## 3 Applications and Results
### Reliability of Reinforced Concrete Beam under Flexure Moment  
In this section the approaches described before are applied to a reinforced concrete beam (RCB) subjected to flexure moment. The example is the same investigated in a previous study [23] but focused only in one design code [24] and three ratios of the mean live to the mean dead load effect for the beam. The rectangular beam section information, LSF, and statistics are succinctly reproduced below. The LSF is  
\[g_{ACI}=BA_{s}f_{y}d\left(1-0.59\frac{A_{s}f_{y}}{f_{c}^{\prime}hd}\right)-D-V \tag{21}\]  
where \(B\) is the modeling error, \(f_{c}\) is the concrete compressive strength, \(A_{s}\) is the reinforcement steel area, \(f_{y}\) is the yielding stress of the reinforcement steel, \(b\) is the section width, \(h\) is the effective depth, and \(D\) and \(V\) are the dead load and live load effect, respectively (flexure moment). The information of all the independent variables in Equation (21) is summarized in Table 3. \(A_{s}\) is assumed deterministic and equal to 3000 mm\({}^{2}\). The PDFs of the random variables in Table 3 are based on previous literature, which in turn reflects results from experimental projects, field information, observed phenomena, and even the engineers experience to characterized these variables properly, since such PDFs have a direct impact in the computed reliabilities, code calibration tasks, and ultimately in the safety of real structures. More details can be found in [23] and the references therein. Mean values of \(D\) and \(V\) are not defined in Table 3, but they are derived by considering given mean live load effect (\(m_{V}\)) to mean dead load effect (\(m_{D}\)) ratios (\(r_{V/D}\) = 0.4, 1.0 and 2.0 are used in this study) and the assumption that the RCB just meets the code requirement; thus, the following expression is used to determine the mean values. \[1.2m_{D}+1.6m_{V}=\phi A_{s}m_{f_{y}}\left(m_{d}-0.59\frac{A_{s}m_{f_{y}}}{m_{ f_{c}}^{\prime}m_{b}}\right) \tag{22}\]  
where \(m\) denotes the mean values of the variables in the corresponding subscripts, and \(\Phi\) = 0.9. Using the previous information, the normality polynomial approach is applied to Equation (21) and the results are shown in Figures 3 and 4. These figures are analogous to Figures 1 and 2, except that the reference reliability indices (dashed lines) correspond to the values computed using FORM, that three cases of the ratio \(r_{V/D}=m_{V}/m_{D}\) are depicted (the largest \(\beta\)s correspond to \(r_{V/D}\) = 0.4 and the smallest to \(r_{V/D}\) = 2.0, as shown in Figure 3a) and that the error in Figure 4 is shown for \(r_{V/D}\) = 0.4 \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Random Variable** & **Mean** & **coeff. of var.** & **PDF** \\ \hline \(B\) & 1.01 & 0.06 & Normal \\ \(f_{c}\) **(MPa)** & 31.6 & 0.145 & Normal \\ \(f_{y}\) **(MPa)** & 474 & 0.05 & Lognormal \\ \(b\) **(mm)** & 303 & 0.04 & Normal \\ \(d\) **(mm)** & 990 & 0.04 & Normal \\ \(D\) **(kN-m)** & \({}^{*}\) & 0.05 & Normal \\ \(V\) **(kN-m)** & \({}^{*}\) & 0.18 & Gumbel \\ \hline \hline \end{tabular} \({}^{*}\) denotes that these values are not determined until a \(r_{V/D}\) value is selected and used together with Equation (22).  
\end{table} 
Table 3: Random variables for the limit state functions (LSF) of the reinforced concrete beam (RCB) considered. (Figure 4a) and for \(r_{V/D}=2.0\) (Figure 4b), which is computed by considering in Equation (18) \(\beta_{bench}\) as the average of the 1000 runs for \(n_{sim}=2\times 10^{7}\) (assumed as the exact value). MCS results are depicted with dashed-doted lines. To perform the MCS, \(m_{D}\) and \(m_{V}\) are defined using Equation (22) and \(r_{V/D}\) as mentioned before. Once they are determined, MCS can then be performed to obtain samples of \(D\) and \(V\) (and all other random variables in Table 3) and the probability of failure as per Equation (17) can be computed.