## 4 Case Study--Safety Assessment of the Rockfill Specim in Triaxial Test
### Variability Modeling  
#### 4.5.1 Mohr-Coulomb Parameters for FEM Model  
The first step of the Monte Carlo analysis consists of modeling the variability of the input parameters, i.e., defining the functions that characterize their statistical distributions. In this case, the effective friction angle \(\varphi^{\prime}\) and the elasticity parameters (E, \(\
u\)) are assumed to follow normal distributions, while the dilation angle \(\psi\) follows a gamma distribution (Table 2). The parameters of these distributions are derived from the twelve calibrated finite-element reconstructions of the laboratory tests and summarized in Table 2. For variables assumed to follow a normal distribution (\(\varphi^{\prime}\), E, \(\
u\)), the mean and standard deviation are computed directly from the calibrated parameters' dataset (Table 1). For the variable \(\psi\) modeled with a gamma distribution, the shape (\(\alpha\)) and scale (\(\beta\)) parameters are calculated using the standard moment-based relations: \(\alpha=2/\sigma^{2}\) and \(\beta=\sigma^{2}/\mu\), where \(\mu\) and \(\sigma\) represent the sample mean and standard deviation, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Sample No.** & \(\sigma^{\prime}{}_{3}\) **(kPa)** & \(\varphi^{\prime}\) **(\({}^{\circ}\))** & **E\({}_{50}\) (MPa)** & \(\psi\) **(\({}^{\circ}\))** & **\(\
u\) **(-)** \\ \hline
1 & & 45.2 & 58 & 2.5 & 0.28 \\
2 & 250 & 45.4 & 66 & 2.3 & 0.24 \\
3 & & 45.5 & 48 & 2.0 & 0.25 \\ \hline
4 & & 45.5 & 105 & 3.4 & 0.27 \\
5 & 500 & 45.2 & 87 & 2.8 & 0.30 \\
6 & & 42.6 & 65 & 0.9 & 0.21 \\ \hline
7 & & 43.1 & 125 & 2.0 & 0.30 \\
8 & 1000 & 42.2 & 100 & 1.2 & 0.22 \\
9 & & 41.2 & 95 & 0.8 & 0.22 \\ \hline
10 & & 40.2 & 155 & 0.5 & 0.22 \\
11 & 2000 & 40.3 & 145 & 0.9 & 0.22 \\
12 & & 41.0 & 160 & 0.8 & 0.21 \\ \hline \end{tabular}
\end{table} 
Table 1: Calibrated Mohr–Coulomb model parameters from 12 triaxial tests. The trends displayed in Figure 13 imply that the key material parameters are not statistically independent, but exhibit systematic interrelations that must be preserved in any probabilistic assessment:  
* Effective friction angle (\(\varphi\)') vs. stiffness (E): As confinement increases, \(\varphi\)' falls while E rises, yielding a pronounced negative correlation. Specimens with a lower elastic modulus consistently mobilize higher peak friction, whereas highly confined (and therefore stiffer) specimens mobilize a reduced \(\varphi\)'. * Effective friction angle (\(\varphi\)') vs. dilation angle (\(\psi\)): Both angles diminish with increasing confinement, resulting in a clear positive correlation where larger friction angles are accompanied by larger dilation angles, and vice versa. * Stiffness (E) vs. dilation angle (\(\psi\)): Because E rises and \(\psi\) falls with confinement, these variables are negatively correlated. These couplings are captured by (i) a bivariate normal distribution for the \(\varphi\)'-E pair, and (ii) a Gaussian copula linking \(\varphi\)' and \(\psi\), which subsequently induces the appropriate secondary dependence between E and \(\psi\). The correlation structure is introduced in the following way:  
* \(\varphi\)'-E pair: A bivariate normal distribution is defined with a Pearson correlation coefficient [39] calculated from the laboratory data (Pearson \(\rho=-0.76\)). * \(\varphi\)'-\(\psi\) pair: A Gaussian copula [40] links the variables in the following manner: \(\varphi\)' is mapped to a uniform variate via its normal cumulative distribution, which is then transformed to \(\psi\) through the inverse gamma CDF (Spearman \(\rho_{\mathrm{s}}\) = 0.87, Kendall \(\tau\) = 0.71). Introducing the parameters in this correlated form ensures that the Monte Carlo sampling considers the physically observed tendencies and yields a realistic spread of input combinations for the safety analysis (Figure 14). A 10,000-sample Monte Carlo set preserves the laboratory-obtained correlation metrics within close range (\(\rho=-0.71\); \(\rho_{\mathrm{s}}\) = 0.94; \(\tau\) = 0.78).