## 2 Methodology
These graphs express the relationship between the selected _IM_ and the debris heap width resulting from building damage. As shown in Figure 3, using the assigned vulnerability functions, it is possible to define the _IM_ value in which a certain performance level is reached, thanks to the _DF_ values provided by the Italian guidelines for the classification of the seismic risk of buildings [13; 14], the so-called SISMABONUS. These \(DF\) values are shown in Table 1. They are valid for typical RC and masonry buildings with residential use, and can conventionally be used for buildings with different uses or structural materials. According to the current Italian seismic code [15] (NTC 2018 hereafter), in the event of an earthquake, any building meeting the operative limit state (OLS) is considered to have no damage to its structural elements and only minor damage to its non-structural elements. Therefore, for this damage state, the building should not suffer significant damage, so as to produce appreciable debris on the road. Consequently, consistent with Table 1, for each \(IM\) in which \(DF\leq 7\%\), the structure retains \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Performance Level:** & IDLS [2] & OLS [3] & DLS [4] & SLS [5] & CLS [6] & RLS [7] \\ \hline
**Damage Factor (DF) 1:** & 0\% & 7\% & 15\% & 50\% & 80\% & 100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Threshold values provided by the Italian guidelines [9; 10]. Figure 3: Construction of debris graphs for (**a**) masonry buildings and (**b**) reinforced concrete (RC) buildings. peak ground acceleration (PGA) is peak ground acceleration. its footprint area (\(A_{f}\)), and the debris width can be assumed equal to zero (\(w_{d}=0\)). For damage beyond the OLS, the structure instead has a different behavior. For masonry buildings (Figure 3a), it is assumed that the maximum debris area (\(A_{d,max}\)) is reached in the collapse limit state (CLS). Such debris area encompasses the building footprint area and the relative debris heap on the road. Meeting this limit state, \(A_{d,max}\) remains constant, while the angle of inclination of the debris heap (therefore the debris volume \(V_{b}\)) increases up to the maximum \(DF\) value (100%). In the transition from OLS to CLS, it is assumed that the \(A_{d}\) grows linearly with the increasing \(IM\). In order to evaluate the maximum debris area in the SLC, it is possible to use the methodology developed by [PERSON] et al. [16]. Based on the geometric and topological characteristics of the masonry buildings of central Italy (building size, wall thickness, arrangement of openings, indoor spaces distribution, etc. ), in [16], the debris area of collapsed buildings was determined by using an amplification factor (\(\varepsilon\)) of the building footprint area, which depended on the geometric characteristics of the structure. In particular, \(A_{d,max}\) and \(\varepsilon\) can be calculated as follows:  
\[A_{d,max}=A_{f}\cdot\varepsilon^{2} \tag{2}\]  
\[\varepsilon=1.228+0.07869\cdot\left(\frac{a}{b}\right)+0.05626\cdot\left(\frac {A_{f}}{V_{b}}\cdot\frac{h_{b}^{2}}{a}\right) \tag{3}\]  
where \(a\), \(b\), \(h_{b}\), \(A_{f}\) = \(a\cdot b\), and \(V_{b}\) = \(a\cdot b\cdot h_{b}\) can be assumed to be the geometric characteristics of the structural aggregates or of the interfering isolated structural unit (see step 1). The debris area can have different distributions around the damaged building. To take into account the uncertainty of the possible distributions, it is possible to use the simplified collapse models proposed by some authors [17; 18]. These models correspond to a collapsed building in one or two directions. Consequently, an equation that estimates the debris width (\(w_{d}\)) can be defined for each debris distribution.