Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI

Abstract

The structural pounding caused by an earthquake may damage structures and lead to their collapse. This study is focused on the pounding between two adjacent asymmetric structures with different dynamic properties resting on the surface of an elastic half-space. An exploration of the relationship between the effects of the seismic analysis with the impact response to the torsional pounding between adjacent buildings under different SSI effects has been presented. In this paper, the authors have proposed a procedure for analyzing the response for adjacent buildings subjected to the pounding effects, considering systems with multiple degrees of freedom and modal equations of motion with four types of soil. All the calculations have been performed based on the fourth-order Runge–Kutta method. The novelty of the present study is related to the fact that the rigorous and approximate methods are used to examine the effects of pounding and SSI simultaneously. As a result, these two methods have been thoroughly investigated for both effects and the results have been compared. The results show that the approximate method produces results that are slightly different from those obtained by the rigorous direct integration method in the case of small SSI effects due to an increase in the pounding force. The efficiency of the method is also validated using numerical examples.

1. Introduction

Seismic researchers have often observed collisions between adjacent buildings that are not sufficiently separated. This phenomenon, known as earthquake-induced structural pounding, 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 pounding 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 pounding responses of adjacent buildings. Earthquake-induced structural pounding 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, pounding 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 be considered. 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]. Jui-Liang and Keh-Chyuan [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 Uz and Hadi [33], the pounding 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 pounding-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 pounding, although some studies [38,39,40,41] investigated the pounding 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 pounding between adjacent buildings under different SSI effects. Both the SSI and pounding effects have not been adequately researched in the past in terms of rigorous and approximate methodologies. In most cases, they were evaluated independently. The novelty of the present study is related to the fact that the rigorous and approximate methods are used to examine the effects of pounding and SSI simultaneously. As a result, these two methods have been thoroughly investigated for both effects and the results have been compared. The response analysis procedure has been developed for adjacent buildings subjected to pounding effects using multi-DOF modal equations of motion with four types of soil. The fourth-order Runge–Kutta method has been applied in all the calculations. The numerical examples have been used to validate the efficiency of the proposed method.

2. Theoretical Model Framework

The basic model for SSI that considers the effects of the pounding force on adjacent buildings has been used in this study.

2.1. Equation of Motion

Figure 1 shows a structural model of two adjacent buildings on a half-space with elastic properties.

Subscripts $i$ and $j$ in Figure 1 indicate the numbers of stories in the buildings. These are $1, 2, \dots , N$ for Building $A$ and $1, 2, \dots , S$ for Building $B$. In this study, the following values have been determined: the mass, stiffness, damping coefficients, and moment of inertia of the floor with respect to the axes parallel to the $x$ and $y$ axes of Building $A$ and in relation to the center of mass. In addition, Buildings $A$ and $B$ are represented either by the subscript or superscript $a$ and $b$, respectively. A building with $N$ stories has $3N+5$ DOF; hence, Buildings A and B have $3N+5$ and $3S+5$ equations, respectively. Here, only the equations for Building $A$ are highlighted. The equation of motion noted from previous studies [30,35,42,43,44] is briefly presented herein for completeness. The three equations of motion of each floor of the building may be written in a matrix form as (3 N):

$$\begin{gathered} \left[M_a\right]\left\{{\ddot{x}}_{ic}^t\right\}+\left[C_{ax}\right]\left\{{\dot{x}}_i\right\}+\left[K_{ax}\right]\left\{x_i\right\}+\left[F_{xij}^p\left(t\right)\right]=\left\{0\right\} \\ \left[M_a\right]\left\{{\ddot{y}}_{ic}^t\right\}+\left[C_{ay}\right]\left\{{\dot{y}}_i\right\}+\left[K_{ay}\right]\left\{y_i\right\}+\left[F_{yij}^p\left(t\right)\right]=\left\{0\right\} \\ {r}_a^2\left[M_a\right]\left\{{\ddot{\theta }}_i^t\right\}+f_a\left[C_{ax}\right]\left\{{\dot{x}}_i\right\}-e_a\left[C_{ay}\right]\left\{{\dot{y}}_i\right\}+\left[C_{\theta R}^a\right]\left\{{\dot{\theta }}_{ic}\right\}+f_a\left[K_{ax}\right]\left\{x_i\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-e_a\left[K_{ay}\right]\left\{y_i\right\}+\left[K_{\theta R}^a\right]\left\{{\theta }_{ic}\right\}+\left[F_{\theta ij}^p\left(t\right)\right]=\left\{0\right\} \end{gathered}$$

(1)

