# Flexural Behavior of an RC Beam Externally Strengthened with a Steel- and CFRP-Based Method
## 4 Prediction of the Flexural Capacity  
For a new kind of retrofitting method, it is necessary to accurately predict the flexural capacity of the beam strengthened using the collaborative method. Regarding the structural safety, applicability, and durability, the experimental retrofitted beam is considered to fail when the maximum crack width is reached. Therefore, during the test, the strains of the concrete, steel reinforcement, angle steel, CFRP plate, and steel plate are considered to meet the plane section assumption. According to the material test, the CFRP can be regarded as a linear elastic material. To simplify the constitutive relationship of the material, the steel is assumed to be a material linear elastic in the elastic stage and a perfect plastic in the yield stage. The tensile strength of the concrete is ignored, while the compressive strength is defined as a curve consisting of a quadratic parabola followed by a horizontal line. With respect to the deformation compatibility, as there was no bonding failure between the CFRP plate, the steel plate, the angle steel, and the concrete, it is concluded that the retrofitting system worked effectively with the original beam during the whole loading process. The thickness of the epoxy adhesive was assumed to be zero during the calculation. The resultant point of the angle steel was set as the centroid, and the distance from the centroid to the edge of the tension zone was determined as has been described. Given the low thickness of the CFRP plate and the steel plate, the resultant points are simplified at the bottom of the beam. Therefore, the strain of the material can be calculated as follows:  
Figure 9: Crack development. \[\frac{\varepsilon_{\text{cp}}}{\varepsilon_{\text{cu}}}=\frac{\varepsilon_{\text{ sp}}}{\varepsilon_{\text{cu}}}=\frac{h-x_{\text{a}}}{x_{\text{a}}} \tag{1}\] \[\frac{\varepsilon_{\text{as}}}{\varepsilon_{\text{cu}}}=\frac{h-x_{ \text{a}}+h_{\text{as}}}{x_{\text{a}}} \tag{2}\]  
where \(\varepsilon_{\text{cp}}\), \(\varepsilon_{\text{sp}}\), \(\varepsilon_{\text{as}}\), and \(\varepsilon_{\text{cu}}\) denote the strain of the CFRP plate, the steel plate, the angle steel, and the concrete at the edge of the compression zone, respectively. According to Code GB50010-2010 [45], \(\varepsilon_{\text{cu}}\) can be taken as 0.0033 for those concretes with a concrete strength grade less than C50. According to the crack distribution concentrated around the pure bending section, the retrofitted beam should be assumed to present an adequate shear capacity; consequently, the corresponding shear and axial deformations are ignored. Based on the aforementioned assumptions, the calculating diagram is plotted as shown in Figure 10. Regarding the contribution of the slab on the beam, the cross-section of the beam is considered to be T-shaped. Therefore, according to the relationship between the actual compression zone height \(x_{\text{a}}\) and the flange height, there are two occasions in total, namely \(x_{a}<h_{f}^{\prime}\) and \(x_{a}\geq h_{f}^{\prime}\), where \(h_{f}^{\prime}\) denotes the height of the flange. The equivalent compression zone height \(x\) is taken to be 0.8 times that of \(x_{\text{a}}\). Regarding the equivalent rectangular compression \(\alpha_{\text{f}}c_{\text{c}}\) is the compressive strength of the concrete, and \(\alpha_{1}\) is the shape coefficient of the equivalent rectangular stress. In addition, \(f_{y}^{\prime}\) and \(f_{y}\) denote the yield strength of the compressive and tensile steel bars, and \(A_{\text{s}}^{\prime}\) and \(A_{\text{s}}\) denote the corresponding area. With respect to the retrofitting material, \(\sigma_{\text{cp}}\), \(\sigma_{\text{sp}}\), and \(\sigma_{\text{as}}\) represent the tensile strength of the CFRP plate, the steel plate, and the angle steel, which could be obtained according to the corresponding strain. \(A_{\text{cp}}\), \(A_{\text{sp}}\), and \(A_{\text{as}}\) denote the cross-sectional area of the corresponding materials.