## 3 Results
### Simulated Case Study
#### 3.1.4 Tracking DSFs in the Time Domain to Detect Nonlinearity  
The detection of damage has been found to yield good results when deriving DSFs from AVs after a damaging earthquake, which has induced a residual shift in the structural stiffness. The performance of OMA-based DSFs thus strongly depends on the stiffness degradation that results from cyclic loading, as indicated by the high correlation between the _PAAC_ and the residual stiffness drop (Figure 7 right). Therefore, the capacity to detect damage is re-evaluated for two new models, for which the stiffness degradation parameter of the Bouc-Wen spring formulation is either reduced from \(\delta\eta=3.610^{-5}\) to \(\delta\eta=10^{-5}\) or increased to \(\delta\eta=10^{-4}\), respectively. When reducing the stiffness degradation effect, the OMA-based DSFs (\(\Delta f_{n}\), \(MAC\), \(\Delta MSC\), and _PAAC_) fail to detect damage, given the very low residual effect of damage in this case. Given the reduced effect of nonlinearity on residual damage, DSFs are evaluated during the GM with short data windows of 4 s, with an overlap of 3.5 s. Thus, temporary nonlinearities may be picked up. The results for one earthquake instance (Montenegro GM with PGA = 5 m/s\({}^{2}\)) are shown in Figure 13. The left column contains the simulated results with low stiffness degradation \(\delta\eta=10^{-5}\), the center column contains the simulated results for high stiffness degradation \(\delta\eta=10^{-4}\), and the right column contains the results for PGA = 1.5 m/s\({}^{2}\), during which the structure remains in the linear elastic range. Each line contains the Mahalonobis distance (see Equation (17)), with respect to the healthy reference distribution, for five DSFs, which can be formulated for short time windows (see Table 1): _TAC_, _Tcnt_, _MER_, _KPRX_, and _WLC_. The _TAC_, shown in the first row of Figure 13, clearly identifies the onset of nonlinearity for both models, with and without stiffness degradation, and correctly identifies the residual loss of stiffness, when this occurs (second column of Figure 13), or the absence of a residual effect, when the structure returns to the initial stiffness (first column in Figure 13). Such a capacity to detect temporary stiffness loss and residual stiffness loss is essential when separating failure modes. For instance, unreinforced masonry walls that fail in bending show rocking behavior with minor residual stiffness loss while showing residual stiffness loss when failing due to shear. In a similar manner to the _TAC_, the _Tcnt_ identifies temporary and residual stiffness loss (see Figure 13 second row). Tracking DSFs over time also depends on the window-size, as shorter windows are more noisy and longer time windows less sensitive to temporary nonlinearity (see Appendix B Figure A1). In this case, a window length of 4 s is chosen. The _MER_, reported in the third row of Figure 13, shows potential to scale with instantaneous stiffness. However, the strong dependence of this DSF on the spectral content of the input motion undermines the detection of damage, as the threshold for detection is not crossed. A similar observation can be made for the _KPRX_ (fourth row of Figure 13), which scales better with the residual stiffness than with the instantaneous stiffness. Finally, the _WLC_, shown in the last row of Figure 13, is very sensitive to temporary stiffness  
Figure 12: Correlation coefficient between floorwise evaluation of DSFs and a damage index (\(DI_{PA}\)). Except for \(TS\) and \(CE\), which evaluate the energy accumulation over time, a good correlation between the locally evaluated DSFs and the floorwise damage is observed. reduction, and it also contains information about the residual stiffness drop. However, this high sensitivity makes it prone to false detection of damage for short time windows, as can be seen for the linear model (last column of Figure 13). The four other DSFs do not detect damage for the linear model.