## 3 Dynamic Modified Numerical Model of RC Columns  
### Establishment of Numerical Model  
The developed numerical model of RC columns was composed of three sub-elements in series (see Figure 2), including the fiber beam-column element, the zero-length shear element and the bond-slip element to simulate the flexural deformation, the shear deformation and the bond-slip deformation, respectively. Based on the limit state material theory [37], the mechanical behaviors of column members under different failure modes could be simulated. Moreover, dynamic modifications were made to the three sub-elements to consider the influences of loading rate on the mechanical behavior and deformation mechanism of RC column members. Figure 1: Illustration of procedure to develop the _DMC_ predictive models. The flexural sub-element was developed by using the force-based beam-column element on the OpenSees platform. The cross section of the flexural sub-element was divided by fibers and the uniaxial material constitutive models of reinforcing steel and concrete 04 were employed to depict the material properties of reinforcement and concrete, respectively. For reinforcement, the strength degradation under cyclic loading [38] and the buckling effect [39] were considered. As for concrete, the degradation of strength and stiffness during cyclic loading, and the enhancement of concrete strength and ductility [40] due to the constraint effect of stirrups were fully taken into account. To consider the strain rate effect of materials on the flexural behavior of columns, the constitutive model parameters of reinforcement and concrete were dynamically modified by using the available empirical DIF equations [41; 42]. The dynamic modified constitutive models and the static constitutive models of reinforcement and concrete are plotted in Figure 3. The parameters modified were the yielding strength \(f_{y}\), the ultimate strength \(f_{u}\) and the initial hardening strain \(\varepsilon_{h}\) of reinforcement, and the compressive strength \(f^{\prime}_{c}\), the tensile strength \(f_{t}\), the elastic modulus \(E_{c}\) and the ultimate compressive strain \(\varepsilon_{cd}\) of concrete. The employed DIF equations are tabulated in Table 1.  
Figure 3: Comparisons of static and dynamic constitutive models for RC materials. The star denotes the fracture failure point of reinforcement. Figure 2: Illustration of the developed numerical model for RC columns. In the numerical model, the shear stress-shear strain relationship obtained by the modified compression field theory (i.e., MCFT) [44] was used to derive the constitutive model of force-deformation for the shear sub-element. To consider the influence of dynamic effect on shear bearing capacity and shear deformation of RC columns, the following MCFT calculation equations were modified using the DIF equations of materials. The material parameters modified were the compressive strength \(f_{c}^{\prime}\), the tensile strength \(f_{t}\) and the elastic modulus \(E_{c}\) of concrete, and the yielding strength of longitudinal reinforcement and stirrups, i.e., \(f_{y}\) and \(f_{yt}\). The MCFT equations considering the dynamic modification are listed in Table 2. The schematic plot of dynamic modified and the static constitutive models of shear force-deformation are compared in Figure 4.  
To accurately describe the dynamic behavior of the bond-slip sub-element, the constitutive model of bond stress-slip amount (\(\tau-s\)) in Equation (6) derived from the dynamic loading pull-out test results [44] was employed  
\[F^{\intercal}(s)= \begin{cases}\tau_{u}\Big{(}2\sqrt{\frac{s}{s_{0}}}-\frac{s}{ s_{0}}\Big{)}&,0\leq s\leq s_{0}\\ \tau_{u}\Big{(}k_{r}+(1-k_{r})\exp\Bigl{(}\beta\Bigl{(}\frac{s}{s_{0}}-1\Bigr{)} ^{2}\Bigr{)}\Big{)}&,s>s_{0}\end{cases}  \tag{6}\]