## 4 Considerations for the Reliability Analyses of the Structural System
### Stochastic Models for the Basic Variables
The test hypothesis evaluates whether the data sample fits this assumed distribution function by comparing the deviation values with a target value, which depends on the significance level \(a\) adopted and the sample size. The assumed lognormal distribution for the model uncertainty was confirmed for all the investigated resistance models (in this case, a resistance model for each of the failure modes identified in Section 1). Second, the David-Hartley-Pearson test was used to detect potential outliers in the dataset [46]. The test checks whether it is likely that an observed extreme value belongs to a normally distributed population or whether it is an extreme observation. If an outlier is detected, the test is repeated, excluding the test result with the greatest distance from the mean value. This process is repeated until no further outliers are identified. Due to the lognormality of the data, before applying the David-Hartley-Pearson test, the data values were previously transformed to normality: \(y_{\text{i}}=\ln(x_{\text{i}})\). The test statistic was then determined for the transformed data values \(y_{\text{i}}\). Once this evaluation was concluded, the database was limited to 37 tests for failure inside the punching shear reinforced zone (failure mode (i)), 44 tests for failure outside the punching shear reinforced zone (failure mode (ii)) and 41 tests for failure at the maximum punching shear capacity (failure mode (iii)). Third, the estimators for the real parameters were determined by means of point estimators and interval estimators (Table 2). As the name implies, a point estimator is a procedure used to produce an estimate of a parameter of interest and it is usually a \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Concrete Class**} & \multicolumn{3}{c}{**Prior Parameters (*) [38]**} & \multicolumn{3}{c}{**Parameters for a Lognormal Distribution**} & \multirow{2}{*}{\(f_{\text{cm}}\)} & \multirow{2}{*}{\(\sigma_{\text{fem}}\)} & \multirow{2}{*}{\(f_{\text{ck}}\)} \\ \cline{2-2} \cline{6-9}  & \(m^{\prime}\) & & & & & & **Standard** & \\ \cline{2-2} \cline{6-9}  & \(m^{\prime}\) & \(n^{\prime}\) & \(s^{\prime}\) & \(v^{\prime}\) & **Mean Value \(\mu_{\text{LND}}\)** & **Deviation \(\sigma_{\text{LND}}\)** & & \\ \hline C25 & 3.65 & 3.0 & 0.12 & 10 & 3.65 & 0.164 & 39.00 & 6.45 & 28.36 \\ C35 & 3.85 & 3.0 & 0.09 & 10 & 3.85 & 0.123 & 47.35 & 5.86 & 38.37 \\ C45 & 3.98 & 3.0 & 0.07 & 10 & 3.98 & 0.096 & 53.76 & 5.17 & 45.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Stochastic values for the concrete compressive strength \(f_{\text{c}}\) variable. function of only sample data values when these are available [47]. For the estimation of the expected value, the arithmetic mean of the population was used and for the estimation of the standard deviation, the unbiased empirical standard deviation of the population was adopted. The point estimators were then contrasted with the interval estimators; in this case, a confidence level of 95% was considered due to the limited size of the dataset. The lower estimates of the expected values and the upper estimates of the standard deviations were initially determined for a normal distribution. Then, the lognormal distributed parameters (i.e., expected value, standard deviation and coefficient of variation) were also determined for the point estimators and interval estimators. Finally, by following the recommendation of [PERSON] [48], the coefficient of variation was corrected with a value of 0.05. It should be clarified here that the geometric and strength variables included in the tests showed some scatter, which was included in the model uncertainty. If the model uncertainty is combined with such variables, the overall scatter can be overestimated. Therefore, the coefficient of variation of the model uncertainty is recommended to be reduced by 0.05. The entire set of test results cannot be presented in this paper due to space limitations.