# Hilbert-Huang-Transform-Based Seismic Intensity Measures for Rocking Response Assessment
## 3 Results
### Fragility Analysis
As long as the modern PBEE implies the definition of performance criteria and the estimation of each annual rate of exceedance (risk estimation), the adoption of a hazard-computable IM is necessary [29]. From this point of view, only a limited number of IMs can be deployed in fragility analysis in order to be used in risk assessment. Thus, hazard maps should be constructed in terms of IMs that optimally describe the rocking response. In order to reduce the seismic demand uncertainty, bivariate IMs are also examined. Specifically, a thorough investigation has been conducted in which all possible couples of IMs are examined. In Table 3, the parameters of the PSDM (Equation (31)) and the rocking overturn fragility (Equations (30) and (32)) of the five most optimal combinations are listed. In four out of five cases, combining the HHT-based parameter A\({}_{\text{HHT}}\) with a conventional IM resulted in enhanced performance compared to any univariate IM. It has to be noted that the combination of a frequency parameter, in our case, T\({}_{\text{m}}\), with an acceleration amplitude one, such as PGA, has emerged as an optimal bivariate IM in past studies [31,32]. The parameter \(\lambda\) denotes the influence of each component of the bivariate IM on the collapse probability regarding Equation (32). It should be mentioned that a value \(\lambda=1\) implies an equal effect by both components. In special cases where \(\lambda=0\) or \(\lambda\)\(\rightarrow\)\(\infty\), the probability of collapse is defined only by the second and the first component of the bivariate IM, respectively. Based on the values listed in Table 3, among the most optimal vector-valued IMs, the overturning fragility primarily depends on the second component. In most cases, the rocking overturn is governed by frequency-based IMs (T\({}_{\text{m}}\), PGV/PGA), which are known for their moderate to high correlation with the rocking response. On the other hand, the regression slope parameters b\({}_{1}\) and b\({}_{2}\) express the grade of direct correlation of each IM component on the finite rocking response estimation. Interestingly, in four out of five most optimal bivariate IMs, their first components present higher interdependencies with the developed finite rocking rotations. Figure 5: Performance evaluation of conventional IMs in predicting the rocking overturn. Figure 6: Performance evaluation of proposed HHT-based IMs in predicting the rocking overturn. Subsequently, the fragility surfaces of the bivariate IMs listed in Table 3 are provided in Figure 7. Obviously, in the first limit state (LS\({}_{\text{I}}\)), which corresponds to a stable rocking response with negligible overturning potential, the inclination of the fragility surfaces is mainly affected by the first component of each bivariate IM. Meanwhile, in the case of the third limit state (LS\({}_{\text{III}}\)), the inclination of the fragility curves is driven by the second IM component. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**IMs**} & \multicolumn{4}{c}{**PSDM**} & \multicolumn{4}{c}{**MLE**} \\  & **a** & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{\beta_{D\,\text{IM}}}\) & \(\mathbf{\mu}\) & \(\mathbf{\sigma}\) & \(\mathbf{\lambda}\) \\ \hline
**PGA—T\({}_{\text{m}}\)** & 0.58 & 2.28 & 1.87 & 1.49 & \(-\)0.06 & 0.19 & 0.49 \\
**A\({}_{\text{HHT}}\)—PGD** & 5.54 & 1.37 & 0.53 & 1.48 & \(-\)1.98 & 0.29 & 0.64 \\
**A\({}_{\text{HHT}}\)—T\({}_{\text{m}}\)** & 5.21 & 1.38 & 0.92 & 1.48 & \(-\)1.14 & 0.22 & 0.40 \\
**A\({}_{\text{HHT}}\)—PGV/PGA** & 6.12 & 1.41 & 0.59 & 1.51 & \(-\)2.48 & 0.27 & 0.59 \\
**A\({}_{\text{HHT}}\)—I\({}_{\text{F}}\)** & 2.97 & 1.01 & 1.19 & 1.44 & 0.26 & 0.32 & 0.13 \\ \hline \end{tabular}
\end{table} 
Table 3: Proposed bivariate fragility surface parameters. Figure 7: Fragility surfaces using vector-value IMs.