The mass, damping, and stiffness of Building A in the submatrices are denoted as M_a, C_ax, C_ay, K_ax, and K_ay, in relation to both directions. These matrices are sized with N × N dimensions. As described by Equation (2), the parameters $x_{ic}^t$, $y_{ic}^t$, and ${\theta }_{ic}^t$ are the displacements of the center of mass of the floors along the longitudinal and transverse axes, and their twists about the upward axis (z) in Building A, respectively. The forces used for pounding in the longitudinal direction $F_{xij}^p(t)$ are obtained from the nonlinear viscoelastic model developed by Jankowski [45]. $F_{yij}^p(t)$ and $F_{\theta ij}^p(t)$ are derived by the Coulomb friction model used by Chopra [46]. The impact model (nonlinear viscoelastic model) is used in this study for the rigorous model as it is. On the other hand, the impact model for the approximate method is deeply examined. x_i, y_i, and θ_ic are the displacement vectors of the center of resistance (CR) in the x and y directions and the twist of each floor with respect to the base, respectively. Additionally, two-way asymmetric buildings have been considered such that the CRs and center of masses (CMs) are not symmetrical along two axes of the horizontal plane as shown in Figure 2. The center of rigidity is the centroid of stiffness in a floor-diaphragm layout. The floor diaphragm experiences translational displacement in two directions and rotation when the center of stiffness is subjected to lateral loading.

The static eccentricity of the center of rigidity from the center of mass ($e$ and $f$) for each story level does not differ, although the CR varies from floor to floor. Hence, the CR lies at the co-ordinates $e_a$, $f_a$ of Building A and $e_b$, $f_b$ of Building B. The radius of gyration of the floor mass is taken from the center of mass. The torsion stiffness matrix defined about the CR is denoted as $K_{\theta R}^a$, whereas $K_{\theta M}^a$ in Equation (1) is defined in relation to CM. Furthermore, the damping matrices of Building A are given as $C_{ax}$, $C_{ay}$, and $C_{\theta R}^a$, in Equation (1), considered proportionally to the stiffness matrices [47]. The displacement vectors of Building A in the related directions without the effects of SSI can be calculated as

$$\begin{gathered} \left\{x_{ic}^t\right\}=x_o^a\left\{1\right\}+x_g\left\{1\right\}+{\phi }_o^a\left\{h_i\right\}+\left\{x_i\right\}-f_a\left\{{\theta }_{ic}\right\} \\ \left\{y_{ic}^t\right\}=y_o^a\left\{1\right\}+y_g\left\{1\right\}+{\psi }_o^a\left\{h_i\right\}+\left\{y_i\right\}+e_a\left\{{\theta }_{ic}\right\} \\ \left\{{\theta }_{ic}^t\right\}={\theta }_o^a\left\{1\right\}+\left\{{\theta }_{ic}\right\} \end{gathered}$$

(2)

where $x_i$ and $y_i$ are vectors of displacement with relation to the CM of the superstructure; $x_o^a$, $y_o^a$, ${\psi }_o^a$, and ${\varphi }_o^a$ represent the DOF at the foundation that is related to translations and rocking about the $x$ and $y$ axes; and ${\theta }_o^a$ is the rotation around the $z$ axis. In each building, the five equations of motion of the foundation are described as translational and rocking distances in the x and y directions as well as a torsional mode of vibration.

2.2. Pounding Forces

The equation of motion for Building A can be expressed in terms of translation and rocking along the $x$ and $y$ axes and twist along the $z$ axis, as described by Richart et al. [30], Sivakumaran and Balendra [23], and Jui-Liang et al. [35]. The static impedance functions have been used in this study (see [30]). Equation (3) shows the pounding-involved equation of motion for buildings with SSI:

$$\begin{gathered} \left[\begin{array}{cc}{M}^a& 0\\ 0& M^b\end{array}\right]\left\{\begin{array}{c}{\ddot{U}}^a\left(t\right)\\ {\ddot{U}}^b\left(t\right)\end{array}\right\}+\left[\begin{array}{cc}{C}^a& 0\\ 0& C^b\end{array}\right]\left\{\begin{array}{c}{\dot{U}}^a\left(t\right)\\ {\dot{U}}^b\left(t\right)\end{array}\right\}+\left[\begin{array}{cc}{K}^a& 0\\ 0& K^b\end{array}\right]\left\{\begin{array}{c}{U}^a\left(t\right)\\ U^b\left(t\right)\end{array}\right\}+\left\{\begin{array}{c}{F}^p\left(t\right)\\ -F^p\left(t\right)\end{array}\right\} \\ =-\left\{\begin{array}{c}{P}^a\left(t\right)\\ P^b\left(t\right)\end{array}\right\} \end{gathered}$$

(3)

where $F^p\left(t\right)$, $P^a\left(t\right)$, and $P^b\left(t\right)$ are the vectors including the impact forces between floor masses $\left(m_i,m_j\right)$.

In this study, the pounding force was simulated based on the nonlinear viscoelastic model [45,48,49,50].

