## 3 Damage Mechanism Analysis Based on Damage Index  
Previous studies have shown that the damage index \(D_{k}\) increases monotonically from 0 to 1 with the increase in total energy dissipation \(E_{\rm C}\). Therefore, the total energy dissipation \(E_{\rm C}\) can be used as a carrier to study the relationship between the damage index \(D_{k}\) and the ductility coefficient \(\mu\), the period \(T(m)\), the seismic intensity (\(A_{\rm PG}\) and \(V_{\rm PG}\)), and the earthquake duration \(t_{\rm d}\).  
Figure 3 shows the influence of the ductility coefficient \(\mu\), the period \(T(m)\), the seismic intensity (\(A_{\rm PG}\) and \(V_{\rm PG}\)), and the earthquake duration \(t_{\rm d}\) on the total energy dissipation \(E_{\rm C}\). The x-coordinate is the earthquake duration \(t_{\rm d}\), and its range is 0 to 30 s. The y-coordinate is the ductility factor \(\mu\), and its range is 1 to 5. The z-coordinate is the total energy dissipation \(E_{\rm C}\). The variation range of the fundamental period \(T\) is 0.1 s-0.7 s. The seismic intensity is divided into three levels: 6 degrees (0.05 \(g\)), 7 degrees (0.1 \(g\)), and 8 degrees (0.2 \(g\)).  
As shown in Figure 3a, the total energy dissipation \(E_{\rm C}\) increased with the extended earthquake duration \(t_{\rm d}\), and the larger the ductility coefficient \(\mu\), the faster the total energy dissipation \(E_{\rm C}\) increased. With the increase in ductility factor \(\mu\), the total energy dissipation \(E_{\rm C}\) increased continuously. Therefore, an effective way to decrease the damage is by controlling the ductility coefficient. The total energy dissipation \(E_{\rm C}\) increased with the increase in the fundamental period \(T\), and the greater the fundamental period \(T\), the faster the total energy dissipation \(E_{\rm C}\) increased. Similar phenomena can be seen in Figure 3b,c. Comparing Figure 3a-c, the higher the seismic intensity, the more the total energy dissipation \(E_{\rm C}\) increased. Among all the influencing factors, the fundamental period \(T\) and seismic intensity contributed more significantly to the total energy dissipation \(E_{\rm C}\). Since the damage index \(D_{k}\) increased monotonically with the increase in the total energy dissipation \(E_{\rm C}\), the influence of the ductility coefficient \(\mu\), the period \(T(m)\), the seismic intensity (\(A_{\rm PG}\) and \(V_{\rm PG}\)), and the earthquake duration \(t_{\rm d}\) on the damage index \(D_{k}\) can be obtained.  
Substitute Formula (14) into Formula (6)  
\[D_{k}=f(H,\rho_{\text{sv}},F_{y},u_{y},\mu,m,n,T,A_{\text{PG}},V_{\text{PG}},t_{ \text{d}}) \tag{15}\]  
In summary, the energy-based damage index \(D_{k}\) showed a corresponding relationship with the height \(H\), stirrup ratio \(\rho_{\text{sv}}\), longitudinal reinforcement (\(F_{y}\), \(u_{y}\)), ductility coefficient \(\mu\), period \(T(m)\), number of column members \(n\), seismic intensity (\(A_{\text{PG}}\) and \(V_{\text{PG}}\)), and earthquake duration \(t_{\text{d}}\). The increase in the stirrup ratio \(\rho_{\text{sv}}\) can slow down the damage, and the slowing process is initially fast and then slows. When the reinforcement ratio is doubled, the damage index decreased by 0.063. The longer the earthquake duration \(t_{\text{d}}\) is, the more serious the damage is, and this phenomenon is more obvious when the ductility coefficient \(\mu\) is larger. With the increase in the ductility coefficient \(\mu\), the damage increased continuously. Therefore, an effective way to decrease the damage is by controlling the ductility coefficient. Among all the influencing factors, the fundamental period \(T\) and seismic intensity contributed more significantly to the damage indicators.