## 2 Background of Scale Model Similitude Theories
### Reference Case Study  
A series of physical shaking table tests comprising flexible wall barrel containers was conducted by [PERSON] et al. [PERSON] et al. (2018) at the UC Berkeley PEER Center Earthquake Simulator Laboratory [PERSON] et al. (2018), and the data from Phase II of this study (namely Test 1.15) was employed herein for validation of the numerical modelling. The physical specimen comprised soil that was embedded with a single pile supporting the superstructure. The experimental set-up was able to physically model the entire seismic soil-pile-superstructure interaction (SSPSI). The main aim of the experimental campaign was to provide an insight into specific SSPSI issues such as the frequency response of the structure, multidirectional excitation, kinematic and inertial responses, and pile soil contact. The individual model piles were tested simultaneously and arranged in the test container in a manner that minimised element interactions [PERSON] (2018). In terms of instrumentation, there were twenty-three accelerometers arranged in two vertical arrays embedded in the soil deposit that were attached to the head masses of the piles to capture translation and rocking motions. In addition, seven pairs of strain gauges were attached to each pile. Based on the soil strength and shear wave velocity profiles, the  
Figure 5: Definition of model soil properties utilising the steady-state line. Modified from [PERSON] (1997), [PERSON] (1958). soil in the tests was defined as being lightly overconsolidated soft to medium stiff clay. The shaking table employed in the physical reference case study tests was 6.1 m \(\times\) 6.1 m in plan and had a load capacity of 580 kN, a frequency range of 0-20 Hz, and six controlled degrees of freedom [2]. In a geotechnical scaled/SSPSI problem, a container is typically used to confine the soil and to impose boundary conditions that may not occur in the prototype full-scale field scenario. Accordingly, in these tests, a suitable container was designed that could minimise the effect of free boundary conditions on the overall system response and also enable the model to replicate the seismic behaviour at the level ground. On this basis, a laterally flexible and radially stiff cylindrical container was selected for the quasi-free field response. This design extended the centrifuge testing laminar box concept to permit multidirectional excitation [22]. The container constrained a soil column that was 2.3 m in diameter and 2.0 m in height and installed on the surface of the shaking table, as shown in Figure 6. The top steel ring was supported by four steel pipe columns connected by heavy-duty universal joints, which allowed the ring full translational freedom but prevented overturning rotations. The flexible wall of the container comprised a neoprene rubber membrane that was 6.4 mm thick that was suspended from the top ring and fastened at the base. The flexible wall was confined by a set of woven Kevlar straps that were 45 mm in width and arrayed in circumferential bands around the exterior of the membrane and arranged at intervals of 60 mm. The elastic modulus \(E\) of the combined membrane was designed to be identical to that of the soil to ensure that the free-soil boundary condition was not affected. The combination of the rubber membrane and the set of straps provided lateral flexibility and radial stiffness for the container boundary conditions. A plastic sheet sealed the top of the soil specimen during the period between tests, and water was sprayed on top of the soil to prevent the soil surface from drying out. For each test series, the pile was driven into the soil. The model was left for approximately five days before performing the next test due to the beneficial effects of soil thixotropy. In Test 1.15, a set of single-pile models with head masses of 3.0, 11.40, 45.4, and 72.70 kg were studied, and the system was subjected to unidirectional shaking.