2.3. Approximate Normal Modes of Adjacent Buildings

Equation (4) can be expressed in the following form, as proposed by Chopra and Goel [51]:

$$P^a\left(t\right)={\displaystyle \sum }_{n=1}^{3N+5}{s}_n^a\left({\mathsf{\Gamma }}_{xn}^a{\ddot{x}}_g+{\mathsf{\Gamma }}_{yn}^a{\ddot{y}}_g\right);P^b\left(t\right)={\displaystyle \sum }_{n=1}^{3S+5}{s}_n^b\left({\mathsf{\Gamma }}_{xn}^b{\ddot{x}}_g+{\mathsf{\Gamma }}_{yn}^b{\ddot{y}}_g\right)$$

(4)

The $n^{th}$-mode modal inertia force distribution can be calculated as $M^a{\phi }_n^a$ for vector $s_n^a$ and $M^b{\phi }_n^b$ for vector $s_n^b$. The ${\phi }_n^a$ calculated from $K^a$ and $M^a$ is the $n^{th}$-mode shape without damping. ${\phi }_n^b$ is also obtained in the same way as ${\phi }_n^a$. ${\mathsf{\Gamma }}_{xn}^a$, ${\mathsf{\Gamma }}_{yn}^a$, ${\mathsf{\Gamma }}_{xn}^b$, and ${\mathsf{\Gamma }}_{yn}^b$ in the longitudinal and perpendicular directions for the $n^{th}$ modal contribution values for both buildings. The output values of $U\left(t\right)$ contain translations in the $x$ and $y$ directions, rotations for each floor of the superstructure and the foundation, and the rocking angles for the foundation only in both directions. The $n^{th}$ mode of the contribution factor in the modal analysis is calculated as

$$\begin{gathered} {\mathsf{\Gamma }}_{xn}^a=\frac{{\phi }_n^a{}^T\times M^a\times {\left[{\mathbf{1}}^{\mathit{T}} {\mathbf{0}}^{\mathit{T}} {\mathbf{0}}^{\mathit{T}}1 0 0 0 0\right]}^T}{{\phi }_n^a{}^T\times M^a\times {\phi }_n^a}; \\ {\mathsf{\Gamma }}_{yn}^a=\frac{{\phi }_n^a{}^T\times M^a\times {\left[{\mathbf{0}}^{\mathit{T}} {\mathbf{1}}^{\mathit{T}} {\mathbf{0}}^{\mathit{T}}0 1 0 0 0\right]}^T}{{\phi }_n^a{}^T\times M^a\times {\phi }_n^a} \end{gathered}$$

(5)

where $\mathbf{1}$ and $\mathbf{0}$ are the unit and zero vectors, respectively, sized as $N\times 1$. Equation (5) proves that the $n^{th}$ modal participation factors rely on the path of the horizontal ground motion excitations. The 1940 El Centro (117 El Centro Array-9 station) and the 1995 Kobe (KJMA station) ground motions have been used for the analysis of both buildings, which is shown in Figure 3. The top accelerations of the related excitations were scaled to 0.3 $\mathrm{g}$ and 0.2 $\mathrm{g}$ for the 1940 El Centro NS-EW and 0.8 $\mathrm{g}$ and 0.6 $\mathrm{g}$ for the 1995 Kobe NS-EW, respectively.

$U_n^a$ and $U_n^b$ are defined in Equation (4) by the time variations in ${\ddot{x}}_g\left(t\right)$ and ${\ddot{y}}_g\left(t\right)$. In the analysis, the vertical motion of the ground was not in use. The force distribution can be expressed with the sum of the modal inertia forces, as given in Equation (4). $T_n^a$ and $T_n^b$ are $\left(3N+5\right)\times 8$ and $\left(3S+5\right)\times 8$ diagonal matrices for both buildings, respectively. ${\phi }_{xn}^a$, ${\phi }_{yn}^a$, and ${\phi }_{\theta n}^a$ in $T_n^a$ are subvectors of the $n^{th}$ natural vibration mode in the superstructure, sized as $N\times 1$. They are related to the displacement in both directions and the rotating DOF. ${\varphi }_{x_{o}n}^a$, ${\varphi }_{y_{o}n}^a$, ${\varphi }_{{\theta }_{o}n}^a$, ${\varphi }_{{\psi }_{o}n}^a,$ and ${\varphi }_{{\varphi }_{o}n}^a$ in $T_n^a$ are five subvectors of the mode shapes of the SSI system for Building A. With generalized modal co-ordinates for both buildings, the $n^{th}$ undamped modal displacement responses, $U_n^a$ and $U_n^b$, can be redefined as

$$\left\{\begin{array}{c}{U}_n^a\left(t\right)\\ U_n^b\left(t\right)\end{array}\right\}=\left[\begin{array}{cc}{T}_n^a& 0\\ 0& T_n^b\end{array}\right]\left[\begin{array}{c}{D}_n^a\left(t\right)\\ D_n^b\left(t\right)\end{array}\right]$$

