## 4 Analytical Approach  
Based on the parametric study performed in this study, an analytical approach to predict the flexural capacity of CFRP prestressed spun PHC pile was proposed and verified against experimental and numerical results. In the proposed approach, it was assumed that the sections remained plane under bending, and no debonding occurred between the CFRP and concrete. For simplicity, the discrete reinforcement was converted into a layer of annulation. There are two potential failure modes for CFRP prestressed PHC piles: concrete crushing and CFRP rupture. The proposed analysis approach involved performing flexural capacity calculations for both cases and comparing the calculation results; the lower one would be the ultimate capacity of the pile. The stress and strain distributions of the analyzed section at failure are shown in Figures 16 and 17, in which, N.A. indicates the neutral axis. For the concrete crushing failure mode, the force equilibrium can be written as:  
\[ \begin{array}{c}\alpha_{1}\mathrm{f}_{\mathrm{cu}}\mathrm{A}_{\mathrm{c}}=2 \mathrm{r}_{\mathrm{p}}\mathrm{f}_{\mathrm{f}}(\int_{\frac{\pi}{2}}^{0}\,( \sigma_{\mathrm{f0}}+\frac{\mathrm{y}+\mathrm{r}_{1}-\mathrm{x}}{\mathrm{x}} \varepsilon_{\mathrm{cu}}\mathrm{E}_{\mathrm{f}})\mathrm{d}\theta+\int_{0}^{ \left(\frac{1}{2}-\alpha\right)\pi}(\sigma_{\mathrm{f0}}-\frac{\mathrm{y}- \mathrm{r}_{1}+\mathrm{x}}{\mathrm{x}}\varepsilon_{\mathrm{cu}}\mathrm{E}_{ \mathrm{f}})\mathrm{d}\theta+\\ \int_{\left(\frac{1}{2}-\alpha\right)\pi}^{\frac{\pi}{2}}(\sigma_{\mathrm{f0} }-\frac{\mathrm{y}-\mathrm{r}_{1}+\mathrm{y}}{\mathrm{x}}\varepsilon_{\mathrm{ cu}}\mathrm{E}_{\mathrm{f}})\mathrm{d}\theta)\end{array}  \tag{2}\]  
where \(\alpha_{1}\) is the the ratio of the stress of the equivalent rectangular compression zone to the axial compressive strength of concrete, which was 0.94 in this study; \(\mathrm{f}_{\mathrm{cu}}\) is the compressive strength of the concrete; \(\sigma_{\mathrm{f0}}\) is the initial prestressing stress; \(\mathrm{A}_{\mathrm{c}}\) is the area of concrete in the compression zone; \(\mathrm{\varepsilon}_{\mathrm{cu}}\) is the ultimate compressive strain of the concrete; \(\mathrm{E}_{\mathrm{f}}\) is the elastic modulus of the CFRP; \(\mathrm{r}_{\mathrm{p}}\) is the distance between the center of the whole section and the center of each CFRP rod; \(\mathrm{t}_{\mathrm{f}}\) is the equivalent thickness of the CFRP; \(\mathrm{r}_{1}\) is the outer radius of the section; \(\mathrm{r}_{2}\) is the inner radius of the section; \(\mathrm{x}\) is the height of the compression zone; \(\mathrm{y}\) is the vertical distance between the investigated point and the center line of the section; and \(\mathrm{\theta}\) is the angle between the center line of the section and the line from the investigated point to the center of the section. Figure 16: Stress and strain distribution of pile that failed due to concrete crush. The left part of Equation (2) represents the compressive force developed by the concrete. Since a concrete crushing failure mode was assumed, the ultimate strength of the concrete was adapted here. The right part of Equation (2) represents the tensile force developed by the CFRPs, which were divided into three parts: the part below the centerline; the part between centerline and the neutral axis; and the part above the neutral axis. \[\mathrm{y}=\mathrm{r_{p}sin}\theta \tag{3}\]  
\[\mathrm{\beta_{1}x}=\mathrm{r_{1}}-\mathrm{r_{p}cos}\alpha\tau \tag{4}\]  
\[\mathrm{t_{f}}=\frac{\mathrm{A_{P}}}{2\pi\mathrm{r_{p}}} \tag{5}\]  
The depth of compression zone \(\mathrm{x}\) and the corresponding angle \(\alpha\) can be solved, and the ultimate capacity can be calculated as:  
\[