# An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties
## 5 Numerical Examples
### Time-Variant Reliability Analysis of the Structure of a Mechanical Part  
The change rule of the resistance of a mechanical part [10] with time is \(R(t)=R_{0}\)exp(-at) MPa. Where a is the attenuation coefficient and interval variable, and \(R0\) is the initial resistance, obeying the lognormal distribution of \(\left(\mu_{R_{0}},\sigma_{R_{0}}\right)\) MPa. \(\mu_{R_{0}}\) and \(\sigma_{R_{0}}\) are 336.3 MPa and 33.63 MPa, respectively. It is worth noting that the permanent load effect \(A\) follows the normal distribution of \(\left(\mu_{A},\sigma_{A}\right)\) MPa. \(\mu_{A}\) and \(\sigma_{A}\) are 66.25 MPa and 4.738 MPa. Taking \(\tau=1000\) h as a period, the statistical value \(Q_{i}\) of the dynamic load \(Q(t)\) in the \(i\)th period follows the extreme value I distribution. Parameters \(\alpha_{\tau}\) and \(u_{\tau}\) equal \(\alpha_{\tau}=0.04493\) MPa\({}^{-1}\) and \(u_{\tau}=49.77\) MPa, respectively. Attenuation coefficient \(a\)\({}^{I}\) is interval variables. \(a\)\({}^{I}\) \(\in\) [1.9,2.1] \(\times\) 10\({}^{-5}\). As can be seen from the Table 1, the reliability index when the sample number is 20,000, 50,000 and 100,000 is gradually close to the result when the sample number is 3,000,000. When the sample is 500,000, 1,000,000, 2,000,000 and 3,000,000, the calculation results in convergence. The sample number is 100,000, which can meet the accuracy requirement. However, in order to make the results more accurate and stable, this paper selected 1,000,000 times. With the increase of samples, the calculation time increases gradually. The calculation time is proportional to the number of samples. For example, when the number of samples is 3,000,000 and 1,000,000, the calculation time is 870.69 s and  
Figure 3: Algorithm flow chart. 300.82 s, respectively. The number of samples is three times larger, and the calculation time is nearly three times longer. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{8}{c}{**Design Reference Period (\(h\))**} & \multicolumn{8}{c}{**Time**} \\ \cline{2-11}  & **1000** & **2000** & **3000** & **4000** & **5000** & **6000** & **7000** & **8000** & **9000** & **10,000** & **(s)** \\ \hline
20,000 & 3.481 & 3.390 & 3.156 & 2.989 & 2.903 & 2.776 & 2.669 & 2.628 & 2.530 & 2.457 & 5.26 \\ \hline
50,000 & 3.673 & 3.470 & 3.279 & 3.179 & 3.062 & 2.921 & 2.802 & 2.708 & 2.624 & 2.556 & 15.78 \\ \hline
100,000 & 3.719 & 3.414 & 3.291 & 3.102 & 2.970 & 2.870 & 2.794 & 2.694 & 2.604 & 2.519 & 30.37 \\ \hline
500,000 & 3.580 & 3.350 & 3.211 & 3.104 & 2.999 & 2.901 & 2.793 & 2.697 & 2.603 & 2.515 & 149.68 \\ \hline
1,000,000 & 3.622 & 3.393 & 3.219 & 3.094 & 2.979 & 2.876 & 2.781 & 2.690 & 2.600 & 2.510 & 300.82 \\ \hline
2,000,000 & 3.612 & 3.388 & 3.223 & 3.099 & 2.984 & 2.877 & 2.782 & 2.688 & 2.597 & 2.508 & 561.31 \\ \hline
3,000,000 & 3.599 & 3.370 & 3.215 & 3.094 & 2.981 & 2.880 & 2.786 & 2.692 & 2.597 & 2.509 & 870.69 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation results of Monte Carlo method under different sample numbers. \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 (\(h\))**} & \multicolumn{8}{c}{**Time**} \\ \cline{2-11}  & **1000** & **2000** & **3000** & **4000** & **5000** & **6000** & **7000** & **8000** & **9000** & **10,000** \\ \hline MCS & 3.622 & 3.393 & 3.219 & 3.094 & 2.979 & 2.876 & 2.781 & 2.690 & 2.600 & 2.510 \\ Proposed method & 3.600 & 3.379 & 3.227 & 3.102 & 2.991 & 2.889 & 2.793 & 2.699 & 2.608 & 2.518 \\ Deviation/\% & 0.62 & 0.41 & 0.25 & 0.27 & 0.40 & 0.45 & 0.43 & 0.34 & 0.31 & 0.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reliability index of the mechanical part in each design reference period. Figure 4: The reliability index curve of a mechanical part in the design reference period. MCS is a more accurate method to solve structural reliability problems, which is generally adopted as an accurate result to verify the accuracy of other methods.