## 5 Case Study
### Material Parameters and Basic Random Variables  
The constitutive model for the dam construction materials and dam foundation cover layer uses the ideal elastoplastic model, with the yield and failure criteria based on the Mohr-Coulomb criterion; the bedrock is modeled using a linear elastic model. Due to the large geometric dimensions of the dam section as well as the small variability of the upstream reservoir water level and the unit weight of the dam construction materials, these are not considered as random variables. Existing works [24; 25] show that if spatial variations are not considered, deformation parameters do not significantly impact slope stability and thus slope reliability. In addition, there is a lack of experimental data on the deformation parameters of the dam materials, and they are also not considered as random variables in this study. This paper primarily considers the random characteristics of the shear strength parameters of the dam construction materials. The physical and mechanical parameters of the dam body and dam foundation cover layer materials, as well as the basic random variables for reliability analysis, are shown in Table 2. The means and standard deviations of friction angle and cohesion for the dam construction materials are estimated based on triaxial test data and engineering experience.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Deterministic Material**} & \multicolumn{4}{c}{**Random Material Parameters**} \\ \cline{2-10}  & \multicolumn{4}{c}{**Parameters**} & & \multicolumn{2}{c}{**\(\sigma\)**} & \multicolumn{2}{c}{**\(\sigma_{0}\)**} & \multicolumn{2}{c}{**\(\Delta\sigma\)**} \\ \cline{2-10}
**Material** & & & & & **(Normal Distribution)** & **(Normal Distribution)** & **(Lognormal Distribution)** \\
**Category** & \(\rho\) & \(P_{\text{sat}}\) & **(t/m\({}^{3}\))** & **(MpA)** & **Mean** & **Standard** & **Mean** & **Standard** & **Mean** & **Standard** \\  & & & & **Value** & **Deviation** & **Value** & **Deviation** & **Value** & **Value** & **Value** & **Standard** \\  & & & & & **(\(\circ\))** & **(\(\circ\))** & **(\(\circ\))** & **(\(\circ\))** & **(\(\circ\))** & **(\(\circ\))** & **Deviation** (\(\circ\))** \\ \hline Cover Layer\({}^{\textregistered}\) & 2.05 & 2.25 & 96.10 & 0.30 & 31.00 & 4.65 & — & — & — & — \\ Cover Layer\({}^{\textregistered}\) & 2.03 & 2.23 & 81.00 & 0.30 & 27.00 & 4.05 & — & — & — & — \\ Cover Layer\({}^{\textregistered}\) & 2.06 & 2.25 & 105.00 & 0.30 & 31.00 & 4.65 & — & — & — & — \\ Cuffardan and & 2.07 & 2.27 & 80.00 & 0.30 & 36.49 & 1.82 & — & — & — & — \\ Core Wall with & & & & & & & & & \\ Gravelly Soil & 2.10 & 2.33 & 44.70 & 0.30 & — & — & 41.67 & 6.13 & 4.70 & 1.22 \\ Filter Layer I & 2.00 & 2.25 & 114.10 & 0.30 & — & — & 43.38 & 2.17 & 3.74 & 0.77 \\ Filter Layer II & 2.02 & 2.26 & 139.60 & 0.30 & — & — & 45.83 & 2.59 & 5.45 & 2.19 \\ Transition Layer & 2.09 & 2.29 & 96.00 & 0.30 & — & — & 46.11 & 2.31 & 5.04 & 1.13 \\ Upstream Gravel Fill & 2.12 & 2.33 & 105.00 & 0.30 & — & — & 42.75 & 2.26 & 3.59 & 1.24 \\ Downstream Main & 2.09 & 2.29 & 123.40 & 0.30 & — & — & 48.15 & 2.95 & 6.35 & 1.41 \\ Downstream & & & & & & & & & \\ Secondary Gravel Fill & 2.07 & 2.27 & 103.40 & 0.30 & — & — & 42.75 & 2.26 & 3.59 & 1.24 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material parameters and basic random variables.  
Figure 1: Typical cross section of the dam and the associated 2D finite element mesh.