# 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
### Critical Failure Paths and Performance Functions  
As seen in Figure 3, there are many structural planes due to the complexity of the rock foundation; thus, it is important to determine the critical failure paths. Therefore, the potential failure paths are searched using the load increment method, and seven potential critical failure paths in the dam-foundation system are obtained according to the progressive failure process of elements under step-by-step loading (as shown in Figure 4), in which failure paths 1-3 are in the dam body and failure paths 4-2 are in the foundation.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Location** & **Materials** & **Distribution** & **Mean** & **CV** & \(\mathbf{L_{x}}\) & \(\mathbf{L_{y}}\) \\ \hline \multirow{4}{*}{Foundation} & Elastic modulus & Log-normal & 18 (GPa) & 0.1 & 40 & 5 \\  & Cohesion & Log-normal & 1.75 (MPa) & 0.36 & 40 & 5 \\  & Friction coefficient & Normal & 1.3 & 0.2 & 40 & 5 \\  & Poisson’s ratio & - & 0.22 & - & - & - \\  & Density & - & 2700 (kg/m\({}^{3}\)) & - & - & - \\ \hline \multirow{4}{*}{Structural plane} & Elastic modulus & Log-normal & 1 (GPa) & 0.1 & 40 & 5 \\  & Cohesion & Log-normal & 0.15 (MPa) & 0.4 & 40 & 5 \\  & Friction coefficient & Normal & 0.75 & 0.25 & 40 & 5 \\  & [PERSON]’s ratio & - & 0.3 & - & - & - \\  & Density & - & 2700 (kg/m\({}^{3}\)) & - & - & - \\ \hline \multirow{4}{*}{Dam concrete} & Elastic modulus & Log-normal & 37.8 (GPa) & 0.1 & 10 & 5 \\  & Cohesion & Log-normal & 1.3 (MPa) & 0.25 & 10 & 5 \\ \cline{1-1}  & Friction coefficient & Normal & 1.35 & 0.15 & 10 & 5 \\ \cline{1-1}  & [PERSON]’s ratio & - & 0.2 & - & - & - \\ \cline{1-1}  & Density & - & 2552 (kg/m\({}^{3}\)) & - & - & - \\ \hline \hline \end{tabular} Note: CV, coefficient of variation.  
\end{table}
Table 1: Mechanical parameters of materials (including the spatial variability parameters).  
Figure 4: Distribution of potential critical failure paths for the system.  
For these determined critical failure paths, the performance functions are constructed according to the failure criteria.  
For the strength failure  
\[G_{1}(m,k)(H,E,c,f)=k-aI_{1}-\sqrt{f_{2}}=\mathbf{A}\mathbf{X}^{\text{T}} \tag{22}\]  
For the stability failure  
\[G_{2(k)}(X)=\sum_{i=1}^{n}\left(f_{i}^{\prime}\cdot\sigma(X)_{i}+c_{i}\right) \cdot b_{i}-\sum_{i=1}^{n}\tau(X)_{i}\cdot b_{i} \tag{23}\]  
\[\sigma(X)_{i}=\mathbf{A}\mathbf{X}^{\text{T}} \tag{24}\]  
\[\tau(X)_{i}=\mathbf{A}\mathbf{X}^{\text{T}} \tag{25}\]  
\[\mathbf{A}=[a_{1},a_{2},\dots,a_{21}] \tag{26}\]  
\[\mathbf{X}=\left[1,H,E_{1},E_{2},E_{3},c_{1},c_{2},c_{3},f_{1},f_{2},f_{3},H^{ 2},E_{1}^{2},E_{2}^{2},E_{3}^{2},c_{1}^{2},c_{2}^{2},c_{3}^{2},f_{1}^{2},f_{2}^ {2},f_{3}^{2}\right] \tag{27}\]  
where \(\mathbf{A}\) is the undetermined coefficient matrix of the RSEs; matrix \(\mathbf{X}\) is determined by the number of random variables and the forms of the RSEs; the other symbols retain their original meanings, as above.  
To improve the speed of numerical simulation, the hardware conditions for this simulation are as follows: (1) CPU: 13 th Gen Intel(R) Core (TM) i7-13700F; (2) RAM: 32.0 GB; (3) NVIDIA Geforce RTX 3060. Given the above hardware and the ABAQUS software 6.14, the processing time of each simulation takes less than 2 min, and then, the RSE sequences can be quickly constructed through finite element simulations with 27 sets of samples that are generated by using the orthogonal design method [35]. The results of solving the coefficients of the RSEs indicate that the multiple correlation coefficients for structural strength failure and dam stability failure are almost always greater than 0.98, implying that the fitting effect of the RSEs is good.