## 5 Examples  
This section elucidates the application of the proposed FFP-IM index computed via the efficient SDP-based method through three typical examples. The analysis results demonstrate the effectiveness of the method in evaluating the impact of fuzzy uncertainties on structural reliability, while also confirming the computational efficiency of the proposed SDP-based solving algorithm and its practicality in real-world engineering applications. Collectively, these examples affirm the robustness and versatility of the FFP-IM index in addressing complex reliability analysis problems under fuzzy uncertainty conditions. In this study, the FFP-IM index is integrated into the structural analysis workflow. Specifically, in the step of calculating the IM index based on failure probability, the efficient SDP method is employed. This approach aims to enhance computational efficiency while maintaining a high level of accuracy in the IM index, thereby enabling the ranking of the impact of fuzzy input variables on structural failure probability and providing a comprehensive evaluation of the influence of fuzzy uncertainties on structural reliability. **Example 1**. : _Consider the highly nonlinear and non-monotonic Ishigami function that is usually used in IM: \(Y=\sin(Z_{1})+a\sin^{3}(Z_{2})+bZ_{3}^{4}\sin(Z_{1})+c\), where the constants \(a\), \(b\), and \(c\) are 2, 0.1, and 3, and \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\) represent fuzzy input variables characterized by triangular membership functions, defined as follows:_  
\[\mu_{Z_{i}}(z_{i})=\left\{ \begin{array}{l}\left(z_{i}+\frac{\pi}{2}\right)/ \frac{\pi}{2}\ -\frac{\pi}{2}\leq z_{i}\leq 0\\ \left(z_{i}-\frac{\pi}{2}\right)/-\frac{\pi}{2}\ 0\leq z_{i}\leq\frac{\pi}{2} \end{array} \right.,\ (i=1,2,3).\]  
Figure 3: Flowchart of FFP-IM index analysis. Table 1 shows the analysis results for the FFP-IM index acquired via the MCS and SDP methods. To present the results more clearly, Figure 4 illustrates a comparison. It can be seen from Table 1 and Figure 4 that the ordering of FFP-IM is as follows: \(Z_{2}>Z_{3}>Z_{1}\). This indicates that the fuzzy variable \(Z_{2}\) exerts the most significant influence on the FFP for the structural system and should receive greater consideration in reliability design. Utilizing the MCS method, we compute the conditional failure probability based on a robust sample size of \(N_{\text{inner}}=10000\), and \(N_{\text{outer}}=5000\) samples are used for the mathematical expectation. For this example, the LSF is non-monotonic, and optimization algorithms are needed to calculate the NPR index, so the sequential quadratic optimization approach is used in this paper. As illustrated in Figure 4, the SDP method attains a level of accuracy comparable to that of the MCS method; yet, it achieves this with a substantially lower computational burden. Specifically, the MCS method necessitates a cumulative sample count of 4 \(\times\) 10\({}^{8}\), whereas the SDP method accomplishes the task with merely 1500 samples. Upon contrasting the results procured from both methodologies, as illustrated in Table 1 and Figure 4, it is apparent that the SDP method has good calculation accuracy, which further illustrates its efficiency and high accuracy. **Example 2**. : _In practical engineering problems, the input-output response relationship usually cannot be formulated explicitly, and FEA is often used to obtain input-output._  
A ten-bar structure, illustrated in Figure 5, has been chosen for analysis. The extent and cross-sectional area of the horizontal and vertical bars are designated by \(L(\text{m})\) and \(A_{i}(i=1,\cdots,6)(\text{m}^{2})\), correspondingly. For bars oriented diagonally, the length and cross-sectional area are given as \(\sqrt{2}L\) and \(A_{i}(i=7,\cdots,10)\), respectively. The elastic modulus applicable to all bars is denoted as \(E(\text{GPa})\), whereas \(P_{1}(\text{N})\), \(P_{2}(\text{N})\), and \(P_{3}(\text{N})\) stand for the external loads.