## 2 Dynamic Model of SFSI for Coupled Horizontal and Rocking Motions  
Consider a superstructure modeled as a damped, one-degree-of-freedom soil-interaction system. This system consists of a mass \(m\), a dashpot with damping coefficient \(c\), and a static spring with stiffness \(k\). The mass-spring-dashpot system is connected to a rigid bar of height \(H\), as shown in Figure 1.  
The connection between the soil and the superstructure is provided by a foundation block. This rigid foundation block has dimensions of length (\(2a\)), width (\(2b\)), and thickness (\(h\)), forming a prismatic shape, as shown in Figure 2. The mass of the foundation block is \(m_{0}\), and its mass moment of inertia relative to the rotation axis passing through the center of gravity is \(I\) (as illustrated in Figure 1). The corresponding mass moment of inertia is  
\[I=\frac{1}{12}m_{0}\Big{[}\Big{(}2a\big{)}^{2}+h^{2}\Big{]} \tag{1}\]  
The explicit computation of wave displacements propagating through unbounded soil is generally unnecessary for practical applications. Instead, the focus is placed on the  
Figure 1: Dynamic model of the SFSI system (including rocking and horizontal motions). Figure 2: Geometry of the foundation block. interaction force-displacement relationship at the foundation block. In the translational cone model, a dashpot with coefficient \(C\) and a spring with a static stiffness coefficient \(K\) are arranged in parallel. In the rotational cone model, the high-frequency limit of radiation damping \(c_{v}\) of the rotational dashpot and the rotational spring with static stiffness coefficient \(k_{v}\) are also connected in parallel. In this model, the foundation node, with a rocking degree of freedom \(v_{0}\), is connected to a roller support. A rotational dashpot with coefficient \(-c_{v}\) connects the roller support to an additional internal rotational degree of freedom \(v_{1}\), which is linked to the foundation by a rotational spring with coefficient \(-k_{v}/3\). The dynamic SFSI system depicted in Figure 1 accounts for the coupling between rocking and horizontal motions. Since the center of the foundation block-soil interface is located at a vertical distance \(e=0.5\ h\) from the foundation block's center of gravity, the rocking and horizontal motions are coupled [24]. The amplitudes of the relative displacement between the base and the mass, referred to as the free-field motion (\(u_{g}\)), and those in a moving frame of reference attached to the rigid base are denoted as \(u_{0}\) and \(u\), respectively. In Figure 1, the total lateral displacements of the mass and the foundation are denoted as \(u^{t}\) and \(u^{t}_{0}\), respectively. The unlayered soil is idealized as a homogeneous, linearly elastic, semi-infinite medium with mass density \(\rho\), and material damping is not considered for the soil. The cones addressed are analyzed under the a priori assumption that the soil is a perfectly elastic medium, which dissipates energy only through the radiation of waves toward infinity. The second important source of energy dissipation--material damping--is also generally present. Material damping affects the spring and damping coefficients of the dynamic stiffness. It causes a pseudo-hyperbolic increase in the low-frequency damping coefficient and a pseudo-linear decrease in the high-frequency spring coefficient. In engineering practice, material damping is typically negligible and does not need to be considered for vibration problems that usually induce rotation angles several orders of magnitude smaller [24]. In contrast, large-amplitude rocking occurs during strong-motion earthquakes. For such cases, it is appropriate to include material damping in the analysis and update its value at each time step. In this study, it is assumed that the spring coefficient is unaffected by material damping and that material damping does not modify radiation damping in the damping coefficient.