# Safety Evaluation of Existing R.C. Buildings: Uncertainties Due to the Location of In Situ Tests
## 3 Finite Element Model of the Building  
A 3D finite element model of the building was implemented with beam elements and floors acting as rigid diaphragms, assuming the columns to be fixed at the base (Figure 1a). In order to represent the possible variation in the mechanical properties, corresponding, for example, to different conditions of the preparation, casting, or curing of the concrete or to different batches of steel, the value of the concrete compressive strength (f\({}_{\text{c}}\)) was considered to be in the range of 9 MPa to 29 MPa, while for the steel strength (f\({}_{\text{s}}\)), the range of variability was considered to be between 300 MPa and 500 MPa. On this basis, each floor of the building was divided into five parts (see Figure 1b). Each part was considered representative of a different casting day or batch of steel, and, for each part and for each floor, the values of the concrete and steel strengths were randomly chosen in the mentioned ranges, as summarised in Tables 1 and 2, respectively. The aforementioned procedure aimed to fix the \"real\" distribution of strengths in the examined structure and, on that basis, to evaluate the \"true\" seismic safety index of the \"real\" structure, which represents the final scope of seismic vulnerability assessment. For the \"real\" structure, the randomly extracted concrete strength values were characterised by a mean value of 19.6 MPa, a standard deviation equal to 6.1 MPa, and a coefficient of variation equal to 0.3, while the chosen steel strengths had a mean value of 396 MPa, a standard deviation of 63 MPa, and a coefficient of variation equal to 0.16. As it is reasonable to expect, the steel strength was characterised, therefore, by a variability lower than that of the concrete. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Concrete Strength [MPa]** & & & \\ \hline  & **A** & **B** & **C** & **D** & **E** \\ \hline First storey & 16 & 28 & 16 & 13 & 10 \\ Second storey & 16 & 23 & 29 & 22 & 10 \\ Third storey & 28 & 22 & 22 & 20 & 17 \\ Fourth storey & 17 & 22 & 24 & 28 & 9 \\ Fifth storey & 22 & 18 & 21 & 28 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reference model M\({}_{0}\) (actual configuration): concrete compressive strength for each part of the building in Figure 1b. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Steel Strength [MPa]** & & & \\ \hline  & **A** & **B** & **C** & **D** & **E** \\ \hline First storey & 360 & 440 & 360 & 440 & 460 \\ Second storey & 460 & 340 & 300 & 380 & 300 \\ Third storey & 360 & 380 & 480 & 500 & 400 \\ Fourth storey & 320 & 460 & 500 & 460 & 360 \\ Fifth storey & 340 & 460 & 340 & 380 & 320 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reference model M\({}_{0}\) (actual configuration): steel strength for each part of the building in Figure 1b. These values were assigned to the beams and columns of each of the different regions (Figure 1b) of the reference model of the structure, from now on denoted as M0 (i.e., model 0) to distinguish it from alternative realisations (Section 4) of the same \"true\" model, and represent the actual mechanical strengths of these elements. The perfect knowledge of these--never attainable in practice, but here introduced as a hypothesis--allowed evaluation of the \"true\" level of seismic safety of the building once all the other geometrical and loading details were also known, as assumed in the following, in order to focus attention on the random variation of mechanical properties. The dynamic analyses of the finite element model, obtained by introducing the values in Tables 1 and 2, were performed considering the spectral acceleration corresponding to the Life-Safety Limit State and assuming a behaviour factor (q) equal to 2, and the seismic safety index (\(\alpha\)) of the model was evaluated as the ratio between the peak ground acceleration capacity (PGA\({}_{\text{C}}\)) and the peak ground acceleration demand (PGA\({}_{\text{D}}\)).