Control Performances of Friction Pendulum and Sloped Rolling-Type Bearings Designed with Single Parameters

applied sciences

Article Control Performances of Friction Pendulum and Sloped Rolling-Type Bearings Designed with Single Parameters

Shiang-Jung Wang 1,* , Yi-Lin Sung 1, Cho-Yen Yang 2, Wang-Chuen Lin 2 and Chung-Han Yu 2 1 Department of Civil and Construction Engineering, National Taiwan University of Science and Technology, Taipei 106335, Taiwan; [email protected] 2 National Center for Research on Earthquake Engineering, Taipei 106219, Taiwan; [email protected] (C.-Y.Y.); [email protected] (W.-C.L.); [email protected] (C.-H.Y.) * Correspondence: [email protected]; Tel.: +886-2-27303650

Received: 9 September 2020; Accepted: 13 October 2020; Published: 15 October 2020

Abstract: Owing to quite different features and hysteretic behavior of friction pendulum bearings (FPBs) and sloped rolling-type bearings (SRBs), their control performances might not be readily compared without some rules. In this study, first, on the premise of retaining the same horizontal acceleration control performance, the effects arising from different sloping angles and damping forces on the horizontal maximum and residual displacement responses of SRBs are numerically examined. For objective comparison of passive control performances of FPBs and SRBs, then, some criteria are considered to design FPBs with the same horizontal acceleration control performance by referring to the designed damping force and the maximum horizontal displacement response of SRBs under a given seismic demand. Based on the considered criteria, the passive control performances of FPBs and SRBs under a large number of far-field and pulse-like near-fault ground motions are quantitatively compared. The numerical comparison results indicate that the FPB models might potentially have better horizontal acceleration and isolation displacement control performances than the SRB models regardless of whether they are subjected to far-field or near-fault ground motions, while the opposite tendency is observed for their self-centering performances, especially when the SRB model designed with a larger sloping angle or a smaller damping force.

Keywords: friction pendulum bearing; sloped rolling-type bearing; numerical comparison; control performance; self-centering

1. Introduction

1.1. 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 [3–6], as illustrated in Figure1a, undoubtedly, possess a very simple and reliable mechanism together with idealized bilinear hysteretic behavior, as shown in Figure2. 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,

