# A Method for Evaluating Systematic Risk in Dams with Random Field Theory
## 5 Conclusions  
(1) A log-normal random field simulation method is proposed to simulate the spatial variability characteristics of concrete and foundation materials in a dam-foundation system. Through in-depth research on the finite element simulation of the spatial variability of gravity dams and foundation materials, as well as by using the failure risk analysis method of structural systems, a composite response surface equation for dam structure and foundation stability functions considering parameter spatial variation is constructed, which solved the problem of the inaccurate consideration of parameter spatial variability in failure path performance functions. (2) A Monte Carlo simulation method and a process for determining failure risk in a dam-foundation system have been proposed, which solve the problem wherein the multi-element sliding path performance function cannot accurately consider the influence of parameter spatial variability on failure risk, as well as the problems of multiple failure paths in large block structures, complex relationships between parameters and paths, and difficulty in solving the failure risk of series-parallel systems. (3) An engineering application shows that the proposed method has good practicability and high precision, which overcomes the defects of the overestimation of failure probability caused by high variance estimation and the homogenization of spatial random characteristics in large-volume concrete and rock foundations. The method proposed in this article is meaningful for those engaged in structural safety risks. What is more, this article proposes a universal method for other gravity dams; this method is applicable, with the main difference being the search method for critical failure paths, performance functions, and parameter distribution characteristics, which vary depending on the project. It can be concluded that the proposed method provides an effective way to reasonably evaluate the reliability of a system consisting of complex foundations and overlying structures. Conceptualization, C.R., Z.Z. and B.G; methodology, C.R., Z.Z., L.P. and X.L. ; software, Z.Z., B.G. and K.H. ; formal analysis, Z.Z. and K.H. ; investigation, C.R. ; resources, L.P, and B.G. ; data curation, Z.Z. and B.G. ; writing--original draft preparation, C.R. ; writing--review and editing, L.P.; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (No. 52309162) and the Fundamental Research Funds for the Central Universities (YJ202287). Not applicable. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Failure Paths**} & \multicolumn{3}{c}{**Considering the Spatial Variability**} & \multicolumn{3}{c}{**Without Considering the Spatial**} \\  & & \multicolumn{3}{c}{**of Parameters**} & \multicolumn{3}{c}{**Variability of Parameters**} \\ \cline{3-6}  & & **Path Failure Risk** & **System Risk** & **Path Failure Risk** & **System Risk** \\ \hline Dam & \(\backslash\)1 & \(4.50\times 10^{-7}\) & & \(5.51\times 10^{-7}\) & \\  & \(\backslash\)2 & \(8.10\times 10^{-7}\) & & \(9.16\times 10^{-7}\) & \\ \hline Dam–Foundation & \(\backslash\)3 & \(5.37\times 10^{-5}\) & & \(2.19\times 10^{-4}\) & \\  & \(\backslash\)4 & \(6.11\times 10^{-4}\) & & \(9.49\times 10^{-4}\) & \\  & \(\backslash\)5 & \(4.13\times 10^{-4}\) & & \(9.06\times 10^{-4}\) & \\  & \(\backslash\)6 & \(2.52\times 10^{-6}\) & & \(1.66\times 10^{-5}\) & \\  & \(\backslash\)7 & \(9.53\times 10^{-8}\) & & \(2.53\times 10^{-7}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Risk results for the dam–foundation system. Data Availability Statement: The data presented in this study are available on request from the corresponding author. [PERSON], [PERSON] and [PERSON] were employed by the company PowerChina Chengdu Engineering Corporation Limited.