## 2 Methods
### Finite Element Model  
A 3D FEA of the corroded spun pile was implemented using the 3D-NLFEA package. The package was verified to analyze various high-strength and steel-fiber reinforced concrete columns [30; 31; 32; 33]. The analysis model was constructed as one quarter of the spun pile section with a height of 250 mm because it had a horizontally axisymmetric shape (Figure 2). Solid elements were used to model the concrete, PC-bars, and transverse rebar elements (Figure 3). A hexahedral formulation was employed for the solid-element mesh [34]. The modified initial stiffness method accelerated convergence in the global nonlinear iteration process [35]. The rust was modeled as part of a PC-bars and transverse rebars element with a 1 mm thickness. Lateral (x and y-directions) and vertical (z-direction) constraints were employed on the side surface and the bottom surface of the model, respectively. A zero-thickness element [36] was used to simulate the bond-slip behavior of the concrete and PC-bars in three (i.e., two transverse and one longitudinal) directions. The PC-bars were assumed to only slip longitudinally, and the occurrence of the slip was determined using a nonassociated flow rule and the hyperbolic Mohr-Coulomb plasticity model. The stress integration algorithm applied the tangent cutting plane method [35]. Figure 4 shows the hyperbolic Mohr-Coulomb plasticity failure model and the Mohr  
Figure 2: Three-dimensional finite element model of one quarter of the spun pile section. Coulomb plasticity failure model as a reference. The failure surface (\(f\)) of the hyperbolic Mohr-Coulomb plasticity model is expressed as follows [37]:  
\[f=\tau_{1}^{2}+\tau_{2}^{2}-(c-\sigma_{n}\tan\phi)^{2}+(c-\xi\tan\phi)^{2} \tag{9}\]  
where \(\tau_{1}\) is the shear stress in the first principal direction; \(\tau_{2}\) is the shear stress in the second principal direction; \(c\) is the cohesive shear strength at the onset of slip occurrence; \(\phi\) is the friction angle, which is related to the roughness of the surface between the concrete and the rebar; \(\xi\) is the bond strength between the concrete and the rebar; and \(\sigma_{n}\) is the stress rate vector defined in the expected directions of the interface. As shown in Figure 5, three different types of analyses were carried out in this study to discuss the impacts of corrosion on the deterioration of the mechanical properties of spun pile materials. Each involved two loading steps. To model the state of a corroed spun pile, the volumetric strain expansion of rust was initially applied to the PC-bar and transverse rebar as a standard loading phase. The magnitude of volumetric strain expansion was determined following the nonuniform rust distribution for the corresponding years, and the strain was applied in an equal increment of 10 steps. Figure 4: Failure surface of hyperbolic Mohr-Coulomb plasticity model illustration. Figure 3: PC-bars and transverse rebars embedded in spun pile model. In the first analysis case (Case 1), concrete was subjected to a displacement-controlled axial compression load to assess the impacts of corrosion on the deterioration of the compressive strength of concrete. In order to determine the stress-strain relationships between the corroded cover and core concrete, the compression load was uniformly applied in an increment of 0.01 mm in 100 steps to the top edge of the model (Figure 5a). In the second analysis case (Case 2), one of the PC-bar elements was subjected to a displacement-controlled axial tensile load with 100 steps of 0.1 mm increment to observe the corrosion effect on the deterioration of the bond strength between the PC-bar and concrete (Figure 5b). Corroded PC-bar and concrete bond-slip relationships were obtained. Moreover, in the third analysis case (Case 3), the displacement-controlled axial tensile load with 100 steps of 0.1 mm increment was applied to one of the PC-bar elements to observe the corrosion effect on the decrease in the yield strength of the PC-bar (Figure 5c).