# Crack Propagation Law of Reinforced Concrete Beams
## 2 Comparison of Finite Element Model and Experiment Results
### Material Constitute of Concrete in the FE Model  
The stress-strain relationship employed in the finite element model corresponds to the constitutive relation of concrete, as specified in the Code for the Design of Concrete Structures (GB 50010-2010) [22]. During the elastic phase, the stress-strain relationship of concrete is determined by its Young's modulus \(E\) and Poisson's ratio \(\lambda\). In the inelastic phase, the stress-strain relationship of concrete is determined by the aforementioned code. The stress-strain relation of concrete under axial compression can be described by Equations (1)-(5):  
\[\sigma=(1-d_{c})E_{c}\varepsilon \tag{1}\]  
\[d_{c}=\left\{\begin{array}{ll}1-\frac{\rho_{c}n}{n-1}\frac{\rho_{c}n}{\frac{ 1}{\rho_{c}}\kappa^{\alpha}}&x\leq 1\\ 1-\frac{\rho_{c}n}{\kappa_{c}(x-1)^{2}+x}&x>1\end{array}\right. \tag{2}\]  
\[\rho_{c}=\frac{f_{c,r}}{E_{c}\varepsilon_{c,r}} \tag{3}\]  
\[n=\frac{E_{c}\varepsilon_{c,r}}{E_{c}\varepsilon_{c,r}-f_{c,r}} \tag{4}\]  
\[x=\frac{\varepsilon}{\varepsilon_{c,r}} \tag{5}\]  
where \(f_{c,r}\) represents the representative value of the axial compressive strength of concrete, which is measured at 22.4 MPa according to [19]; \(E_{c}\) represents the modulus of elasticity of concrete, which is taken as 30,904.1 MPa; \(\alpha_{c}\) represents the descending parameter value of the axial compression stress-strain curve of concrete, set as 1.563; \(\varepsilon_{c,r}\) is the peak compressive strain of concrete corresponding to the representative value of the axial compressive strength \(f_{c,r}\) of concrete, set as 0.001696; and \(d_{c}\) is the propagation parameter of concrete axial compression damage.  
Figure 5 shows the stress-strain curves and axial compressive damage-strain curves of concrete under axial compression, respectively.  
The axial tensile stress-strain curve of concrete can be calculated according to Equations (6)-(9):  
\[\sigma=(1-d_{t})E_{c}\varepsilon \tag{6}\]  
\[d_{t}=\left\{\begin{array}{ll}1-\rho_{t}(1.2-0.2\kappa^{5})&x\leq 1\\ 1-\frac{\rho_{t}}{\kappa_{t}(x-1)^{1/\alpha}+x}&x>1\end{array}\right. \tag{7}\]  
\[x=\frac{\varepsilon}{\varepsilon_{t,r}} \tag{8}\]  
Figure 5: Stress–strain and damage relationship curves of concrete under axial compression. (**a**) stress–strain curve; (**b**) damage–strain curve.  
\[\rho_{t}=\frac{f_{t,r}}{E_{c}\epsilon_{t,r}} \tag{9}\]  
where \(f_{t,r}\) represents the representative value of the axial tensile strength of concrete, which is taken as 2.2 MPa according to the code [22]; \(\alpha_{t}\) represents the descending parameter value of the axial tensile stress-strain curve of concrete, set as 1.53; \(\epsilon_{t,r}\) is the peak tensile strain of concrete corresponding to the representative value of axial tensile strength \(f_{t,r}\), set as 0.0000998; and \(d_{t}\) is the propagation parameter of concrete axial tensile damage.  
Figure 6 shows the axial tensile stress-strain curve of concrete and the axial tensile damage-strain curve of concrete, respectively.  
By combining the data calculated from the above formulas with the plastic damage constitutive model provided by Abaqus [23], the shear failure test results of the simply supported beams are verified accordingly. The hardening model in Abaqus follows the von Mises yield criterion and has a good ability to simulate the nonlinear material of concrete.