## 3 Dynamic Modified Numerical Model of RC Columns
### Establishment of Numerical Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Parameter** & **Expression of DIF Model** & **Relevant Parameters** \\ \hline Concrete [42] & \(f_{t}\) & \(DIF_{f_{t}}=\frac{f_{yt}}{f_{m}}=\Big{(}\frac{\dot{x}}{\dot{x}_{0}}\Big{)}^{1,016\alpha},\dot{x}_{t}\leq 30\text{s}^{-1}\) & \(\alpha=\frac{1}{10+\frac{6\alpha}{60}}\) \\  & \(DIF_{f_{t}}=\frac{f_{yt}}{f_{m}}=\gamma\Big{(}\frac{\dot{x}}{\dot{x}_{0}}\Big{)} ^{1},\dot{x}_{t}>30\text{s}^{-1}\) & \(\log\gamma=7.11\alpha-2.33\) \\  & \(E_{c}\) & \(DIF_{E_{c}}=\frac{E_{d}}{E_{\alpha}}=\Big{(}\frac{\dot{x}}{\dot{x}_{c0}}\Big{)} ^{0,026}\) & - \\  & \(\varepsilon_{cd}\) & \(DIF_{c_{c}}=\frac{\varepsilon_{cd}}{\varepsilon_{\alpha}}=\Big{(}\frac{\dot{x }}{\dot{x}_{c0}}\Big{)}^{0,02}\) & - \\  & \(f_{y}\) & \(DIF_{f_{y}}=\frac{f_{yt}}{f_{m}}=1+c_{t}f\lg\frac{\dot{x}}{\dot{x}_{0}}\) & \(c_{f}=0.1709-3.289\times 10^{-4}f_{ys}\) \\ Reinforcement [43] & \(f_{u}\) & \(DIF_{f_{x}}=\frac{f_{yt}}{f_{m}}=1+c_{u}\lg\frac{\dot{x}}{\dot{x}_{0}}\) & \(c_{u}=0.02783-2.982\times 10^{-5}f_{ys}\) \\  & \(\varepsilon_{h}\) & \(DIF_{\varepsilon_{h}}=\frac{f_{yt}}{f_{m}}=1+c_{h}\lg\frac{\dot{x}}{\dot{x}_{0}}\) & \(c_{h}=0.9324-0.00212f_{ys}\) \\ \hline \hline \end{tabular} Note: The subscripts ’\(s\)’ and ’\(d\)’ denote the static and dynamic material parameters, respectively; \(\dot{x}_{c}\) and \(\dot{x}_{0}=3\times 10^{-5}/s\) represent the strain rates of compressive concrete under dynamic and static loading, respectively; \(\dot{x}_{t}\) and \(\dot{x}_{0}=3\times 10^{-6}/s\) represent the strain rates of tensile concrete under dynamic and static loading, respectively; \(\dot{x}\) and \(\dot{x}_{0}=2.5\times 10^{-4}/s\) denote the strain rates of reinforcement under dynamic and static loading, respectively.  
\end{table} 
Table 1: DIF equations for dynamic modification to flexural sub-element. \begin{table}
\begin{tabular}{c c c} \hline \hline
**MCFT Equations [43]** & **Description** & **Modified Parameters** \\ \hline \(f_{2}\leq\frac{f_{t}^{\prime}\cdot DIF_{f_{t}}}{08-0.34\frac{1}{\dot{x}_{c}}} \Big{[}2\Big{(}\frac{\dot{x}_{2}}{\dot{x}_{c}}\Big{)}-\Big{(}\frac{\dot{x}_{ 2}}{\dot{x}_{c}}\Big{)}^{2}\Big{]}\) & The average compression stress\(f_{2}\) should not exceed its allowable value. & \(f_{c}^{\prime}\) \\ \(f_{1}=\varepsilon_{1}\cdot E_{c}\cdot DIF_{E_{c}}\leq\frac{f_{t}\cdot DIF_{f_{t}}}{ 1+\sqrt{500}\varepsilon_{1}}\) & The average tensile stress\(f_{1}\) should not exceed its allowable value. & \(f_{t}\) and \(E_{c}\) \\ \(f_{yx}=E_{s}\cdot\varepsilon_{xx}\leq f_{y}\cdot DIF_{f_{y}}\) & The stress of longitudinal reinforcement should not exceed its dynamic yielding strength & \(f_{y}\) \\ \(f_{yy}=E_{s}\cdot\varepsilon_{xy}\leq f_{yt}\cdot DIF_{f_{yt}}\) & The stress of stirrups should not exceed their dynamic yielding strength & \(f_{yt}\) \\ \(v_{ci}\leq v_{ci,\max}=\frac{\sqrt{f_{c}\cdot DIF_{c}}}{0.31+\frac{2\alpha}{6 \alpha+12}}\) & The shear stress \(v_{ci}\) should not exceed the maximum allowable value \(v_{ci,\max}\) & \(f_{c}^{\prime}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dynamic modified MCFT equations for shear sub-element. where \(\tau_{u}\) and \(s_{0}\) represent the ultimate bond stress and the corresponding slip amount; \(\beta\) denotes the control coefficient for the descending branch of model; \(k_{r}\) is the ratio of residual bond stress to ultimate bond stress. The parameters \(\tau_{u}\), \(s_{0}\) and \(\beta\) are relevant to the loading rate and the corresponding formulas are tabulated in Table 3.  
Figure 5 illustrates the comparison results of static and dynamic bond-slip constitutive models. It can be observed that with the increase of loading rate, the ultimate bond stress \(\tau_{u}\) is magnified and the slip amount \(s_{0}\) is reduced, while the model shape is slightly changed. The bond-slip material [45] defining the relationship between longitudinal reinforcement stress and pull-out amount (\(\sigma_{s}-S\)) was employed to simulate the end rotation of columns caused by the bond-slip effect [46; 47].