# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 3 Numerical Verification
### A 1220-Bar Steel Structure  
The second example used to verify the proposed SFE-KS method is the 1220-bar structure shown in Figure 7. The structure has 15 spans horizontally (each span being 0.2 m) and 20 layers vertically (each layer being 0.2 m). To conserve space on the graphic, the dashed lines in Figure 6 represent the omitted members of the structure, while the solid lines represent some of the members. The elastic modulus of the material used in this structure is 206 GPa, and the density is 7850 kg/m\({}^{3}\). The cross-sectional area of each member is assumed to be a random variable that obeys normal distribution with a mean of 59.66 mm\({}^{2}\) and a standard deviation of 5.966 mm\({}^{2}\). Without loss of generality, 1000 samples were gathered to obtain the statistical characteristics of the static response of the structure and the failure probability. Structural failure is defined as the horizontal displacement at the loading point exceeding a certain threshold. This threshold is taken as 1.02 times the displacement of the structural vertex without considering randomness. The truss structure under consideration has 640 degrees of freedom. Utilizing the flops estimation formula detailed in the previous section, it was found that the flops counts for a complete analysis, the NSFE method, and the proposed method are 262,553,600, 4,096,000, and 2,460,172, respectively. The proposed method emerges as the most efficient option, requiring the lowest number of flops. Furthermore, Table 9 provides the calculation time of the complete analysis, NSFE method, and the proposed method to compare the calculation efficiencies of various methods. Figure 8 gives the calculated horizontal displacements at the loading point using the three methods. Table 10 presents the range, mean, standard deviation, and failure probability of the horizontal displacement at the loading point. As shown in Table 9, the time required for the NSFE method is approximately 13% of the full analysis, while the SFE-KS approach requires approximately 8% of the full analysis time and 68% of the NSFE algorithm time. Clearly, the proposed SFE-KS method exhibits the highest computational efficiency. From Table 10, one can find that the calculation results of the proposed method are closer to the exact solution obtained using the complete analysis than those of the NSFE method.  
Figure 6: Calculation results of displacement at mid-span point using SFE-KS method with \(m=3\) and \(m=4\).  
\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 & [8.1852, 8.6521] & [8.1840, 8.6502] & [8.1847, 8.6513] \\ \hline \multirow{2}{*}{Mean, mm} & \multirow{2}{*}{8.3907} & 8.3900 & 8.3903 \\  & & (0.008\%) * & (0.005\%) \\ \hline \multirow{2}{*}{Standard deviation} & \multirow{2}{*}{0.0726} & 0.0725 & 0.0725 \\  & & (0.138\%) & (0.138\%) \\ \hline \multirow{2}{*}{Failure probability} & \multirow{2}{*}{0.052} & 0.049 & 0.051 \\  & & (5.769\%) & (1.923\%) \\ \hline \hline \end{tabular}  
* The data in brackets indicate the relative error between the calculated value and the exact solution.  
\end{table}
Table 10: Statistical characteristics of horizontal displacement for example 2.  
Figure 8: Horizontal displacement of 1000 samples for example 2.  
Figure 7: A 1220-bar structure.