## 4 Numerical Comparisons between FPBs and SRBs
### FPB Models  
To quantitatively and objectively discuss the potential advantages of FPBs and SRBs in diverse aspects, the seismic responses of SRB-3 and SRB-6, as detailed in Section 3.2, were referred by the design criteria for the FPB models first and then were compared with those of the FPB models designed. Ground motions of Groups 2 and 3, as detailed in Section 2 and listed in Table 1, were also employed herein. More precisely, first, the average of maximum horizontal displacement responses of each SRB model, respectively, subjected to ground motions of Groups 2 and 3 was considered as the design displacement \((D_{D})_{i}\) (\(i=2\) and 3) for the FPB model, as given in Equation (8). At the design displacement, the FPB model had the same horizontal acceleration response as the SRB model, i.e., 0.08\(\cdot\)g. Through referring to SRB-3 subjected to ground motions of Group 2, the friction damping force designed for the FPB model (denoted as FPB-3-2M hereafter) was the same as that for SRB-3, as illustrated in Figure 10a. In other words, at any horizontal displacement, FPB-3-2M and SRB-3 had the same energy dissipation area caused by the friction damping force. Similarly, through referring to SRB-3 subjected to ground motions of Group 3, SRB-6 subjected to ground motions of Group 2, and SRB-6 subjected to ground motions of Group 3, the friction damping forces designed for the FPB models (correspondingly denoted as FPB-3-3M, FPB-6-2M, and FPB-6-3M hereafter) were identical to those for SRB-3, SRB-6, and SRB-6, respectively, as illustrated in Figure 10. The curved sliding surfaces designed for FPB-3-2M, FPB-3-3M, FPB-6-2M, and FPB-6-3M, as well as the sloped rolling surfaces designed for SRB-3 and SRB-6, are conceptually illustrated in Figure 11 for a better comparison of their different geometrical dimensions. Based on the design criteria aforementioned, the design parameters of FPB-3-2M, FPB-3-3M, FPB-6-2M, and FPB-6-3M can be determined, as also listed in Table 2. Note: based on the design criteria, when subjected to ground motions of Group 3, i.e., near-fault pulse-like ground motions, the FPB models were designed with a considerable curvature radius, which might not often be seen for practical application when simply considering far-field ground motions. \[\left(D_{D}\right)_{i}=\frac{\sum_{j=1}^{n}D_{i,j}^{\max}}{n},\ \ i=2\ \text{and}\ 3, \tag{8}\]  
where \(i\) represents the ground motions of Group \(i\) (\(i=2\) and 3); \(j\) represents the \(j\)th ground motion of Group 2 or 3; \(n\) represents the number of ground motions of Group 2 or 3 (for Groups 2 and 3, \(n=25\) and 41, respectively); \(D_{i,\ j}^{\max}\) is the maximum horizontal displacement response of SRB-3 or SRB-6 subjected the \(j\)th ground motion of Group \(i\). Figure 9: Idealized composition of analytical models for FPBs. The design criteria for the FPB models aforementioned were based on the mean response subjected to ground motions of Group 2 or 3, which is called Benchmark I hereafter. For more comprehensive comparison and considering the effects caused by diverse ground motions, different FPB models designed based on the maximum horizontal displacement response of each SRB model subjected to each ground motion, i.e., \((D_{D})_{i,j}\) (\(i=2\) and \(3\); \(j=1\)-\(n\)), as calculated by Equation (9), were also considered herein, which is called Benchmark II hereafter. In short, Benchmark II is basically and conceptually the same as Benchmark I, except that the maximum horizontal displacement response of each SRB model subjected to each ground motion was considered, in lieu of the average of maximum horizontal displacement responses of each SRB model subjected to ground motions of Group 2 or 3. Thus, a total of 132 FPB models based on Benchmark II were designed accordingly (considering two SRB models and 66 sets of ground motions).