## 6 Stochastic Analysis
### Stochastic Analysis  
Based on the probabilistic models presented in Table 3, 10 000 Latin Hypercube Samples (LHS) are generated. Considering that the standard LHS may bring undesired spurious correlations into the sample scheme, correlation reduced Latin Hypercube Sampling (CLHS) is used to avoid this unwanted effect [74]. It is worth noting that all the parameters are assumed to be independent, except for the yield stress \(f_{y}\) and the tensile strength \(f_{\rm u}\) of the reinforcement, for which a correlation coefficient of 0.86 is considered [66]. This is an assumption adopted in many studies in relation to the consideration of material uncertainties for RC structures [44; 75; 76; 77; 78]. Subsequently, both a pushdown analysis and an IDA are carried out for every sample for every column removal case, i.e., Cases A, B, and C. The loading schemes are identical to the corresponding deterministic analyses in the previous section. In order to reduce the computational demand in the stochastic dynamic analyses, a resolution level of 0.5 kN/m (compared to 0.05 kN/m used for the previous deterministic dynamic analysis) is used for the IDA when determining the ultimate load-bearing capacity. The damping ratio DR = 5% is considered in the stochastic dynamic analysis. The histograms of the ultimate load-bearing capacity from both static and dynamic analyses are shown in Figure 17 and a summary of the results is presented in Table 4. Note that the resistances obtained through the pushdown analyses are considered to correspond to first failure as discussed before. Cases A, B, and C are presented in Figure 17a-c, respectively, in which both the results from the pushdown analyses and IDA are shown in order to compare with each other. Lognormal distributions are found to fit the histograms well for the three cases. The resistance from the pushdown analysis is found to be larger than that of the IDA, as also already evidenced in the previous sections. Mean values and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Units** & **Distribution** & **Mean** & **COV** \\ \hline \(c\) & mm & Beta & 30 & 0.17 \\ \(f_{\rm c}\) & MPa & Lognormal & 28 & 0.18 \\ \(\varepsilon_{\rm c1}\) & \% & Lognormal & 0.21 & 0.15 \\ \(Y_{2,j}\) & – & Lognormal & 1 & 0.30 \\ \(E_{\rm s}\) & GPa & Normal & 205 & 0.08 \\ \(f_{y}\) & MPa & Lognormal & 560 & 0.05 \\ \(f_{u}\) & MPa & Lognormal & 655 & 0.06 \\ \(\epsilon_{u}\) & \% & Lognormal & 12 & 0.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Probabilistic models for random variables. standard deviation (St.D.) for the stochastic pushdown and IDA analyses are summarized in Table 4. Furthermore, it was found that the resistance in the case of the external column loss scenario (Case A) is significantly lower than those for the internal column loss scenarios (Case B and Case C), as it is more difficult to redistribute the unbalanced load in the former case. Next, the EBM is adopted to calculate the dynamic capacity curves from the pushdown curves. It is important to indicate that here, as well, the EBM curve is calculated based on the pushdown curve up to the first failure (i.e., the first load peak as it is the ultimate load-bearing capacity) in the frame. The predicted dynamic load-bearing capacity curves are similar to those of the deterministic analyses. The histograms and PDFs of the ultimate load-bearing capacities from EBM are presented in Figure 18 for the three cases, for which lognormal distributions are found to be appropriate to fit the histograms. Only a small discrepancy between the EBM result and the IDA result is observed for Case A (Figure 18a), while little difference is observed between the EBM results and the IDA results for both Cases B and C (Figure 18b,c).