## 5 Examples
### Example 3: Aircraft Wing Structure with Results and Discussion  
The wing structure of a long-range transport aircraft [42] is considered, as shown in Figure 9. In the example, time parameter \(t\) denotes coordinate position on the x-axis.  
The chord length \(c(t)\) of the wing is  
\[c(t)=c_{r}\bigg{[}\left(\frac{c_{0}}{c_{r}}\right)+\left(1-\frac{c_{0}}{c_{r}} \right)\frac{t}{b/2}\bigg{]} \tag{42}\]  
where \(\frac{c_{0}}{c_{r}}\) is the leaf-root-tip ratio.  
The thickness \(h(t)\) of the wing is  
\[hc(t)=c_{r}\bigg{(}\frac{h}{c(t)}\bigg{)}\bigg{[}\left(\frac{c_{0}}{c_{r}} \right)+\left(1-\frac{c_{0}}{c_{r}}\right)\frac{t}{b/2}\bigg{]} \tag{43}\]  
Figure 8: Failure probability trend of corroded simply supported beams.  
Figure 9: Scheme of wing structure: (**a**) cross-sectional view of the wing, with T, h, and c representing the thickness, depth, and chord length of the wing, respectively; (**b**) the top view of the wing, where b denotes the span; (**c**) the loads acting on the wing.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & \(N_{call}\) & \(N\) \\ \hline MCS-FCTP & \(3.06\times 10^{7}\) & \(10^{5}\) \\ EDFK-FCTP & \(4657\) & \(42\) \\ ASVR & \(7466\) & \(49\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of calculation results of various methods for Corroded Simply Supported Beams.  
The moment of inertia \(I_{z}(t)\) is  
\[I_{z}(t)=\frac{1}{2}c(t)Th^{2}(t) \tag{44}\]  
The wing loading \(P(t)\) is  
\[P(t)=P_{r}\bigg{[}1-\bigg{(}\frac{t}{b/2}\bigg{)}^{2}\bigg{]} \tag{45}\]  
The cross-sectional area \(A(t)\) of the wing is  
\[A(t)=\frac{1}{2}[c(t)+c_{0}]t \tag{46}\]  
Therefore, the wing bending moment \(M(t)\) can be expressed as follows:  
\[M(t)=4P_{r}c_{r}b\bigg{[}\bigg{(}\frac{c_{0}}{c_{r}}\bigg{)}\frac{t^{4}}{12b^{ 2}}\bigg{]}+4P_{r}c_{r}b\bigg{[}\bigg{(}1-2\frac{c_{0}}{c_{r}}\bigg{)}\frac{t^{ 5}}{20b^{3}}+\bigg{(}\frac{c_{0}}{c_{r}}-1\bigg{)}\frac{t^{6}}{30b^{4}}\bigg{]} \tag{47}\]  
When the maximum stress in the wing structure is greater than the allowable stress \(\sigma_{f}\), the wing fails. Therefore, the LSF of the wing structure can be expressed as follows:  
\[g(\textbf{Z},t)=\sigma_{f}-\frac{M(t)[h(t)/2]}{I_{z}(t)} \tag{48}\]  
where \(\textbf{Z}=\Big{[}P_{r},\sigma_{f},c_{r},\frac{c_{0}}{c_{r}},\frac{h}{c},T \Big{]}\) is input random variable. The relevant parameters are shown in Table 6.  
EDFK-FCTP, ASVR, and MCS-FCTP are used to calculate the trend of the failure probability of the wing structure with the x-axis. As shown in Figure 10, the estimation accuracy of EDFK-FCTP is high. Comparing the three curves in Figure 11, both EDFK-FCTP and ASVR have higher accuracy in solving failure probability. However, the fitting accuracy of EDFK-FCTP is more advantageous in low failure probability region. Further, Table 7 demonstrates the relative errors of the six-moment points, indicating that the accuracy of the proposed method can guarantee the engineering requirements. Further, as shown in Table 7, the relative errors at six moment points indicate that the accuracy of the proposed method can guarantee engineering requirements.  
Comparing the computational efficiency of various methods, the results are shown in Table 8. The \(N_{call}\) and \(N\) used in the EDFK-FCTP are significantly less than the ASVR. The EDFK-FCTP method greatly improves computational efficiency.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Distribution** & **Mean** & **Cov (\%)** \\ \hline \(P_{r}\) & Normal & \(4.53\times 10^{3}\) & 0.05 \\ \(\sigma_{f}\) & Normal & \(4\times 10^{8}\) & 0.05 \\ \(c_{r}\) & Normal & 7.62 & 0.01 \\ \(\frac{c_{0}}{c_{r}}\) & Normal & 0.4 & 0.01 \\ \(\frac{c}{c}\) & Normal & 0.12 & 0.01 \\ \(\frac{c}{T}\) & Normal & \(