## 2 Methods
### Finite Element and Frame Analysis Modeling  
The PLAXIS 2D [23] program was used to create a 2D FEA model (Figure 3). PSW models were created with 15 node point triangular elements as plane strain models. This analysis used two consecutive loading phases: static loading (i.e., accounting solely for the PSW and soil self-weights) and seismic loading phases. In the static phase, the side boundary conditions were set as normally fixed, while the bottom boundary conditions  
Figure 2: Scenario 2 (with a soil slope) pile-supported wharf configurations. were set as fully fixed. For the seismic loading phase, the side and bottom boundaries were set as free field and compliant base boundaries, respectively [24]. The seismic load was applied as a ground motion waveform at the base of the soil layer and in-plane seismic excitation was applied. To transmit ground motions in the frequency range of up to 10 Hz, the fine mesh option was chosen for mesh coarseness with a maximum mesh height of 2 m.  
In the FEA, the wharf deck was modeled using plate elements and the piles were represented using an embedded beam row element. The embedded beam row element enables the 2D modeling of piles with out-of-plane spacing, which has been validated by comparisons with a three-dimensional (3D) FEA simulation and data from actual testing. In the embedded beam row element, the pile is separated from the 2D soil model, which makes the soil mesh continuous. Special interface elements between the soil mesh and embedded pile are employed to consider the soil-pile interaction. [PERSON] et al. [25] compared the 2D embedded beam row model with the 3D embedded pile model for axial and lateral loading, and it was found that the displacement of an embedded beam row model was comparable to the 3D pile displacements. [PERSON] [26] also demonstrated the applicability of the embedded beam row model by comparing the response of the embedded beam row model and experimental results. The hardening soil small strain (HSS) model was implemented to account for the nonlinearity of soil. The HSS model is a variation of the hardening soil model that considers the increased stiffness of soils at small strains [24], which is described by two additional material parameters: the small strain shear modulus (\(G_{0}\)\({}^{n\prime}\)) and the shear strain level at which the small strain shear modulus decreases to 70% of its initial value (\(\gamma^{0.7}\)). To discuss the structural seismic response, various ground motions should be considered [27; 28; 29; 30; 31] together with site-specific ground motions [32; 33]. The simulations used two waveforms with different characteristics: the 1968 Tokachi-Oki earthquake (TE) and the 2020 Mentawai earthquake (ME). The ME waveform was modified to match the target Indonesian site class SA [34]. These waveforms were deconvoluted at the bedrock, and the peak acceleration and duration were 0.210 g and 19 s for the TE and 0.215 g and 130 s for the ME, as illustrated in Figure 4. The structure's inelastic behavior may have catastrophic consequences [35; 36]; thus, many vital structures (e.g., PSW) were designed to remain in an elastic state during earthquakes. We focused on the elastic response of the PSW; therefore, we multiplied the waveforms by 0.8 to reduce their amplitude so that the structural members in the analysis remained in an elastic state during the earthquake. Figure 3: Typical FEA model of the PSW. The SAP 2000 [37] software was used to create a 2D FA model (Figure 5). Piles and beams were modeled as rigidly connected frame elements. To describe the relationship between the soil reaction (P) and the lateral displacement (Y) around the piles for each soil layer with depth, we used soil spring elastic-plastic P-Y curves. The P-Y curve for the spring properties was calculated using the L-PILE [38] computer program based on a study by [PERSON] [39]. Figures 6-8 show the P-Y curves at different depths below the seabed.