# Attenuation Zones of Two-Dimensional Periodic Foundations Including the Effect of Vertical Loads
## 2 Computational Method  
Figure 1 shows the cross section of a 2D periodic foundation with vertical loads. The foundation was made of periodic arrays of square steel cylinders coated with rubber in a concrete matrix, where the length of the cylinders in the \(y\)-direction was assumed to be infinite. Hence, wave propagation in the periodic foundation could be reduced to a 2D problem. A typical unit cell can be defined, as shown in Figure 1, where the side length of square cylinder is \(l\), the thickness of coating layer is \(t_{k}\), and the periodic constant is \(a\). According to the periodic structure theory [18], wave propagation in periodic structures can be investigated by the dynamic analyses of a typical unit cell. In the present paper, the materials of the cores, the coating layers, and the matrix were assumed to be linearly elastic and isotropic. Furthermore, for simplicity, the effect of vertical loads from the upper structure was reduced as homogeneous initial stress, \(\sigma_{zz}^{0}\), in the 2D periodic foundation. Figure 1: Cross section of a 2D periodic foundation and its typical unit cell. According to the weak form quadrature element method, the typical unit cell can be decomposed into several quadrature elements [19]. Let \(u\) and \(w\) be the displacement components in the \(x\)- and \(z\)-directions, respectively. Thus, the strain energy of the quadrature element \(e\) can be expressed as  
\[U^{(e)}=\iint\limits_{\Omega^{(e)}}\frac{1}{2}\Big{(}\epsilon^{T}E\epsilon+ \epsilon^{*T}\mathbf{\sigma}^{0}\mathbf{\epsilon}^{*}\Big{)}\ dxdz=\int_{-1}^{1}\int_{- 1}^{1}\left(A^{(e)}+A^{0(e)}\right)\ d\xi d\eta, \tag{1}\]  
where \(\Omega^{(e)}\) is the area in the \(x\)-\(z\) plane of the quadrature element \(e\), \(\epsilon=\left[ \begin{array}{cc}\frac{\partial u}{\partial x}&\frac{ \partial v}{\partial y}&\frac{\partial v}{\partial x}+\frac{\partial u}{ \partial y}\\ \end{array} \right]^{T}\) is the strain vector, \(\mathbf{E}=\frac{\overline{E}}{1-\overline{V}^{2}}\left[ \begin{array}{cc}1 &\overline{v}&0\\ \overline{v}&1&0\\ 0&0&(1-\overline{v})/2\\ \end{array} \right]\) in which \(\left\{ \begin{array}{cc}\overline{E}=\frac{E}{1-\overline{v}^{2}}\\ \overline{\mu}=\frac{V}{1-\overline{v}}\\ \end{array} \right.,E\) is the Young modulus, \(v\) is the Poisson ratio, \(\epsilon^{*}=\left[ \begin{array}{cc}\frac{\partial u}{\partial z}&\frac{ \partial w}{\partial z}\\ \end{array} \right]^{T}\), \(\mathbf{\sigma}^{0}=\left[ \begin{array}{cc}\sigma_{zz}^{0}&0\\ 0&\sigma_{zz}^{0}\\ \end{array} \right]\), \(A^{(e)}\) is the strain energy density of the linear strains, and \(A^{0(e)}\) is the strain energy density due to the initial stress terms, respectively. By using the quadrature and differentiation scheme in the weak form quadrature element method [20; 21; 22], Equation (1) can be obtained as  
\[U^{(e)}=\frac{1}{2}\mathbf{d}^{(e)^{T}}\Big{(}\mathbf{K}^{(e)}+\mathbf{K}^{0( e)}\Big{)}\mathbf{d}^{(e)}, \tag{2}\]  
where \(\mathbf{d}^{(e)}\), \(\mathbf{K}^{(e)}\) and \(\mathbf{K}^{0(e)}\) are the nodal displacement vector, stiffness matrix and geometric stiffness matrix of the quadrature element \(e\), respectively. The potential energy of external loads and the kinetic energy of the quadrature element \(e\) can be written as  
\[W^{(e)}=-\int_{\partial\Omega^{(e)}}u^{T}\mathbf{f}\ ds=-\mathbf{d}^{(e)T} \mathbf{F}^{(e)},T^{(e)}=\frac{1}{2}\mathbf{\dot{d}}^{(e)T}\mathbf{M}^{(e)} \mathbf{\dot{d}}^{(e)}, \tag{3}\]  
where \(\mathbf{F}^{(e)}\) and \(\mathbf{M}^{(e)}\) denote the nodal force vector and the mass matrix of the quadrature element \(e\), respectively; the superposed dot denotes derivative with respect to the time.