## 2 Seismic Damage Indexes for RC Buildings  
The growing interest of the scientific community in damage detection and control in buildings has led to the definition of a variety of damage indexes that can express the condition of structural elements and components or the entire building after a seismic event by a scalar measure [1]. Since the 1970s, systematic attempts have been made to quantitatively estimate the local or global extent of seismic damage to structural systems. In the past, the damage indexes for reinforced concrete (RC) buildings proposed in the literature have been extensively discussed and studied [2; 3]. This section presents the main damage indexes that have been developed to assess local damage to RC structural members, as well as a global index suitable for assessing the overall damage of the whole building. As for the local damage in a single element, [PERSON] and [PERSON] (1977) [4] introduced the following index:  
\[DI_{\rm L-S}=\frac{k_{0}}{k_{m}} \tag{1}\]  
where \(k_{0}\) and \(k_{m}\) are the elastic and the secant stiffness at the maximum displacement, respectively. Assuming a monotonically increasing load path, this index starts at a value of 1.0 in the undamaged state and gradually increases until it reaches a maximum value of 3.5 in the final state. In the same year, a flexural damage index was introduced by [PERSON] et al. (1977) [5]:  
\[DI_{\rm FDR}=\frac{\frac{\phi_{m}}{M_{m}}-\frac{\phi_{y}}{M_{y}}}{\frac{\phi_{x }}{M_{u}}-\frac{\phi_{y}}{M_{y}}} \tag{2}\]  
where \(\phi_{m}\), \(\phi_{y}\) and \(\phi_{u}\) are the maximum, yield and ultimate curvature, respectively, and \(M_{m}\), \(M_{y}\) and \(M_{u}\) are the maximum, yield and ultimate bending moment, respectively. The index is designed to provide a value of zero in the absence of damage and to gradually increase to a value of 1.0 when the structure reaches its final state. [PERSON] and [PERSON] (1982) [6] introduced the first damage index based on the dissipated energy, which is particularly relevant for cyclic loads, such as those experienced during seismic events. This index is calculated for each loading cycle:  
\[DI_{\rm B-V}=\sqrt{\left(\frac{d_{m}}{d_{y}-1}\right)^{2}+\left[\left(\frac{2 E_{h}}{F_{y}d_{y}}\right)^{0.38}\right]^{2}} \tag{3}\]  
where \(d_{m}\) and \(d_{y}\) are the maximum and the yield displacement, respetively, \(F_{y}\) is the yield force and \(E_{h}\) is the dissipated energy. This damage index is calculated for each cycle and is defined as the maximum value among the indexes determined. It assumes values greater than zero even before the yield point and exceeds 9.0 under failure conditions. [PERSON] and [PERSON] (1984) [7] defined the following index based on the dissipated energy:  
\[DI_{\rm H-W}=\sum_{i=1}^{n}\frac{k_{i}{d_{i}}^{2}}{k_{e}{d_{y}}^{2}} \tag{4}\]  
where \(k_{e}\) and \(k_{i}\) are the element's initial elastic flexural stiffness and the element's flexural stiffness at the i-th cycle, respectively, \(d_{y}\) and \(d_{i}\) are the element's yield displacement and the element's displacement at the i-th cycle, respectively, and \(n\) is the number of cycles. As can be seen from its mathematical formulation, the index represents the cumulative damage across all cycles. This results in values that are already greater than zero before the yield point and exceed 4.0 in the final state of the structural element. The index proposed by [PERSON] (1985) [8] is  
\[DI_{\rm P-A}=\frac{d_{m}}{d_{u}}+\frac{\beta}{F_{y}d_{u}}\int dE \tag{5}\]  
in which  
\[\beta=\left(-0.447+0.073\frac{L}{d}+0.24n+0.314\rho_{t}\right)\cdot 0.7^{\rho_{ \infty}} \tag{6}\]where \(d_{u}\) is the ultimate displacement under monotonic loading, \(d_{m}\) is the maximum displacement experienced during the earthquake, \(dE\) is the differential dissipated energy, \(L\) is the structural element length, \(d\) is the effective depth of the cross-section, \(n\) is the normalized axial load, \(\rho_{t}\) is the percentage of longitudinal reinforcement and \(\rho_{\omega}\) is the confinement ratio.