# Experimental Assessment of the Effect of Vertical Earthquake Motion on Underground Metro Station
## 2 Shaking Table Tests
### Model  
In view that the similitude ratio placed a severe limitation on the choice of suitable model materials, the similitude ratio design would be conducted before choosing the model materials. Based on the capability of the existing facility and equipment, the similitude ratio design of the experiment was accomplished on the basis of the dimensional analysis, which was adopted extensively in the design of the shaking table tests by other researchers. [PERSON] [26] designed and performed a series of scale model shaking table testing of the piles in clay in a 1 g scale model environment [27], in which he summarized that the method of the governing equations involved the differential equation describing the process and the formation of the similarity variables that related the model to the prototype. The similitude ratio design for this research herein closely follows the aforementioned method. \begin{table}
\begin{tabular}{c c} \hline \hline
**Table Size/(m \(\times\) m)** & **4 \(\times\) 4** \\ \hline Maximum specimen mass/ton & 25 \\ Operating frequency range/Hz & 0.1–50 \\ \(X\) direction & \(\pm\)100 \\ Maximum displacement/mm \(Y\) direction & \(\pm\)50 \\ \(Z\) direction & \(\pm\)50 \\ \(X\) direction & \(\pm\)1000 \\ Maximum velocity/(mm/s) \(Y\) direction & \(\pm\)600 \\ \(Z\) direction & \(\pm\)600 \\ \(X\) direction & \(\pm\)4.0 \\ Maximum acceleration (empty table)/g \(Y\) direction & \(\pm\)2.0 \\ \(Z\) direction & \(\pm\)4.0 \\ \(X\) direction & \(\pm\)1.2 \\ Maximum acceleration (with mass 15 ton)/g \(Y\) & \(\pm\)0.8 \\ direction & \(Z\) direction & \(\pm\)0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main technical parameters of the shaking table system. Figure 1: Photos of the shaking table and flexible-wall container. With respect to dynamics problems, the differential equation of motion expressed through displacement can be stated as (take the equation of x-direction as an example)  
\[(\lambda+G)\frac{\partial\varepsilon}{\partial x}+G\
abla^{2}u+X=\rho\frac{ \partial^{2}u}{\partial t^{2}} \tag{1}\]  
where \(u\), \(t\), \(\rho\), and \(\varepsilon\) are displacement, time, density, and strain, respectively. \(x\) and \(X\) refer to the coordinate position and the force per unit volume in the x direction, respectively. \(\lambda\) and \(G\) are the Lame constant and shear modulus, respectively. Hence, Equation (1) is also called the Lame equation. The problem-solving process of general dynamics problems is reduced to solve the Lame equation under the given boundary conditions. Then, Equation (1) can be expressed as  
\[G = \frac{\rho\frac{\partial^{2}u}{\partial t^{2}}-\lambda\frac{ \partial\varepsilon}{\partial x}-X}{\
abla^{2}u+\frac{\partial\varepsilon}{ \partial x}} \tag{2}\]  
The most common sets of basic quantities are those of mass M, length L, and time T [28]. In Equation (2), \(\varepsilon\) is a dimensionless quantity and thus the dimension of \(\frac{\partial\varepsilon}{\partial x}\) is L\({}^{-1}\). The dimension of \(\rho\) is ML\({}^{-3}\). The dimension of \(\frac{\partial^{2}u}{\partial t^{2}}\), which represents acceleration, \(a\), is LT\({}^{-2}\). The three items in the numerator have the same dimensions. The two items in the denominator also have the same dimensions. Since \(\varepsilon\) is a dimensionless quantity for both prototype and model soil, the similitude ratio for \(\varepsilon\) must equal unity (1). The similitude ratios for \(\
abla^{2}u+\frac{\partial\varepsilon}{\partial x}\) and \(\rho\frac{\partial^{2}u}{\partial t^{2}}-\lambda\frac{\partial\varepsilon}{ \partial x}-X\) are \(S_{l}^{-1}\) and \(S_{\rho}S_{a}\), respectively. The similitude ratio for \(G\) is \(S_{G_{d}}\), where the subscript \(G_{d}\) represents dynamic shear modulus. Then, the similitude relations between the model soil and the prototype soil can be deduced from Equation (2) as:  
\[\frac{G_{d}^{m}}{G_{d}^{p}}=S_{G_{d}}=S_{\rho}S_{a}S_{l} \tag{3}\]  
where \(G_{d}^{m}\) and \(G_{d}^{p}\) are dynamic shear modulus of model and prototype, respectively.