## 5 Examples
**Example 3:** _Figure 8 shows a simplified model of a leading-edge stiffener rib of a civil aircraft [35], with detailed size parameters depicted in Figure 9. The stiffener rib features six circular holes, the largest of which is utilized for engine mounting to facilitate the control of the slat's movement. The top hole serves as a conduit for pipelines and cables, while the remaining four holes are designed to support the slideway of the slat. The stiffener ribs are constructed from 7050-T7451 aluminum alloy, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & \(\tau_{L}\) & \(\tau_{E}\) & \(\tau_{P_{1}}\) & \(\tau_{P_{2}}\) & \(\tau_{P_{3}}\) & \(\tau_{A_{1}}\) \\ \hline MCS & \(1.637\times 10^{-2}\) & \(2.121\times 10^{-2}\) & \(2.698\times 10^{-3}\) & \(2.973\times 10^{-4}\) & \(2.181\times 10^{-5}\) & \(1.167\times 10^{-2}\) \\ \hline SDP & \(1.609\times 10^{-2}\) & \(2.269\times 10^{-2}\) & \(2.453\times 10^{-3}\) & \(2.896\times 10^{-4}\) & \(1.791\times 10^{-5}\) & \(1.216\times 10^{-2}\) \\ \hline Error & \(1.71\%\) & \(6.98\%\) & \(9.09\%\) & \(2.58\%\) & \(17.88\%\) & \(4.20\%\) \\ \hline Method & \(\tau_{A_{2}}\) & \(\tau_{A_{3}}\) & \(\tau_{A_{4}}\) & \(\tau_{A_{5}}\) & \(\tau_{A_{6}}\) & \(\tau_{A_{7}}\) \\ \hline MCS & \(5.848\times 10^{-6}\) & \(1.103\times 10^{-2}\) & \(5.682\times 10^{-6}\) & \(1.411\times 10^{-5}\) & \(4.012\times 10^{-7}\) & \(9.267\times 10^{-3}\) \\ \hline SDP & \(4.041\times 10^{-6}\) & \(1.216\times 10^{-2}\) & \(3.355\times 10^{-6}\) & \(9.583\times 10^{-6}\) & \(2.716\times 10^{-7}\) & \(9.916\times 10^{-3}\) \\ \hline Error & \(30.90\%\) & \(10.24\%\) & \(40.96\%\) & \(32.07\%\) & \(24.82\%\) & \(7.00\%\) \\ \hline Method & \(\tau_{A_{6}}\) & \(\tau_{A_{9}}\) & \(\tau_{A_{10}}\) & \(N^{\circ}\) & \(t\) & \\ \hline MCS & \(5.780\times 10^{-3}\) & \(2.470\times 10^{-4}\) & \(5.727\times 10^{-7}\) & \(5\times 10^{8}\) & \(1.424\times 10^{9}\) & \\  & & & & & & \\ \hline SDP & \(5.319\times 10^{-3}\) & \(2.561\times 10^{-4}\) & \(3.598\times 10^{-7}\) & \(3200\) & \(\begin{array}{c}9.110\times 10^{3}\\ \text{s}\end{array}\) & & & \\ \hline Error & \(7.97\%\) & \(3.69\%\) & \(37.17\%\) & — & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Analysis results of the FFP-IM index, for example, 2. Figure 7: Example 2: Comparison of FFP-IM results obtained using the MCS and SDP methods. characterized by a specific Poisson's ratio. The left end of the stiffener rib is fixed, and aerodynamic loads \(P_{1}\) and \(P_{2}\) are applied at the edges of the stiffener rib, as illustrated in Figure 10.  
The input variables include web thickness \(d(\mathrm{mm})\), the modulus of elasticity of aluminum alloy \(E(\mathrm{MPa})\), and aerodynamic loads \(P_{1}(\mathrm{MPa})\) and \(P_{2}(\mathrm{MPa})\). The membership functions for these variables are presented as follows:  
\[\mu_{d}(d)= \begin{cases}\frac{(d-4.75)}{(d.025)},4.75\leq d\leq 5\\ \frac{(d-5.25)}{-0.25},5\leq d\leq 5.25\end{cases} \]  
Figure 8: The stiffener rib is on the leading edge of a civil aircraft. Figure 10: Force analysis of the stiffener rib. Figure 9: Simplified stiffener rib structure. \[\mu_{E}(E)=\left\{ \begin{array}{l}\frac{(E-9.5\times 10^{10})}{5\times 10^{9}},9.95\times 10^{10}\leq E\leq 10^{11}\\ \frac{(E-1.05\times 10^{11})}{-5\times 10^{9}},10^{10}\leq E\leq 1.05\times 10^{11} \end{array} \right.\]  
\[\mu_{P_{1}}(P_{1})=\left\{ \begin{array}{l}\frac{(P_{1}-4750)}{280},4750\leq P_{1}\leq 5000\\ \frac{(P_{1}-5250)}{-250},5000\leq P_{1}\leq 5250\end{array} \right.\]  
\[\mu_{P_{2}}(P_{2})=\left\{ \begin{array}{l}\frac{(P_{2}-4750)}{280},4750\leq P _{2}\leq 5000\\ \frac{(P_{2}-5250)}{-250},5000\leq P_{2}\leq 5250\end{array} \right.\]  
The LSF \(Y=g=0.061-\max(u)\) is constructed under the constraint that the maximum longitudinal displacement of the structure does not exceed 0.061 mm. In this context, \(\max(u)\) is defined as an implicit function of the displacement \(d\), the elastic modulus \(E\), and the external loads \(P_{1}\) and \(P_{2}\).