# Attenuation Zones of Two-Dimensional Periodic Foundations Including the Effect of Vertical Loads
## 4 Results and Discussion  
### Periodic Foundation with Infinite Unit Cells  
Figure 4 shows the dispersion curves in the 2D locally-resonant periodic foundation in Section 3.2, where the initial stress due to the vertical loads was taken as \(\sigma_{zz}^{0}=-40\) kPa. Figure 5 shows the mode shape of frequencies A\({}_{1}\), A\({}_{2}\), B\({}_{1}\), and B\({}_{2}\). Combing Figures 3-5, it can be found that the unit cell was symmetric in terms of both the geometric and mechanical aspects, without considering the effect of vertical load. Thus, the bound frequencies of the AZ, which were corresponding to the vibration modes of the steel core in the \(x\)- and \(z\)- directions, were identical. However, after including the effect of vertical loads, the bound frequency corresponding to the vibration modes of the steel core in the \(x\)-direction decreased more than that in the \(z\)-direction, which led to the discrepancy between the AZ bounded by A\({}_{1}\) and B\({}_{1}\) and that by A\({}_{2}\) and B\({}_{2}\), as shown in Figure 4. For vibration modes of the steel core in the \(x\)-direction, there were two AZs below 10 Hz, which were 4.24-6.09 and 8.44-9.58 Hz, respectively. For vibration modes of the steel core in the \(z\)-direction, there was only one AZ below 10 Hz, which was 4.37-6.85 Hz. According to the definition of complete AZ, the complete AZ with \(\sigma_{zz}^{0}=-40\) kPa was 4.37-6.09 Hz, which was lower and narrower than that without vertical load.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Materials**} & **Mass Density** & **Young Modulus \(E\)** & **Poisson Ratio \(v\)** \\  & \(\rho\) **(kg/m\({}^{3}\))** & **(GPa)** & **Poisson Ratio \(v\)** \\ \hline Steel & 7850 & 210 & 0.330 \\ Rubber & 1300 & \(1.37\times 10^{-4}\) & 0.463 \\ Concrete & 2300 & 25 & 0.330 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material parameters.  
Figure 3: Dispersion curves of in-plane waves in a 2D locally-resonant periodic foundation.  
The width of the complete AZ was defined as \(\text{WAZ}\ =\ \text{UBF}\ -\ \text{LBF}\), where UBF and LBF are the upper- and lower-bound frequencies of the complete AZ, respectively. The influence of the initial  
Figure 4: Dispersion curves of in-plane waves in the 2D periodic foundation with \(\sigma_{zz}^{0}=-40\ \text{kPa}\).  
Figure 5: Mode shapes of the unit cell at the bound frequencies at (**a**) A\({}_{1}\), (**b**) A\({}_{2}\), (**c**) B\({}_{1}\) and (**d**) B\({}_{2}\).  
stress on the complete AZ is shown in Figure 6. It could be found that both the LBF and UBF decrease as the compressive initial stress increases. Moreover, the decrease of the UBF was more than that of the LBF, resulting in the decrease of WAZ. The reason is that the stiffness of the unit cell to resist displacement in the \(x\)-direction was reduced more than that in the \(z\)-direction. It resulted in the UBF, corresponding to the vibration mode of the steel core in the \(x\)-direction, being more sensitive to the change of compressive initial stress.  
Figure 7 shows the attenuation coefficients in complete AZs with different initial stresses. It can be seen from Figure 7a that the complete AZ became lower and narrower with the increase of the compressive initial stress, which was in accordance with the results in Figure 6. In addition, the attenuation coefficients in the complete AZ decreased with the increase of the compressive initial stress. From Figure 7b, it can be further seen that the compressive initial stress mainly affected the attenuation coefficients below the center frequency of the complete AZ.