# Flexural Behavior of an RC Beam Externally Strengthened with a Steel- and CFRP-Based Method
## 4 Prediction of the Flexural Capacity
As all the angle steel yielded during the test, the tensile strength of the angle steel is taken as its yield strength\(f_{y,\text{as}}\). Based on the conditions of the equilibrium of forces, the ultimate flexural capacity can be calculated. When \(x_{a}<h_{f}^{\prime}\), as the tensile strength of the concrete is neglectable, the rectangle with a width equal to the effective flange width \(b_{f}^{\prime}\) can be used directly in the calculation. According to Code JTG 3362-2018 [44], \(b_{f}^{\prime}\) is taken as the minimum of the following two values: (1) one-third of the calculated span \(l\); (2) \(b+2b_{\text{h}}+12h_{f}^{\prime}\), where \(b\) and \(b_{\text{h}}\) represent the width of the web and the length of the bridge support, respectively. In this study, \(b_{\text{h}}\) is considered to be zero. Therefore, the formulas are listed as follows:  
\[\alpha_{1}f_{\text{c}}b_{f}^{\prime}x+f_{y}^{\prime}A_{\text{s}}^{\prime}=f_{ y}A_{\text{s}}+\sigma_{\text{cp}}A_{\text{cp}}+\sigma_{\text{sp}}A_{\text{sp}}+f_{y,\text{as}}A_{\text{as}} \tag{3}\]  
\[ \begin{array}{l}M_{\text{u}}=\alpha_{1}f_{\text{c}}b_{f}^{\prime}x(h-\frac{ x}{2})+f_{y}^{\prime}A_{\text{s}}^{\prime}(h_{0}-a_{\text{s}}^{\prime})+\sigma_{ \text{cp}}A_{\text{cp}}(h-h_{0})+\sigma_{\text{sp}}A_{\text{sp}}(h-h_{0})+\\ f_{y,\text{as}}A_{\text{as}}(h-h_{0}+h_{\text{as}})\end{array}  \tag{4}\]  
Figure 10: Calculating diagram for the retrofitted beam: (a) \(x_{a}<h_{f}^{\prime}\); (b) \(x_{a}\geq h_{f}^{\prime}\). where \(a_{\rm s}^{\prime}\) and \(a_{\rm s}\) denote the distance from the resultant points of the compressive or tensile steel bars to the edge of the concrete in the compression/tension zone, respectively. As for \(x_{a}\geq h_{f}^{\prime}\), \(M_{\rm u}\) can be predicted according to the following formulas:  
\[\alpha_{1}f_{\rm c}b_{\rm f}^{\prime}x+\alpha_{1}f_{\rm c}(b_{\rm f}^{\prime}- b)h_{\rm f}^{\prime}+f_{\rm y}^{\prime}A_{\rm s}^{\prime}=f_{\rm y}A_{\rm s}+ \sigma_{\rm cp}A_{\rm cp}+\sigma_{\rm sp}A_{\rm sp}+f_{\rm y,as}A_{\rm as} \tag{5}\]  
\[ \begin{array}{l}M_{\rm u}=\alpha_{1}f_{\rm c}b_{\rm f}^{\prime}x(h_{0}- \frac{x}{2})+\alpha_{1}f_{\rm c}(b_{\rm f}^{\prime}-b)h_{\rm f}^{\prime}(h_{0} -\frac{h_{\rm f}^{\prime}}{2})+f_{\rm y}^{\prime}A_{\rm s}^{\prime}(h_{0}-a_{ \rm s}^{\prime})+\\ \sigma_{\rm cp}A_{\rm cp}(h-h_{0})+\sigma_{\rm sp}A_{\rm sp}(h-h_{0})+f_{\rm y,as}A_{\rm as}(h-h_{0}+h_{\rm as})\end{array}  \tag{6}\]  
Using the aforementioned formulas, the ultimate load-bearing capacity of the five retrofitted specimens was calculated. The results are listed in Table 4, where \(M_{\rm u,exp}\) and \(M_{\rm u,pre}\) denote the ultimate flexural capacity obtained from the experimental and the prediction methods, respectively. As there was a limited contribution to the bearing capacity from the separated steel plate in Specim L6-C-Ss-A, it was not considered to add to the length of the moment arm of the angle steel. In comparison, most of the calculated results underestimate the ultimate flexural capacity, because the strengthening stage of the steel material is ignored during the process. Generally, as the errors ranged from \(-7.09\%\) to \(3.72\%\), with an average error of \(4.03\%\), the prediction method can be considered to be valid.