# Safety Evaluation of Existing R.C. Buildings: Uncertainties Due to the Location of In Situ Tests
## 4 Effects of Uncertainties Related to Test Locations
\(f_{c}\) and \(f_{s}\) are the mean concrete compressive strength and the mean steel strength, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **300 MPa** & **340 MPa** & **380 MPa** & **420 MPa** & **460 MPa** \\ \hline
**15 MPa** & 0.242 & 0.268 & 0.283 & 0.293 & 0.315 \\
**20 MPa** & 0.253 & 0.278 & 0.288 & 0.302 & 0.323 \\
**25 MPa** & 0.258 & 0.283 & 0.297 & 0.310 & 0.327 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Seismic safety index (\(\alpha_{KL2}\)) for 15 different combinations of mean concrete and steel strength values corresponding to knowledge level KL2. six cores per storey for the investigated building (about 800 m\({}^{2}\) plan). For the five-floor building considered here, therefore, the strengths (\(f_{c}\) and \(f_{s}\)) to be considered in the safety evaluation were obtained for each analyst as the average of the strengths of 30 cores and 10 steel bars extracted by columns. A Monte Carlo approach was adopted to define 100 different cases. For each case, the procedure simulated the extraction of six values of concrete strength and two values of steel strength for each floor, corresponding to a random subset of the columns, and, based on such values, the mean values of the concrete compressive strength (\(f_{c}\)) and steel strength (\(f_{s}\)) were evaluated. The obtained values are plotted in Figure 3.  
By means of the interpolating surface in Figure 2, the seismic safety index was then evaluated for each combination of strength values in Figure 3, and their cumulative distribution is shown in Figure 4. Moreover, in Figure 4, the cumulative distribution of the Gaussian curve which fits the evaluated safety indexes is plotted. The red broken line in Figure 4 corresponds to the safety index evaluated by means of the reference model (M\({}_{0}\)) (Section 2), which assumed perfect knowledge (never attainable in practice) of the effective strength of each structural element. The safety index of the reference model can therefore be assumed as an ideal target, whose comparison with the indexes evaluated by different analysts may allow estimation of the reliability of the safety level attainable when a finite (and usually small) number of in situ tests are available, as in real cases. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **300 MPa** & **340 MPa** & **380 MPa** & **420 MPa** & **460 MPa** \\ \hline
**15 MPa** & 0.230 & 0.247 & 0.263 & 0.283 & 0.293 \\
**20 MPa** & 0.236 & 0.253 & 0.268 & 0.288 & 0.297 \\
**25 MPa** & 0.242 & 0.258 & 0.273 & 0.293 & 0.302 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Seismic safety index (\(\alpha_{KL1}\)) for 15 different combinations of mean concrete and steel strength values corresponding to knowledge level KL1. Figure 3: Mean concrete compressive strength (**a**) and mean steel strength (**c**) for 100 different subsets of structural elements. Cumulative distribution functions of concrete (**b**) and steel (**d**) mean strengths and their Gaussian interpolation (red lines). Figure 4 highlights that for the randomly selected locations of in situ tests corresponding to the average values of \(f_{c}\) and \(f_{s}\) in Figure 3, the probability of overestimating the seismic safety is negligible once the confidence factor \(\text{CF}_{\text{KL2}}=1.2\) is adopted in the analysis, as suggested by the Eurocodes [4] and the Italian codes [2] for the level of knowledge KL2.