## 2 Theoretical Concepts
### Damage Quantification  
Worldwide there is an increasing concern regarding the resilience of the existing buildings in order to avoid major damage [18] or even collapse [19] during next significant seismic events. Therefore, damage quantification is a central subject in current structural design. In recent papers, ref. [20] investigated the cumulative damage of a 20-story high-rise steel building equipped with buckling-restrained braces subjected to earthquake and wind events using [PERSON]'s damage rule, ref. [21] introduced a multiple-input convolutions neural network model in order to estimate the seismic damage buildings. Furthermore, ref. [22] studied the post seismic behavior of a reinforced concrete building which presented structural damage due to flexural and axial actions in one of its walls on the first floor. Several solutions to improve the seismic behavior were discussed. At the same time, ref. [23] proposed a novel definition of damage states for reinforced concrete members to be used for structural vulnerability assessment. Moreover, ref. [24] utilized the seismic fragility method based on incremental dynamic analysis in order to evaluate the damage levels of high-rise structures under long-period ground motions. Furthermore, some theoretical concepts used in this paper for the implementation of structural damage assessment are presented. In order to quantify the damage induced by earthquake excitation and sustained by the structural elements through plastic deformations, a damage index approach can be used. A well-documented and highly used damaged index was given in [25] and later in [26]. The general relation for the damage index of a plastic hinge model was given by Equation (26). \[DI=\frac{\theta_{m}-\theta_{r}}{\theta_{u}-\theta_{r}}+\frac{\beta}{M_{g}\theta _{u}}Eh \tag{26}\]  
where \(\theta_{m}\) is the maximum rotation obtained during loading history, \(\theta_{r}\) is the recoverable rotation at unloading, \(\theta_{u}\) is the ultimate rotation capacity of the cross-section, \(\beta\) is a strength degrading parameter, \(M_{y}\) is the yield moment of cross-section, and \(Eh\) is the dissipated hysteresis energy. Having been introduced in the above section, it is important to say that the hysteresis model further utilized in this paper is based on the dependence between plastic rotation and bending moment. Therefore, the damage index evolution for a plastic hinge is given by Equation (27)  
\[DI_{PH_{i}}(t)=\frac{\theta_{m}^{{}^{\prime\prime}}(t)}{\theta_{u}}+\frac{ \beta}{M_{y}\theta_{u}}Eh_{PH_{i}}(t) \tag{27}\]  
where \(Eh_{PH_{i}}(t)\) is the cumulative dissipated hysteresis energy up to time step \(t\) for the \(i\)th plastic hinge. Furthermore, in order to quantify the damage record at each of the stories the following expression can be used. \[DI_{j,storey}(t)=\sum_{i=1}^{n}\lambda_{i}^{j}(t)\;DI_{PH_{i}}(t)\quad i= \overline{1,n},\quad j=\overline{1,p} \tag{28}\]  
with \(n\) being the number of plastic hinges corresponding to \(j\)th story while \(p\) represents the number of stories. Moreover \(\lambda_{i}^{j}(t)\) is the normalized dissipated hysteresis energy of plastic hinge \(i\)th corresponding to \(j\)th story with respect to the dissipated hysteresis energy of the \(j\)th story (Equation (29)). \[\lambda_{i}^{j}(t)=\frac{Eh_{PH_{i}}^{j,storey}(t)}{\sum_{i=1}^{n}Eh_{PH_{i}}^ {j,storey}(t_{end,curr})}\quad i=\overline{1,n},\quad j=\overline{1,p} \tag{29}\]  
where \(Eh_{PH_{i}}^{j,storey}(t)\) is the cumulative dissipated energy up to time step \(t\) for the \(i\)th plastic hinge corresponding to \(j\)th story, and \(\sum_{i=1}^{n}Eh_{PH_{i}}^{j,storey}(t_{end,curr})\) is the maximum (or current) value attained during loading history as a sum of all the cumulative dissipated hysteresis energy by the plastic hinges corresponding to \(j\)th story. In a similar manner, the damage index for the whole building can be defined by Equation (30).