# Fuzzy Multivariate Regression Models for Seismic Assessment of Rocking Structures
## 2 Methods
### Seismic Intensity Measures  
The seismic excitation records are considered to be described using 18 intensity measures to assess the maximum rocking response of the examined structures. Specifically, the following parameters were considered: peak ground acceleration (PGA), peak ground velocity (PGV) and peak ground displacement (PGD), the mean period (Tm), the ratios \(\mathrm{PGV}/\mathrm{PGA}\) and \(\mathrm{PGD}/\mathrm{PGV}\), acceleration's root mean square (ARMS), velocity's root mean square (VRMS), displacement's root mean square (DRMS), characteristic length scale of acceleration (\(\mathrm{L_{Ca}}\)) and velocity (\(\mathrm{L_{cv}}\)), Arias intensity (\(\mathrm{I_{A}}\)), characteristic intensity (\(\mathrm{I_{C}}\)), Fajfar index (\(\mathrm{I_{F}}\)), specific energy density (SED), significant duration (\(\mathrm{t_{sig}}\)), and cumulative absolute velocity (CAV) and displacement (CAD). To result in a robust predictive model, dimensionless, structure-specific IMs are calculated. To this end, the gravitational acceleration g, the frequency parameter p, and the slenderness \(\alpha\) of each structural system are appropriately used according to [PERSON] et al. [18]. Table 1 presents the dimensionless IMs along with their respective definitions. The IMs are calculated using the time histories of acceleration (a(t)), velocity (v(t)), and displacement (d(t)). The seismic excitation's total duration is denoted as t\({}_{\text{tot}}\). The Fourier amplitudes of the accelerograms are represented by C\({}_{i}\), and the corresponding Fourier transform frequencies between 0.25 and 20 Hz are denoted as f\({}_{i}\).  
\begin{table}
\begin{tabular}{c c c c} \hline
**i** & **IM\({}_{i}\)** & **Description** & \\ \hline
1 & PGA/(g tan(\(\alpha\))) & PGA = max( \(\mid\alpha_{\text{g}}(\text{t})\mid\)) & [46] \\
2 & p PGV/(g tan(\(\alpha\))) & PGV = max( \(\mid\
u_{\text{g}}(\text{t})\mid\)) & [46] \\
3 & p\({}^{2}\) PGD/(g tan(\(\alpha\))) & PGD = max( \(\mid\) d\({}_{\text{g}}(\text{t})\mid\)) & [47] \\
4 & p Tm & Tm = \(\sum\)(C\({}_{i}\)\({}^{2}\)/f\({}_{i}\))/\(\sum\)C\({}_{i}\)\({}^{2}\) & [47] \\
5 & p PGV/ PGA & - & [48] \\
6 & p PGD/PGV & - & [49] \\
7 & A\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & A\({}_{\text{RMS}}\) = \(\sqrt{1/t_{\text{tot}}\int_{0}^{t_{\text{tot}}}\alpha_{\text{g}}(\text{t})^{2} \text{dt}}\) & [50] \\
8 & p V\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & V\({}_{\text{RMS}}\) = \(\sqrt{1/t_{\text{tot}}\int_{0}^{t_{\text{tot}}}\alpha_{\text{g}}(\text{t})^{2} \text{dt}}\) & [50] \\
9 & p\({}^{2}\) D\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & D\({}_{\text{RMS}}\) = \(\sqrt{1/t_{\text{tot}}\int_{0}^{t_{\text{tot}}}\text{d}_{\text{g}}(\text{t})^{2} \text{dt}}\) & [50] \\
10 & p\({}^{2}\) L\({}_{\text{a}}\)/(g tan(\(\alpha\))) & L\({}_{\text{a}}\) = PGA Tm\({}^{2}\) & [51] \\
11 & p\({}^{2}\) L\({}_{\text{c}}\)/(g tan(\(\alpha\))) & L\({}_{\text{c}}\) = PGV Tm & [24] \\
12 & p I\({}_{\text{A}}\)/(g tan(\(\alpha\))) & I\({}_{\text{A}}\) = \(\pi/(2\text{g})\int_{0}^{t_{\text{tot}}}\alpha_{\text{g}}(\text{t})^{2}\text{dt}\) & [52] \\
13 & p\({}^{0.50}\) I\({}_{\text{C}}\)/(g tan(\(\alpha\))) \({}^{3/2}\) & I\({}_{\text{c}}\) = R\({}_{\text{RMS}}\)\({}^{1.5}\) & tot\({}^{0.5}\) \\
14 & p\({}^{1.25}\) I\({}_{\text{F}}\)/(g tan(\(\alpha\))) & I\({}_{\text{F}}\) = PGV ts\({}_{\text{ig}}\)\({}^{0.25}\) & [54] \\
15 & p\({}^{3}\) SED/(g tan(\(\alpha\))) \({}^{2}\) & SED = \(\int_{0}^{t_{\text{tot}}}\) y\({}_{\text{g}}(\text{t})^{2}\text{dt}\) & [46] \\
16 & p t\({}_{\text{sig}}\) & - & [55] \\
17 & p CAV/(g tan(\(\alpha\))) & CAV = \(\int_{0}^{t_{\text{tot}}}\)\(\mid\)a\({}_{\text{g}}(\text{t})\)\(\mid\)dt & [56] \\
18 & p\({}^{2}\) CAD/(g tan(\(\alpha\))) & CAD = \(\int_{0}^{t_{\text{tot}}}\)\(\mid\)d\({}_{\text{g}}(\text{t})\)\(\mid\)dt & [56] \\ \hline \end{tabular}
\end{table}
Table 1: Dimensionless IMs.  
Figure 1: Examined free-standing rigid structural systems.