## 5 Examples
### Example 2: Corroded Simply Supported Beams with Results and Discussion  
This example considers continuously corroding steel simply supported beam [3], as shown in Figure 6. The beam length \(L=5\) m and the cross-section is rectangular (\(a_{0}=0.2\) m, \(b_{0}=0.04\)m). The beam weight \(p=\rho_{st}a_{0}b_{0}\) (N/m), where \(\rho_{st}=78.5\)kN/m\({}^{3}\). The beam carries load \(F\) in the span.  
Beam cross-sectional area changes after rusting. Assuming that the corroded portion loses all structural properties, the cross-sectional area of the beam at moment t is expressed as follows:  
\[s(t)=a(t)\cdot b(t) \tag{38}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Time (s)** & **Err (\%)** \\ \hline
12.5 & 7.80 \\
13 & 4.70 \\
13.5 & 2.04 \\
14 & 0.30 \\
14.5 & 0.13 \\
15 & 0.03 \\ \hline \hline \end{tabular}
\end{table}
Table 1: EDFK-FCTP relative error of mathematical problem.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & \(N_{call}\) & \(N\) \\ \hline MCS-FCTP & \(2.95\times 10^{7}\) & \(10^{5}\) \\ EDFK-FCTP & \(1643\) & 21 \\ ASVR-FCTP & \(1646\) & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of calculation results of various methods for Mathematical Problem.  
Figure 6: Scheme of corroded simply supported beams.  
where \(a(t)=a_{0}-2\kappa t\), \(b(t)=b_{0}-2\kappa t\), \(\kappa\) is corrosion rate, taken as 0.03 mm/year. The mid-span cross-section of the beam is hazardous cross-section, and the actual bending moment is  
\[M=\frac{FL}{4}+\frac{PL^{2}}{8} \tag{39}\]  
The ultimate bending moment of mid-span section at failure of the structure is  
\[M_{u}(t)=\frac{a(t)b^{2}(t)}{4}\sigma_{s} \tag{40}\]  
where \(\sigma_{s}\) is yield strength of the material.  
Therefore, the LSF of beam is expressed as follows:  
\[g(\mathbf{Z},t)=M_{u}(t)-M=\frac{a(t)b^{2}(t)}{4}\sigma_{s}-\left(\frac{FL}{4} +\frac{\rho_{st}a_{0}b_{0}L^{2}}{8}\right) \tag{41}\]  
where \(\mathbf{Z}=[\sigma_{s},a_{0},b_{0},F]\) is the random input variable, and the relevant parameters are shown in Table 3.  
From the probability distributions in Figure 7 and the relative errors in Table 4, the accuracy of the EDFK-FCTP results can be guaranteed. The cumulative failure probability trend of the three methods is demonstrated in Figure 8. Similarly to the mathematical problem, the overall results are better, and the fitting accuracy of EDFK-FCTP is better than ASVR in the low failure probability region.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Time (Year)** & **Err (\%)** \\ \hline
60 & 0.04 \\
120 & 9.03 \\
180 & 7.03 \\
240 & 0.32 \\
300 & 0.44 \\
360 & 0.45 \\ \hline \hline \end{tabular}
\end{table}
Table 4: EDFK- FCTP relative error of corroded simply supported beams.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Distribution** & **Mean** & **Cov (\%)** \\ \hline \(\sigma_{s}\) & Lognormal & 240 MPa & 0.1 \\ \(a_{0}\) & Lognormal & 0.2 m & 0.05 \\ \(b_{0}\) & Lognormal & 0.04 m & 0.1 \\ \(F\) & Lognormal & 3500 N & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Stochastic parameter distribution of corroded simply supported beams.  
Figure 7: FCTP probability density function of corroded simply supported beams.  
The computational efficiencies of the various methods are compared, and the results are shown in Table 5. From Table 5, the \(N_{call}\) and \(N\) used in EDFK-FCTP are less than ASVR, and the efficiency is improved to some extent.