## 4 Probabilistic Analyses
### Non-Linear Finite Element Analyses
#### 4.8.3 Probabilistic NLFEM-Modeling  
The probabilistic modeling based on the BOKU ATENA 3D NLFEM model and the stochastic information of the selected random variables according to Tables 4 and 5 were \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**X**} & **Variable** & **Dist.** & **Unit** & \(\mathbf{X_{k}}\) & \(\mathbf{\mu}\) & \(\mathbf{\sigma}\) \\ \hline \multicolumn{1}{c}{C100/115} & & & & & \\ \hline X\({}_{1}\) & \(E_{ci}\) & Initial tangent concrete modulus of elasticity & LN & GPa & 48.9 & 48.9 & 6.23 \\ X\({}_{2}\) & \(E_{s}\) & Reinforcing steel modulus of elasticity & D. & GPa & 200 & 200 & - \\ X\({}_{11}\) & \(f_{c}\) & Concrete compressive strength & LN & MPa & 100.0 & 108.0 & 4.99 \\ X\({}_{12}\) & \(f_{ct}\) & Concrete tensile strength & LN & MPa & 3.7 & 5.2 & 1.08 \\ X\({}_{13}\) & \(G_{F}\) & Concrete fracture energy & LN & MPa & 119 & 170 & 35.0 \\ X\({}_{18}\) & \(f_{y}\) & Reinforcing steel yield strength & LN & MPa & 500 & 548 & 40.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input parameters for the probabilistic analyses of the EN 1992-1 Non-Linear Finite Element provisions of the slender column made of C100/115 (details regarding C45/55 are provided in Table 3). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**C45/55 and C100/115**} \\ \hline  & & \(E_{ci}\) & \(E_{s}\) & \(f_{c}\) & \(f_{ct}\) & \(G_{F}\) & \(f_{y}\) \\ \hline X\({}_{1}\) & \(E_{ci}\) & 1 & 0 & 0.7 & 0.6 & 0.8 & 0 \\ X\({}_{2}\) & \(E_{s}\) & & 1 & 0 & 0 & 0 & 0 \\ X\({}_{11}\) & \(f_{c}\) & &carried out with the Structural Analysis and Reliability Assessment (SARA) program package developed in the project \"Safety and Reliability Assessment of Structure\" [36,37] in which the Latin Hyper Cube Sampling (LHS) method with a sample number of \(n=305\) was used for the development of the input sampling sizes. The significantly increased number of samples compared to the previously described analytical probabilistic analysis results from the fracture processes that occur during the loading process and the clearly pronounced physical and material non-linearities that are also taken into account in the NLFEM analysis. A corresponding convergence analysis in the statistical structure responses already shows correspondingly stable results with an LHS sample number of 105, however, the number of samples was increased to 305 due to the trustworthiness of the results. In total, around 1450 simulations were carried out for all deterministic and probabilistic considerations on the cross-sectional level and on the structural level for the materials C45/55 and C100/115. The \(N\)-\(M\) structural response curves from the 305 NLFEM probabilistic analyses are shown in Figure 12a with the maximum values indicated by pink dots. As can be seen from this bundle of results and the associated maximum values (pink points in Figure 12a, the P-NLFEM results in terms of axial forces are well above the experimental blue values which can also be seen in the corresponding PDF shown in Figure 12a on the right. These P-NLFEM results, which lie above the experimental results, are to be classified as dangerous because the P-NLFEM results lie above the real N-load of the experimental results and, as shown in the previous sections, the experimental failure points. Similar to the analytical predictions based on the interaction diagrams, the NLFEM failure points are also clearly located within interaction curves, and as can be seen in Figure 12a on the right, the NLFEM results are even below the experimental moment lines for some modeling groups. In addition to the probabilistic modeling based on the BOKU ATENA 3D NLFEM model, the probabilistic analyses were also carried out based on the findings of the LHS simulations on the BOKU model and the models from:  
_STUBA:_ Slovak University of Technology in Bratislava, Bratislava, Slovakia. _UNIZG-FCE:_ University of Zagreb Faculty of Civil Engineering, Department of Structural Engineering.