(6)

where $D_n^a$ and $D_n^b$ are the $n^{th}$ generalized modal co-ordinate of both buildings. By substituting Equation (6) and rearranging each side of the equation of motion by ${\left[\begin{array}{cc}{T}_n^a& 0\\ 0& T_n^b\end{array}\right]}^T$, we obtain

$$\begin{gathered} \left[\begin{array}{cc}{M}_n^a& 0\\ 0& M_n^b\end{array}\right]\left[\begin{array}{c}{\ddot{D}}_n^a\left(t\right)\\ {\ddot{D}}_n^b\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{C}_n^a& 0\\ 0& C_n^b\end{array}\right]\left[\begin{array}{c}{\dot{D}}_n^a\left(t\right)\\ {\dot{D}}_n^b\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{K}_n^a& 0\\ 0& K_n^b\end{array}\right]\left[\begin{array}{c}{D}_n^a\left(t\right)\\ D_n^b\left(t\right)\end{array}\right]+\left[\begin{array}{c}{F}_n^{ap}\left(t\right)\\ -F_n^{bp}\left(t\right)\end{array}\right] \\ =-\left\{\begin{array}{c}{M}_n^a\iota \left({\mathsf{\Gamma }}_{xn}^a{\ddot{x}}_g+{\mathsf{\Gamma }}_{yn}^a{\ddot{y}}_g\right)\\ M_n^b\iota \left({\mathsf{\Gamma }}_{xn}^b{\ddot{x}}_g+{\mathsf{\Gamma }}_{yn}^b{\ddot{y}}_g\right)\end{array}\right\} \end{gathered}$$

(7)

where $M_n^a=T_n^a{}^T{M}^a{T}_n^a$, $C_n^a=T_n^a{}^T{C}^a{T}_n^a$, $K_n^a=T_n^a{}^T{K}^a{T}_n^a$, $M_n^b=T_n^b{}^T{M}^b{T}_n^b$, $C_n^b=T_n^b{}^T{C}^b{T}_n^b$, and $K_n^b=T_n^b{}^T{K}^b{T}_n^b$ are sized 8 $\times$ 8; $\iota$ is a vector sized as an 8 $\times$ 1 column with all elements the same as unity; the $3N+5$ and $3S+5$ multi-DOF modal equations of motion, as given in Equation (7) for both buildings, comprise a nonproportionally damped system.

$$\begin{gathered} C^a=T_n^a{}^T\left(\alpha M^a+\beta K^a\right)T_n^a=\alpha M_n^a+\beta K_n^a \\ {C}^b=T_n^b{}^T\left(\alpha M^b+\beta K^b\right)T_n^b=\alpha M_n^b+\beta K_n^b \end{gathered}$$

(8)

Cruz and Miranda [52] found that the Rayleigh damping model underestimates the damping of higher modes that contribute to the seismic response, resulting in an overestimation of the seismic response. This manuscript does not address the validity of the mass-proportional and stiffness-proportional assumptions in Rayleigh damping. The approximation approach and rigorous methods are contrasted in the first three modes of this study. If $C^a$ and $C^b$ are proportionally damped, i.e., $C^a=\left(\alpha M^a+\beta K^a\right)$ and $C^b=\left(\alpha M^b+\beta K^b\right)$, the $n^{th}$ modal damping matrices for each building can be described as Equation (8). In here, $\alpha$ and $\beta$ are coefficients found by the damping ratios of the two specific modes, since the original SSI system is not proportionally damped, i.e., $C^a\ne \left(\alpha M^a+\beta K^a\right)$, $C^b\ne \left(\alpha M^b+\beta K^b\right)$. In approximate method, Equation (8) converts $C^a\ne \alpha M_n^a+\beta K_n^a$ and $C^b\ne \alpha M_n^b+\beta K_n^b$ based on [35]. The nonlinear viscoelastic model is only modified for the approximate method used in this study, as given in Equation (9). Notably, the elements of $D_n^a$ and $D_n^b$ are not the same, even in an elastic state.

