## 4 Numerical Investigation
### Constraint Conditions and Interaction  
The interaction between members was considered using proper constraints and interactions, as shown in Figure 12. The shear connector was divided into two parts: the lower part was connected to the upper flange of the steel girder by a tie constraint, and an embedded constraint was used to connect the upper part to the connection grout. The grout-steel interaction was defined utilizing surface-to-surface contact, hard contact was adopted in the normal direction of the interface, a penalty function was adopted in the tangential direction of the interface, and the friction factor was 0.3 [14]. Reinforcements were also considered in the normal direction of the interface, and the friction factor was 0.3 [14].  
Figure 11: Finite-element type and mesh.  
Figure 10: The reinforcement arrangement of cases.  
were placed inside the connection grout and concrete deck, and were connected to them by the embedded constraint.  
For these analytical models, the contact surfaces between the connection grout and the precast concrete deck play a key role. In the present approach, a surface-based cohesive behavior initially defines a traction-separation model, followed by the initiation and evolution of damage, which is used to model the cohesive effect at the interface. Therefore, the contact surface is assumed to indicate linear elastic behavior in terms of a constitutive matrix that relates tractions to separations by:  
\[\left\{\begin{array}{c}t_{n}\\ t_{s}\\ t_{t}\end{array}\right\}=\left[\begin{array}{ccc}K_{nn}&0&0\\ 0&K_{ss}&0\\ 0&0&K_{tt}\end{array}\right]\left\{\begin{array}{c}\delta_{n}\\ \delta_{s}\\ \delta_{t}\end{array}\right\} \tag{9}\]  
For the uncoupled traction-separation type, \(t_{nr}\), \(t_{s}\), and \(t_{t}\) represent normal, shear, and tangential tractions (along the global \(Z\)-axis, \(X\)-axis, and \(Y\)-axis, respectively), while \(\delta_{n}\), \(\delta_{s}\), and \(\delta_{t}\) denote the corresponding separations. Subsequently, damage modeling describes degradation and failure of the bond at the interface, where the damage initiation describes the beginning of degradation of the cohesive response at each contact point, while the damage evolution refers to the rate at which the cohesive stiffness is degraded once the corresponding initiation criterion is reached.  
For the damage initiation at the interface, the quadratic traction criterion is considered among some criteria available in the ABAQUS:  
\[\left\{\frac{\left\langle t_{n}\right\rangle}{t_{n}^{0}}\right\}^{2}+\left\{ \frac{t_{s}}{t_{s}^{0}}\right\}^{2}+\left\{\frac{t_{t}}{t_{t}^{0}}\right\}^{2}=1 \tag{10}\]  
An effective separation (\(\delta_{m}\)) is considered to describe the damage evolution under a combination of normal and other separations across the interface:  
\[\delta_{m}=\sqrt{\left\langle\delta_{n}\right\rangle^{2}+\delta_{s}^{2}+\delta _{t}^{2}} \tag{11}\]  
In general, uncoupled stiffness coefficients (\(K_{nn}\), \(K_{ss}\), \(K_{tt}\)), peak values of traction (\(t^{0}{}_{n}\), \(t^{0}{}_{s}\), \(t^{0}{}_{t}\)), and an effective separation at complete failure (\(\delta^{\prime}_{m}\)) are utilized to simulate cohesive failure in the contact surfaces. In this study, these parameters are selected so that the flexural  
Figure 12: Constraint and interaction surfaces.  
moment strength of the girder reaches the nominal flexural resistance (\(M_{n}\)). The nominal flexural resistance is determined from the equation proposed by [PERSON] et al. [11]. The ability of the flexural moment strength of the girder to reach the nominal flexural resistance was verified in studies of [PERSON] et al. [11] and [PERSON] et al. [15]. As a result, the values of these parameters are as follows: \(K_{m}=K_{ss}=K_{tt}=15\), \(t^{0}{}_{n}=t^{0}{}_{s}=t^{0}{}_{t}=5\), and \(\delta^{\prime}{}_{m}=4\).