## 6 Numerical Application
### Truss Bridge Structure  
In this section, the proposed reliability analysis framework is employed to assess the probability of failure in a truss structure under the influence of multiple point loads. These point loads are categorized into two distinct groups: \(F_{1}\) and \(F_{2}\) forces, which are applied to the joints on the bottom chords, and the \(F_{3}\) force group, which is applied to the joints on the top chords. A visual representation of this configuration can be found in Figure 9. The force groups, denoted as \(F_{1}\), \(F_{2}\), and \(F_{3}\), are modeled as random variables and are assumed to follow a lognormal distribution. Additionally, the three moduli of elasticity, \(E_{1}\) for the bottom chords, \(E_{2}\) for the top chords, and \(E_{3}\) for the diagonal members, are also treated as random variables and are assumed to follow a Gumbel distribution. The statistical characteristics of these moduli, including their means and moments, are succinctly presented in Table 5. Moreover, Figure 10 provides a visual representation of the correlation matrix, illustrating the inter-relationships between the various random variables involved in the reliability analysis. In the initial phase, a deterministic structural analysis is performed on the truss structure, incorporating the cross-sectional areas outlined in Table 6. This analysis utilizes the mean values of the forces and modulus of elasticity with the objective of determining both the maximum vertical displacement and normal stress within the structure. The results of this deterministic analysis are depicted in Figure 11. Then, probabilistic constraints are established for both the maximum displacement and stress. These constraints are characterized by specific threshold values, where the maximum displacement is limited to \(\Delta_{p}=1.0\) inch, and the allowable stress is set at \(\sigma_{aw}=45\) ksi. Therefore, the limit-state function is written as:  
\[ \begin{split} g(\mathbf{X}):\text{displacement}&=\{ \Delta_{p}-\text{max}(u(\mathbf{X}))\leq 0\}\\ g(\mathbf{X}):\text{stress}&=\{\sigma_{aw}-\text{ max}(|\sigma(\mathbf{X})|)\leq 0\}\end{split}  \tag{42}\]  
Figure 10: Correlation matrix of random variables for the truss bridge structure. Figure 9: A truss bridge structure geometry and loading and boundary conditions. The results of reliability analysis presented in Table 7, utilizing FORM, SORM, MCS, AK-MCS, IS, and the proposed HAK method, indicate that the overall computational cost is lower for FORM, but it yields less accurate results. In contrast, the other approaches produce results that closely align with each other. In the exploration of noise impact on the truss structure problem, noise, denoted as \(\varepsilon\) with a variance of \(\sigma_{\text{noise}}^{2}\), is introduced. This study assumes that observed responses, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variables** & **Distribution** & **Mean** & Moments \\ \hline \(E_{1}\) (BC) & Gumbel & 29,000 ksi & 2900 ksi \\ \(E_{2}\) (TC) & Gumbel & 29,000 ksi & 5800 ksi \\ \(E_{3}\) (DG) & Gumbel & 29,000 ksi & 8700 ksi \\ \(F_{1}\) & Lognormal & 30 kips & 9 kips \\ \(F_{2}\) & Lognormal & 50 kips & 10 kips \\ \(F_{3}\) & Lognormal & 60 kips & 6 kips \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Cross sectional areas of the truss bridge structure and threshold values of probabilistic constraints.