# The Multistate Reliability Assessment of Ship Hull Girders
## 4 Results of Reliability Analysis of Longitudinal Watertight
### Estimation of Reliability and Risk
Table 6 presents the number of measurements in the reliability state subsets \(\{l,l+1,\ldots,2\}\), \(l\in\{0,1,2\}\). Next, using the data in Table 5, the coordinates of reliability function values \(R(t,l)\), \(l\in\{0,1,2\}\) after five, ten, fifteen, twenty, and twenty-five years were estimated. These values are defined as the probability that the lower side of the girder's plates is in reliability state subset \(\{0,1,2\}\), \(\{1,2\}\), or \(\{2\}\), and they are presented in Table 7.  
From the data contained in Table 6, and from Figure 7 it can be concluded that sometime between twenty and twenty-five years of operation, the reliability function' component \(R(t,1)\) fell below the acceptable level of 90%. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Time (Years)**} & **Number of Measurements** & **Number of Measurements in Reliability State** & **Number of Measurements in Reliability State** \\  & **Number of Measurements** & **Measurements in Reliability State** & **Measurements in Reliability State** \\  & & **Subset \{0,1,2\}** & **Subset \{1,2\}** & **Subset \{2\}** \\ \hline
0–5 & 90 & 90 & 90 \\
5–10 & 100 & 100 & 100 \\
10–15 & 190 & 190 & 190 \\
15–20 & 400 & 400 & 367 & 364 \\
20–25 & 180 & 180 & 131 & 130 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Number of measurements of the lower side of the girder’s steel plates in a subset of reliability states. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Time (Years)**} & **R(t, 0)** & **R(t, 1)** & **R(t, 2)** \\ \hline
5 & 1 & 1 & 1 \\
10 & 1 & 1 & 1 \\
15 & 1 & 1 & 1 \\
20 & 1 & 0.918 & 0.910 \\
25 & 1 & 0.728 & 0.722 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The estimated values of the reliability function in reliability state subsets \(\{0,1,2\}\), \(\{1,2\}\), and \(\{2\}\) for the lower side of the girder’s plates. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Time (Years)**} & **Number of Measurements That Are** & **Number of Measurements** & **Number of Measurements** \\  & **\textless{}0.15\%) of the Original Thickness** & **That Are \textless{}15\%, 20\%) of the Original Thickness** & **That Exceed 20\% of the Original Thickness** \\ \hline
0–5 & 90 & 0 & 0 \\
5–10 & 100 & 0 & 0 \\
10–15 & 190 & 0 & 0 \\
15–20 & 364 & 3 & 33 \\
20–25 & 130 & 1 & 49 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The number of measurements for the steel plates of the lower side of the girder in reliability states \(l\), \(l\in\{0,1,2\}\). For the upper side of the girder, we have to determine the time when the risk function exceeds an acceptable level 0.1. This will help us answer the question of when the reliability of the lower side of the girder should be improved (when the steel plates should be replaced). Using Lagrange polynomial interpolation, the reliability function coordinate \(R(t,1)\) is given by the formula below:  
\[ \begin{split} R(t,1)\approx\overset{\sim}{R}(t,1)=R_{0}(t,1)p_{ 0}(t)+R_{1}(t,1)p_{1}(t)+R_{2}(t,1)p_{2}(t)\\ +R_{3}(t,1)p_{3}(t)R_{4}(t,1)p_{4}(t)\\ =p_{0}(t)+p_{1}(t)+p_{2}(t)+0.918p_{3}(t)+0.728p_{4}(t),\end{split}  \tag{13}\]  
where \(p_{0}(t),\ p_{1}(t),\ p_{2}(t),\ p_{3}(t),\ \text{and}\ p_{4}(t)\) are given by (8)-(12). The next step is to determine the moments when the steel plates' risk function \(r(t)=1-R(t,1)\) exceeds the acceptable level of 0.1. The approximately risk and reliability functions are given in Figure 8.  
Figure 8: Approximate functions obtained from Lagrange interpolation. Figure 7: Graphical interpretation of function \(R(t,l)\) coordinates. Using Formulas (3), (4), and (13), the moment \(\tau\) when the risk function of the steel plates exceeds a permitted threshold \(\delta\) = 0.1 is \(\tau\) = 20.6 years. The above analysis indicates that the risk function for the steel plates of the lower side of girder will exceed the acceptable level of 0.1 after 20.6 years.