## 4 Seismic Performance of Isolation Systems Equipped with SLDTFPB and SLDRFPB  
In order to illustrate seismic performance of an isolation structure equipped with SLDRFPB or SLDTFPB subjected to rare and very-rare earthquakes, the high-rise frame structure with 20 stories is established, as shown in Figure 13. Length and width of each story are 25 m, and height of the structure is 60 m. Natural period and damping ratio of the structure are 1.48 s and 0.05, respectively. A multi-degree of freedom model is established in MATLAB, and time-history analysis of the isolation structure is simulated. The superstructure of the isolation structure is simplified to an elastic shear model based on Equation (9). The mode results in MATLAB are similar to those in SAP2000. Assume that the structure with large deformation has constant stiffness. This assumption is only used for simplifying structures which have not been equipped with isolation bearing. We note that in the model of the structure equipped with SLDRFPB, the stiffness between the isolation layer and the upper structure of each layer is not zero. The damping matrix is determined according to the Rayleigh damping model. The mass matrix and the stiffness matrix of the whole isolation structure and the first- and second-order mode frequencies of the superstructure are used to simulate the effective damping matrix of the superstructure. The results of time-history analysis are reliable because only the damping force of the isolation layer is an approximate result, which is much less than that of the friction force in the equation of the isolation layer. Friction force of the sliding surface follows the Coulomb friction principle. On the basis of Equations (26) and (27), the models of SLDRFPB and SLDTFPB are idealized. The horizontal stiffness of the isolation layer is simulated by the concentrated mass of the horizontal translational spring connection. The classical 4-order Runge-Kutta method is used to carry out the  
Figure 13: Frame structure of 20 stories. nonlinear numerical test of SLDRFPB and SLDTFPB in order to consider the stick-slip motion of the isolation layer. The main characteristics of five natural ground motions used in the time-history analysis are listed in Table 1. Results of two artificial seismic waves are also used to evaluate seismic performance. Each earthquake motion is adjusted by the reference of different peak ground accelerations (PGA) and follows Chinese standards. Seismic precautionary intensity of the structure is 8 degrees. Accordingly, an earthquake wave whose PGA is 0.7 m/s\({}^{2}\) represents a frequent earthquake, and earthquake wave whose PGA is 4 m/s\({}^{2}\) represents a rare earthquake and an earthquake wave whose PGA is 6.2 m/s\({}^{2}\) represents a very-rare earthquake [9]. As a three-dimensional structure, each story of the superstructure rotates around the center of the sphere under a unidirectional earthquake, and the radius equals the distance from the center of the sphere to the center of each layer. For the whole floors whose thickness is much less than the length and width, moment of inertia around the longitudinal axis is equal to the sum of the moment of inertia around two vertical axes in the plane, as shown in Equation (31):  
\[I_{z}=I_{x}+I_{y}=I_{a}+I_{b} \tag{31}\]  
The inertia moment of a regular quadrangle around any set of vertical axes in the plane has the same value so that incidence angle between the earthquake and the overall coordinate of the structure has no influence on the motion equation of the isolation layer. Since mass is distributed uniformly, a three-dimensional structure can be simplified to a two-dimensional in-plane problem. Based on the analysis in Section 3 and structural requirements, depth of SLDTFPB is selected as 2/9 and 4/9 m, and depth of SLDRFPB is selected as 1 m for comparison. The friction coefficient is selected as 0.01 [41,42], which can represent the friction coefficient of liquid or graphite lubricants.