# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 4 Examples Verification
### Unequal Span Three-Span Continuous Beam  
Example 5 is a simple finite element model, that is, a three-span continuous beam with unequal spans, which is common in engineering structures. The specific layout is shown in Figure 8. The side span is \(L=20\) m, and the middle span is 30 m. The rectangular beam section is adopted, with a height of 1.4 m and a width of 0.8 m. The middle span acts as a uniform load \(Q\), and the middle span bears a concentrated load \(P\). See Table 16 for the random variable distribution law of the unequal span three-span continuous beam.  
The maximum allowable deflection of the main beam is \(1.5\)L\(/360=1/12\) m, so the structural performance function of the unequal span three-span continuous beam is defined as:  
\[G(X)=\frac{1.5L}{360}-w_{max} \tag{17}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Random Variables** & **Distribution** & **Mean** & **Standard Deviation** \\ \hline \(E\) (kN/m\({}^{2}\)) & Normal & \(3\times 10^{7}\) & \(0.6\times 10^{7}\) \\ \(I\) (m\({}^{4}\)) & Normal & 0.1829 & 0.03 \\ \(q\) (kN/m) & Normal & 10.5 & 2.1 \\ \(P\) (kN) & Normal & 320 & 64 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Random variable distribution of Three-unequal-span continuous girder.  
Figure 8: Three-unequal-span continuous girder.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(N_{\text{FEM}}\) & \(N_{\text{\it call}}\) & \(P_{\text{\it f}}\) & \(COV_{\text{\it Pf}}\) & \(\epsilon\) \\ \hline MCS [34] & \(5\times 10^{6}\) & \(5\times 10^{6}\) & \(1.57\times 10^{-4}\) & 0.0357 & \\ AK-IS [34] & 108 & \(5\times 10^{6}\) & \(1.5\times 10^{-4}\) & 0.0365 & 4.46\% \\ SS [35] & \(3.7\times 10^{5}\) & \(5\times 10^{6}\) & \(1.6\times 10^{-4}\) & 0.0354 & 1.91\% \\ AK-SS [35] & \(12+55.5\) & \(5\times 10^{6}\) & \(1.533\times 10^{-4}\) & 0.0361 & 2.36\% \\ Direct BP-MCS & 127 & \(5\times 10^{6}\) & \(1.592\times 10^{-4}\) & 0.0354 & 1.40\% \\ Improved BP-MCS & 127 & 132,466 & \(1.55\times 10^{-4}\) & 0.0359 & 1.27\% \\ \hline \hline \end{tabular}
\end{table}
Table 14: Reliability calculation results of multivariable response surface polynomials.  
where: \(w_{max}\) is the maximum deflection of the three-span continuous beam with unequal span, which is calculated by the ANSYS finite element model (version 2022R1). Beam188 element is used to simulate the continuous beam in ANSYS finite element model, with a total of 30 elements. The beam section is rectangular, and the constraints at the support are consistent with those in Figure 8.  
For the implicit structural performance function with four random variables, the calculation results of different methods are shown in Table 17. The Improved BP-MCS only needs 93 sets of finite element sample points to get 0.7% relative error failure probability, and its \(N_{FEM}\) is roughly the same as AK-MCS-U. AK-MCS-IRS has fewer \(N_{FEM}\), but its relative error is 1.38% and only applies to low-dimensional problems [28]. Thus, the Improved BP-MCS calculation accuracy is higher, the calculation times of finite element model is less, and the call times of BP model is reduced from \(5\times 10^{6}\) to 122,767 times.  
In the calculation process, the calculation time of failure probability of Improved BP-MCS and Direct BP-MCS is counted. When using Direct BP-MCS, the calculation time is as high as 35,939.7 s (about 10 h). When Improved BP-MCS is used, the calculation time is shortened to 1116.2 s, which is about 18.6 min. Compared with Direct BP-MCS, the calculation time of Improved BP-MCS is obviously shortened, which verifies the efficiency of the algorithm.