## 2 Finite Element Model
### Model Overview  
ABAQUS finite element software was used to develop the finite element model of the SC composite beam. The behavior of the concrete was simulated by a solid element (C3D8R); the behavior of the steel plate was simulated by a shell element (S4R); the behavior of the tie-bars and studs in concrete was simulated by a truss element (T3D2); and the interfacial slip of the tie-bars and studs was simulated by a spring element. Tie-bars and studs were embedded in concrete through an embedded function. The interaction between the steel plate and concrete was simulated by a surface-to-surface contact algorithm. The normal behavior was defined through \"Hard\" contact pressure-overclosure, the tangential  
Figure 2: Placement of tie-bars and studs: (**a**) SCB1; (**b**) SCB2. behavior was defined through Penalty friction formulation, and the friction coefficient was 0.6. Due to the symmetries of specimen and loading, the finite element model of half of the SC composite beam was developed. On the symmetry plane, the normal displacement was confined. One end of specimen was fixed, and the other end was free. The cyclic loading was carried out on the free end. The mesh size of the C3D8R element and S4R element was 0.04 m, and the length of the T3D2 element was 0.044 m. The finite element model of SC composite beam is shown in Figure 4.  
Figure 4: Finite element model of SC composite beam. Figure 3: Test set-up. The constitutive behavior of concrete was simulated by a concrete damaged plasticity (CDP) model. The compressive stress-strain relationship of concrete was calculated based on the following functions proposed by [PERSON] and [PERSON] [21]. \[\sigma_{c}=\frac{f_{c}^{\prime}\beta(\varepsilon_{c}/\varepsilon_{c}^{\prime})}{ \beta-1+(\varepsilon_{c}/\varepsilon_{c}^{\prime})^{\beta}} \tag{1}\]  
\[\beta=\frac{\varepsilon_{c}/\varepsilon_{c}^{\prime}}{1-f_{c}^{\prime}/( \varepsilon_{c}/\varepsilon_{c}^{\prime})} \tag{2}\]  
where \(\sigma_{c}\) represents the compressive stress of concrete, \(\varepsilon_{c}\) represents the compressive strain of concrete, \(f_{c}^{\prime}\) is the compressive strength of cylindrical concrete, \(\varepsilon_{c}^{\prime}\) represents the strain corresponding to \(f_{c}^{\prime}\), which is taken as 0.002, and \(\beta\) is a function of secant modulus. In the CDP model, the dilation angel was assumed as 30\({}^{\circ}\), the flow potential eccentricity was assumed as 0.667, the ratio of the biaxial compressive strength versus uniaxial compressive strength was assumed as 1.16, the eccentricity was assumed as 0.1, and the Poisson's ratio was assumed as 0.2. The elasticity modulus of concrete was calculated based on the following equation stipulated in ACI-314-14:  
\[E_{c}=\sqrt{4700f_{c}^{\prime}} \tag{3}\]  
where \(E_{c}\) represents elasticity modulus of the concrete. In the quasi-static test, the compressive strength of cube concrete (\(f_{cu}\)) and compressive strength of prismatic concrete (\(f_{c}\)) were measured. According to the relationship between \(f_{c}^{\prime}\) and \(f_{cu}\), the conversion factor between \(f_{c}^{\prime}\) and \(f_{c}\) could be 1.1. The ultimate tensile strength of the cylindrical concrete (\(f_{l}^{\prime}\)) was assumed as 0.1\(f_{c}^{\prime}\), and the tension behavior of the concrete was calculated based on the stress-fracture energy cracking model. The following equation was used to calculate the fracture energy (\(G_{l}\)), which is stipulated in CEB-FIP [22]:  
\[G_{f}=a_{f}\big{(}f_{c}^{\prime}/10\big{)}^{0.7} \tag{4}\]  
where \(a_{f}\) is related to the maximum size of the coarse aggregate, and it can be assumed as 0.3. The CDP model has two failure mechanisms, including tensile cracking and compressive crushing. Two damage variables, \(d_{l}\) and \(d_{c}\), were used to represent the degrees of tensile cracking and compressive crushing. When \(d_{l}\) or \(d_{c}\) reached 1, the stiffness of the element could be negligible.