## 2 Materials and Methods
### Finite Element Modeling
#### 2.2.2 Material Model of Concrete  
In this research, the concrete material has been modeled via the use of a concrete damage plasticity (CDP) material model. The stress-strain diagram (\(\sigma_{\mathrm{c}}-\epsilon_{\mathrm{c}}\)) of concrete under compression (Figure 5a) was defined based on recommendations [36], such as the following equation:  
\[\sigma_{\mathrm{c}}=\frac{\mathrm{k}\eta-\eta^{2}}{1+(\mathrm{k}-2)\eta}\xi_ {\mathrm{c}};\ \ \eta=\frac{\epsilon_{\mathrm{c}}}{\epsilon_{\mathrm{c}1}};\ \ \mathrm{k}=1.05\mathrm{E}_{\mathrm{c}}\frac{ \epsilon_{\mathrm{c}1}}{\xi_{\mathrm{c}}}, \tag{1}\]  
where the following is the case:  
\(\epsilon_{\mathrm{c}1}=0.00205\), which is the strain at the maximal stress.  
Figure 4: FE model B-G3: (**a**) beam geometry and boundary conditions; (**b**) geometry of reinforcement and GFRP bars.  
\(\epsilon_{\rm c\
u 1}=0.00350\), which is the ultimate strain.  
\(\rm f_{c}=31.60\) MPa, which is the compressive strength.  
\(\rm E_{c}=32.80\) GPa, which is the modulus of elasticity.  
The elastic behavior of the analyzed concrete under compression was \(\rm 0.4f_{c}=12.64\) MPa (with a correspondent strain of \(\rm 0.000385\)).  
The characteristics of the concrete under tension (\(\rm\sigma_{t}-\epsilon_{t}\) diagram) have been defined via the use of the relation given in [37]:  
\[\rm\sigma_{t}=f_{t}\left(\frac{\epsilon_{cr}}{\epsilon_{t}}\right)^{0.40}, \tag{2}\]  
where the following is the case:  
\(\rm f_{t}\) is the concrete tensile strength.  
\(\rm\varepsilon_{cr}\) is the strain at the concrete tensile strength.  
The tensile strength of concrete (\(\rm f_{t}\)) has been defined according to the relationship proposed in [38], and by compressive strength (\(\rm f_{c}\)):  
\[\rm f_{t}=0.34\sqrt{f_{c}}=0.34\sqrt{31.60}=1.911\ MPa. \tag{3}\]  
The strain that corresponds to the tensile strength was \(\rm\varepsilon_{cr}=0.00005826\). This value defines the boundary of the elastic behavior of the analyzed concrete under tension. A stress-strain diagram of the concrete under tension is presented in Figure 5b.  
The plastic behavior of concrete is described as the damage of concrete due to both compression and tension in the adopted CDP model. This model has been selected as adequate for modeling quasi-brittle materials [39; 40; 41]. Several input parameters need to be defined for the CDP material model, and these values are adopted according to the recommendations [39; 40; 41] and presented in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Dilation angle, \(\psi\) & \(\rm 30^{\circ}\) \\ Flow potential eccentricity, \(\epsilon\) & \(\rm 0.1\) \\ Ratio of the biaxial compressive and uniaxial & \(\rm 1.16\) \\ compressive yield stress, \(\rm\sigma_{bb}/\sigma_{c0}\) & \\ Ratio of the second stress invariant on the & \\ tensile meridian to that on the compressive & \(\rm 0.666\) \\ meridian at initial yield, K & \\ Viscosity parameter, \(\rm\mu\) & \(\rm 0.0001\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input parameters for the CDP material model.  
Figure 5: Stress–strain diagram for concrete: (**a**) compression; (**b**) tension.  
The CDP model enables the reducing of the initial modulus of elasticity that arises owing to concrete damage. This is realized by means of damage parameters for compression, \(\mathrm{d_{c}}\), and for tension, \(\mathrm{d_{t}}\)[39; 42], and they are determined by the following relations:  
\[\mathrm{d_{c}}=1-\frac{\sigma_{\mathrm{c}}}{\mathrm{f_{c}}};\ \mathrm{d_{t}}=1-\frac{ \sigma_{\mathrm{t}}}{\mathrm{f_{t}}}. \tag{4}\]  
Damage parameters have been defined with respect to inelastic strains, \(\varepsilon_{\mathrm{c}}\)in, and with respect to cracking strains, \(\varepsilon_{\mathrm{c}}\)cr, for compression and tension, respectively (Figure 6). Based on these data, the plastic strain in compression and tension have been calculated by software [39].