## 2 Materials and Methods  
Curvature evaluation method is based on the evaluation of modal curvature variation over time, [1]. The method uses a narrow-band-time-variable filter [10] to isolate the fundamental modal shape over time, as well as during the phase of maximum nonlinearity (strong motion phase). This makes it possible to estimate and localize the occurring damage during the seismic event by evaluating the modal curvature evolution over time. In other words, this filter is able to extract from a nonstationary and/or nonlinear signal only the energy content of interest, preserving both the amplitude and phase in the region of interest for each time instant [8; 10]. The band-variable filter is based on the Stockwell Transform [43], a time-frequency representation of a time-series that overcomes the limitations due to the assumptions of stationary behavior. Under the assumption of linear behavior over a short time interval, the S-transform allows analysis of the nonlinear behavior of any dynamic system, also during strong motion phases [44]. The Stockwell Transform operates simultaneously in both the time and frequency domains. The novelty of the approach presented in this present paper consists of the evaluation of the mode curvature difference. Firstly, the modal shape variation has been evaluated between different stories before and after the strong shaking phases and, secondly, at different time instants. The Procedure for Damage Localization (PDL) involves several steps:  
* Identification of the fundamental frequency of the structure in the time-frequency domain by analyzing the top floor's relative accelerometric recordings (difference between top and bottom instant acceleration values, see Figure 1);
* definition of a filtering matrix, using the band-pass-time-variable filter, fitting the evolution over-time of the fundamental frequency of the monitored structure (evaluated in the previous step);
* convolution of the filtering matrix with the Stockwell Transform of the signal recorded at each floor, along the same direction for all floors;
* evaluation of the variation of the fundamental mode shape and related mode curvature over time (time instants A and B, see Figure 1);
* computation of the mode curvature differences among floors at different time steps (see Equations (2) and (3));
* selection of two different time instants (one immediately before the earthquake and one corresponding to the minimum fundamental frequency) and evaluation of the mode curvature difference between them (time instants A and B, see Figure 1). Figure 1: **Top**: Top floor relative accelerometric recording. **Bottom**: An example of the Normalized S-Transform of the signals associated with the first mode of vibration of the monitored structure and selection of the time instants (**A**) and (**B**). In order to completely automatize the procedure, the Short Time Impulse Response Function (STIRF) procedure, as described in [45], has been used to filter the matrix selection. Using a combined approach based on the FT and on the seismic interferometric analysis, it can be a useful tool for the automatic evaluation of both fundamental frequency and related equivalent viscous damping factors of nonlinear structures. Modal curvature can be calculated by modeling a framed structure as a beam:  
\[W^{\prime\prime}(v,t)=-\frac{M(v,t)}{EI(v,t)}=\frac{W(v-h,t)-2W(h,t)+W(v+h,t)}{ h^{2}} \tag{1}\]  
where \(v\) is the position along the height of the monitored structure, \(h\) is a finite increment of \(v\), \(M(v,t)\) is the bending moment, \(EI(v,t)\) is the bending stiffness, \(W_{(v,t)}\) is the displacement, and \(W^{\prime\prime}_{(v,t)}\) is the related curvature. Considering the fundamental mode shape of a framed structure as a beam displacement, it is possible to localize structural damage by analyzing the singularity on the curvature of the fundamental mode shape [46]. In the first version of the method, the damage feature was evaluated by only considering the curvature variation over time.