# Hilbert-Huang-Transform-Based Seismic Intensity Measures for Rocking Response Assessment
## 3 Results  
### Fragility Analysis  
For the scope of this study, a set of 100 natural strong ground motion records [79] has been used to perform the time history analysis of the examined structural system. The selected excitations cover a wide range of intensities and, consequently, structural damage. The parameters of the PSDMs using the conventional and the proposed HHT-based IMs are presented in Figures 3 and 4, respectively. Specifically, the coefficient of determination \(\mathrm{R_{r}}^{2}\), the linear regression coefficient \(\mathrm{b}\), the standard deviation \(\mathrm{\beta_{D}}\mathrm{{}_{IM}}\), and the parameter \(\zeta\) are reported. The parameters provided in every subfigure are adequately normalized to the value of the best-performing IM to visualize the relative performance between the examined IMs. A more optimal IM is characterized by higher normalized values given a performance criterion. Figure 3: Performance evaluation of conventional IMs in developing PSDM. Regarding the conventional IMs, Housner's Intensity (I\({}_{\rm H}\)), followed by PGV, are the most optimal IMs to develop PSDMs considering every performance criterion. Although PGA level determines rocking initiation, it moderately affects the rocking response. It is evident that PGA governs the dynamic response of free-standing rigid blocks of small size, while larger-size blocks are mainly affected by velocity-based IMs [38]. Considering the dimensions of the examined structural system and the equivalence between rocking frames and single blocks, the moderate performance of PGA was expected. It should be noted that the height of the equivalent single block of the examined rocking frame is 2H\({}_{\rm eq}\) = 14.6 m. Conventional IMs present a varying perforce, while in the case of HHT-based IMs, A\({}_{\rm HHT}\) outperforms every criterion examined. It can be noted that the performance of A\({}_{\rm HHT}\) is similar to the most optimal conventional IMs. Especially in terms of the efficiency criterion, both IMs present almost identical values. Except for the finite rocking response, the estimation of the collapse probability is of particular interest. To this end, the standard deviation \(\sigma\) obtained by the MLE is presented in Figures 5 and 6 for the conventional and the examined HHT-based IMs, respectively. Interestingly, although the I\({}_{\rm H}\) and A\({}_{\rm HHT}\) outperform in predicting the finite rocking response, this is not the case in rocking overturn. Based on the standard deviation criterion, I\({}_{\rm F}\) and A\({}_{\rm Pos,HHT}\) are the proper IMs to estimate the probability of collapse. It has to be mentioned that the optimal IMs in the case of rocking overturn estimation present a satisfactory performance in finite rocking prediction and vice versa. Specifically, regarding the conventional IMs, I\({}_{\rm H}\) and PGV, which are the most optimal IMs regarding the finite rocking estimation, are the second and the fifth most optimal IMs in the case of rocking overturn estimation, respectively. Moreover, I\({}_{\rm F}\), which is the best IM in rocking overturn estimation, is the third most optimal in the case of finite rocking prediction. Considering the proposed IMs, A\({}_{\rm Pos,HHT}\), which outperforms in the case of finite rocking response estimation, is clearly the second most optimal for estimating the rocking overturn. On the other hand, A\({}_{\rm Pos,HHT}\), given its performance in terms of proficiency criterion, is the second most optimal IM to be used for the PSDM. Figure 4: Performance evaluation of the proposed HHT-based IMs in developing PSDM. Despite the performance criteria that an optimal IM should fulfil, regarding its interdependency with the seismic demands, hazard computability is of significant importance [39]. The combination of fragility and hazard analysis consists of the components of risk assessment.