## 6 Numerical Application
### Truss Bridge Structure
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Limit-State Function** & **Method** & \(P_{f}\) & \(\Delta p_{f}\) & \(\beta\) & \(N_{\text{e}all}\) \\ \hline \multirow{5}{*}{Stress} & FORM & \(3.76\times 10^{-3}\) & - & 2.673 & 48 \\  & SORM & \(3.95\times 10^{-3}\) & - & 2.656 & 128 \\  & MCS & \(3.92\times 10^{-3}\) & 0.05 & 2.659 & 10\({}^{5}\) \\  & AK-MCS & \(4.11\times 10^{-3}\) & 0.05 & 2.643 & 52 \\  & IS & \(4.22\times 10^{-3}\) & 0.05 & 2.634 & 1048 \\  & HAK & \(4.02\times 10^{-3}\) & 0.02 & 2.650 & 43 \\ \hline \multirow{5}{*}{Displacement} & FORM & \(9.19\times 10^{-2}\) & - & 1.329 & 40 \\  & SORM & \(9.35\times 10^{-2}\) & - & 1.320 & 120 \\ \cline{1-1}  & MCS & \(9.43\times 10^{-2}\) & 0.01 & 1.315 & 10\({}^{5}\) \\ \cline{1-1}  & AK-MCS & \(9.51\times 10^{-2}\) & 0.01 & 1.311 & 166 (10\({}^{5}\)) \\ \cline{1-1}  & IS & \(9.44\times 10^{-2}\) & 0.04 & 1.314 & 1040 \\ \cline{1-1}  & HAK & \(9.41\times 10^{-2}\) & 0.01 & 1.316 & 68 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Reliability analysis results for the truss bridge structure obtained through various approaches without noise. Figure 11: Illustration of deformed shape and truss member stresses with mean values of random variables. Tensile stresses are denoted in blue, and compressive stresses are shown in red. The maximum vertical displacement node, its corresponding value, and the maximum stress are highlighted. \begin{table}
\begin{tabular}{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 5: Distribution type and parameters of the random variables considered for the reliability analysis of the truss bridge structure. including normal stress, are subject to noise. Specifically, varying variances of noise are added to a truss member where the maximum stress is observed. The proposed framework, outlined in Section 4, is applied to estimate the failure probability of the stress limit-state function in the presence of noise. Four different variances (\(\sigma_{\text{noise}}^{2}=\{5.06,9.92,20.25,45.56\}\)) kips\({}^{2}\)) of Gaussian noise corresponding to coefficients of variation (c.o.v) of 0.05, 0.07, 0.1, and 0.15, respectively, are considered. After five implementations of the proposed framework for each noise variance case, the average failure probabilities are computed. The minimum and maximum failure probabilities for each variance case are also recorded to demonstrate the probability range. For a comprehensive comparison, reliability analysis results of AK-MCS, which does not consider regression progress to account for noise in the reliability analysis, are presented. It is worth noting that methods employing linearization (FORM) and curvature (SORM) to approximate the limit-state function face challenges in estimating failure probability. Specifically, the FORM method resulted in a 50% failure probability, and SORM failed to converge to a stable result. Figure 12 illustrates the ranges of estimated failure probability using the proposed method. The proposed framework demonstrates commendable ability in estimating failure probability under varying noise conditions introduced to stress responses. However, the failure probability range computed by AK-MCS for each variance case significantly increases as the variance rises. When noise is considered in the Kriging modeling and active learning process, a more stable outcome (lower range of failure probability) is observed with the proposed method.