## 5 Engineering Application
### Negative Reaction Risk Assessment under the Live Load  
Due to errors in the manufacturing and construction processes, the material properties, tension properties, and structural geometric parameters may be affected by uncertainties. Hence, these parameters are supposed to be treated as random variables rather than deterministic variables. The random variables for the Yongjiang Bridge are listed in Table 11, including the tension errors of stay cables A1-A24, B1-B24, C1-C16, and D1-D16 and the weight errors of pavements P1-P32 and concrete girder G. In this study, all the random variables were assumed to be independent of each other to simplify the reliability and sensitivity analyses. The sensitivity analysis results are presented in Figure 10.  
As shown in Figure 10, the weight error of concrete girder G is the most significant factor affecting the RAP. Among the weight errors of the pavements, those of P2-P7 and P15-P20 have the greatest impact on the RAP, followed by those of P1, P13, P14, and P21, and the others have little impact on the RAP. Stay cables A10-A20 have significant impacts on the RAP, and the other stay cables have minimal influence on the RAP. Therefore, the weight error of the concrete girder G, the tension errors of stay cables A10-A20, and the weight errors of pavements P1-P7 and P13-P21 were treated as random variables to assess the NRRAP under the live load.  
According to the procedure proposed in Figure 6, 500 groups of input side samples were obtained by LHST [41], and the corresponding output side samples were obtained through finite element analysis and reliability analysis. The input and output side samples were trained by the GRNN to obtain the objective function and reliability constraint function. The goodness-of-fit \(R^{2}\) was used to evaluate the prediction accuracy of the GRNN surrogate model (\(R^{2}\) approaching 1.0 indicates a higher fitting accuracy). The GRNN surrogate model of RAP \(R_{0}\) was validated by FEM. As shown in Figure 11, the \(R^{2}\) value of \(R_{0}\) is 0.9897. The Monte Carlo method based on FEM was employed to validate the GRNN surrogate model of the reliability index \(\beta\), with a sample size of 5000. As shown in Figure 12, the \(R^{2}\) value of \(\beta\) is 0.958. The above results show that the prediction accuracy of the GRNN model is sufficient for engineering applications.  
Figure 10: Sensitivity analysis results of the negative reactions.  
It is necessary to assess the NRRAP under the live load because of the increased negative reaction risk described in Section 2.2. The assessment results are shown in Figure 13. The RAP under the dead load at the design stage is 1059 kN. Considering the live load and parameter uncertainty, the reliability of the auxiliary pier is 2.605, and the failure probability is 0.67%. At the operation stage, the RAP under the dead load is 256 kN. Considering the live load and parameter uncertainty, the reliability of the auxiliary pier is 0.785, and the failure probability is 18.33%.  
The above results show that: 1 the auxiliary pier bearing has a large reserve of reaction force and a low probability of failure at the design stage. 2 However, due to the increased negative reaction risk, the auxiliary pier bearing has a higher failure probability under the live load at the operation stage, although the RAP under the dead load is still positive.  
Figure 11: Structural response values from FEM and GRNN.  
Figure 12: Reliability index from Monte Carlo and GRNN.