$$\begin{gathered} F_{xijn}^p\left(t\right)=0\mathrm{for}{\delta }_{ijxn}\left(t\right)\le 0; \\ {F}_{xijn}^p\left(t\right)=\overline{\beta }{\left({\delta }_{ijxn}\left(t\right)\right)}^{3/2}+{\overline{c}}_{ijn}\left(t\right){\dot{\delta }}_{ijxn}\left(t\right)\mathrm{for}{\delta }_{ijxn}\left(t\right)>0\mathrm{and}{\dot{\delta }}_{ijxn}\left(t\right)>0; \\ {F}_{xijn}^p\left(t\right)=\overline{\beta }{\left({\delta }_{ijxn}\left(t\right)\right)}^{3/2}\mathrm{for}{\delta }_{ijxn}\left(t\right)>0\mathrm{and}{\dot{\delta }}_{ijxn}\left(t\right)\le 0; \\ {\delta }_{ijxn}\left(t\right)={\delta }_{ijyn}\left(t\right)=D_{xn}^a{\phi }_{ixn}^a\left(t\right)-D_{xn}^b{\phi }_{jxn}^b\left(t\right)-D;{\delta }_{ij\theta n}\left(t\right)=D_{\theta n}^a{\phi }_{i\theta n}^a\left(t\right)f^a- \\ {D}_{\theta n}^b{\phi }_{j\theta n}^b\left(t\right)f^b-D; \\ {\dot{\delta }}_{ijxn}\left(t\right)={\dot{D}}_{xn}^a{\phi }_{ixn}^a\left(t\right)-{\dot{D}}_{xn}^b{\phi }_{jxn}^b\left(t\right);{\dot{\delta }}_{ijyn}\left(t\right)={\dot{D}}_{yn}^a{\phi }_{iyn}^a\left(t\right)-{\dot{D}}_{yn}^b{\phi }_{jyn}^b\left(t\right); \\ {\dot{\delta }}_{ij\theta n}\left(t\right)={\dot{D}}_{\theta n}^a{\phi }_{i\theta n}^a\left(t\right)f^a-{\dot{D}}_{\theta n}^b{\phi }_{j\theta n}^b\left(t\right)f^b; \\ {F}_{ijn}^{ap}={\left[F_{xijn}^{ap}{F}_{yijn}^{ap}{F}_{\theta ijn}^{ap}00000\right]}_{\left(3N+5\right)\times 1}^T \end{gathered}$$

(9)

$\mathrm{Here},{\delta }_{ij}\left(t\right)$ is the total displacement between the buildings with respect to the foundation; ${\dot{\delta }}_{ij}\left(t\right)$ is their velocity; $\overline{\beta }$ is the damping; and ${\overline{c}}_{ij}\left(t\right)$ is the stiffness of the impact element. The damping ratio $\left(\overline{\xi }\right)$ in relation to the coefficient of restitution $\left(e\right)$ provides the dissipation of energy during impact [53]. $D$ is the distance between the buildings. $F_{\theta ijn}^{ap}$ can be calculated as $F_{xijn}^{ap}$ using the related ${\delta }_{ij\theta n}\left(t\right)$ and ${\dot{\delta }}_{ij\theta n}\left(t\right)$ in Equation (9). $F_n^{ap}\left(t\right)=T_n^a{}^T{F}_{ijn}^{ap}$ in Equation (7) is an 8 $\times$ 1 vector with pounding forces. As a result, the approximate method substantially reduces the size of the matrices from 3 N + 5 to 8 for each building. The modal responses with regard to the displacement of each building, $D_n^a\left(t\right)$ and $D_n^b\left(t\right)$, are derived using the method of direct integration in Equation (7). The following equation is derived to obtain all the responses of the two nonproportionally damped asymmetric buildings on the top of an elastic half-space:

$$U^a\left(t\right)={\displaystyle \sum }_{n=1}^{3N+5}{U}_n^a\left(t\right)\approx {\displaystyle \sum }_{n=1}^{3N+5}{T}_n^a{D}_n^a\left(t\right);U^b\left(t\right)={\displaystyle \sum }_{n=1}^{3S+5}{U}_n^b\left(t\right)\approx {\displaystyle \sum }_{n=1}^{3S+5}{T}_n^b{D}_n^b\left(t\right)$$

(10)

The first few modal responses in Equation (10), to acquire a satisfying result, should be summed similarly to the displacement analysis.

2.4. Equation of Motion for SSI System

Based on Figure 1, the equation of motion for the entire foundation system for Building A can be derived from Equation (11) for the translation along the x and y axes, rotation around the z axis, and rocking along the x and y axes.

$$\begin{gathered} m_o^a\left({\ddot{x}}_g+{\ddot{x}}_o^a\right)+{\left\{1\right\}}^T\left[M_a\right]\left\{{\ddot{x}}_{ic}^t\right\}+P_{xa}\left(t\right)=0 \\ {m}_o^a\left({\ddot{y}}_g+{\ddot{y}}_o^a\right)+{\left\{1\right\}}^T\left[M_a\right]\left\{{\ddot{y}}_{ic}^t\right\}+P_{ya}\left(t\right)=0 \\ {r}_a^2{m}_o^a{\ddot{\theta }}_o^a+r_a^2{\left\{1\right\}}^T\left[M_a\right]\left\{{\ddot{\theta }}_i^t\right\}+T_a\left(t\right)=0 \\ {\displaystyle \sum }_{i=0}^N{I}_{xi}{\ddot{\psi }}_o^a+{\left\{h_i\right\}}^T\left[M_a\right]\left\{{\ddot{y}}_{ic}^t\right\}+Q_{xa}\left(t\right)=0 \\ {\displaystyle \sum }_{i=0}^N{I}_{yi}{\ddot{\varphi }}_o^a+{\left\{h_i\right\}}^T\left[M_a\right]\left\{{\ddot{x}}_{ic}^t\right\}+Q_{ya}\left(t\right)=0 \end{gathered}$$

