## 3 Evaluation of Ultimate Flexural Limit State
### Simplified Evaluation Procedure  
The iterative method presented above includes the convergence procedure and requires excessive computational costs, especially in a design practice involving large-scale or complex structures. Consequently, a simplified method is also proposed here based on the approximation of \(\Delta_{c,ex}\). The flexural ultimate strength, deformation, and the corresponding tendon strains can be determined directly by the simplified method. Similar to the previous study by the authors [21; 22], it is assumed that along the PC zone (\(l^{\prime}\)), the concrete strains \(\varepsilon_{nx}\) at the extreme compression fiber distribute in a logarithmic form, as illustrated in Figure 9. The approximation of concrete axial shortening can thus be calculated as Equation (21) and denoted by \(\Delta^{*}_{c,ex}\). \[\Delta^{*}_{c,ex}=\varepsilon_{\text{cu}}l/3 \tag{21}\]  
Then, substituting \(\Delta^{*}_{c,ex}\) into Equations (5) and (6) gives the expressions for the tendon strain \(\varepsilon_{t,c}\) and \(\varepsilon_{t,t}\), which contain the unknown neutral axis depth \(x_{n}\). Substituting the expressions into Equation (1) and assuming that \(\varepsilon_{t,c}\) and \(\varepsilon_{t,t}\) are between the elastic limit \(\varepsilon_{te}\) and the yield limit \(\varepsilon_{ty}\), the expressions for \(T_{t,c}\) and \(T_{t,t}\) can be obtained. Next, by substituting the expressions of \(T_{t,c}\) and \(T_{t,t}\) into Equation (2), Equation (22) is derived, which is a quadratic equation with respect to \(x_{n}\). \[\frac{\sigma_{B}bl_{t}}{2E_{t2}A_{t}}x_{n}^{2}+2\left[\frac{\varepsilon_{cu}}{ 3}l^{\prime}-\varepsilon_{c0}l^{\prime}-\varepsilon_{t0}l_{t}+\varepsilon_{ te}l_{t}-\frac{\sigma_{te}}{E_{t2}}l_{t}\right]\,x_{n}-\frac{\varepsilon_{cu}}{3}l^{ \prime}D=0 \tag{22}\]  
By solving for \(x_{n}\) and substituting it into Equations (5) and (6), values of \(\varepsilon_{t,c}\) and \(\varepsilon_{t,t}\) can be calculated based on \(\Delta^{*}_{c,ex}\) given by Equation (21). Then, \(P_{0,u}\) and \(R_{0,u}\) can be calculated using the similar steps as in the iterative method. It is noted that if the tendon length is extremely short, the tendon might yield before the beam reaches the ultimate flexural state, which was actually found in the specimen PCJ05, as shown later in Section 4.2. In this specimen, the tendon length inside the beam was set to be only \(1.0D\), where \(D\) is the beam depth, and the strain (= 7085 \(\mu\)) of the tensile side tendon just exceeded the yielding limit \(\varepsilon_{ty}\) (=7010\(\mu\)) at the ultimate limit. However, it is judged to be just slightly exceeding the yielding limit, and in practice, it would be an exceptional case, since an unyielding tendon is favorable. In addition, limiting the strain value of the tendon to \(\varepsilon_{ty}\) can provide conservative estimation of the ultimate deformation of the beam. The simplified method proposed above is based on the approximation of concrete axial shortening \(\Delta^{*}_{c,ex}\). Because the beam deflection angle of the PC zone (\(R_{pc,u}\)) is directly related to \(\Delta^{*}_{c,ex}\), the approximation might result in lower accuracy of the evaluation result of beam deflection angle, as shown in Section 4.3. Therefore, two coefficients are introduced  
Figure 9: Concrete strain distribution at extreme compression fiber in PC zone. as Equation (23), and the ratio of the accurate \(\Delta_{c,ex}\) and the approximated \(\Delta^{*}_{c,ex}\) can be given as Equation (24), based on Equations (7), (21) and (23). \[\gamma=x_{n}/D,\quad\chi=(d_{t}-\beta d)/D \tag{23}\]  
\[\frac{\Delta_{c,ex}}{\Delta^{*}_{c,ex}}=\frac{3\gamma}{3\chi-\gamma}\frac{l}{l^ {\prime}}ln\left|\frac{3\chi-\gamma}{3\gamma}\frac{l^{\prime}}{l}+1\right|=\xi \tag{24}\]  
The ratio is denoted by correction factor \(\xi\), and by multiplying \(\xi\), the evaluation result \(R_{pc,u}\) through the simplified method could be corrected to have higher accuracy.