## 3 Dynamic Modified Numerical Model of RC Columns
### Influences of Model Complexity on the Simulated Results  
To investigate the influences of dynamic modification to sub-elements on the numerical results, six models with different complexities were established, as shown in Figure 7. By analyzing the results of Model-1/Model-2 (single sub-element), Model-3/Model-4 (two sub-elements) and Model-5/Model-6 (three sub-elements), the difference in simulated hysteretic curves using numerical models with different sub-elements was investigated. By comparing the simulated results of Model-1/Model-3/Model-5 (without dynamic modification) and Model-2/Model-4/Model-6 (with dynamic modification), the impact of dynamic modification on the hysteretic behaviors of RC columns was revealed. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Expression of Dynamic Modification** \\ \hline Yielding strength \(f_{y}\), & \(f_{y\text{\tiny{}}}\cdot Df_{f_{y}}\) \\ Strain at yielding strength \(\varepsilon_{y}\) & \(f_{ud}=f_{u}\cdot Df_{f_{u}}\) \\ Ultimate strength \(f_{u}\) & \(\varepsilon_{y\text{\tiny{}}}\cdot\frac{f_{ud}}{f_{y}}\varepsilon_{y}=\text{ DIF}_{f_{x}}\cdot\varepsilon_{y}\) \\ Slope of hardening branch \(E_{h}\) & \(E_{hd}=\frac{f_{y}\cdot DIF_{f_{y}}-f_{y}\cdot DIF_{f_{y}}}{\varepsilon_{u}- \text{DIF}_{y}\cdot\varepsilon_{y}}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Dynamic modified parameters for bond-slip sub-element [42]. Figure 6: Comparisons of static and dynamic constitutive models in bond-slip sub-elements. Figure 7: _Cont_. Figure 8-10 show the influences of the modeling approach on simulated hysteric behavior of three represented RC columns. Case-1, Case-2 and Case-3 were predicted to be under flexural, shear-flexural and shear failure modes, respectively. The design parameters of RC columns for case study are listed in Table 5. The critical mechanical behavior points, i.e., the yielding point, the ultimate point and the failure point, are also marked in the figures. The ductility coefficient is defined as the ratio of ultimate displacement to the yielding displacement. The yielding point was obtained by using the average results of the energy equivalent method, the geometric method and the R-PARK method. The failure point was taken as the point when the bearing capacity dropped to 85% of the ultimate bearing capacity. Figure 8: _Cont._  
Figure 7: Numerical models with different complexities. Figure 8: Comparisons between hysteretic curves using different numerical models for Case-1. Figure 9: Comparisons between hysteretic curves using different numerical models for Case-2. For Case-1, by comparing the group of static models shown in Figure 8a-c, it was found that the inclusion of bond-slip sub-element had a considerable impact on the simulated result of ductility while the influence of the shear sub-element was minor. By comparing the group of dynamic models shown in Figure 8d-f, it can be observed that the influences of sub-elements were more significant when considering the dynamic effect. For Case-2 and Case-3, by comparing Figure 9a-c (or Figure 10a-c), it was found that the inclusion of the shear sub-element had a considerable impact on the simulated result of hysteretic behavior as compared with the bond-slip sub-element. As the bond-slip sub-element was considered (Figures 9e and 10e), the enhancement of bearing capacity induced by dynamic  
Figure 10: Comparisons between hysteretic curves using different numerical models for Case-3.