## 5 Examples
Figure 11 presents the schematic representation of the FEA conducted on the stiffener rib, wherein the membership levels of the fuzzy variables have been established at 1. The LSF of this example is implicit, and the response output needs to be obtained by FEA. The traditional MCS method requires double-layer sampling, which requires many calls to the finite element model. The calculation cost is huge and time-consuming, and practical engineering cannot afford it. Therefore, the presented SDP method is utilized to calculate the FFP-IM index, with the analytical findings presented in Table 3. Figure 12 illustrates the convergence trend of the FFP-IM index for the fuzzy variable \(E\), demonstrating that the results stabilize when the sample count reaches 2800. Figure 11: FEA of the stiffener rib. (**a**) Mesh grids and loads and constraints. (**b**) Deformation maps. Figure 12: The convergence process of the FFP-IM index of fuzzy variable \(E\). It can also be observed from Table 3 that the ranking of the influences of the input fuzzy variables on the FFP is as follows: \(E>P_{2}>P_{1}>d\). Therefore, according to the ranking results, engineers can more effectively prioritize the collection of information for input variables with higher IM index, thereby reducing their associated uncertainties, such as \(E\), while for the variable with a smaller IM index, such as \(d\), they can reduce their attention to save the cost of designing and analyzing. Analysis of the examples demonstrates the limitations of the MCS method in solving the FFP-IM index. Its high computational cost and slow convergence rate make it unfeasible for large-scale or real-time applications, especially when handling complex structural systems. In contrast, the proposed method demonstrates superior accuracy while requiring significantly lower computational resources, thereby establishing itself as a more viable and efficient alternative for the FFP-IM index. The FFP-IM index is crucial for analyzing fuzzy structural models, enabling analysts to reduce output uncertainty efficiently and cost-effectively. Additionally, it enhances the robustness of model predictions and diminishes the FFP of structural systems. Furthermore, disregarding the uncertainty of less significant variables can decrease the dimensionality of design parameters. This approach also assists with handling the over-parameterization issue that can arise during the optimization of uncertain structural systems. By ranking the influence of fuzzy variables on failure probability, engineers can identify and prioritize the most critical variables for further analysis or optimization, simplifying the design process and reducing computational costs. In this study, three typical examples were employed to validate the effectiveness of the proposed method. These examples encompass numerical models, typical engineering structures, and complex components in practical aerospace engineering, demonstrating the applicability and universality of the method across diverse scenarios. The Ishigami function, as a classical numerical example, effectively validates the applicability of the proposed method in nonlinear problems. The ten-bar truss structure and the reinforced rib of a civil aircraft, as representative examples in the field of engineering, verify the reliability of the method in structural mechanics analysis. Through the validation of these three examples, the proposed method has been thoroughly tested across multidimensional scenarios, ranging from numerical models to real-world engineering applications. The results of this study indicate that the proposed method exhibits broad applicability and significant value for engineering applications.