## 1 Introduction  
In the performance-based design process, the structural safety assessment targets should be quantified according to the damage index [1; 2; 3]. A reasonable damage index not only reflects the damage caused by the three elements of the earthquake (amplitude, frequency spectrum, and earthquake duration) [4; 5; 6; 7], but also establishes a corresponding relationship with the construction member parameters to improve the mechanical properties of the structure [8]. The existing damage index can be classified into three aspects: (a) degradation-based damage index; (b) deformation-based damage index; and (c) energy-based damage index. The degradation-based damage index describes structural damage by using the changes of structural characteristics, such as stiffness [9], frequency [10], and strength [11]. Although the degradation-based damage index does not directly include the three elements of the earthquake, it reflects the structural damage caused by the three elements. Therefore, the degradation-based damage index is applied to describe the structural damage. However, the deficiency of the degradation-based damage index is that it cannot establish a corresponding relationship with the construction member parameters. The deformation-based damage index considers that the structure damage is caused by the maximum displacement deformation; the ratio of the maximum displacement deformation to the limit displacement deformation is used to define the damage [12]. The deformation-based damage index takes the deformation demand as the design target, which can directly understand the deformation state of the structure under the earthquake actions, but the disadvantage is that the impact of earthquake duration has not been effectively considered. There is a lack of correlation between the maximum displacement deformation of the structure and the earthquake duration [13], but there is a good correlation between the earthquake duration effect and the accumulated hysteretic energy of the structure under the earthquake actions [14]. With the intensification of the decay process, the correlation between the earthquake duration effect and the accumulated hysteretic energy is higher [15]. Therefore, the accumulated hysteretic energy is widely used to express the earthquake duration effect [16]. The energy-based damage index not only includes the construction member parameters, but also reflects the effect of the earthquake duration (accumulated hysteretic energy), which can comprehensively reflect the structural damage caused by the earthquake actions. Therefore, the energy-based damage index is the main research focus. In 1985, the damage index with displacement deformation term and accumulated hysteretic energy term was established by [PERSON] and [PERSON] [17,18]. This intuitive expression (a simplified linear relationship of displacement deformation term and accumulated hysteretic energy term) is easy to accept, and the Park-Ang damage index establishes a quantitative relationship with construction member parameters (yield load, reinforcement conditions, and ductility conditions), which makes it possible to reverse design based on damage targets. Therefore, this damage index has been used to describe damage by many scholars, but the simplification in form cannot truly reflect the damage mechanism. [PERSON] and [PERSON] et al. [19,20] found that the influence of ultimate deformation capacity of RC columns was mainly due to the plastic cumulative damage of longitudinal reinforcements and stirrups. [PERSON] et al. [21] found that the ultimate deformation capacity was related to the accumulated hysteretic energy, while [PERSON] et al. [15] found that the earlier the maximum displacement occurs during the loading process, the greater the accumulated hysteretic energy damage of the structure. It can be seen that there is a coupling relationship between the accumulated hysteretic energy term and the displacement deformation term.