## 3 Case Study
### Cracking Risk Analysis of the Core Wall  
#### 3.3.1 Analysis of the Sample Point Change Law for Independent and Joint Sampling of Parameters  
Through the independent (IS) and joint sampling (JS) of the Duncan-Chang model parameters, a sample scatter plot was compiled, as shown in Figure 9. For the method of independent sampling, the sample points of each parameter are random, without any associated features, and do not conform to the true law of material parameters, such as the negative correlation for \(c\)-\(\varphi\) in the core wall, the positive correlation for \(\varphi_{0}\)-\(\Delta\varphi\) in the rockfill, etc. For the method of joint sampling, the sample points of each parameter exhibit good correlation characteristics, which conform to the inherent correlation law of material mechanical parameters. For example, \(c\)-\(\varphi\) of the core wall, \(\varphi_{0}\)-\(\Delta\varphi\) of the rockfill, and \(K\)-\(K_{b}\) of the overburden all exhibit correlation laws that are consistent with their statistical characteristics. This implies that in the safety risk analysis of high core rockfill dams, if the relevant laws of material parameters are not considered, it will cause physical distortion in parametric sampling and form unreasonable evaluation results. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & **Mean** & **VC** & **Type** \\ \hline \multirow{3}{*}{Cutoff Wall} & \(c\) (MPa) & \(1.13\) & \(0.25\) & Lognormal \\ \cline{2-8}  & \(\varphi\) (\({}^{\circ}\)) & \(42.3\) & \(0.15\) & Normal \\ \cline{2-8}  & \(E\) (GPa) & \(28.5\) & \(0.10\) & Lognormal \\ \hline \multirow{3}{*}{Bedrock} & \(c\) (MPa) & \(1.30\) & \(0.30\) & Lognormal \\ \cline{2-8}  & \(\varphi\) (\({}^{\circ}\)) & \(49.0\) & \(0.20\) & Normal \\ \cline{2-8}  & \(E\) (GPa) & \(9.0\) & \(0.15\) & Lognormal \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Random characteristics of parameters in the dam and foundation (2). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & \multicolumn{4}{c}{**Internal Friction Angle**} & \multirow{2}{*}{\(\mathbf{R_{f}}\)} & \multirow{2}{*}{\(\mathbf{K}\)} & \multirow{2}{*}{\(\mathbf{K_{b}}\)} & \multirow{2}{*}{\(\mathbf{m}\)} & \multirow{2}{*}{\(\mathbf{n}\)} \\ \cline{3-3} \cline{5-10}  & & & \(\varphi\) & & & & \(0.68\) & \(411.0\) & \(185.0\) & \(0.30\) & \(0.68\) \\ \cline{2-10} \multirow{3}{*}{Core wall} & Mean & \(0.06\)(MPa) & \(35^{\circ}\) & & & \(0.07\) & \(0.19\) & \(0.27\) & \(0.39\) & \(0.27\) \\ \cline{2-10}  & VC & \(0.57\) & \(0.14\) & / & & Weibull & Weibull & Lognormal & Weibull & Weibull \\ \cline{2-10}  & Type & Weibull & Weibull & & & & & & & \\ \hline \multirow{3}{*}{Main rockfill} & Mean & / & / & \(30.76\) & \(10.0\) & \(0.63\) & \(1184.0\) & \(489.0\) & \(0.33\) & \(0.35\) \\ \cline{2-10}  & VC & / & / & \(0.05\) & \(0.22\) & \(0.10\) & \(0.27\) & \(0.32\) & \(0.41\) & \(0.29\) \\ \cline{2-10}  & Type & / & / & Truncated normal & Weibull & Lognormal & Weibull & Weibull & Weibull & Lognormal \\ \hline \multirow{3}{*}{Secondary rockfill} & Mean & / & / & \(28.6\) & \(8.3\) & \(0.64\) & \(1027.0\) & \(380.0\) & \(0.36\) & \(0.36\) \\ \cline{2-10}  & VC & / & / & \(0.05\) & \(0.22\) & \(0.10\) & \(0.27\) & \(0.32\) & \(0.41\) & \(0.29\) \\ \cline{2-10}  & Type & / & / & Truncated normal & Weibull & Lognormal & Weibull & Weibull & Weibull & Lognormal \\ \hline \multirow{3}{*}{Overburden} & Mean & / & \(38^{\circ}\) & & & \(0.64\) & \(780.0\) & \(400.0\) & \(0.18\) & \(0.42\) \\ \cline{2-10}  & VC & / & \(0.15\) & / & & \(0.09\) & \(0.44\) & \(0.44\) & \(0.37\) & \(0.16\) \\ \cline{2-10}  & Type & / & Truncated normal & & Truncated normal & Weibull & Lognormal & Weibull \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Random characteristics of parameters in the dam and foundation (1). Compared to the results obtained using the independent sampling method, the coefficients of variation for dam deformation and stress using the joint sampling method are relatively smaller.