## 2 Dynamic Centrifuge Tests
### Scaling Law and Physical Models  
To reproduce the same overburden stress in the scaled model as that in the prototype, the conventional scaling law for dynamic centrifuge tests is commonly established using the Buckingham \(\pi\) theorem. However, the shortcoming of the conventional scaling law is apparent: the set of scaling factors is built on the predetermined gravitational acceleration scaling factor with a relatively fixed relationship. Consequently, the large size engineering prototype could be difficult to model when the experiments are faced with restrictions of facility capacity. To overcome the limitation, [PERSON] et al. proposed the concept of two-stage scaling and developed the generalized scaling law (GSL) in dynamic centrifuge tests [35]. As shown in Figure 3, \(\mu\) is the dimensional scaling factor at the first stage, and \(\eta\) is the gravitational acceleration scaling factor at the second stage. The two-stage scaling and GSL extend the limits of the existing facilities and allow the dynamic centrifuge tests with a dimensional scaling factor reaching up to 100-1000 [35].  
Due to the stress dependency of soil material, separate scales for stress and strain are needed to secure the kinetic and kinematic similarity between the model and the prototype. [PERSON] and [PERSON] [36] developed three different types of GSL ([PERSON]'s Type I, II, III GSL) within  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Indictors** & **Specifications** \\ \hline Maximum acceleration & 20 g \\ Maximum velocity & 38 cm/s \\ Maximum displacement & \(\pm\)6 mm \\ Frequency range & 20–200 Hz \\ Bearing capacity & 300 kg \\ Maximum shaking duration & 1 s \\ Shaking table dimensions & 1400 \(\times\) 1000 mm \\ Input loading & Sinusoidal or earthquake waves \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of TLJ-150 centrifuge shaking table.  
Figure 3: Concept of two-stage scaling.  
Figure 2: Test facilities: (**a**) electro-hydraulic shaking table; (**b**) laminar shear model box.  
the framework forementioned, each based on a different strain scaling factor determination method. [PERSON]'s type II GSL assumes that the tangential stiffness of sand is proportional to the square root of the confining pressure, and the relations between different scaling factors is presented in Equations (1)-(3):  
\[\mu_{\sigma}=\mu\mu_{\rho} \tag{1}\]  
\[\mu_{D}=\mu_{\sigma}/\mu_{\epsilon}=\mu_{\sigma}^{0.5} \tag{2}\]  
\[\mu_{\epsilon}=\mu_{\sigma}/\mu_{D}=\mu^{0.5} \tag{3}\]  
where \(\mu_{\sigma}\), \(\mu_{\rho}\), \(\mu_{D}\), \(\mu_{\epsilon}\) are the scaling factors of stress, density, tangent stiffness, and strain, respectively.  
The applicability of the two-stage scaling has been investigated and validated in previous research. [PERSON] et al. (2017) and [PERSON] et al. (2018) adopted [PERSON]'s type II GSL in dynamic centrifuge tests of saturated sand and found that the model behaviors well mimic the prototype. In order to model the thick liquefiable calcareous sand layer in the engineering site, considering the limited amount of model material, the dimensions of the model container, and the capacity of the centrifuge, the two-stage scaling with [PERSON]'s Type II GSL was adopted in the dynamic centrifuge tests, the scaling factors of which are shown in Table 2. During the experiments, rotational acceleration was set to be 40 g, and \(\mu\) and \(\eta\) were 2 and 40, respectively. The length, width, and depth of the saturated level ground model are 500 \(\times\) 400 \(\times\) 250 mm, corresponding to the prototype of 40 \(\times\) 32 \(\times\) 20 m. Additionally, two free-field models, medium dense (M1) and dense (M2), are tested.