# Experimental and Parametric Study on the Pull-Out Resistance of a Notched Perfobond Shear Connector
## 4 Finite Element Analysis
### Material Constitution of Concrete  
The nonlinear behavior of the concrete material in compression and tension was characterized by a uniaxial compressive stress-strain curve and a tensile stress-crack width relationship, as shown in Figure 10.  
The concrete material constitutions in compression are described by Equation (1) [26; 27]. As shown in Figure 10a, the initial branch of the stress-strain curve is assumed to be elastic. The next two branches are a nonlinear parabolic portion and a descending branch, respectively.  
\[\sigma_{c}=\left\{\begin{array}{ll}E_{c}\epsilon_{c}&(0\leq\varepsilon_{c} \leq 0.4f_{c}/E_{c})\\ \frac{k\cdot\eta-\eta^{2}}{1+(k-2)\cdot\eta}f_{c}&(0.4f_{c}/E_{c}<\varepsilon _{c}\leq\varepsilon_{cp})\\ \left(1-0.15\frac{\epsilon-\varepsilon_{cp}}{\varepsilon_{cu}-\varepsilon_{cp }}\right)f_{c}&\left(\varepsilon_{cp}<\varepsilon_{c}\leq\varepsilon_{cu} \right)\end{array}\right. \tag{1}\]  
where \(\sigma_{c}\) is the compressive stress (MPa); \(\varepsilon_{c}\) is the compressive strain; \(E_{c}\) is the Young's modulus (MPa); \(f_{c}\) is the compressive strength of concrete (MPa); \(k\) is the plasticity number, \(k=E_{c}\cdot\varepsilon_{cp}/f_{c}\); \(\eta\) is the ratio of strain to peak strain, where \(\eta=\varepsilon_{c}/\varepsilon_{cp}\); \(\varepsilon_{cp}=0.002\); and \(\varepsilon_{cu}=0.0033\).  
A linear stress-strain relationship was adopted to depict the uncracked concrete in tension. For a cracked section, as shown in Figure 10b, a nonlinear approach for the stress-crack width relationship can be determined by using Equation (2) [28].  
\[\frac{\sigma_{t}}{f_{t}}=\left[1+\left(c_{1}\cdot\frac{w}{w_{c}}\right)^{3} \right]\cdot\exp\left(-c_{2}\cdot\frac{w}{w_{c}}\right)-\frac{w}{w_{c}}\cdot \left(1+c_{1}^{3}\right)\cdot\exp(-c_{2}) \tag{2}\]  
Figure 10: Material constitution of concrete. (**a**) Compression; (**b**) tension.  
Figure 9: Interaction and boundary conditions.  
where \(\sigma_{t}\) is the tensile stress (MPa); \(f_{t}\) is the tensile strength of concrete (MPa); \(w\) is the crack width (mm); \(w_{c}\) is the crack width at the complete release of stress, \(w_{c}=5.14\)\(G_{F}/f_{t}\) (mm); \(G_{F}\) is the fracture energy required to create a unit area of stress-free crack, \(G_{F}=0.073\,f_{c}^{0.18}\) (N/mm); and the constants are \(c_{1}=3\) and \(c_{2}=6.93\).  
The concrete damaged plasticity model was adopted to describe the degraded response of the concrete material. Two independent uniaxial damage variables, \(d_{c}\) and \(d_{t}\), were used to evaluate the damage of concrete due to compressive crushing and tensile cracking [25].  
For concrete in compression, the evolution of \(d_{c}\) is related to the plastic strain \(\varepsilon^{pl}\), determined to be proportional to the inelastic strain \(\varepsilon^{in}=\varepsilon_{c}\)-\(\sigma_{c}/E_{c}\), using a constant factor \(b_{c}\) (0 \(<b_{c}<\) 1) in Equation (3) [29].  
\[d_{c}=1-\frac{\sigma_{c}}{E_{c}\cdot\varepsilon_{c}^{pl}\cdot(1/b_{c}-1)+ \sigma_{c}} \tag{3}\]  
where \(d_{c}\) is the concrete compressive damage component; and \(b_{c}\) is the ratio of plastic strain to inelastic strain, \(b_{c}=\varepsilon_{c}pl/\varepsilon_{c}^{in}\), where \(b_{c}\) is taken as 0.7 [29].  
For concrete in tension, the damage evolution component \(d_{t}\) is associated with the \"plastic\" crack width \(w^{pl}\), which is proportional to the crack width \(w\), using a constant factor \(b_{t}\) (0 \(<b_{t}<\) 1) in Equation (4), referring to [29].  
\[d_{t}=1-\frac{\sigma_{t}\cdot l_{0}}{E_{c}\cdot w^{pl}\cdot(1/b_{t}-1)+\sigma_ {t}\cdot l_{0}} \tag{4}\]  
where \(d_{t}\) is the tensile damage variable of concrete; \(l_{0}\) is assumed to be unit length; and \(b_{t}\) is the ratio of the \"plastic\" crack width to the crack width, \(b_{t}=w^{pl}/w\), where \(b_{t}\) is set as 0.1 [29].