# A Method for Evaluating Systematic Risk in Dams with Random Field Theory
## 3 Risk Analysis Method for Gravity Dams Considering the Spatial Variability of Parameters
### MC Method for Calculating System Failure Probability  
For a complex dam-rock foundation system, there are many failure paths, which leads to great difficulties in determining the correlation among failure paths and solving the reliability of series and parallel systems (Figure 1). To solve these problems, an agent model of the performance function for strength and stability failure is established. According to the spatial variability of the mechanical parameters of rock mass and concrete, the failure probability of the rock foundation system of the gravity dam and each critical failure path is calculated through the overall simulation using the MC method. For a dam-rock foundation system with \(n\) failure paths, the failure events of \(n\) failure paths are defined as \(A_{1}\), \(A_{2}\), \(\ldots\), and \(A_{n}\), and the failure probability, \(P_{fi}\), of the \(i^{\text{th}}\) failure path is  
\[P_{fi}=P(A_{i}) \tag{15}\]  
If \(n\) failure paths belong to the parallel system, then the parallel system failure probability, \(P_{f}^{p}\), can be expressed by  
\[P_{f}^{p}=P(A_{1}\cap A_{2}\cdots\cap A_{n}) \tag{16}\]  
For the convenience of calculation, an indicative function is pre-set:  
\[I[F(x)]= \begin{cases}1,&\overset{n}{\underset{i=1}{\cap}}A_{i}\\ 0,&\overset{n}{\underset{i=1}{\cup}}\overline{A_{i}}\end{cases}  \tag{17}\]  
Therefore, the parallel system failure probability, \(P_{f}^{p}\), is  
\[P_{f}^{p}=\frac{1}{N}\underset{j=1}{\overset{N}{\underset{j=1}{\sum}}}I\Big{[} F(x)_{j}\Big{]} \tag{18}\]  
where \(N\) is the total amount of MC sampling; \(F(x)_{j}\) is the value of performance function corresponding to the \(j^{th}\) sampling. If \(n\) failure paths belong to the series system, then the system failure probability, \(P_{f}^{S}\), can be expressed by  
\[P_{f}^{S}=P(A_{1}\cup A_{2}\cdots\cup A_{n})=\underset{i=1}{\overset{n}{\underset {i=1}{\sum}}}P(A_{i})-\underset{1\leq i<j\leq n}{\underset{j\leq i}{\sum}}P(A _{i}A_{j})+\underset{1\leq i<j<k\leq n}{\underset{j\leq n}{\sum}}P(A_{i}A_{j}A _{k})+\cdots+(-1)^{n-1}P(A_{1}A_{2}\cdots A_{n}) \tag{19}\]  
Figure 1: Schematic diagram of failure paths of gravity dam–rock foundation system. the foundation stress and deformation behavior, both the foundation system and the dam structure are meshed with 76,096 elements using the ABAQUS software 6.14. Based on a geological survey, the design report of this project, and the statistical results of the relevant literature [32; 33; 34], the physical and mechanical parameters of the rock mass and concrete are shown in Table 1. The uncertainty and spatial variability of the elastic modulus, cohesion, and friction coefficient of the bedrock, structural plane, and dam concrete are considered, which have a significant influence on the stress and strain of the dam-foundation system. Based on the results of compressive elastic modulus drilling tests at 22 monitoring points in an arch dam, the autocorrelation distance of the vertical compressive modulus obtained by using the autocorrelation function method is about 6.0 m, and the correlation between compressive modulus at further measurement points is significantly reduced [35]. At the same time, it is advisable to take a vertical correlation distance of 3-6 m considering the close relationship between the spatial variability of material parameters in the vertical direction of concrete dams and their pouring height. Similar to soil materials, the autocorrelation function of concrete materials in the horizontal direction is generally greater than that in the vertical direction. Due to temperature control and pouring strength, the size of the pouring surface is generally controlled between 15 m and 20 m. Therefore, it is recommended to take 3-5 times the autocorrelation distance in the vertical direction for the horizontal direction. Additionally, the reservoir water is selected to be a random variable with an average value of 136.46 m and a coefficient of variation of 0.05.