## 1 Introduction  
### Literature Review  
On a theoretical basis, the satisfactory seismic performances of friction pendulum bearings (FPBs) and sloped rolling-type bearings (SRBs) have been demonstrated separately in many past numerical and experimental studies [1; 2]. Well-developed analytical models, including simplified and sophisticated ones, have also been proposed and verified to be capable of predicting the dynamic behavior of FPBs and SRBs, accordingly. Currently, FPBs and SRBs, the former in particular, have been widely applied in engineering practice for seismically protecting critical structures, facilities, and equipment to guarantee their desired functionality during and after earthquakes. FPBs designed with a single curvature radius and friction coefficient [3; 4; 5; 6], as illustrated in Figure (a)a, undoubtedly, possess a very simple and reliable mechanism together with idealized bilinear hysteretic behavior, as shown in Figure 2. The details about the unique features of FPBs and the notations presented in the figures will be further explained in Section 1.2. In the past few decades,in addition to adjusting the design parameters of FPBs for better estimating and enhancing their control performances [7], there have also been many studies on mechanically modifying and upgrading FPBs to offer designers greater flexibility to optimize their performance as much as possible at different design stages. The FPBs, which can individually or simultaneously present variable energy dissipation and different stiffness (or restoring force) capabilities by correspondingly designing different friction coefficients and varying curvature radii, such as the so-called double, triple, and even quintuple FPBs, were analytically and experimentally studied by [PERSON] and [PERSON] [8,9], [PERSON] and [PERSON] [10], [PERSON] et al. [11], [PERSON] and [PERSON] [12], [PERSON] and [PERSON] [13], and [PERSON] and [PERSON] [14]. Besides concave sliding surface designs, [PERSON] et al. [15] and [PERSON] et al. [16] analytically and experimentally studied some complicated polynomial functions designed for the sliding surfaces of FPBs so as to present variable stiffness properties and thus achieve multiple design objectives. Furthermore, in addition to studying the unilateral behavior of double or triple FPBs, their multi-axial behavior was thoroughly studied by [PERSON] et al. [17], [PERSON] and [PERSON] [18], [PERSON] and [PERSON] [19], and [PERSON] et al. [20]. Some research results indicated that the effect arising from vertical excitations on the horizontal seismic performance of double or triple FPBs might not be negligible for obtaining more accurate and conservative design results [21,22]. Recently, not only far-field ground motions but the so-called pulse-like near-fault ones were also considered in many relevant studies to propose the corresponding design strategies for FPBs [12,23]. The post-earthquake self-centering capability of double or triple FPBs, which is another important performance index for seismic isolation and will significantly affect serviceability and functionality of isolated superstructures and nonstructural components (or systems) crossing isolation layers after earthquakes, was also theoretically and experimentally studied by [PERSON] et al. [24], [PERSON] et al. [25], and [PERSON] et al. [26]. Besides applying FPBs to new construction, their applicability and economic performance for seismically retrofitting existing structures, with consideration to soil deformability, have also been studied [27]. Figure 1: Simplified models for friction pendulum bearings (FPBs) and sloped rolling-type bearings (SRBs). Figure 2: Typical hysteretic behavior of FPBs and SRBs. SRBs designed with a constant sloping angle and damping force [28; 29; 30; 31; 32; 33], as illustrated in Figure 1b, mechanically exhibit the so-called twin-flag hysteretic behavior as also shown in Figure 2. The details about the unique features of SRBs and the notations presented in the figures will be further explained in Section 1.2.