# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 3 Numerical Verification
### A Beam Structure
For the case where the standard deviation is 0.2 times the mean value, the computation times required for the full analysis method, the NSFE algorithm, and the proposed SFE-KS algorithm are listed in Table 5. The calculated vertical displacement at the mid-span point is shown in Figure 5. The range, mean, standard deviation, and failure probability of the displacement at the mid-span point are listed in Table 6. From Table 5, one can see that the proposed SFE-KS method has the highest computational efficiency. In Table 6, for the vertical displacement at the mid-span point, the calculation results of the SFE-KS approach are closer to the results of complete analysis than those of the NSFE method. However, it can also be found in Tables 2, 4 and 6 that with the increase in the standard deviation of the elastic modulus, the error of the calculation results of the proposed method increases accordingly. By comparing Figures 3-5 together, it can be observed that the results obtained using the three methods in Figures 3 and 4 show little difference. However, the results from the full analysis in Figure 5 differ significantly from those obtained using the two other rapid algorithms. This is attributed to the increasing standard deviation of the elastic modulus, which leads to increasingly larger computational errors in the two rapid algorithms. Consequently, the graphs representing the results from the two rapid algorithms in Figure 5 differ markedly from the precise solution graph obtained from the complete analysis. If necessary, the calculation accuracy of the proposed method can be improved by increasing the value of \(m\). Table 7 presents the computational time of the proposed method when \(m=3\) and \(m=4\), while Table 8 and Figure 6 provide the corresponding calculation results for these two cases. It can be seen in Table 7 that the calculation time of the proposed method increases with the increase in \(m\). From Table 8, it can be concluded that the calculation result of the proposed method is closer to the exact solution obtained using the complete analysis with the increase in \(m\). By comparing Figure 6b with Figure 5a, it can be observed that the calculation results of the proposed method when \(m=4\) are almost identical to the exact solution of the complete analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Complete Analysis** & **NSFE (with \(m=4\))** & **SFE-KS (with \(m=2\))** \\ \hline Vertical displacement range, mm & [8.4935, 10.5694] & [8.4616, 10.4601] & [8.4815, 10.5367] \\ \hline Mean, mm & 9.3172 & \begin{tabular}{c} 9.3030 \\ (0.152\%) * \\ \end{tabular} & \begin{tabular}{c} 9.3080 \\ (0.096\%) \\ \end{tabular} \\ \hline Standard deviation & 0.2622 & \begin{tabular}{c} 0.2587 \\ (1.335\%) \\ \end{tabular} & \begin{tabular}{c} 0.2610 \\ (0.458\%) \\ \end{tabular} \\ \hline Failure probability & 0.0998 & \
\begin{tabular}{c} 0.0881 \\ (11.723\%) \\ \end{tabular} &
\begin{tabular}{c} 0.0936 \\ (6.212\%) \\ \end{tabular} \\ \hline \hline \end{tabular}  
* The data in brackets indicate the relative error between the calculated value and the exact solution.  
\end{table} 
Table 4: Statistical characteristics of vertical displacement at the mid-span point when the standard deviation is 0.15 times the mean value. Figure 4: Calculation results of displacement at the mid-span point when the standard deviation is 0.15 times the mean value. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Complete Analysis** & **NSFE (with \(m\) = 4)** & **SFE-KS (with \(m\) = 2)** \\ \hline Vertical displacement range, mm & [8.3197, 18.2692] & [8.2519, 11.2314] & [8.3259, 11.4614] \\ \hline Mean, mm & 9.5103 & 9.4589 & 9.4768 \\  & (0.540\%) * & (0.352\%) \\ \hline Standard deviation & 0.3955 & 0.3670 & 0.3748 \\  & (7.206\%) & (5.234\%) \\ \hline Failure probability & 0.3277 & 0.2824 & 0.2981 \\  & (13.824\%) & (9.033\%) \\ \hline \hline \end{tabular}  
* The data in brackets indicate the relative error between the calculated value and the exact solution.  
\end{table}