# An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties
## 5 Numerical Examples
### A Wing Structure  
The reliability problem of the wing structure [45] is adopted to verify the application of the presented approach in actual engineering. Figure 8 shows that the wing structure consists of three ribs and a skin. The load \(F\) is applied to the left side of the wing, and the other side is constrained in three directions. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**ns**} & \multicolumn{8}{c}{**Design Reference Period (Year)**} & \multirow{2}{*}{**Time**} \\ \cline{2-2} \cline{5-11}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline
30,000 & 3.41 & 3.20 & 3.10 & 2.94 & 2.88 & 2.82 & 2.76 & 2.72 & 2.68 & 2.66 & 9.9 \\ \hline
100,000 & 3.34 & 3.15 & 3.04 & 2.94 & 2.85 & 2.78 & 2.73 & 2.69 & 2.65 & 2.62 & 38.4 \\ \hline
500,000 & 3.33 & 3.13 & 3.02 & 2.92 & 2.83 & 2.78 & 2.73 & 2.68 & 2.65 & 2.62 & 205.5 \\ \hline
1,000,000 & 3.30 & 3.11 & 2.99 & 2.90 & 2.83 & 2.78 & 2.73 & 2.68 & 2.64 & 2.61 & 384.4 \\ \hline
2,000,000 & 3.31 & 3.11 & 2.99 & 2.91 & 2.83 & 2.78 & 2.73 & 2.68 & 2.64 & 2.61 & 763.8 \\ \hline
3,000,000 & 3.31 & 3.11 & 2.99 & 2.90 & 2.83 & 2.77 & 2.72 & 2.68 & 2.64 & 2.61 & 1153.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Calculation results of Monte Carlo method under different sample numbers. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{8}{c}{**Design Reference Period (Year)**} & \multirow{2}{*}{**Time**} \\ \cline{2-2} \cline{5-11}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline MCS & 3.30 & 3.11 & 2.99 & 2.90 & 2.83 & 2.78 & 2.73 & 2.68 & 2.64 & 2.61 \\ Proposed method & 3.35 & 3.15 & 3.04 & 2.95 & 2.88 & 2.83 & 2.78 & 2.74 & 2.70 & 2.67 \\ Deviation/\% & 1.52 & 1.29 & 1.67 & 1.72 & 1.77 & 1.80 & 1.83 & 2.24 & 2.27 & 2.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Reliability index of the roof truss structure in each design reference period. Figure 7: Reliability index curve of roof structure in design reference period. The young's modulus of the rib as an interval variable, the Poisson's ratio and the density, are \(E_{1}\), 0.35 and 4.0 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\), respectively. Moreover, the young's modulus of the skin as an interval variable, the Poisson's ratio and the density are \(E_{2}\), 0.35 and 1.8 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\), respectively. It is worth noting that the rib thickness and skin thickness are recorded as \(th\) and \(s\), respectively. The variables [\(th\), \(s\), \(E_{1}\), \(E_{2}\)]\({}^{T}\) are independent from each other: \(th\)-\(N\)(3.0, 0.02\({}^{2}\)), \(s\)-\(N\)(3.0, 0.02\({}^{2}\)), \(E_{1}\)\(\in\)[42,750 MPa, 4,7250 MPa] \(E_{2}\)\(\in\)[28,500 MPa, 31,500 MPa]. The wing structure requires that during the design baseline period \(T\) = 10 years, the vertical deformation \(D\)(\(t\)) of the wing surface during the loading process should not exceed \(D\)(\(t\)) = \(D_{0}\)[1 + ln(1-0.0002\(t\))], and \(D_{0}\) is the initial displacement. \(D_{0}\)-\(N\)(85, 7.5\({}^{2}\)).When the design life is 5 years, the time-variant load is the \(F\), and the maximum variable load \(F_{T}\) is subjected to an extreme value I distribution, \(\mu_{F_{T}}=3000N\), \(\sigma_{F_{T}}=350N\). Therefore, the limit state function of the wing structure can be defined as:  
\[g(t)=D(t)-D_{\max}(E_{1},E_{2},F(t),s,th) \tag{43}\]  
where \(D_{\max}\)() denotes the maximum displacement of the structure, which can be achieved through the finite element method (FEM). Figure 9 shows that the wing structure adopts the finite element model composed of hexahedron elements and four-node shell elements. The model has 2899 elements and 2869 nodes, respectively. Opstruct software is used to calculate this finite element model.