# An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties
## 5 Numerical Examples
### A Roof Truss Structure  
Figure 6 shows that the bottom and tension members are made of steel, and the top and pressure members are reinforced with concrete [44; 45]. \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{2-11}  & **5** & **10** & **15** & **20** & **25** & **30** & **35** & **40** & **45** & **50** \\ \hline
30,000 & 3.72 & 3.59 & 3.54 & 3.40 & 3.38 & 3.35 & 3.31 & 3.29 & 3.23 & 3.18 & 8.5 \\ \hline
100,000 & 4.10 & 3.81 & 3.69 & 3.61 & 3.54 & 3.49 & 3.43 & 3.37 & 3.32 & 3.22 & 32.4 \\ \hline
500,000 & 3.94 & 3.78 & 3.68 & 3.58 & 3.50 & 3.46 & 3.40 & 3.34 & 3.27 & 3.18 & 157.4 \\ \hline
1,000,000 & 3.94 & 3.77 & 3.67 & 3.59 & 3.52 & 3.46 & 3.40 & 3.34 & 3.26 & 3.17 & 673.8 \\ \hline
2,000,000 & 3.94 & 3.78 & 3.68 & 3.60 & 3.53 & 3.47 & 3.41 & 3.35 & 3.27 & 3.17 & 734.8 \\ \hline
3,000,000 & 3.94 & 3.77 & 3.66 & 3.58 & 3.52 & 3.46 & 3.39 & 3.34 & 3.26 & 3.16 & 1153.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation results of Monte Carlo method under different sample numbers. Figure 5: Reliability index curves of reinforced concrete short column of a structure in design reference period. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{8}{c}{**Design Reference Period (Year)**} \\ \cline{2-11}  & **5** & **10** & **15** & **20** & **25** & **30** & **35** & **40** & **45** & **50** \\ \hline MCS & 3.94 & 3.77 & 3.67 & 3.59 & 3.52 & 3.46 & 3.40 & 3.34 & 3.26 & 3.17 \\ Proposed method & 3.98 & 3.82 & 3.72 & 3.65 & 3.59 & 3.53 & 3.47 & 3.40 & 3.33 & 3.23 \\ Deviation/\% & 1.02 & 1.33 & 1.36 & 1.67 & 1.99 & 2.02 & 2.06 & 1.80 & 2.15 & 1.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Reliability index of each design reference period of the short column. The roof is subjected to evenly distributed load \(q(t)\), where \(q(t)\) can be converted to the equivalent nodal force \(P=q(t)l/4\). Moreover, the vertical displacement at node C can be expressed as follows:  
\[\Delta C=\frac{q(t)l^{2}}{2}\bigg{(}\frac{3.81}{A_{C}E_{C}}+\frac{1.13}{A_{S}E_{ S}}\bigg{)} \tag{41}\]  
where \(A_{C}\) and \(A_{S}\) denote the cross-sectional areas of the cement and steel rods, respectively. It should be indicated that \(A_{C}\) and \(A_{S}\) are interval variables with the ranges of [0.0323 m\({}^{2}\), 0.0357 m\({}^{2}\)] and [8.93 \(\times\) 10\({}^{-4}\) m\({}^{2}\), 9.87 \(\times\) 10\({}^{-4}\) m\({}^{2}\)], respectively. Moreover, \(E_{C}\) and \(E_{S}\) denote their elastic modulus, respectively. During the loading process, the vertical displacement of node C should be less than \(D(t)\) that decays with time. The decay rule is: \(D(t)\)=\(D_{0}\)[\(1+\ln(1-0.0002t)\)], and \(D_{0}\) is the initial displacement. Table 6 lists the specific situation of the random variable in this problem. The design base period is assumed to be \(T=10\) years. The limit state function is defined as:  
\[g(t)=D(t)-\Delta C \tag{42}\]  
It can be seen again from Table 7 that the calculation results based on MCS method are convergent, and the calculation time increases linearly with the increase of the number of samples. As shown in Figure 7 and Table 8, the reliability indices are 3.35, 3.15, 3.04, 2.95, 2.88, 2.83, 2.78, 2.74, 2.70 and 2.67, which decreases significantly with time. Table 7 presents that the maximum error between the results obtained by this method. It can be seen that the errors are 1.52%, 1.29%, 1.67%, 1.72%, 1.77%, 1.80%, 1.83%, 2.24%, 2.27% and 2.30%, respectively. Compared with MCS method, the maximum error is 2.30%. Once again, it should be indicated that the results obtained by this method are very accurate. From the