# The Dynamic Response of an Arch Dam During a Record Low-Intensity Earthquake
## 3 Numerical Model
### Model Calibration  
The numerical model was calibrated with the results of the two forced vibration tests performed in the dam. Forced vibration testing consists of the application to the structure of a dynamic sinusoidal load by means of a vibrator, with a continuously varying frequency. The determination of the structural dynamic response is achieved by the calculation of the FRF (Frequency Response Function), between the series recorded in the sensors installed in the body of the dam, and the record of the applied load. The natural frequencies of the structure are expected to be close to the maximum FRF values occurring in the frequency interval of the applied excitation, which correspond to the vibration modes of the dam. The mechanical properties of the materials and joints were based on the values obtained from the material characterization tests [9]. Table 1 shows the mechanical properties of the material and interfaces adopted in the numerical model for the full reservoir situation. Figure 7: Numerical models of the Baixo Sabor dam: (**a**) reservoir level at 195.5 m; (**b**) reservoir level at 234.0 m; (**c**) details of the mesh of the dam and reservoir. Table 2 shows the natural frequencies determined experimentally based on the results of the forced vibration tests, the natural frequencies calculated using numerical models, and the differences between the two methodologies for the Baixo Sabor dam. The 3 DEC model with fluid elements and viscous boundaries does not allow for a direct calculation of the eigenfrequencies. Thus, a white noise loading with frequency contents between 0 and 30 Hz is applied at the crest, and the resulting velocity history of about 120 s long is recorded. Then, modal identification methodologies (SSI-COV, Covariance-driven Stochastic Subspace Identification) are employed to determine the natural frequencies and modal configurations from the time series of the response. Checking the correspondence between the experimental and numerical modes can initially be performed by visually comparing the modal configurations, which gives a direct idea of the correlation between the shapes of the vibration modes. However, modal configurations are generally quite complex, which is why it is advisable to use the MACij (Modal Assurance Criterion) coefficient matrix [27]. There was excellent agreement between the experimental and numerical values, with a maximum difference of 1.33%, and it can be concluded that the numerical models developed reproduce the site conditions when the tests were carried out. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Material** & **Young’s Modulus [GPa]** &
\begin{tabular}{c} **Poisson** \\ **Ratio** \\ \end{tabular} \\ \hline Dam & 42.0 & 0.20 \\ Rock mass & 35.0 & 0.20 \\ \hline \hline \multirow{2}{*}{**Interface**} & **Normal stiffness** & **Shear stiffness** \\  & **[GPa/m]** & **[GPa/m]** \\ \hline Dam contraction joints & 50.0 & 20.8 \\ Dam–rock & 50.0 & 20.8 \\ Dam–water & 2.0 & 0.0 \\ Rock–water & 2.0 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of the material and interfaces adopted in the numerical model for the full reservoir situation [9].