# Hilbert-Huang-Transform-Based Seismic Intensity Measures for Rocking Response Assessment
## 1 Introduction  
During intense seismic excitations, modern earthquake-resistant structures are expected to respond inelastically. Although this practice is commonly accepted and adopted by the current seismic design codes [1], it may lead to significant losses related to potential rehabilitation in the case of excessive damage occurrence [2]. Since the 1960s, two alternative earthquake-resistant design practices have emerged: seismic isolation and rocking-based structural systems [3]. Structures that can uplift and rock under seismic excitations present remarkable displacement capacities without exhibiting damage. Due to negative stiffness, their response differs substantially from fixed-base or seismic isolated structures [4; 5].  
The stability of rigid blocks standing free on rigid ground has been a subject of study since [PERSON]'s seminal work [6], in which the factors influencing the rocking response under horizontal excitations were identified. Since then, various rocking-based systems have been investigated, and their unique properties have been highlighted [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. Along with theoretical studies, rocking-based seismic isolation has been implemented in bridge construction technology [23; 24; 25; 26].  
Small changes in structural and ground motion characteristics significantly affect the rocking response. Given the uncertainties regarding the characteristics of seismic excitations, the probabilistic treatment of rocking response emerged [27; 28]. Several studies have focused on the seismic fragility assessment of rocking structural systems [29; 30; 31; 32; 33; 34; 35; 36]. Additionally, the performance of numerous intensity measures (IMs) in estimating the dynamic response of freestanding blocks and frames has been investigated in order to reduce the variability introduced by seismic excitations [37; 38; 39; 40; 41]. It has emerged that rockingmotion is strongly governed by size-frequency-scaled phenomena [6; 10], which means that it is affected both by the amplitude and the frequency content of the seismic excitation. Even though numerous conventional intensity measures (IMs) present high correlation with the seismic performance of nonrocking oscillators [42; 43; 44; 45; 46; 47], only a limited number of them capture the size-frequency effect of the rocking response [31; 32].  
Due to this fact, a robust signal-processing procedure based on the Hilbert-Huang Transform (HHT) [48; 49] which qualitatively and quantitatively describes the amplitude and frequency properties of a signal over time is adopted in this study to introduce novel ground motion IMs. HHT-based IMs have been proven promising alternatives, over conventional ones, in the seismic response prediction of reinforced concrete building structures [50; 51; 52; 53; 54]. Considering the high nonlinearity of the rocking response and the probabilistic basis of seismic design, seismic response uncertainty reduction is of great importance. As such, in the present research, the performance of conventional and HHT-based IMs in predicting the seismic response of a typical rocking bridge is comparatively evaluated based on optimal IM selection criteria. Fragility analysis is performed using the most optimal bivariate IMs, and analytical expressions are proposed. Finally, discriminant analysis is utilized to provide a reliable and rapid estimation of the maximum seismic demand [55].