(11)

The soil–structure interface has been modelled using a parallel set of frequency-independent springs and dashpots considered in the study of Richart et al. [30]. $P_{xa}$, $P_{ya}$, $T_a$, $Q_{xa}$, and $Q_{xa}$ in Equation (11) are the interaction forces of Building A as given in Equation (12).

$$\begin{gathered} P_{xa}\left(t\right)=C_T{\dot{x}}_o^a+K_T{x}_o^a \\ {P}_{ya}\left(t\right)=C_T{\dot{y}}_o^a+K_T{y}_o^a \\ {T}_a\left(t\right)=C_{\theta }{\dot{\theta }}_o^a+K_{\theta }{\theta }_o^a \\ {Q}_{xa}\left(t\right)=C_{\psi }{\dot{\dot{\psi }}}_o^a+K_{\psi }{\psi }_o^a \\ {Q}_{ya}\left(t\right)=C_{\varphi }{\dot{\dot{\varphi }}}_o^a+K_{\varphi }{\varphi }_o^a \end{gathered}$$

(12)

where $K_{T,\theta ,\psi ,\varnothing }$ and $C_{T,\theta ,\psi ,\varnothing }$ are the spring and dashpot coefficients of translations along both x and y axes, the torsion and rocking movements along both x and y axes, respectively. These constants of the static impedance functions are shown in

Table 1. Spring and dashpot constants used by Richart et al. [30].

	Sliding	Torsion	Rocking
Spring	$K_T=\frac{32\left(1-\upsilon \right)G{r}_o}{7-8\upsilon }$	$K_{\theta }=\frac{16G{r}_o^3}{3}$	$K_{\psi ,\varphi }=\frac{8G{r}_o^3}{3\left(1-\upsilon \right)}$
Mass Ratio	$B_T=\frac{\left(7-8\upsilon \right)M_T}{32\left(1-\upsilon \right)\rho r_o^3}$	$B_{\theta }=\frac{I_{\theta }}{\rho r_o^5}$	$B_{\psi ,\varphi }=\frac{3\left(1-\upsilon \right)I_{\psi ,\varphi }}{8\rho r_o^5}$
Damping Ratio	$D_T=\frac{0.288}{\sqrt{B_T}}$	$D_{\theta }=\frac{0.5}{1+2B_{\theta }}$	$D_{\psi ,\varphi }=\frac{0.15}{\left(1+B_{\psi ,\varphi }\right)\sqrt{B_{\psi ,\varphi }}}$
Coefficient	$C_T=2D_T\sqrt{K_T{M}_T}$	$C_{\theta }=2D_{\theta }\sqrt{K_{\theta }{I}_{\theta }}$	$C_{\psi ,\varphi }=2D_{\psi ,\varphi }\sqrt{K_{\psi ,\varphi }{I}_{\psi ,\varphi }}$

with many subscripts, as seen in Equation (12) [30].

Where $M_T$, $I_{\theta },\mathrm{and}{I}_{\psi ,\varphi }$ are the total mass, polar moment of inertia, and moment of inertia of the rigid body for rocking, respectively. $G,\rho ,\upsilon$, and ${\nu }_s$ are the shear modulus, mass density of half-space, Poisson’s ratio, and shear velocity of the elastic medium, respectively. $r_o$ is the radius of the massless disc on the surface of an elastic homogeneous half-space.

3. Numerical Study

In this study, five- and four-story asymmetric buildings are placed on an elastic half-space as Buildings A and B, respectively.

3.1. Structure Properties

Buildings A and B had the dimensions of 20 $\mathrm{m}$ by 15 $\mathrm{m}$ and 25 $\mathrm{m}$ by 20 $\mathrm{m}$, respectively, with the longer lengths in the longitudinal direction ($x$) for each building. The ratio of the foundation mass to the floor mass was 3 for each building. Each story in each building was 2.85 $\mathrm{m}$ high.

Table 2. Building details [31,51].

