## 4 Probabilistic Analyses
### Elements of the Limit State Formulations  
As a first step, the resistance of the cross-section was calculated according to standard EN1992-1-1 which enabled the drawing of the \(N\)-\(M\) gradient lines for the given cross-section for the characteristic, design and the mean values of the resistance parameters.  
The resistance mechanism of the reinforced concrete cross-section must be known in order to make the load capacity assessment, as shown in Figure 7. The calculation algorithm was used to obtain the interesting cross-section moments from different normal forces. When assessing a slender column according to EN 1992-1-1 [8], creep is accordingly taken into account by a global factor. The relationship between the normal force applied to a slender column and the resulting bending moment is defined as:  
\[M_{ed}=N_{ed}.^{\prime}(e_{0}+e_{i}+e_{2}) \tag{15}\]  
where \(M_{ed}\) is the design bending moment, \(N_{ed}\) is the design normal force, \(e_{0}\) is the initial eccentricity of the applied load, \(e_{i}\) is the eccentricity caused by geometric imperfection and \(e_{2}\) is the eccentricity caused by second-order deformation of the system (i.e., slender column). The limits of the concrete pressure zone \(x_{lim}\) and \(F_{cd,lim}\) are computed by:  
\[x_{lim}=\frac{700+d_{[m]}}{f_{yd[N/mm^{2}]}+700} \tag{16}\]  
Since the analyses on the component consisting of the material C100/115 are of the same interest, as the influence of the material properties on the stability problem of interest can be shown, the input parameters for the probabilistic analyses for the material C100/115 were developed in an analogous way as for the C45/55 in Table 3.  
\[F_{cd,lim}=0.8095\cdot x_{lim}\,b_{Zug}\cdot f_{cd} \tag{17}\]  
where \(b_{Zug}\) is the section height under tension. For the control as to whether predominantly bending failure takes place can be tracked by:  
\[x=1.202\cdot\left(d-\sqrt{d^{2}-\frac{2.055\cdot M_{sl}}{b\cdot f_{cd}}}\right) \tag{18}\]  
if  
\[0\leq x\leq x_{lim} \tag{19}\]  
then the failure can be considered predominantly determined by bending.  
The integrated pressure zone then results in:  
\[F_{cd}=0.8095\cdot x\cdot b\cdot f_{cd} \tag{20}\]  
and with the obtained force, it is now possible to determine the reinforcement for a cross-sectional side (with uniaxial bending).  
\[A_{s1}=\frac{F_{cd}-N_{ed}}{\sigma_{s1}} \tag{21}\]  
Using the equations presented, it is now possible for different \(M\) to determine the associated \(N\) for a given cross-section and reinforcement. The derivation was carried out for the design, characteristic and the mean resistance parameters. For the probabilistic calculation, 30 samples were generated from the mean values, the corresponding standard deviations and the distribution of the main random variables (see Table 3). These samples yielded 30 random values for \(X_{1}\) to \(X_{11}\). For these randomly generated values, the cross-sectional resistance was then calculated (see Table 3, bottom part) allowing the creation of a point cloud around the mean _I-D_ as shown in figure in Section 4.7.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**X**} & \multicolumn{2}{c}{**Variable**} & **Dis.