# Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI
## 1 Introduction  
Seismic researchers have often observed collisions between adjacent buildings that are not sufficiently separated. This phenomenon, known as earthquake-induced structural bounding, may lead to minor damage at the points of interaction during moderate ground motions [1, 2]. It may also cause serious damage to colliding structures and even result in their collapse during major earthquakes. Examples include the Loma Prieta earthquake in 1989 [3] or the Athens earthquake in 1999 [4], when many cases of structural bounding of buildings occurred. Three important aspects of a complex seismic analysis of structure-foundation systems (see [5]) include the frequency-dependent interaction forces, nonproportional damping of soil-structure interaction (SSI), liquefaction [6, 7, 8, 9, 10], and bounding responses of adjacent buildings. Earthquake-induced structural bounding between symmetric buildings has been investigated [11, 12, 13, 14, 15]. Numerous researchers have studied the effects of structural interactions by applying different structural models and using different models of collisions [16, 17, 18, 19, 20]. Numerous researchers have also analyzed the SSI of asymmetric buildings exposed to seismic excitations [21, 22, 23, 24, 25]. In contrast, bounding between adjacent asymmetric structures with different dynamic properties and incorporating SSI has not been sufficiently explored. SSI problems (see [5, 26]) have been addressed in the frequency domain by means of either a Fourier or a Laplace transform [27, 28, 29] to consider the frequency-dependent interaction forces. However, in frequency-domain analysis, only linear responses can beconsidered. The soil springs and dashpots associated with the translational rocking and torsional modes of vibration can be satisfactorily calculated for typical multistory buildings in the time domain using frequency-independent expressions [30]. Employing traditional modal analysis methods to derive equivalent modal damping from the diagonal terms of the transformed damping matrix without examining the off-diagonal elements is one of the common approximate approaches for analyzing nonclassically damped systems. Certain conditions may lead to unacceptable errors in the response when off-diagonal terms are ignored in the transformed damping matrix [31]. [PERSON] and [PERSON] [32] verified the accuracy of the two degree-of-freedom (DOF) modal equations of motion for conserving nonproportional damping compared with that of the damped one-way asymmetric buildings. A simplified modal response analysis was developed for engineering applications that do not require complicated calculations of the equivalent modal damping. The outcomes of a parametric investigation, conducted by varying the structural parameter values, have also been investigated in the previous studies of the authors [33; 34]. The parametric study is beyond the scope of this paper, which focuses on considering the SSI and impact effects for rigorous and approximate methods. According to the results of the response analysis conducted by [PERSON] and [PERSON] [33], the bounding of buildings during ground motion excitation has a considerable influence on the longitudinal behavior of the lighter structure. Simplifications of the modal analysis and investigations on bounding-involved structural responses have been proposed [31; 33; 35; 36; 37]. Studies have also been conducted on the earthquake-induced responses of colliding symmetric structures incorporating SSI. However, to the best of our knowledge, our study is the first exploration of the effect of SSI on the response of adjacent asymmetric buildings exposed to bounding, although some studies [38; 39; 40; 41] investigated the bounding effect or SSI effects individually on adjacent buildings. Therefore, our main objective in this study was to investigate the relationship between the effects of the seismic analysis with the response to the torsional bounding between adjacent buildings under different SSI effects.