Story No.	Story Height ${\mathit{h}}_{\mathit{i}}$, ${\mathit{h}}_{\mathit{j}}$ $(\mathbf{m})$	$\mathbf{Building}\mathit{A}$		$\mathbf{Building}\mathit{B}$
		${\mathit{m}}_{\mathit{i}}\times {10}^6$ $\left(\mathbf{k}\mathbf{g}\right)$	${\mathit{k}}_{\mathit{i}}\times {10}^8$ $\left(\mathbf{N}/\mathbf{m}\right)$	${\mathit{m}}_{\mathit{j}}\times {10}^6$ $\left(\mathbf{m}\right)$	${\mathit{k}}_{\mathit{j}}\times {10}^8$ $\left(\mathbf{N}/\mathbf{m}\right)$
1F	2.85	0.30	3.46	0.4065	5.06
2F	5.7	0.30	3.46	0.4065	3.86
3F	8.55	0.30	3.46	0.4065	3.86
4F	11.4	0.30	3.46	0.4065	3.86
5F	14.25	0.30	3.46	-	-

provides the basic values describing the structural characteristics that have been used for this study.

Based on the reported results [34,54,55], we used $\overline{\beta }$ = 2.75 $\times {10}^9$ $\mathrm{N}/{\mathrm{m}}^{3/2}$ and $\overline{\xi }$ = 0.35 for the pounding force parameters in the nonlinear viscoelastic model, with an established coefficient of friction $\left({\mu }_f\right)$ of 0.5. In Equation (13), the translational and torsional stiffness at the center of mass for each story of a building are longitudinally in proportion to the stiffness of the same story [56].

$${\beta }_y=\frac{k_{yi}}{k_{xi}}=\frac{k_{yj}}{k_{xj}}; {\beta }_t=\frac{k_{\theta i}}{r_a^2{k}_{xi}}=\frac{k_{\theta j}}{r_b^2{k}_{xj}}$$

(13)

The values of ${\beta }_y$ and ${\beta }_t$ were 1.32 and 1.69 for both buildings, respectively. Both buildings were characterized by 2$\%$ of the critical damping as a constant of proportionality ($\alpha$) in the first vibration mode. Poisson’s ratio ($\upsilon$) was 0.333, and the density of the soil ($\rho$) was 1922 $\mathrm{kg}/{\mathrm{m}}^3$. Specifically, four types of soil at shear wave velocities ranging from 65, 130, 200, to 300 $\mathrm{m}/\mathrm{s}$ as Case I to IV have been examined, respectively. These ranges are described as soft to hard soil, based on Abdel Raheem, et al. [57] and Sulistiawan et al. [58]. Soft soil (Case I) indicates the large SSI effect on the building’s response, while hard soil (Case IV) indicates the small SSI effect. The original distance, $D$, between the buildings was 0.04 $\mathrm{m}$. The selection of this value has been based on the previous studies [48,59,60]. A rigorous method using the direct integration method has been applied to compute the equation of motion for the responses of the SSI system of each building shown in Equation (1).

3.2. Response Analysis

The results have been compared with those reported by Balendra et al. [42], Sivakumaran and Balendra [23], and Jui-Liang et al. [35], who did not address the effect of pounding. As the reference building (RB) in this study, another four-story building without considering SSI has been chosen.

The first six natural frequencies of the eight-story building-foundation system used by Balendra et al. [42] are shown in

Table 3. First six natural frequencies of an eight-story building-foundation system considered in [38] compared with those in the current study.

Cases Frequency	Case I	Case I *	Case IV	Case IV *	RB	RB *	$\frac{\mathbf{Case}\mathbf{I}*}{\mathbf{RB}*}$	$\frac{\mathbf{Case}\mathbf{IV}*}{\mathbf{RB}*}$
${\mathrm{W}}_1$ (Hz)	0.698	0.685	0.788	0.787	0.794	0.792	0.865	0.994
${\mathrm{W}}_2$ (Hz)	0.783	0.787	0.932	0.931	0.941	0.940	0.837	0.990
${\mathrm{W}}_3$ (Hz)	1.120	1.107	1.224	1.222	1.233	1.231	0.899	0.993
${\mathrm{W}}_4$ (Hz)	1.896	1.874	1.942	1.937	1.943	1.941	0.965	0.998
${\mathrm{W}}_5$ (Hz)	2.227	2.175	2.301	2.297	2.306	2.303	0.944	0.997
${\mathrm{W}}_6$ (Hz)	2.905	2.795	3.013	3.007	3.019	3.015	0.927	0.997

* Obtained by current studies; RB: reference building.

. As can be seen from the table, the x-direction displacement component is dominant in the first and fourth modes, while the y-direction displacement component is dominant in the second and fifth modes. The major component in the third and sixth modes corresponds to the rotation around the vertical axis. Building B has a modal response that is less than that of Building A, but with the same trend; therefore, the results of Building B are not provided here. Figure 4 shows the mode shapes of Building A placed on rigid ground as well as the mode shapes of each case using the approximate method. The thin lines in Figure 4 represent the translations and rotations of the base, as shown by the offsets and slopes. In Figure 4, the first to third mode shapes of the dominant motions are the $x$- and $y$-translation and rotation around the vertical axis for both the RB without the effects of SSI and for all cases.

