## 5 Probabilistic Analysis  
The structural reliability of the existing reinforced concrete beam cross-section (Figure 4) exposed to the permanent and variable loads specified in Section 3 has been analyzed by using the FORM method. When using probabilistic methods, the reliability of a structure shall be verified in terms of the failure probability according to the following condition [2; 12]:  
\[P_{f}=P\{g(x_{i})<0\}<P_{ft} \tag{5}\]  
where \(g\) (\(x_{i}\)) is the limit state function of the basic variables \(x_{i}\) and \(P_{th}\) is the target probability of failure for a given reference period, here \(t_{\text{ref}}\) = 50 years. In the example of the reinforced concrete beam, the limit state function for the bending resistance in the mid-span cross-section is given as:  
\[g(x_{i})=\zeta_{R,M}\cdot\left[f_{y}\cdot A_{s}\cdot d-0.5\cdot\frac{\left(f_{ y}\cdot A_{s}\right)^{2}}{b\cdot\eta_{c}\cdot f_{c}}\right]-\zeta_{E,M}\cdot \left[M_{gc}+M_{gp}+M_{q}\right]=0 \tag{6}\]  
where \(M_{gc}\), \(M_{gp}\) and \(M_{q}\) are the bending moments due to the self-weight of the beam, the other permanent loads and the variable (imposed) load, respectively. Other variables are explained in previous sections. In a first analysis, the probabilistic models given in Table 3 are used. The resulting design values and sensitivity factors for the basic variables are indicated in Table 12. A probability of failure of \(p_{f}\) = 2.86 \(\times\) 10\({}^{-3}\) has been obtained, which corresponds to a reliability index \(\beta\) = 2.76. Considering the assumed target reliability index  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Approaches** & & \(M_{E\text{a,2}}\) **[kNm]** & \(M_{R\text{a,2}}\) **[kNm]** & \(M_{E\text{a,2}}\leq M_{R\text{a,2}}\) \\ \hline  & NSB1 & 230.0 & 216.5 & No \\  & NSB2 & & Not carried out & \\  & NSB3 & 223.2 & 265.0 & Yes \\  & NSB4 & 223.0 & 163.0 & No \\  & Member states [2] & NSB5 & 217.0 & 250.0 & Yes \\  & NSB6 & 212.0 & 230.0 & Yes \\  & NSB7 & 229.5 & 288.7 & Yes \\  & NSB8 & 207.0 & 214.0 & Yes \\ \hline
[9] procedure & & 223.1 & 218.4 & No \\ \hline \multirow{2}{*}{_fib_ Bulletin 80 [10]} & DVM & 225.6 & 217.0 & No \\ \cline{2-5}  & APFM & 205.4 & 213.3 & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 11: Second stage assessment values for bending moments and resistance.  
of \(\beta_{t}\) = 3.8 (Table 5), the reliability of the beam is insufficient. This confirms the results obtained in first stage of the safety verification by applying the different partial factor approaches (Table 5).  
To improve the estimate of \(\beta\), the updated values for the reinforcing steel yield strength, \(f_{y}\), and for the concrete compressive strength, \(f_{c}\), indicated in Table 9 are adopted. The probabilistic analysis of the RC beam results in a probability of failure \(p_{f}\) = 1.6 \(\times\) 10\({}^{-5}\) and the corresponding reliability index increases to \(\beta\) = 4.16, which is higher than the required reliability level expressed by \(\beta_{t}\) = 3.8. Therefore, for the remaining working life the beam can be judged sufficiently reliable for bending resistance. The obtained assessment values, \(X_{\text{a}}\), and sensitivity factors, \(\alpha_{X}\), for the basic variables are also indicated in Table 12. The comparatively low value obtained for the sensitivity factor of \(f_{c}\) shows that the concrete compressive strength is not among the most relevant variables with regard to bending failure and more important variables, as \(f_{y}\), should be preferably updated. Most effective would be updating the parameters of the variable load \(q\), in case corresponding information would be available.