## 2 Numerical Simulations of Vibratory Sheet Pile Driving  
### Simulation Model  
As displayed in Figure 1, a full-scale finite element (FE) model was established using the commercial ABAQUS software. The model consisted of two parts--the soil domain and the sheet pile pre-inserted in the soil domain. The simulated soil domain was 120 m \(\times\) 120 m \(\times\) 100 m (X \(\times\) Y \(\times\) Z). The element type selected for soil was C3D8R, eight-node quadrilateral elements with reduced integration and hourglass control. At the four surrounding sides, the external soil domain with 30 m was modeled and discretized into infinite elements (CIN3D8), which are typically used to model non-reflective boundaries to transmit energy outside the mesh boundaries (Figure 1b). In the vertical direction, a large depth of 100 m was modeled, which is enough to avoid wave reflections from the bottom side of the model. Figure 1: Numerical modeling of vibratory sheet pile driving. (**a**) Isometric view of the full model; (**b**) top view of the full model; (**c**) sheet pile pre-inserted at any target depth. Element dimensions should be selected on the basis of the highest frequency \(f_{\max}\) and the lowest wave velocity \(v_{r}\)[25]. A rational element dimension \(g\) is determined by:  
\[g\leq\chi\frac{v_{r}}{f_{\max}} \tag{4}\]  
where the constant \(\chi\) must be less than 0.5 due to the Nyquist limit, and it is further determined by whether the mass matrices are consistent (\(\chi\) = 0.25) or lumped (\(\chi\) = 0.2). Thus, according to the simulated soil properties and vibratory driving frequency, a uniform mesh size of 0.35 m was used in three directions of the soil domain, except for soil elements near the pile axis, where a finer mesh size was defined to guarantee suitable contact between pile and soil. For simplicity, in this study, we assumed the sheet pile as a medium only to transmit the vibratory driving force to the surrounding soil through the pile toe force and the shaft friction. Consequently, the sheet pile was modeled as a discrete rigid body whose motion was governed by a pre-defined rigid body reference node [26]. The pre-defined reference node was assigned with the total mass of the sheet pile, which was calculated according to its geometry and material properties defined in Table 1, wherein the modeled sheet pile has a standard U-type section supplied by a local corporation in Hong Kong and follows the specification of BS EN 10248:1996 [27]. Soil domain was typically modeled as a homogeneously elastic-plastic medium using the Mohr-Coulomb model to consider the nonlinear behavior of soil in the vicinity of the pile. As listed in Table 2, we adopted the same soil parameters as those used in previous studies [10; 11; 12] to define the soil behavior. It should be noted that the modeled soil properties correspond to a shear wave velocity consistent with the tested construction site presented in Section 4. In addition, an appropriate material damping ratio of 7% was defined to consider the anelastic property of soil. The surface-to-surface contact discretization formulation in ABAQUS was used to define the interaction between sheet pile and soil, wherein the normal and tangential behavior were simulated using the hard contact model and the Coulomb friction model, respectively. The friction coefficient was set as \(\mu\) = 0.25 for the steel sheet piles against silty sand, gravel, or sand mixed with silt or clay, according to NAVFAC standards [28]. The loading procedure consists of two steps. In the first step, gravity was added to the whole pile-soil system to establish an initial stress state of the whole soil domain and contact state between pile and soil. In the second step, a vibratory driving force was applied to the pre-defined rigid body reference node of the pile to generate the spatial vibration propagations in the soil.