However, the third mode shape in Case I is characterized by a rocking movement in the $y$-direction. In Cases I to IV, the first to third modes of shapes are consistent with those of the reference building. Figure 5 shows that the modal dislocation–time interactions for each of the eight DOFs for Case I are different.

In Case IV, the modal responses for the eight DOFs display a similar pattern to those of the second and third modes. Case I shows that the modal responses for the first three modes are mainly affected by SSI effects, and they are out of phase. Figure 6 shows the response of the first mode in Case IV without any pounding.

As shown in Figure 6, when the necessary distance is present between the buildings to avoid pounding, the modal responses of the eight DOFs for each vibration mode are similar to those in Case IV. Figure 7 illustrates the response histories of Cases I and IV obtained with the proposed method under the 1995 Kobe earthquake. The dashed line in Figure 7 denotes the approximate method (App.) given by Equation (7), whereas the solid line denotes the rigorous method (Rig.) obtained by Equation (3).

For Cases I and II, excellent agreement between the peak and phase responses for both methods has been found for the selected earthquakes. Notably, both methods produced slightly different response histories for the fourth story due to the larger forces under the small SSI effects, as shown in Figure 7. As a result of the high pounding forces in the first through third modes, the responses at the foundation obtained by the rigorous method in Figure 8 are not fully in agreement with those achieved by the approximate method in Case IV. Based on the comparison between the conventional rigorous method and the approximate method, it could be concluded that the approximate method markedly improves the accuracy of the analytical results without increasing the computational effort and by reducing the dimensions of the matrix.

Regarding the large SSI effect (Case I), both methods produced the same results for the 1995 Kobe earthquake in terms of translation, twisting, and rocking. Figure 9 compares both methods based on the story shears and torque with and without pounding. Figure 9 shows the maximum story shears in the x- and y-directions, together with the maximum story torque in relation to the upright direction for shear waves traveling at 65 and 300 $\mathrm{m}/\mathrm{s}$. This is based on simulations by Balendra et al. [42,43]. For validation purposes, these two methods used in this study are compared with the findings of Balendra et al. [42,43] without considering the pounding effect.

Based on the results for the large SSI case, the approximate and rigorous methods provided in Figure 9 for adjacent buildings are completely in agreement. Figure 9 also shows that, in Case IV, the maximum responses to the story shear and torque are reduced due to reduced pounding.

Abdel Raheem [61] has studied the bi-directional excitation and biaxial interaction of the base isolation system, although there is still debate on the base isolation system’s behavior under the SSI and hammering impacts. Two approaches utilized in this study will be applied to base-isolated buildings on various soil types, taking impact effects into account.

4. Conclusions

In this study, the seismic behavior of multistory asymmetric adjacent buildings by considering SSI and structural pounding has been examined. The effects of the pounding force on the adjacent structures in a simplified model have been considered. The parallel set of frequency-independent springs and dashpots to simulate the interaction forces at the SSI has been applied. The fourth-order Runge–Kutta differential equations for two-way asymmetric shear buildings have been derived, which have been solved both with and without impacts. The approximate method using multi-DOF modal equations of motion and the rigorous method using direct integration have been compared.

The SSI has been incorporated to generate a set of modal equations of motion for the pounding responses for each building based on the frequency-independent equation of motion. The results of this study indicated that pounding detrimentally affects the dynamic properties of a building. Because the shear wave velocity is very low, the approximate method produces the same impact force as the rigorous method. However, these findings are not valid for small SSI effects due to the increased number of collisions. The response to small SSI effects at the foundation is less than that to large SSI effects. The top floor deformations of adjacent buildings are somewhat conservative at a high shear wave velocity. Finally, buildings are considerably affected by increased shear wave velocity. All the vibration modes, rather than only the first few vibration modes, should be considered to achieve satisfactory results.

The simultaneous effects of SSI and the pounding of adjacent buildings under seismic loading have not been assessed so far by the use of the approximate and rigorous method. Both methods only evaluated the cases showing the SSI effect in the past. By comparing the SSI and pounding effects in these two techniques, the originality of this work has been demonstrated. It is vital to simulate colliding structures as inelastically as feasible in order to limit the consequences of pounding between buildings. By increasing the shear wave velocity, the responses, based on the deformation vectors for each structure, are drastically decreased while the SSI forces at the building’s foundation are enhanced. In order to determine the efficacy of these approaches in different circumstances, further studies should compare them to standard isolated buildings using a sensitivity analysis.