## 4 Conclusions  
An interval estimation of the statistical parameters of the Duncan-Chang E-B model for high core rockfill dam materials is proposed. The bootstrap method was used to estimate the mean value, standard deviation, and coefficient of variation for the Duncan-Chang E-B model parameters of the core wall, rockfill, and overburden based on 155 groups of test data. The maximum and minimum variation coefficients of the parameters are as follows: the cohesion \(c\) and the failure ratio \(R_{f}\) in the core wall, the exponent of the volume modulus \(m\) and the internal friction angle \(\varphi\) in the rockfill, and the exponent of the volume modulus \(m\) and the failure ratio \(R_{f}\) in the overburden. The mean value of the internal friction angle \(\varphi\) in the core wall material is the smallest, while the mean value of the internal friction angle \(\varphi_{0}\) in the rockfill material is the largest, with the minimum variation coefficient, while the variation coefficient of \(\Delta\varphi\) is large. The marginal distribution types of Duncan-Chang E-B model parameters for high core rockfill dam materials are proposed. Since the marginal distribution is not always consistent when using the K-S and A-D test methods, AIC optimization is employed to accurately estimate the marginal distribution types of these parameters. In addition to \(K_{b}\) of the core wall material following a lognormal distribution, the other parameters of the core wall follow a Weibull distribution. The internal friction angle \(\varphi_{0}\) of the rockfill follows a truncated normal distribution, \(\Delta\varphi\), \(K\), \(K_{b}\), and \(m\) follow a Weibull distribution, and the other parameters of the rockfill follow a lognormal distribution. \(\varphi\), \(R_{f}\), and \(K\) of the overburden obey a truncated normal distribution, \(m\) obeys a lognormal distribution, and the other parameters obey a Weibull distribution. Furthermore, the results obtained through bootstrap sampling are consistent with those from the statistical sample tests. Based on the correlation analysis and marginal distribution results of the Duncan-Chang model parameters, a multi-dimensional joint distribution model for the parameters of the core wall, rockfill, and overburden materials in high core rockfill dams was established using the \(t\) Copula function. The samples generated by this method demonstrate that the correlation laws effectively represent the original samples and reasonably describe the correlation and joint probability distribution of the Duncan-Chang E-B model parameters for high core rockfill dam materials. A method for analyzing the cracking risk of the core wall using the multi-dimensional joint distribution of Duncan-Chang model parameters is proposed, leading to a more reasonable security assessment. The results for the PB dam demonstrate that joint sampling effectively captures the internal correlation between material mechanical parameters. In contrast, independent sampling can introduce physical distortions in parametric sampling and does not align with the inherent random correlations of material parameters. The cracking risk of the core wall using the proposed model is \(2.14\times 10^{-7}\). It is noted that this paper focuses on the statistical analysis of only three main material parameter zones--the core wall, rockfill, and overburden--using bootstrap sampling to obtain their statistical values. However, actual core wall rockfill dams include additional material zones, such as the transition layer and filter layer. Due to the lack of statistical samples for these zones, the analysis was approximated and simplified. Furthermore, high core rockfill dams are characterized by large volumes, multiple material zones, and complex structures, which exhibit significant spatial variability in parameters. This spatial variability was not considered in the parameter statistical analysis and requires further research. Conceptualization, Q.G., X.L. and J.C.; methodology, Q.G., H.H., X.L. and J.C.; software, Q.G., X.Z. and Z.Z.