## 3 Applications and Results
### Reliability of Reinforced Concrete Beam under Flexure Moment
Note that the order of the coefficients of the normality polynomials for the RCB problem can be very small (compared to the classical LSF problem and to a coastal engineering application shown later); this could be due to the units employed, and should not be understood as if the order of the polynomial could be decreased, whereas obtaining comparable precision, because the use of at least third-order normality polynomials was illustrated and found adequate in [11]. To end this section, the results of using the multi-linear regression approach for the LSF defined in Equation (21) are listed in Table 4 for \(r_{V/D}=1.0\). The subscripts in Table 4 (and the units of the design point) are associated to the random variables in Table 3. The reported values correspond to the last of the 1000 runs used to develop Figure 4. As an example of the coefficients obtained by multi-linear regression, the ones from the last of the 1000 runs (for deriving Figure 4) corresponding to \(r_{V/D}=1.0\) and \(1\times 10^{6}\) simulations resulted in \(c=3.3635\times 10^{6}\), \(b_{1}=4.0861\times 10^{5}\), \(b_{2}=2.7767\times 10^{5}\), \(b_{3}=1.1215\times 10^{5}\), \(b_{4}=-1.1888\times 10^{5}\), \(b_{5}=-9.1330\times 10^{5}\), \(b_{6}=3.1471\times 10^{4}\), and \(b_{7}=2.8878\times 10^{5}\). The previous information indicates that a similar conclusion to that of the previous example (i.e., for the case of Equation (15)) can be drawn, i.e., at least a sufficiently large number of simulations is required for a successful multi-linear regression. However, unlike in the previous example, the design point and sensitivity factors are not invariant by varying \(n_{sim}\). The differences are not so significant though; therefore, once a minimum number of simulations is ensured (around \(8\times 10^{4}\) simulations) a very precise \(\beta\) is obtained; it is also observed that the required number of simulations for adequately carrying out the multi-linear regressions decreases with increasing \(r_{V/D}\) (this could be attributed to the same reason argued before about the larger errors obtained for smaller \(r_{V/D}\)). If the tolerance \(e\) is increased, the number of simulations can be reduced, but not as significantly as for the classical LSF case (i.e., Equation (15)). For instance, an increment to \(e=0.35\) reduces \(n_{sim}\) to around \(5\times 10^{4}\); this also changes the values of the design point and sensitivity factors, but not substantially. From Table 4, it is also observed that the values are in very good agreement with the FORM results, with even higher precision from the regression approach for the reliability index. Therefore, it is concluded that the multi-linear regression by itself can be a very attractive alternative to compute \(\beta\), if a minimum \(n_{sim}\) (similar to those mentioned above) is feasible; it is emphasized once more that an additional advantage is that the design point and sensitivity factors are also determined. One final application of the revisited described methods is performed for a coastal structure in the following section.