## 2 Methods
### Frame Analysis  
In this study, FA-1 and FA-2 were used to observe the nonlinear response of PSWs. Each 2D FA model was simulated against the effects of corrosion for up to 75 years. The 2D PSW model, which was developed using SAP2000 software [53], considered hinge property settings to characterize the inelastic behavior of the PSW [30]. The 2D FA models used the elastic-plastic \(p\)-\(y\) curve of the soil spring as applied in the FB-FEA model (Figure 4). The inelastic behavior of the PSW can be modeled in SAP2000 using either default (auto-generated) or user-defined hinges [53]. In FA-1, the auto-generated hinge properties of spun piles based on ASCE 41-13 [32] were considered, while the user-defined hinge properties of spun piles were implemented in FA-2. Figure 7a shows the moment-curvature curve adopted in the 2D FA-1 model. Five points (I, II, III, IV, and V) define the hinge behavior of the reinforced concrete column or pile members. No plastic deformation occurs up to point II, where the hinge yields. The rotation at point II was determined using Equation (10) [53]. Point II is followed by a yield plateau or strain hardening behavior until point III, representing the hinge's ultimate capacity. The ultimate capacity of the hinge was set at 1.1 times the hinge yield capacity, based on FEMA 356 [31]. After point III, the curve drops to point IV, corresponding to the hinge's residual strength. The magnitude of the residual strength and the rotations at points III and IV follow ASCE 41-13. Point V represents the ultimate rotation capacity of the hinge, and the magnitude of the final rotation also follows ASCE 41-13 [32]. \[\phi_{II}=\frac{M_{y}}{\left(6\ E_{p}I_{p}\right)}L \tag{10}\]  
where \(\phi_{II}\) is the rotation at point II (rad), \(M_{y}\) is the yield bending moment of the spun pile (N\(\cdot\)mm), \(I_{p}\) is the moment of inertia of the spun pile (mm\({}^{4}\)), \(E_{p}\) is the modulus of elasticity of the spun pile (MPa), and \(L\) is the spun pile length (mm). In the proposed FA method (FA-2), the hinge property applied to the model was defined based on the moment-curvature analysis of the spun pile section. A trilinear approximation is commonly used to develop the moment-curvature curve for a reinforced concrete section. However, trilinear idealization does not adequately represent the moment-curvature relationships for pile and column sections [54] because the plastic moment occurs at a very low curvature compared to the ultimate curvature. Thus, this study used a multilinear idealization for the spun pile sections through points I, II, III, IV, and V (Figure 7b). Specifically, point I represents the origin, point II is the point of first yield, point III is the maximum moment point (plastic moment), point IV is an intermediate point on the horizontal branch of the multilinear idealization, and point V is the residual point. The moment-curvature curves were developed to be symmetric so that the absolute value of each point's coordinate in the negative direction is the same as its co  
Figure 7: Typical nonlinear deformation curve of the spun pile. (a) FA-1; (b) FA-2. direction. The moment-curvature curve for a spun pile under pure bending conditions can be determined using Equations (11)-(19) based on ACI 318 [55] and the PCI Design Handbook [56].