## 4 Analytical Approach
\begin{array}{c}\mathrm{M}=2\mathrm{r_{p}t_{f}(\int_{-\frac{\alpha}{2}}^{0} \left(\sigma_{\mathrm{f0}}+\frac{\mathrm{y+r_{1}-x}}{\mathrm{x}}\varepsilon_{ \mathrm{cu}}\mathrm{E_{f}}\right)(\mathrm{r_{1}-y})\mathrm{d}\theta+\int_{0}^ {(\frac{1}{2}-\alpha)\pi}\left(\sigma_{\mathrm{f0}}+\frac{\mathrm{r_{1}-x-y}} {\mathrm{x}}\varepsilon_{\mathrm{cu}}\mathrm{E_{f}}\right)(\mathrm{r_{1}+y}) \mathrm{d}\theta+\\ \int_{(\frac{1}{2}-\alpha)\pi}^{\frac{\alpha}{2}}\left(\sigma_{\mathrm{f0}}- \frac{\mathrm{y-r_{1}+x}}{\mathrm{x}}\varepsilon_{\mathrm{cu}}\mathrm{E_{f}} \right)(\mathrm{r_{1}+y})\mathrm{d}\theta)-\alpha_{1}\mathrm{f_{cu}}\mathrm{A_{ c}}(2\mathrm{r_{2}}-\frac{\mathrm{x}\mathrm{B_{1}}}{2})\end{array}  \tag{6}\]  
In Equation (6), the moment was calculated to be about the very bottom of the section. The right side of the equation was divided into four parts: the moment of the CFRP below the centerline; the moment of the CFRP between the centerline and the neutral axis; the moment of the CFRP above the neutral axis; and the moment of the concrete in the compression zone. For the CFRP rupture failure mode, the force equilibrium can be written as:  
\[\alpha_{1}\mathrm{f_{c}}\mathrm{A_{c}}=2\mathrm{r_{p}t_{f}(\int_{-\frac{\alpha }{2}}^{(\frac{1}{2}-\alpha)\pi}\left(\sigma_{\mathrm{f0}}+\frac{\mathrm{y+r_{1 }-x}}{\mathrm{r_{1}-x+r_{p}}}\varepsilon_{\mathrm{fu}}\mathrm{E_{f}}\right) \mathrm{d}\theta+\int_{(\frac{1}{2}-\alpha)\pi}^{\frac{\alpha}{2}}\left( \sigma_{\mathrm{f0}}-\frac{\mathrm{y-r_{1}+x}}{\mathrm{r_{1}-x+r_{p}}} \varepsilon_{\mathrm{fu}}\mathrm{E_{f}}\right)\mathrm{d}\theta) \tag{7}\]  
\[\mathrm{f_{c}}=\mathrm{E_{c}}\varepsilon_{\mathrm{c}} \tag{8}\]  
\[\varepsilon_{\mathrm{c}}=\frac{\mathrm{x}}{\mathrm{r_{1}-x+y}}\varepsilon_{ \mathrm{fu}} \tag{9}\]  
The left part of Equation (7) represents the compressive force developed by the concrete. The right part of Equation (7) represents the tensile force developed by the CFRPs, which are divided into three parts: the part below the centerline; the part between the centerline and the neutral axis; and the part above the neutral axis. Since a CFRP rupture failure mode was assumed, the full strength of the CFRP was adapted in Equation (6). Figure 17: Stress and strain distribution of pile that failed due to CFRP rupture. The depth of compression zone x and the corresponding angle \(\alpha\) can be solved, and the ultimate capacity can be calculated as:  
\[ \begin{array}{c}\mathrm{M}=2\mathrm{r}_{\mathrm{p}}\mathrm{t}_{\mathrm{f}}( \int_{-\frac{\alpha}{2}}^{0}\left(\sigma_{\mathrm{\theta}0}+\frac{\mathrm{y}+ \mathrm{r}_{1}-\mathrm{x}}{\mathrm{r}_{1}-\mathrm{x}+\mathrm{r}_{\mathrm{p}}} \mathrm{e}_{\mathrm{fu}}\mathrm{E}_{\mathrm{f}})(\mathrm{r}_{1}-\mathrm{y}) \mathrm{d}\theta+\int_{0}^{(\frac{1}{2}-\alpha)\pi}(\sigma_{\mathrm{\theta}0}+ \frac{\mathrm{y}+\mathrm{r}_{1}-\mathrm{x}}{\mathrm{r}_{1}-\mathrm{x}+\mathrm{r }_{\mathrm{p}}}\mathrm{e}_{\mathrm{fu}}\mathrm{E}_{\mathrm{f}})(\mathrm{r}_{1} +\\ \mathrm{y})\mathrm{d}\theta+\int_{(\frac{1}{2}-\alpha)\pi}^{\frac{\alpha}{2}}( \sigma_{\mathrm{\theta}0}-\frac{\mathrm{y}-\mathrm{r}_{1}+\mathrm{x}}{\mathrm{ r}_{1}-\mathrm{x}+\mathrm{r}_{\mathrm{p}}}\mathrm{e}_{\mathrm{fu}}\mathrm{E}_{ \mathrm{f}})(\mathrm{r}_{1}+\mathrm{y})\mathrm{d}\theta)-\alpha_{1}\mathrm{f}_{ \mathrm{c}}\mathrm{A}_{\mathrm{c}}(2\mathrm{r}_{2}-\frac{\mathrm{x}\mathrm{x} \mathrm{y}}{2})\end{array}  \tag{10}\]  
In Equation (10), the moment was calculated to be about the very bottom of the section. The right side of the equation was divided into four parts: the moment of the CFRP below the centerline; the moment of the CFRP between the centerline and the neutral axis; the moment of the CFRP above the neutral axis; and the moment of the concrete in the compression zone. The flexural capacity of the piles modeled in a previous study was calculated using the proposed approach, and the results are summarized in Table 5. The proposed approach offered a conservative prediction compared to the numerical and finite element results.