# The Multistate Reliability Assessment of Ship Hull Girders
## 4 Results of Reliability Analysis of Longitudinal Watertight
### Estimation of Reliability and Risk
Since we chose state 1 as the reliability critical state, we determine the moment when the steel plates risk exceeds the permitted threshold \(\delta\) = 0.1. This will help us answer the question when the reliability of the upper side of the girder should be improved (when the steel plates should be replaced). In the reliability analysis performed, the reliability function coordinates in reliability state subsets [1, 2] and [2] are not at our disposal because the system lifetimes in these subsets were not available. Only the values of the reliability function at five, ten, fifteen, twenty, and twenty-five years were designated based on the thickness reduction measurements conducted during the surveys. From a mathematical analysis, we know that, if the values of a function at a finite number of points are known, then for any distribution, such a function can be interpolated using Lagrange polynomial interpolation. Thus, using Lagrange polynomial interpolation, the reliability function coordinate \(R(t,1)\) for the upper side of the girder is given by the following formula:  
\[ \begin{split} R(t,1)\approx\overset{\sim}{R}(t,1)=R_{0}(t,1)p_ {0}(t)+R_{1}(t,1)p_{1}(t)+R_{2}(t,1)p_{2}(t)\\ +R_{3}(t,1)p_{3}(t)R_{4}(t,1)p_{4}(t)\\ =p_{0}(t)+0.991_{1}(t)+0.884p_{2}(t)+0.780p_{3}(t)+0.372p_{4}(t), \end{split}  \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Time (Years)** & **R(t, 0)** & **R(t, 1)** & **R(t, 2)** \\ \hline
5 & 1 & 1 & 1 \\
10 & 1 & 0.99 & 0.99 \\
15 & 1 & 0.884 & 0.800 \\
20 & 1 & 0.780 & 0.748 \\
25 & 1 & 0.372 & 0.361 \\ \hline \end{tabular}
\end{table} 
Table 4: The estimated values of the reliability function in reliability state subsets {0, 1, 2], [1, 2], and [2] for the upper side of the girder’s plates. Figure 5: Graphical interpretation of function \(R(t,l)\) coordinates. where  
\[ \begin{array}{l}p_{0}(t)=\prod_{j=0}^{4}\frac{t-l_{j}}{l_{0}-l_{j}}=\frac{(t-l_{ 1})(t-l_{2})(t-l_{3})(t-l_{4})}{(t_{0}-l_{2})(t_{0}-l_{3})(t_{0}-l_{4})}\\ j\
eq 0\\ =\frac{(t-10)(t-15)(t-20)(t-25)}{15,000},\\ p_{1}(t)=\prod_{j=0}^{4}\frac{t-l_{j}}{l_{1}-l_{j}}=\frac{(t-l_{0})(t-l_{2})(t -l_{5})(t-l_{4})}{(t_{1}-l_{0})(t_{1}-l_{2})(t_{1}-l_{3})(t_{1}-l_{4})}\\ j\
eq 1\\ =-\frac{(t-5)(t-15)(t-20)(t-25)}{3750},\\ p_{2}(t)=\prod_{j=0}^{4}\frac{t-l_{j}}{l_{2}-l_{j}}=\frac{(t-l_{0})(t-l_{1})(t -l_{3})(t-l_{4})}{(t_{2}-l_{0})(t_{2}-l_{1})(t_{2}-l_{3})(t_{2}-l_{4})}\\ j\
eq 2\\ =\frac{(t-5)(t-10)(t-20)(t-25)}{2500},\\ p_{3}(t)=\prod_{j=0}^{4}\frac{t-l_{j}}{l_{3}-l_{j}}=\frac{(t-l_{0})(t-l_{1})( t-l_{2})(t-l_{4})}{(t_{3}-l_{0})(t_{3}-l_{1})(t_{3}-l_{2})(t_{3}-l_{4})}\\ j\
eq 3\\ =-\frac{(t-5)(t-10)(t-15)(t-25)}{3750},\\ p_{4}(t)=\prod_{j=0}^{4}\frac{t-l_{j}}{l_{4}-l_{j}}=\frac{(t-l_{0})(t-l_{1})( t-l_{2})(t-l_{3})}{(t_{4}-l_{0})(t_{4}-l_{1})(t_{4}-l_{2})(t_{4}-l_{3})}\\ j\
eq 4\\ =\frac{(t-5)(t-10)(t-15)(t-20)}{15,000}\end{array}  \tag{8}\]  
Through further considerations, using (7)-(12), the moment when the steel plates' risk function \(r(t)=1-R(t,1)\) exceeds an acceptable level of 0.1 was found. The approximately risk and reliability functions are given in Figure 6.  
According to Formulas (3), (4), and (7)-(12), when the risk function of the steel plate crossed the acceptable threshold \(\delta\) = 0.1 was at \(\tau\) = 14.2 years. The above analysis shows that the risk function will exceed the acceptable level of 0.1 after only 14.2 years. This indicates  
Figure 6: Approximate function obtained from Lagrange interpolation. that the upper side of the girder plates' reliability should be improved by replacing corroded surfaces after the second special survey/inspection, i.e., after ten years of operation. This replacement could minimize hazardous effects of corrosion, such as environmental pollution or structural damage. A similar reliability analysis was performed for the lower side of girders. Table 5 presents the number of measurements of the lower side of the girders' steel plates in the particular reliability states \(l\), \(l\in\{0,1,2\}\).