# Deep Embedment (DE) FRP Shear Strengthening of Concrete Bridge Slabs under Loads Close to Supports
## 5 Finite Element Analysis
### Modelling of Concrete Material  
In this numerical study, a material model named Concrete Damage Plasticity Model [22; 23] was adopted based on the previous testing of concrete deck slabs [24]. This model assumes non-associated potential plastic-flow where the flow potential is defined by the Drucker-Prager hyperbolic function and the yield function. The equation by [PERSON] et al. [25] combined with the [PERSON]'s [26] assumption on the elastic modulus of concrete were used to model compressive stress vs. strain relationship of concrete material. This model was given by Equations (1) to (5).  
\[\frac{f_{c}}{f_{c^{\prime}}}=\frac{\varepsilon_{c}}{\varepsilon_{c^{\prime}}} \frac{n}{n-1+\left(\varepsilon_{c}/\varepsilon_{c^{  
When \(\varepsilon/\varepsilon_{c^{\prime}}\) is less than 1, k equals 1. When \(\varepsilon/\varepsilon_{c^{\prime}}\) exceeds 1, k is a number larger than 1.  
\[k=0.67+\frac{f_{c^{\prime}}}{62} \tag{4}\]  
\[E_{c}=4723\sqrt{f_{c}^{\prime}} \tag{5}\]  
The tensile response of reinforced concrete is modelled using a non-linear tension stiffening model. Tension stiffening is influenced by the reinforcement ratio, crack spacing, and quality of the bond between the concrete and reinforcement. Before the occurrence of cracks, the tension behaviour of concrete was assumed to be linear. After this stage, the stress and strain relationship of concrete under uniaxial tension was expressed as below [27; 28; 29]:  
\[\frac{\sigma_{t}}{f_{t}}=\Bigg{[}1+\bigg{(}c_{1}\frac{w_{t}}{w_{cr}}\bigg{)}^ {3}\Bigg{]}e^{-(c_{2}\frac{w_{t}}{w_{cr}})}-\frac{w_{t}}{w_{cr}}(1+c_{1}^{3})e ^{(-c_{2})} \tag{6}\]  
\[w_{cr}=5.14\frac{G_{F}}{f_{t}} \tag{7}\]  
where \(w_{t}\) is crack open displacement; \(w_{cr}\) is cracking opening displacement at the complete release of stress or fracture energy; \(\sigma_{t}\) is tensile stress normal to crack direction; \(f_{t}\) is concrete uniaxial tensile strength; \(G_{F}\) is fracture energy required to create a stress-free crack over a unit area; and \(c_{1}=3.0\) and \(c_{2}=6.93=\) constants determined from tensile tests of concrete. In this study, \(G_{F}\) could be estimated from the equation of CEB-FIB [30] as below:  
\[G_{F}=(0.0469d_{a}^{2}-0.5d_{a}+26)(\frac{f_{c}^{\prime}}{10})^{0.7} \tag{8}\]  
Additionally, the damage parameters are defined for compressive failure and tension failure in this material model. The evolution of the compressive damage component \(d_{c}\) is linked to the corresponding plastic strain \(\varepsilon_{c}^{pl}\) which is determined proportional to the inelastic strain \(\varepsilon_{c}^{in}=\varepsilon_{c}-\sigma_{c}E_{c}^{-1}\) using constant bc with the expression of \(\varepsilon_{c}^{pl}=b_{c}\varepsilon_{c}^{in}\).  
\[d_{c}=1-\frac{\sigma_{c}E_{c}^{-1}}{\varepsilon_{c}^{pl}(1/b_{c}-1)+\sigma_{ c}E_{c}^{-1}} \tag{9}\]  
In Equation (9), \(b_{c}\) is equal to 0.7 according to the reported experimental test results [31]. Similar to the Equation (9), the tensile damage \(d_{t}\) depends on \(\varepsilon_{t}^{pl}\) and experimentally determined parameter \(b_{t}=0.1\)[32]. Therefore, the unloading is assumed to return almost back to the origin and to leave only a small residual strain (see Equation (10)).  
\[d_{t}=1-\frac{\sigma_{t}E_{c}^{-1}}{\varepsilon_{t}^{pl}(1/b_{t}-1)+\sigma_{ t}E_{c}^{-1}} \tag{10}\]  
where \(\varepsilon_{t}^{pl}=b_{t}\varepsilon_{t}^{in}\) and \(\varepsilon_{t}^{in}=\varepsilon_{t}-\sigma_{t}E_{t}^{-1}\).