## 3 Results  
### Simulated Case Study  
#### 3.1.1 Model Simulations  
In this section, we exploit a simulated toy example, namely a nonlinear spring-mass model, to offer a first comparison of the DSFs presented in Section 2.1 in terms of their efficacy in detecting the onset of nonlinearity and residual damage as well as localizing and quantifying nonlinearity/damage within elements of a building structure. The model comprises five DOFs, emulating a five-story shear building. The DOFs are connected by means of hysteretic Bouc-Wen springs [85], with the nonlinear time-history analyses carried out in the Opensees software [86]. The masses that are lumped into each of the DOFs are \(\mathbf{M}=\{117.7,124.2,103.7,119.3,116.6\}\) tons, and the linear and nonlinear properties of the springs connecting the DOFs are summarized in Table 2. In addition, the following properties define the Bouc-Wen hysteretic model of the springs: the exponent regulating the transition between linear and nonlinear branch: \(n=5\), stiffness degradation \(\delta\eta=3.610^{-5}\), strength degradation \(\delta\
u=10^{-3}\), and hardening of 2%. In addition, a Rayleigh-type damping (with a damping coefficient of \(\zeta=3\%\) for the first two modes) is adopted. A schematic representation of the model and the behavior of the link 0-1 are shown in Figure 3.  
Given the natural variability of GM signals, changes in spectral content and signal duration may undermine the capacity of DSFs to separate linear from nonlinear behavior. In an effort to cover a realistic range of GMs, 21 ground motions from historical earthquakes are scaled to multiple amplitudes for generating a structural response dataset with the adopted nonlinear model. An overview of the earthquake GMs is shown in Figure 4, with the details of the utilized historical earthquake records summarized in Table 11, which can be found in the Appendix. As the aim of the analysis is to assess the robustness of DSFs to identify earthquake-induced damage, the GMs are deliberately chosen to represent several faulting styles and source-to-site distances, yet their scaling from peak-ground-acceleration (PGA) values of 0.5 m/s\({}^{2}\) to 13.5 m/s\({}^{2}\) may not always reflect realistic earthquake GMs. The variability of the respective frequency spectra is shown in Figure 4 (top right), together \begin{table}
\begin{tabular}{c c c} \hline
**Link** & **Stiffness (kN/mm)** & **Yield Force (kN)** \\ \hline
0-1 & 270 & 500 \\
1-2 & 250 & 433 \\
2-3 & 230 & 350 \\
3-4 & 210 & 300 \\
4-5 & 190 & 225 \\ \hline \end{tabular}
\end{table} 
Table 2: Definition of the nonlinear link properties of the simulated mass-spring model. with the scatter in two intensity measures characterising the earthquakes (Figure 4, bottom right): first, the duration during which 50% of the GM energy is accumulated (\(D_{25-75}\)), and secondly, the ratio between the spectral acceleration at the fundamental frequency (\(S_{a}(T_{1})\)) of the modeled structure and the PGA. With the help of the nonlinear time-history model simulations, the DSFs, described in Section 2.1, are reviewed with respect to the capacity to detect and quantify nonlinearity/damage (Section 3.1.2), to locate damage (Section 3.1.3), and in terms of their sensitivity to noise (Section 3.1.5). Taking advantage of the simulated model, three EDPs (namely maximum transient roof drift ratio, hysteretic work, and residual stiffness drop), and one damage index (DI), namely the \(DI_{Park\&Ang}\) are used for cross-validation. It should, however, be stressed that these EDPs cannot be directly measured or straightforwardly quantified at the required resolution for a reasonable cost, using current sensing solutions. Nevertheless, these EDPs are widely used in defining seismic damage sustained by a structure and its influence on residual performance. Figure 4: Overview of the GM signals used as input for the conducted nonlinear time-history analyses.