## 3 Case Study
### Project Specification
#### 3.2.1 Interval Estimation of the Duncan-Chang Model Parameters for High Rockfill Dams  
The Duncan-Chang E-B model is a widely used soil constitutive model in the numerical simulation of stress and deformation in earth-rock dams. This nonlinear elastic model accounts for the nonlinear characteristics of soil material deformation and uses a hyperbola to fit the triaxial test data. A total of 155 groups of parameter test data for the Duncan-Chang E-B model were collected from several high core rockfill dams and dam foundation overburdens. The material parameter samples included 52 groups for the core wall, 59 groups for the rockfill, and 44 groups for the overburden. The core of the bootstrap sampling method involves randomly sampling the original experimental data with replacement to generate numerous subsamples of the same size as the original data. By repeatedly sampling and estimating, a series of statistical values are obtained, from which confidence intervals and probability distributions can be derived to assess the variability of these values. To determine the optimal sampling frequency for the bootstrap method, sampling tests with different frequencies were conducted. Figure 4 shows the relationship curves between the mean  
Figure 3: The finite element generalized model of a high core rockfill dam. Figure 2: Typical section of PB Hydropower Station. value and the coefficient of variation of the cohesion \(c\) of the core wall, the internal friction angle of the rockfill material, and the deformation modulus \(K\) of the overburden and the bootstrap sampling frequency. As can be seen from Figure 4, when the sampling frequency exceeds 10,000, the mean value and coefficient of variation of parameters \(c\) and \(K\) are stable. Therefore, a sampling frequency of 10,000 was adopted for the bootstrap method in the subsequent research. Based on the statistical characteristic values of the Duncan-Chang E-B model parameters for the core material, rockfill material, and dam foundation overburden of the high core rockfill dam, the interval estimation with a 95% confidence level is presented in Tables 1-3. The results show that the parameters with the highest interval estimation values of the coefficient of variation for the core wall, rockfill, and overburden are the cohesion, the exponent of the volume modulus, and the volume variation modulus, respectively. Among these, the coefficient of variation for the cohesion \(c\) of the core wall is the highest, ranging from 0.51 to 0.63. The mean of \(R_{f}\) of the core wall is the highest, followed by the overburden and rockfill materials, with the variation coefficients of the failure ratios for all three materials being relatively small. The variation coefficients of \(K\) and \(K_{b}\) are similar across the three materials. Notably, the variation coefficient of the deformation modulus of the overburden is the highest, while those of the core wall and rockfill materials are comparable. This may be due to the core wall and rockfill materials being manually compacted, resulting in controlled physical and mechanical properties (e.g., dry density) and less modulus anisotropy compared to the natural overburden. Additionally, the variation coefficients of the exponent of the volume modulus are the same for the three materials. The variation coefficient of the exponent of the initial deformation modulus is smallest in the overburden and identical in the core wall and rockfill.