# The Multistate Reliability Assessment of Ship Hull Girders
## 4 Results of Reliability Analysis of Longitudinal Watertight  
### Estimation of Reliability and Risk  
The reliability analysis of this study is based on the assumption that the permissible wear of steel plates is 20%. This means that the values below 80% of the original thickness are considered unacceptable, i.e., interpreted as failures. Although the remaining thickness ensures the impermeability of steel plates, classification societies still require the replacement of corroded surfaces in these cases. Figure 4: Diagram showing the reliability improvement procedure. Figure 3: Coordinate of exemplary improvement reliability function. A multistate approach to reliability was applied, and three reliability states were adopted, defined as follows:  
* Second state of reliability--The system works and is totally safe; the corrosion wear of steel plates is in the range of <0.15%). * First state of reliability--The system works, but its operation is less safe, which may cause environmental pollution, cargo damage, decrease the ship's safety, etc. ; the corrosion wear of steel plates is in the range of <15%, 20%). * Zeroth state of reliability--The system is damaged (not fulfilling (regulatory) requirements); the corrosion wear of steel plates exceeds 20%. Since the limit of acceptable reliability is 90%, the first reliability state was chosen as critical state \(r\). Empirical data of corrosion values were used to estimate reliability characteristics. All data were obtained during regular special surveys after five, ten, fifteen, twenty, and twenty-five years of operation. In our study, reliability characteristics were determined separately for the upper and lower parts of the girders. First, a reliability assessment was conducted for the upper side of the girders. Table 2 shows the number of steel plate measurements for the upper sides of girders in particular reliability states \(l\), \(l\in\{0,1,2\}\). Further, using the data in Table 3, the coordinates of the reliability function values \(R(t,l)\), \(l\in\{0,1,2\}\), after five, ten, fifteen, twenty, and twenty-five years were estimated. These values were estimated as the probability that the upper side of the girder's plates is in reliability state subset {0, 1, 2}, {1, 2}, or {2}, and they are presented in Table 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Time (Year)**} & **Number of Measurements That Are** & **Number of Measurements** & **Number of Measurements** \\  & **\textless{0.15\%) of the Original Thickness**} & **That Are <15\%, 20\%) of the Original Thickness** & **That Exceed 20\% of the Original Thickness** \\ \hline
0–5 & 90 & 0 & 0 \\
5–10 & 99 & 0 & 1 \\
10–15 & 152 & 16 & 22 \\
15–20 & 299 & 13 & 88 \\
20–25 & 65 & 2 & 113 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The amount of measurements for steel plates of the upper side of girder in reliability states \(l\), \(l\in\{0,1,2\}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Time (Years)**} & **Number of Measurements** & **Number of Measurements** & **Number of Measurements in** & **Number of Measurements in** \\  & **Measurements** & **Reliability State** & **Reliability State** & **Reliability State** \\  & & **Subset {0, 1, 2}** & **Subset {1, 2}** & **Subset {2}** \\ \hline
0–5 & 90 & 90 & 90 & 90 \\
5–10 & 100 & 100 & 99 & 99 \\
10–15 & 190 & 190 & 168 & 152 \\
15–20 & 400 & 400 & 312 & 299 \\
20–25 & 180 & 180 & 67 & 65 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Number of measurements of upper sides of girder’s steel plates in a subset of reliability states \(\{l,l+1,\ldots 2\}\). Because the limit of accepted reliability is 90%, reliability state \(1\) was chosen as a critical state. From the data presented in Table 4 and from Figure 5, it can be seen that. for ships between ten and fifteen years of age, the reliability function component \(R(t,1)\) has fallen below the acceptable level of 90%. Now, we have to determine when exactly the risk function exceeds the acceptable level of 0.1.