## 2 Methods
### Frame Analysis
\[M_{cr}=\frac{\left(f_{r}+\frac{F_{pe}}{M_{g}}\right)I_{p}}{0.5D} \tag{11}\]  
\[[yellow]\varphi_{cr}=\frac{\left(f_{r}+\frac{F_{pe}}{M_{g}}\right)/\ E_{p}}{0.5D} \tag{12}\]  
\[M_{y}\!\!=\Sigma\ A_{s}\ f_{pc}\bigg{(}y_{i}-\frac{0.5D}{2}\bigg{)}+\ \Sigma\ \frac{1}{2}f_{ca}A_{c,comp}y_{c} \tag{13}\]  
\[\varphi_{y}=\frac{\frac{f_{cp}}{E_{\omega}}}{c} \tag{14}\]  
\[M_{p}\!\!=\Sigma\ A_{s}\ f_{y-pc}\bigg{(}y_{i}-\frac{0.5D}{2}\bigg{)}+\ \Sigma\ 0.85f_{ca}A_{c,comp}\bigg{(}\frac{0.5D}{2}-y_{c}\bigg{)} \tag{15}\]  
\[\varphi_{p}=\frac{\varepsilon_{co}}{c} \tag{16}\]  
Residual bending moment and curvature were adopted from [57], and are:  
\[M_{res}=0.85\ M_{p} \tag{17}\]  
\[\varphi_{res}=\varphi_{p}+\big{(}\varphi_{p}-\varphi_{y}\big{)} \tag{18}\]  
Ultimate curvature was adopted from [32], and is:  
Figure 7:0  
where \(M_{cr}\) is the cracking moment of the spun pile section (N\(\cdot\)mm), \(f_{r}\) is the cover concrete tensile strength (MPa), \(F_{pe}\) is the prestress force (N), \(A_{g}\) is the cross-sectional area of the spun pile (mm\({}^{2}\)), \(D\) is the diameter of the spun pile (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), \(\varphi_{cr}\) is the cracking curvature of the spun pile (1/mm), \(M_{y}\) is the yield bending moment of the spun pile (N\(\cdot\)mm), \(A_{s}\) is the PC bar cross-sectional area (mm\({}^{2}\)), \(f_{pc}\) is the PC bar tensile stress (MPa), \(y_{i}\) is the distance from the PC bar axis point to the neutral axis of the spun pile cross-section (mm), \(f_{co}\) is the cover concrete compressive strength (MPa), \(A_{c,comp}\) is the compressed cross-sectional area of the spun pile (mm\({}^{2}\)), \(y_{c}\) is the distance from the compressed area axis point to the neutral axis of the spun pile cross-section (mm), \(\varphi_{y}\) is the yielding curvature of the spun pile (1/mm), \(E_{co}\) is the secant modulus of elasticity of the cover concrete (MPa), \(c\) is the neutral axis of the spun pile cross-section (mm), \(M_{p}\) is the plastic bending moment of the spun pile (N\(\cdot\)mm), \(f_{y-pc}\) is the PC bar yield strength (MPa); \(\varphi_{p}\) is the plastic curvature of the spun pile (1/mm), \(\varepsilon_{co}\) is the strain at the cover concrete's peak strength, \(M_{res}\) is the residual bending moment of the spun pile (N\(\cdot\)mm), \(\varphi_{res}\) is the residual curvature of the spun pile (1/mm), and \(\varphi_{ult}\) is the ultimate curvature of the spun pile (1/mm). To implement the proposed moment-curvature curve in the FA-2 model, points I, II, III, IV, and V must be converted into points in the moment-rotation (M\(\rightarrow\)\(\varphi\)) relationship. Thus, the curvature must be considered to act on the selected plastic hinge length (\(l_{p}\)), where plasticity is expected to occur in the element. In this case, the length of the plastic hinge was assumed to be equal to the depth or diameter of the member section [58]. Hinges were assigned at 0.1 and 0.9 of the relative distance of each spun pile element. Tables 3 and 4 summarize the moment-rotation relationship points for piles A, B, and C, considered in FA-1 and FA-2, respectively. Figure 8 shows the comparison of the  
proposed moment-curvature curve adopted in FA-2 with the curves considered in FA-1 and FB-FEA.