[p_{pe}=\exp\left[-\left(\frac{D_{p}^{2}}{2m_{0,r}}\right)\right] \tag{17}\]  
where \(D_{p}\) is the propeller blade tips' depth, while \(m_{0,r}\) is the variance of the relative motion at the location of the propeller. The transfer function of the relative motion is given by an expression analogous to Equation (5). Transfer functions of heave and pitch are multiplied by corresponding modeling uncertainties. The limit state function for propeller emergence is then given as:  
\[x_{pe}-p_{pe}<0 \tag{18}\]  
where \(x_{pe}\) is normal distribution with a mean value of 0.25, according to Table 2, and with assumed CoV of 0.05. The slamming probability, \(p_{s}\), employed in the slamming limit state function, is given as:  
\[p_{s}=\exp\left[-\left(\frac{T_{e}^{2}}{2m_{0,r}}+\frac{v_{kr}^{2}}{2m_{0,r}} \right)\right], \tag{19}\]  
where \(T_{e}\) is the effective draft; \(m_{0,r}\) is the variance of the relative vertical motion at ship FP; \(v_{kr}\) is the critical vertical velocity; and \(m_{0,p}\) is the variance of the relative velocity at FP. Critical velocity is calculated from [PERSON]'s equation:  
\[v_{kr}=0.093\sqrt{\mathrm{g}L_{p}}, \tag{20}\]  
where LP is a ship length between perpendiculars.