## 4 Probabilistic Analyses
### Sensitivity Analyses
#### 4.6.2 Component Level  
The component sensitivity analysis was performed using the advanced Monte Carlo-based Latin Hyper Cube sampling technique with 60 generated column samples of the non-linear numerical finite element model (P-NLFEM). The P-NLFEM was developed by the University of Natural Resources and Life Sciences using ATENA Scientific software [35]. In the course of the step-by-step axial loading process, the basic variables of the model input data were determined for each loading step with a step size increment of \(\Delta N\) = 10 kN. The development of the sensitivity factors calculated using the Kendal Tau algorithm [36] were analyzed and are plotted as a function of the axial (normal) force \(N\) of the column in Figure 9a and the transverse deformation of the column at half the system length in Figure 9b. The Kendall rank correlation coefficient, commonly referred to as the Kendall's \(r\) coefficient, was used to measure the ordinal association between two measured quantities, e.g., model input variable \(i\) and model response \(k\). The sensitivity analyses according to FORM were not applied here as the Kendal Tau algorithm allows a much simpler handling of the sensitivity considerations for finite element applications. Comparative studies have shown that both methods lead to almost the same results Figure 9d shows the statistical responses of the horizontal column deflections versus the gradually applied normalized axial force \(N\). At the component level, the variables X\({}_{7}\) and X\({}_{8}\) were not taken  
Figure 8: Sensitivity analyses at the cross-section level: (**a**) sensitivity of material parameters with respect to the applied normalized normal force (maximum bearing capacity of N = 335.5 kN); (**b**) strains in compression and tension with respect to the applied normal force, see also Table 3.  
into account in the sensitivity analyses because (a) the exact locations of the reinforcement were quarantined by an extraordinary quality control during the fabrication of the columns and (b) the influence of the material laws, solution algorithms, non-linear fracture processes and slenderness on the instability process were the focus of interest. As can be seen in Figure 9a, the modulus of elasticity of the concrete \(X_{1}(E_{ci})\) as well as the compressive strength \(X_{11}(f_{c})\) dominate in the ultimate failure load in compression. When considering the sensitivity factors with regard to the horizontal deflection in the middle height of the column, see Figure 9b, the tensile strength of the concrete \(X_{12}(f_{ci})\) and the compressive strength \(X_{11}(f_{c})\) both play a significant role initially; however, the modulus of elasticity of the concrete \(X_{1}(E_{ci})\) becomes more important with increasing horizontal deflection. On the basis of the 60 Latin Hyper Cube samples, Figure 9d provides an insight into the scatters in the relationship between the axial force and the horizontal displacement in the middle height of the column, computed in ATENA. As can be seen from the graph in Figure 10a for the C100/C115 concrete column, the modulus of elasticity of the concrete \(X_{1}(E_{ci})\), as well as the compressive strength \(X_{11}(f_{c})\)  
Figure 9: Sensitivity analyses at the component/column level based on the probabilistic non-linear analysis of the considered column made of C45/55, see Section 4.4: (**a**) sensitivity of material parameters with respect to the normalized applied axial force; (**b**) sensitivity of material parameters with respect to the normalized horizontal displacement at middle height of the column; (**c**) basic variables of considered material models; (**d**) normalized horizontal displacement vs. normalized axial force. and fracture energy \(X_{13}(G_{f})\) dominate in the ultimate failure load.