# Strengthening of RC Beams with CFC Panels for Improving Impact Resistance
## 4 Analytical Program
### Material Models  
Reinforcement: The steel of the reinforcing bars was modeled as an elastoplastic material that follows the von Mises yield condition and the simple bilinear stress-strain relationship, shown in Figure 13, for both compression and tension. The hardening stiffness after yielding was set to 1/100 of the initial stiffness. Concrete: The concrete was modeled as an elastoplastic model based on the linear pressure-dependent Drucker-Prager yield criterion, which is expressed as:  
\[f(J_{2},I_{1})=\sqrt{J_{2}}+\alpha I_{1}-k=0, \tag{1}\]  
where \(J_{2}\) is the second invariant of the deviatoric stress, \(I_{1}\) is the first invariant of the stress (\(I_{1}=\sigma_{11}+\sigma_{22}+\sigma_{33}\)), \(\alpha=\frac{\sigma_{c}-\sigma_{1}}{\sqrt{3}(\sigma_{c}+\sigma_{1})}\) and \(k=\frac{2\sigma_{c}+\sigma_{1}}{\sqrt{3}(\sigma_{c}+\sigma_{1})}\) are material constants, and \(\sigma_{c}\) and \(\sigma_{1}\) are the compressive and tensile strengths, respectively. The tensile softening response was assumed as defined by the bilinear function developed by [PERSON] [38], in which the tensile stress \(\sigma_{t}\) decreases after cracking and tends to be zero at a total crack width \(w\) of 3.6 times the first crack opening, as shown in Figure 14. The fracture energy \(G_{f}\) of the concrete was defined using an existing experimental formula based on the compressive strength. Furthermore, the anisotropic damage law was used to compute the tensile cracking of concrete. The tensile damage in each principal direction at each time step was calculated using the relationship between the damage and accumulated plastic strain, as shown in Figure 15.  
The accumulated plastic strain in the current principal direction was calculated by summing the plastic strain increments determined at each step along the global coordinates. According to the tensile softening, the principal accumulated plastic strain and direction must be calculated to evaluate the tensile damage in each principal direction \(d_{t\_pr}\) by applying the following equation:  
\[d_{t\_pr}=d_{t\_lim}\left(e^{a-e^{(b(c-(\sigma^{\prime}-\frac{\sigma^{\prime} _{max})}{2}))}}\right), \tag{2}\]  
Figure 12: Virtual adhesive particles in this study. Figure 13: Stress-strain curve of the bilinear elastic hardening plastic material used for the rebar [37]. where \(d_{t\_lim}\) is the tensile damage limit and \(\varepsilon^{p}\) is the accumulated plastic strain. The constant values that were applied in this study are as follows: \(a=0.02,\ b=230\), and \(c=0.02\), and are the Gompertz parameters, and \(\varepsilon^{p}_{max}=0.025\) is the maximum accumulated plastic strain. The damage in each principal direction is transformed into a global coordinate system as follows:  
\[ \begin{cases}D_{1}\\ D_{2}\\ D_{3}\end{cases} =\left|d_{1,t\_pr} \begin{pmatrix}x_{1}\\ y_{1}\\ z_{1}\end{pmatrix} \right|+\left|d_{2,t\_pr} \begin{pmatrix}x_{2}\\ y_{2}\\ z_{2}\end{pmatrix} \right|+\left|d_{3,t\_pr} \begin{pmatrix}x_{3}\\ y_{3}\\ z_{3}\end{pmatrix} \right|,\ e_{i,t\_pr}=\left\{x_{i},y_{i},\ z_{i}\right\}^{t}, \tag{3}\]  
\[D_{i}=\sum_{i=1}^{3}\bigl{|}d_{i,\ t\_pr}e_{i,t\_pr}\bigr{|}, \tag{4}\]  
where \(D_{i}\) is the damage index in the global coordinates, and \(e_{i,t\_pr}\) is the principal direction vector (\(i=1,2,3\)). To consider the effect of tensile damage in this analysis, the elastic stiffness was multiplied by a reduction factor as follows:  
\[\Phi_{ij}=\sqrt{(1-D_{i})\bigl{(}1-D_{j}\bigr{)}}, \tag{5}\]  
Figure 14: Tensile softening model by [PERSON] [38]. Figure 15: Damage–plastic strain relation. \[E^{\epsilon}= \begin{pmatrix}(\lambda+2\mu)\Phi_{11}&\lambda\Phi_{12}&\lambda\Phi_{13 }&0&0&0\\ \lambda\Phi_{12}&(\lambda+2\mu)\Phi_{22}&\lambda\Phi_{23}&0&0&0\\ \lambda\Phi_{13}&\lambda\Phi_{23}&(\lambda+2\mu)\Phi_{33}&0&0&0\\ 0&0&0&2\mu\Phi_{12}&0&0\\ 0&0&0&0&2\mu\Phi_{23}&0\\ 0&0&0&0&0&2\mu\Phi_{13}\\ \end{pmatrix}