## 2 Methods
### Pushover Analysis  
The pushover analysis was performed in the 2D FB-FEA, FA-1, and FA-2 models, considering the corrosion effect up to 75 years and constant gravity loads (P = 100 kN). Furthermore, the lateral load (F\({}_{\text{push}}\)) was applied as a series of displacement-based loads [59] \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Point**} & \multicolumn{2}{c}{**Pile A with Infilled Concrete**} & \multicolumn{2}{c}{**Pile A**} & \multicolumn{2}{c}{**Pile B\&C with Infilled Concrete**} & \multicolumn{2}{c}{**Pile B\&C**} \\  & \multicolumn{2}{c}{**Bending**} & \multirow{2}{*}{**M/My**} & \multirow{2}{*}{\(\mathbf{\phi}\)**} & \multicolumn{2}{c}{**Bending**} & \multirow{2}{*}{**M/My**} & \multirow{2}{*}{\(\mathbf{\phi}\)**} & \multicolumn{2}{c}{**Bending**} & \multirow{2}{*}{**M/My**} & \multirow{2}{*}{\(\mathbf{\phi}\)**} \\  & \multicolumn{2}{c}{**Moment**} & & & & & & & & & & & \\  & \multicolumn{2}{c}{**(kN-m)**} & \multicolumn{2}{c}{**(rad)**} & \multicolumn{2}{c}{**(kN-m)**} & & & & & & & & & \\ \hline I & 0.00 & 0.00 & 0.0000 & 0.00 & 0.00 & 0.0000 & 0.00 & 0.00 & 0.00 & 0.000 & 0.00 & 0.000 & 0.000 \\ II & 706.36 & 1.00 & 0.0013 & 560.39 & 1.00 & 0.0013 & 547.79 & 1.00 & 0.0014 & 457.80 & 1.00 & 0.0015 \\ III & 882.95 & 1.25 & 0.0136 & 711.70 & 1.27 & 0.0119 & 684.74 & 1.25 & 0.0103 & 581.41 & 1.27 & 0.0114 \\ IV & 750.51 & 1.06 & 0.0258 & 604.94 & 1.08 & 0.0225 & 582.03 & 1.06 & 0.0192 & 494.20 & 1.08 & 0.0212 \\ V & 750.51 & 1.06 & 0.0613 & 604.94 & 1.08 & 0.0613 & 582.03 & 1.06 & 0.0613 & 494.20 & 1.08 & 0.0613 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Nonlinear modeling parameters used in FA-2 models. Figure 8: Moment-curvature relationship of the spun pile. (**a**) Pile A with infilled concrete; (**b**) Pile A\({}_{2}\) (**c**) Piles B and C with infilled concrete; (**d**) Piles B and C.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Point**} & \multicolumn{2}{c}{**Pile A with Infilled Concrete**} & \multicolumn{2}{c}{**Pile A**} & \multicolumn{2}{c}{**Pile B\&C with Infilled Concrete**} & \multicolumn{2}{c}{**Pile B\&C**} \\  & \multicolumn{2}{c}{**Bending**} & \multirow{2}{*}{**Bending**} & \multirow{2}{*}{**Bending**} & \multirow{2}{*}{**Bending**} & \multirow{2}{*}{**M/My**} & \multirow{2}{*}{\(\mathbf{\phi}\)**} & \multicolumn{2}{c}{**Bending**} & \multirow{2}{*}{**M/My**} & \multirow{2}{*}{\(\mathbf{\phi}\)**} \\  & \multicolumn{2}{c}{**Moment**} & & & & & & & & & & \\  & \multicolumn{2}{c}{**(kN-m)**} & \multicolumn{2}{c}{**(rad)**} & \multicolumn{2}{c}{**(kN-m)**} & & & & & & & & \\ \hline I & 0.00 & 0.00 & 0.0000 & 0.00 & 0.00 & 0.000 & 0.00 & 0.00 & 0.000 & 0.00 & 0.00 & 0.000 \\ II & 712.01 & 1.00 & 0.0014 & 566.00 & 1.00 & 0.0015 & 552.17 & 1.00 & 0.0016 & 468.96 & 1.00 & 0.0017 \\ III & 783.21 & 1.10 & 0.0330 & 622.59 & 1.10 & 0.0330 & 667.39 & 1.10 & 0.0330 & 515.85 & 1.10 & 0.0330 \\ IV & 156.64 & 0.20 & 0.0333 & 124.52 & 0 implemented on the top edge of the deck in 1 mm increments until the target displacement was reached. The target displacement was the horizontal displacement of the node at the top of piles A1 and B1 (Figure 9). In general, the following stages were carried out in this study to observe the nonlinear seismic response of corroded PSWs utilizing three methods (FB-FEA, FA-1, and FA-2):  
1. Determining the reduction in PC bar area due to corrosion over 75 years. The corrosion degree was determined for each elapsed year. 2. Developing mechanical properties of PSWs with corroded spun piles. 3. Constructing PSW models based on the step (2) results. FB-FEA models considered the stress-strain curve of the corroded spun pile materials combined with the corroded PC bar area reduction resulting from step (1). Furthermore, FA-1 and FA-2 models considered the reduced PC bar area, deteriorated concrete compressive strength of the corroded spun pile, and reduced PC bar yield strength. 4.