# Combined Attenuation Zones of Combined Layered Periodic Foundations
## 2 Computational Methods  
### Dispersion Curves of SH Waves in Traditional LPFs with Identical Unit Cells  
A traditional LPF consists of an infinite repetition of alternating concrete layer A with thickness \(h_{1}\) and rubber layer B with thickness \(h_{2}\) is schematically shown in Figure 1. The material parameters of concrete and rubber are listed in Table 1. The effect of the self-weight of the superstructure on the traditional LPF is reduced to uniformly distributed initial stress \(\sigma_{zz}^{0}\). A typical unit cell whose thickness is \(h=h_{1}+h_{2}\) can be easily identified. Due to the spatial periodicity in the traditional LPF in Figure 1, it is sufficient to consider the dynamic characteristics of the only one typical unit cell to obtain wave propagation in the whole traditional LPF.  
Figure 1: Traditional layered periodic foundation with identical unit cells.  
SH waves propagating in the \(z\) direction in the traditional and CLPF are considered in the present paper, where the displacement is in the \(x\) direction. The governing equation of SH waves in LPF can be written as [23]  
\[(G^{(e)}+\sigma_{zz}^{0})\frac{\partial^{2}u^{(e)}}{\partial z^{2}}=\rho^{(e)} \ddot{u} \tag{1}\]  
where \(u\) is the displacement in the \(x\) direction; the superscript \(e\) denotes the material of layer A or B; \(G^{(e)}\) and \(\rho^{(e)}\) are shear modulus and mass density of the material of layer \(e\), respectively. The solution of Equation (1) can be expressed as  
\[u^{(e)}(z,t)=U^{(e)}(z)e^{i\omega t} \tag{2}\]  
where \(\omega\) is the angular frequency of the SH wave and \(U(z)\) is the displacement amplitude who is only dependent of the location in the \(z\) direction.  
The connection interface of two adjacent layers is assumed to be perfectly bonded. Thus, the continuity conditions satisfy  
\[U^{(A)}(h_{1})=U^{(B)}(0),\ \tau^{(A)}(h_{1})=\tau^{(B)}(0), \tag{3}\]  
where \(\tau\) denotes the shear stress.  
By using the weak form quadrature element method [24; 25; 26; 27], the dynamic governing equations of a typical unit cell can be expressed as  
\[\Big{[}\mathbf{K}+\mathbf{K}^{0}-\omega^{2}\mathbf{M}\Big{]}\mathbf{d}=0 \tag{4}\]  
where \(\mathbf{K}\), \(\mathbf{K}^{0}\), \(\mathbf{M}\) and \(\mathbf{d}\) are the stiffness matrix, geometric stiffness matrix due to the initial stress, and the matrix of the typical unit cell and the nodal displacement amplitude vector, respectively.  
According to the Bloch-Floquet theory [5; 28], the periodic boundary conditions of the unit cell can be given as  
\[U^{(A)}(0)e^{i\mathbf{k}h}=U^{(B)}(h_{2}),\ \tau^{(A)}(0)e^{i\mathbf{k}h}=\tau^{(B)} (h_{2}), \tag{5}\]  
where \(k\) is the wave number.  
Substituting Equation (5) into Equation (3) yields standard complex eigenvalue equations:  
\[\Big{[}\mathbf{K}^{r}(k)+\mathbf{K}^{0r}(k)-\omega^{2}(k)\mathbf{M}^{r}\Big{]} \left[\begin{array}{c}\mathbf{d}_{1}\\ d_{1}\end{array}\right]=\left[\begin{array}{c}\mathbf{0}\\ 0\end{array}\right] \tag{6}\]  
where \(\mathbf{K}^{r}(k)\), \(\mathbf{K}^{0r}(k)\) and \(\mathbf{M}^{r}\) are the reduced stiffness matrix, geometric stiffness matrix and mass matrix after applying the periodic boundary conditions, respectively; \(\mathbf{d}_{i}\) is the nodal displacement amplitude vector of interior nodes of the typical unit cell and \(d_{1}\) is the nodal displacement amplitude of the bottom boundary, respectively. For a specific wave number \(k\), the eigenfrequencies \(\omega\) can be determined by Equation (6). Dispersion curves can be obtained by varying the wave number \(k\) in the first irreducible Brillouin zone. The gaps between two adjacent branches of the dispersion curves are the so-called AZs.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Materials** & **Mass Density \(\rho\) (kg/m\({}^{3}\))** & **Young Modulus \(E\) (GPa)** & **Poisson Ratio \(v\)** \\ \hline Concrete & 2300 & 25 & 0.330 \\ Rubber & 1300 & \(1.37\times 10^{-4}\) & 0.463 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material parameters.