# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 3 Numerical Verification  
### A Beam Structure  
The first example used to verify the proposed SFE-KS method is the I-beam structure shown in Figure 1. The structure has a span of 6.2 m and is divided into 62 beam elements. The cross-sectional area of the I-beam used is 53.54 cm\({}^{2}\), the moment of inertia is 5280 cm\({}^{4}\), and the density is 7800 kg/m\({}^{3}\). Assuming that the elastic modulus of the material follows a normal distribution with a mean value of 200 GPa, three scenarios are considered for the standard deviation: 0.1, 0.15, and 0.2 times the mean value. The determination of the number of random samples can be estimated by dividing 100 by the failure probability. For example, if the failure probability is 0.01, then the number of samples can be taken as 100/0.01 = 10,000. In the calculation process, all samples need to be computed to obtain the failure probability, and there is no specific convergence criterion used to terminate the computation. Theoretically, the more samples are used in the calculation, the more accurate the resulting failure probability will be. However, due to the limitations of computer hardware, only a subset of samples can be practically utilized in this work. Without loss of generality, \(10^{4}\) random samples are taken from the structure to analyze the random response and failure probability of the beam structure under the illustrated load (each concentrated force is 2.5 kN). The vertical displacement at the mid-span point is taken as the observation object. The displacement threshold is set at 1.06 times the mid-span displacement without considering randomness, and samples exceeding this displacement threshold are considered failures. The diagram in Figure 2 intuitively illustrates the detailed operations of each step in the proposed method. As stated before, the larger the value of \(m\) is, the higher the calculation accuracy will be, but the calculation time will also increase. Therefore, the appropriate choice of \(m\) should strike a balance between calculation accuracy and efficiency. For random analysis scenarios with relatively small variances, setting \(m=4\) for the NSFE method and \(m=2\) for the proposed method can meet the computational requirements. For random analyses with relatively large variances, if necessary, \(m\) can be increased to enhance the computational accuracy at the cost of sacrificing the computational efficiency. This beam structure possesses 122 degrees of freedom. Applying the flops estimation formula discussed in the previous section, it was found that the flops counts for a complete analysis, the NSFE algorithm, and the proposed method are 1,830,732, 148,840, and 89,804, respectively. It is clear that the proposed method demands the lowest number of flops, highlighting its efficiency. For the case where the standard deviation is 0.1 times the mean value, the computation times required for the full analysis method, the NSFE algorithm (with \(m=4\) items), and the proposed SFE-KS algorithm (with the dimension of the Krylov subspace being \(m=2\)) are listed in Table 1. The calculated vertical displacement at the mid-span point is shown in Figure 3. The range, mean, standard deviation, and failure probability of the displacement at the mid-span point are listed in Table 2. As seen in Table 1, the computation times of both the NSFE and SFE-KS algorithms are less than that of the full analysis. Specifically, the time required for the NSFE approach is approximately 20% of the full analysis, while the SFE-KS method requires approximately 17% of the full analysis time and 87% of the NSFE algorithm time. Clearly, the proposed SFE-KS method exhibits the highest computational efficiency. In Table 2, for the vertical displacement at the mid-span point, the calculation results of the SFE-KS method are closer to the results of complete analysis than those of the NSFE method. Figure 1: A beam structure. Figure 2: Operation flow of the proposed method.