## 4 FE Numerical Model of Pre-Cracked Beam  
### Overview of FE Numerical Models  
In this section, ABAQUS is used to establish the FE model of the pre-cracked test T-beam. The FE model was mainly composed of concrete, steel cage and prefabricated cracks. The steel cage was embedded in the T-beam concrete. The translational freedom degrees of the reinforcement joints were consistent with the embedded concrete units. The network division of the T-beams was executed using three-dimensional 8-node reduction integral element (C3D8R). The three-dimensional 2-node truss elements (T3D2) were used to divide the network of the steel cage. The global network size was 20 mm. The freedom  
Figure 11: Cracking load of test beams. Figure 12: Ultimate load of test beam. degree of the T-beam loading surface was coupled to a reference point above the top surface of the beam. Load was applied to the reference point. The calculation and analysis adopted the position control method. The FE model of the pre-cracked T-beam is shown in Figure 13.  
The compressive performance of concrete adopts a uniaxial compressive stress-strain curve. The tensile performance of concrete adopts a tensile stress-crack width curve. The constitutive relation curve of C50 concrete used in this test is shown in Figure 14. The concrete compressive stress-strain curve is divided into three stages. Stage I: the relationship between stress and strain is linear when the compressive stress is less than \(0.4f_{\rm c}\). The slope of the curve is \(E_{\rm c}\). Stage II: the relationship between stress and strain is parabolic when the compressive stress is greater than \(0.4f_{\rm c}\) and less than the compressive strength. Stage III: the stress decreases linearly with the increase in strain when the compressive stress is greater than the compressive strength of the concrete. The ultimate strain of the concrete is taken as 0.034. The constitutive relationship of the concrete can be expressed as [39]  
\[\sigma_{\rm c}=\left\{ \begin{array}{ll}E_{\rm c}\varepsilon_{\rm c}&,&0\leq \sigma_{\rm c}\leq 0.4f_{\rm c}\\ f_{\rm c}(k\eta-\eta^{2})[1+(k-2)\eta]&,&0.4f_{\rm c}\leq\sigma_{\rm c}\leq f _{\rm c}\\ (1-4.76\varepsilon_{\rm c})f_{\rm c}-\varepsilon_{\rm cp}&,&\varepsilon_{\rm cp }\leq\varepsilon_{\rm c}\leq 13.6\varepsilon_{\rm cp}\end{array} \right. \tag{1}\]  
\[\frac{\sigma_{\rm f}}{f_{\rm t}}=\left[1+\left(c_{1}\cdot\frac{w}{w_{\rm t}} \right)^{3}\right]\cdot\rm e^{(-c_{2}\frac{w}{w_{\rm c}})}-\frac{w}{w_{\rm c}} \Big{(}1+c_{1}^{3}\Big{)}\cdot\rm e^{(-c_{2})} \tag{2}\]  
where \(\sigma_{\rm c}\) is the compressive stress of the concrete; \(E_{\rm c}\) is the elastic modulus of the concrete, \(E_{\rm c}=3.45\times\)\(10^{4}\) MPa; \(\varepsilon_{\rm c}\) is the compressive strain of the concrete; \(f_{\rm c}\) is the compressive strength of the concrete cylinder, \(f_{\rm c}=41.5\) MPa; \(k\) is the plasticity value, \(k=E_{\rm c}\times\varepsilon_{\rm cp}/f_{\rm c}\), which is 2.078 in this study; \(\eta\) is the ratio of strain to peak strain, \(\eta=\varepsilon_{\rm c}/\varepsilon_{\rm cp}\), \(\varepsilon_{\rm cp}=0.0025\); \(\sigma_{\rm t}\) is the tensile stress of the concrete; \(f_{\rm t}\) is the tensile strength of the concrete, \(f_{\rm t}=3.4\) MPa; \(w\) is the crack width; \(w_{\rm c}\) is the crack width when the stress is completely released, \(w_{\rm c}=0.216\) mm; and constant items \(c_{1}=3\), \(c_{2}=6.93\). The plastic-damage model is used to consider the nonlinear behavior of concrete. The compression damage factor (\(d_{\rm c}\))and tension damage factor (\(d_{\rm t}\))are used to represent the  
Figure 14: Constitutive relation curve of concrete. (**a**) Compressive stress–strain; (**b**) Tensile stress-crack width. Figure 13: Finite element model of pre-cracked T-beam. (**a**) Concrete T-beam network division; (**b**) Steel cage. degradation response of concrete materials. The damage factors of \(d_{\rm c}\) and \(d_{\rm t}\) range from 0 to 1, where the damage factor of 0 or 1 means no damage or complete failure, respectively.