## 2 Seismic Damage Indexes for RC Buildings
[PERSON] and [PERSON] (2005) [14] defined a general damage index as the ratio between the initial and the ultimate resistance capacity of RC elements. \[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]3  
where \(\beta_{1}=\beta_{3}=0.1\), \(\beta_{2}=2.4\), \(\vartheta_{m}\) is the maximum rotation, \(\vartheta_{u}\) is the ultimate rotation and \(E_{u}^{*}\) is a parameter that considers the dissipated energy. [PERSON] and [PERSON] (2009) [15] proposed the following index considering using earthquake records which did not result in building collapse:  
\[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]4  
in which the energy absorbed in elastic system is  
\[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]5  
where, for a single degree of freedom (SDOF), \(E_{h}\) is the hysteretic energy, \(k_{\theta}\) is the elastic energy absorbed by SDOF system while it achieves \(\theta=\pm 1\), \(\theta_{c}\) is the maximum drift ratio. [PERSON] et al. (2011) [16] suggested a damage index based on displacement history, hysteretic energy and plastic deformation, modifying the existing Park and Ang model [7] to enhance damage assessment accuracy. \[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]6  
in which  
\[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]7  
where \(d_{u}\) is the ultimate displacement under monotonic loading, \(d_{m}\) is the maximum displacement experienced during the earthquake, \(d_{y}\) is the yield displacement, \(\mu_{c}\) (\(\mu_{c}=d_{c,m}/d_{y}\)) is the cyclic ductility depending on the maximum cyclic displacement request \(d_{c,m}\) pending a ground movement [17], \(\mu_{ac}\) is the accumulative ductility depending on the total inelastic displacements, \(\beta\) is the coefficient introduced by Park and Ang [7] defined in Equation (6), \(E_{h}\) is the hysteretic energy. [PERSON] et al. (2016) [18] developed another implementation of the existing Park and Ang index [7], proposing a three-dimensional damage index to account for the biaxial effect due to earthquake events. \[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]8  
where \(\beta\) is the weight factor of energy dissipation, \(\mu_{\theta,m}^{-}\) is the maximum normalized rotation ductility factor, \(dE_{p}\) is the plastic energy dissipation increment, \(E_{mon}\) is the value of plastic energy dissipation due to the effect of monotonic loading at the failure limit of the structure member, \(\alpha_{m}\) is the correction of the energy dissipation term. The aforementioned damage indexes can be classified based on the parameters involved in their formulation, as listed in Table 1, or based on the required knowledge of the structure as represented in \[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]9  
More precisely, the level of knowledge in Table 2 refers to parameters obtained from dynamic identification methods or on construction details, materials and dimensions required to define ductility, displacements and ultimate rotations. To ensure a meaningful comparison, integration into decision-making processes and consistent model calibration, it is important that all index values are within the same range. A review of the literature shows that each index typically operates within its own range, so rescaling is necessary for consistency. In this context, the previous indexes have been rescaled so that they lie between 0, for no damage, and 1, for collapse. The rescaled damage indexes in this section are denoted as \(DI_{*}\).