## 2 Energy-Based Damage Index  
### The Proposed Energy-Based Damage Index  
The random variable amplitude loading history was applied to the reinforced concrete column members with different reinforcement levels, the causes of capability degradation were studied, and the energy-based damage index \(D_{k}\) was proposed [22], as follows:  
\[D_{k}=\left[A_{k}(1-\mathrm{e}^{-0.47B_{k}u_{k}})\right]^{0.09} \tag{1}\]  
with  
\[A_{k}=0.62{\mu_{\mathrm{e}}}^{0.2} \tag{2}\]  
\[B_{k}=\frac{3.64{\mu_{\mathrm{sv}}}^{-0.13}}{(1+{\mu_{\mathrm{e}}})^{5.63{\mu_ {\mathrm{sv}}}^{0.09}}} \tag{3}\]\[\mu_{\rm e}=\frac{0.1H}{u_{\rm y}} \tag{4}\]  
\[n_{k}=\frac{E_{\rm C}}{0.5F_{\rm y}u_{\rm y}} \tag{5}\]  
where the parameter \(A_{k}\) is the peak value of damage index \(D_{k}\), the parameter \(B_{k}\) is the energy dissipation requirements of the reinforced concrete column members, \(\mu_{\rm e}\) is the normalized amplitude, \(\rho_{\rm sw}\) is the stirrup ratio of densification zone, \(H\) is the height of the member, \(n_{k}\) is the normalized accumulated hysteretic energy, \(E_{\rm C}\) is the total energy dissipation, and \(F_{\rm y}\) and \(u_{\rm y}\) are the yield load and yield displacement, respectively. According to Formulas (1)-(5)  
\[D_{k}=f(H,\rho_{\rm sw},F_{\rm y},u_{\rm y},E_{\rm C}) \tag{6}\]  
It can be found from Formula (6) that the corresponding relationship between the damage index \(D_{k}\) and the construction member parameters (\(H\), \(\rho_{\rm sw}\), \(F_{\rm y}\), \(u_{\rm y}\)) and the total energy dissipation \(E_{\rm C}\) is established. Influence of Reinforcement Conditions and Total Energy Dissipation \(E_{\rm C}\) on Damage Index \(D_{k}\)  
When the section and height of the reinforced concrete column member are determined, the yield load \(F_{\rm y}\) and the yield displacement \(u_{\rm y}\) are related to the longitudinal reinforcement. Therefore, according to Formula (6), the influence of the stirrup ratio \(\rho_{\rm sw}\), the longitudinal reinforcement (\(F_{\rm y}\), \(u_{\rm y}\)), and the total energy dissipation \(E_{\rm C}\) on the damage index \(D_{k}\) is discussed. Figure 1 shows the influence of the stirrup ratio \(\rho_{\rm sw}\), the longitudinal reinforcement conditions, and the total energy dissipation \(E_{\rm C}\) on the damage index \(D_{k}\), where the height of the column member \(H\) is 3000 mm. When the longitudinal reinforcement is relatively small (\(\rho_{\rm s,small}\)), \(F_{\rm y}\) and \(u_{\rm y}\) are taken as 51.50 kN and 11.07 mm, respectively, and when the longitudinal reinforcement is relatively large (\(\rho_{\rm s,large}\)), \(F_{\rm y}\) and \(u_{\rm y}\) are taken as 89.02 kN and 16.66 mm, respectively [13]. The x-coordinate is the stirrup ratio \(\rho_{\rm sw}\), and its range is 0.1% to 3%. The y-coordinate is the total energy dissipation \(E_{\rm C}\), and its range is 0 to 100,000 kN\(\cdot\)mm. The z-coordinate is the damage index \(D_{k}\), and its range is 0 to 1. Figure 1: The influence of the stirrup ratio \(\rho_{\rm sw}\), the longitudinal reinforcement conditions, and the total energy dissipation \(E_{\rm C}\) on the damage index \(D_{k}\). It can be found from Figure 1, with the increase in the total energy dissipation \(E_{\rm C}\), the damage index \(D_{k}\) increased monotonically from 0 to 1, that is, a one-to-one correspondence between the total energy dissipation \(E_{\rm C}\) and the damage index \(D_{k}\) can be established. The increase in the stirrup ratio \(\rho_{\rm sv}\) can slow down the damage development, and the slowing process is initially fast and then slows down. The increase in longitudinal reinforcement can significantly decrease the damage. In order to further understand the quantitative description of the damage reduction value caused by the increase in the number of longitudinal reinforcement, the longitudinal reinforcement ratios of 0.587%, 0.971%, and 1.198% were used to study the quantitative relationship.