## 3 Numerical Benchmark: System with Three Interacting Diaphrags  
The lumped mass model of three adjacent interacting diaphrags represented in Figure 2 is now considered. The system, presenting a diaphragmatic behavior with a chain-like interaction, is composed of three masses \(m_{1}\), \(m_{2}\) and \(m_{3}\), and their respective polar moments of inertia \(J_{0,1}\), \(J_{0,2}\) and \(J_{0,3}\). The values of the translational stiffnesses along the x-direction, \(\overline{k}_{1}^{x}\), \(\overline{k}_{2}^{x}\) and \(\overline{k}_{3}^{x}\), the translational stiffnesses along the y-direction, \(\overline{k}_{1}^{y}\), \(\overline{k}_{2}^{y}\) and \(\overline{k}_{3}^{y}\), the torsional stiffnesses \(\overline{k}_{1}^{\gamma}\), \(\overline{k}_{2}^{y}\) and \(\overline{k}_{3}^{\gamma}\) around \(\gamma\), were chosen to represent typical values of square concrete diaphragms of 50 m on each side. The mixed terms of stiffnesses \(\overline{k}_{1}^{xy}\), \(\overline{k}_{2}^{xy}\), \(\overline{k}_{3}^{xy}\) and \(\overline{k}_{1}^{y\gamma}\), \(\overline{k}_{2}^{y\gamma}\), \(\overline{k}_{3}^{y\gamma}\), and the static moments \(S_{1}^{x}\), \(S_{2}^{x}\), \(S_{3}^{x}\) and \(S_{1}^{y}\), \(S_{2}^{y}\), \(S_{3}^{y}\) have been calculated accordingly. The numerical values of masses, polar moments of inertia, static moments, and stiffnesses are reported in Table 1.  
Figure 1: Lumped mass model of the interacting \(i\)-th diaphragm. Figure 2: Lumped mass model of three adjacent interactive diaphragms. The stiffnesses describing the interaction \(k_{2}^{x}\), \(k_{2}^{y}\), \(k_{2}^{y}\), \(k_{2}^{\gamma}\), and \(k_{3}^{x}\), \(k_{3}^{y}\), \(k_{3}^{\gamma}\) are set as a fraction (factor varying between 0 and 2), defined as \(k_{var}\), of the values reported in Table 1, which corresponds to the continuity of the spring. The eigenvalue problem of the above-mentioned system has been then solved to extract the modal parameters, i.e., natural frequencies and mode shapes of the system. Parametric simulations were conducted to study the relative variation of the modal frequencies of the system with respect to \(k_{var}\). A simultaneous uniform variation of \(k_{2}^{x}\), \(k_{2}^{y}\), \(k_{2}^{\gamma}\), and \(k_{3}^{x}\), \(k_{3}^{y}\), \(k_{3}^{\gamma}\) has been considered. To this aim, the modal frequencies of the system, generally called \(f_{r}\) (with \(r\) varying from 1 to 9), were normalized with respect to the fundamental frequency. Figures 3-5 represent the variation of the 9 modes and of the 9 natural frequencies of the system with respect to \(k_{var}\). To have a better visualization, the representation is divided into groups of 3 modes each: Figure 3 represents the modes from 1 to 3, Figure 4 from 4 to 6, and Figure 5 from 7 to 9. It is worth noting that the y-axis scales of Figures 3-5 are different. Considerations can be made concerning the modal parameters of the system. In general, an increasing linear trend can be observed in the case of the natural frequencies. Figure 3 shows that the curve corresponding to the first natural frequency \(f_{1}\) is almost flat, while a clear variation of \(f_{r}\) can be observed for the curves corresponding to the second and the third ones (\(f_{2}\) and \(f_{3}\)). A similar trend is observed for the other two groups reported in Figures 4 and 5. Therefore, it can be said that increasing values of the stiffness characterizing the interaction clearly affect the higher modal frequencies of each group more. Comparing the three figures, it is noticeable that in the case of the groups of frequencies \(f_{1}\), \(f_{2}\), \(f_{3}\) and \(f_{4}\), \(f_{5}\), \(f_{6}\), for values of \(k_{var}\) equal to 0, the numerical value of the frequencies is almost the same. The same behavior is not found for the group of frequencies \(f_{7}\), \(f_{8}\) and \(f_{9}\), where the numerical value of \(f_{9}\) is almost double the numerical values of \(f_{7}\) and \(f_{8}\).