## 4 Case Study--Safety Assessment of the Rockfill Specim in Triaxial Test
### Probability Analysis Results  
The Monte Carlo analysis--10,000 simulations of the triaxial test FE model--is carried out with the full correlation structure observed in the laboratory data (negative \(\varphi^{\prime}\)-E and positive \(\varphi^{\prime}\)-\(\Downarrow\) links). Ten thousand input quadruples (\(\varphi\)', E, \(\psi\), \(\
u\)) are generated using a Python script. These samples provide the stochastic basis for the subsequent Monte Carlo safety assessment. Histogram bin widths are selected with the FD rule and tested \(\pm\)50% around that optimum bin width value. The resulting change in the estimated mean and standard deviation of obtained distributions is less than 0.3%, demonstrating that the bias-variance trade-off is negligible at the 10,000 sample size.  
Figure 15 shows the baseline probability density function (PDF) of calculated axial strain \(\epsilon_{1}\) from the baseline Monte Carlo simulations. Most simulated strains lie below 0.20, although a small number exceed 0.25 owing to samples with very low elastic modulus E.  
Figure 16 compares the \(\epsilon_{1}\) PDFs before and after weighting. For clarity, the plot trims the baseline curve at \(\epsilon_{1}>0.10\); the long right-hand tail (up to \(\epsilon_{1}\approx 0.30\)) remains in the data set but is visually uninformative. After weighting, the \(\epsilon_{1}\) distribution localizes around  
Figure 14: Joint densities for (**a**) the correlated pairs \(\varphi^{\prime}\)-E; (**b**) the correlated pairs \(\varphi^{\prime}\)-\(\Downarrow\). Red points represent data from recreated 12 triaxial tests. Legends list the laboratory and simulated data correlation metrics. Insets show QQ diagnostics; R\({}^{2}\) values above 0.9 indicate excellent agreement between the empirical and simulated marginals.  
Figure 15: Baseline analysis results: PDF of axial strain \(\epsilon_{1}\).  
the monitored value, demonstrating that the procedure pulls the simulation ensemble toward the observed behavior, which is confirmed by a very high Kullback-Leibler (KL) divergence [41] (\(\approx\)2.9 nats) between baseline and weighted PDFs.  
Regarding the factor of safety according to SSR, a baseline analysis yields the FS distribution with a mean \(\mu\) = 1.69 and standard deviation \(\sigma\) = 0.11 (Figure 17). Introducing the monitoring data-normal distribution centered on the measured axial strain (\(\varepsilon_{1}\) = 2.45%, \(\sigma\) = 0.002) and weighting each simulation accordingly shift the outcome markedly (KL divergence is registered \(\approx\)1.6 nats): the mean FS drops to \(\mu\) = 1.48 and the spread narrows to \(\sigma\) = 0.08. The decrease in the mean value of FS from 1.7 in baseline to 1.5 in weighted analysis reflects that the lower friction angle is generally mobilized at the higher confinement level.  
Although the baseline run comprises \(10^{4}\) simulations, the weighting step reduces the weighted simulation count to only about 235 and effective sample size (ESS) [42] to \(\approx\)372 (Figure 17). The resulting histogram is therefore noticeably rougher than the baseline version.  
Figure 16: PDF of axial strain \(\varepsilon_{1}\) (tail of the baseline PDF beyond \(\varepsilon_{1}\) = 0.10 omitted for clarity). The inset reports the baseline (10,000) and effective (\(\approx\)235) weighted simulation counts, as well as ESS (\(\approx\)373) and KL divergence (\(\approx\)2.9 nats) metrics.  
Figure 17: PDF of factor of safety FS. Dashed lines mark the respective means, and the inset reports both the baseline (10,000) and effective (\(\approx\)235) weighted simulation counts, as well as ESS (\(\approx\)373) and KL divergence (\(\approx\)1.6 nats).