Appl. Sci. 2020, 10, 7200; doi:10.3390/app10207200 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 20 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 controlAppl. performances Sci. 2020, 10, x FOR [7],PEER there REVIEW have also been many studies on mechanically modifying2 of 20 and upgradingAppl. Sci. FPBs2020, 10 ,to 7200 offer designers greater flexibility to optimize their performance as 2much of 21 as possiblethe notationsat different presented design in thestages. figure Thes will FPBs, be further which explained can individually in Section 1.2. or In simultaneously the past few decades, present in addition to adjusting the design parameters of FPBs for better estimating and enhancing their variablein addition energy to dissipation adjusting the and design different parameters stiffness of FPBs (or forrestoring better estimating force) capabilities and enhancing by correspondingly their control control performances [7], there have also been many studies on mechanically modifying and designingperformances different [7], friction there have coefficients also been manyand varyin studiesg on curvature mechanically radii, modifying such as and the upgrading so-called FPBs double, upgrading FPBs to offer designers greater flexibility to optimize their performance as much as triple,to oandffer designerseven quintuple greater flexibilityFPBs, were to optimize analytic theirally performance and experimentally as much as possiblestudied atby di ffFenzerent and possible at different design stages. The FPBs, which can individually or simultaneously present Constantinoudesign stages. [8,9], The Kim FPBs, YS which and Yu can CB individually [10], Calvi or et simultaneously al. [11], Panchal present and variable Jangid energy[12], Bao dissipation and Becker variable energy dissipation and different stiffness (or restoring force) capabilities by correspondingly and different stiffness (or restoring force) capabilities by correspondingly designing different friction [13], anddesigning Lee and different Constantinou friction coefficients [14]. Besides and concave varying slidingcurvature surface radii, suchdesigns, as the Lu so-called et al. [15] double, and Saha coefficients and varying curvature radii, such as the so-called double, triple, and even quintuple FPBs, et al. triple,[16] analytically and even quintupleand experimentally FPBs, were studied analytic someally andcomplicated experimentally polynomial studied functions by Fenz designed and were analytically and experimentally studied by Fenz and Constantinou [8,9], Kim YS and Yu CB [10], for theConstantinou sliding surfaces [8,9], Kimof FPBs YS and so as Yu to CB present [10], Calvi variable et al. stiffness[11], Panchal properties and Jangid and [12], thus Bao achieve and Becker multiple Calvi et al. [11], Panchal and Jangid [12], Bao and Becker [13], and Lee and Constantinou [14]. Besides design[13], objectives. and Lee and Furthermore, Constantinou in [14]. addition Besides to concave studyi slidingng the surfaceunilateral designs, behavior Lu et al.of [15]double and Sahaor triple concave sliding surface designs, Lu et al. [15] and Saha et al. [16] analytically and experimentally studied FPBs,et their al. [16] multi-axial analytically behavior and experimentally was thoroughly studied stud someied complicatedby Panchal etpolynomial al. [17], Becker functions and designed Mahin [18], some complicated polynomial functions designed for the sliding surfaces of FPBs so as to present Sah andfor the Soni sliding [19], surfaces and Furinghetti of FPBs so etas toal. present [20]. Some variable research stiffness results properties indicated and thus that achieve the effect multiple arising variable stiffness properties and thus achieve multiple design objectives. Furthermore, in addition to design objectives. Furthermore, in addition to studying the unilateral behavior of double or triple fromstudying vertical theexcitations unilateral on behavior the horizontal of double seismic or triple performance FPBs, their multi-axial of double behavior or triple wasFPBs thoroughly might not be FPBs, their multi-axial behavior was thoroughly studied by Panchal et al. [17], Becker and Mahin [18], negligiblestudied for by obtaining Panchal et more al. [17 accurate], Becker and and conservative Mahin [18], Sah design and Soniresults [19 ],[21,22]. and Furinghetti Recently, et not al. only [20]. far- Sah and Soni [19], and Furinghetti et al. [20]. Some research results indicated that the effect arising field Someground research motions results but indicated the so-called that the pulse-like effect arising near-fault from vertical ones were excitations also onconsidered the horizontal in many from vertical excitations on the horizontal seismic performance of double or triple FPBs might not be relevantseismic studies performance to propose of doublethe corresponding or triple FPBs design might strategies not be negligible for FPBs for [12,23]. obtaining The more post-earthquake accurate negligible for obtaining more accurate and conservative design results [21,22]. Recently, not only far- and conservative design results [21,22]. Recently, not only far-field ground motions but the so-called self-centeringfield ground capability motions of but double the so-called or triple pulse-like FPBs, which near-fault is another ones importantwere also consideredperformance in manyindex for pulse-like near-fault ones were also considered in many relevant studies to propose the corresponding seismicrelevant isolation studies andto propose will thesignificantly corresponding affect design serviceability strategies for FPBs and [12,23]. functionality The post-earthquake of isolated design strategies for FPBs [12,23]. The post-earthquake self-centering capability of double or triple superstructuresself-centering andcapability nonstructural of double orcomponents triple FPBs, which (or issystems) another importantcrossing performance isolation indexlayers for after FPBs, which is another important performance index for seismic isolation and will significantly earthquakes,seismic wasisolation also theoreticallyand will significantly and experimentally affect serviceability studied by Katsarasand functionality et al. [24], of Cardone isolated et al. affect serviceability and functionality of isolated superstructures and nonstructural components [25], superstructuresand Ponzo et al.and [26]. nonstructural Besides applying components FPBs (o tor systems)new construction, crossing isolation their applicability layers after and (or systems) crossing isolation layers after earthquakes, was also theoretically and experimentally economicearthquakes, performance was also for theoretically seismically and retrofitting experimentally existing studied structures, by Katsaras with et al. consideration[24], Cardone et toal. soil studied by Katsaras et al. [24], Cardone et al. [25], and Ponzo et al. [26]. Besides applying FPBs to [25], and Ponzo et al. [26]. Besides applying FPBs to new construction, their applicability and deformability,new construction, have also their been applicability studied [27]. and economic performance for seismically retrofitting existing economic performance for seismically retrofitting existing structures, with consideration to soil structures, with consideration to soil deformability, have also been studied [27]. deformability, have also been studied [27].

(a) FPB. (b) SRB. (a) FPB. (b) SRB. Figure 1. Simplified models for friction pendulum bearings (FPBs) and sloped rolling-type bearings FigureFigure 1. SimplifiedSimpliﬁed models models for friction for friction pendulum pendulum bearings bearings (FPBs) and (FPBs) sloped and rolling-type sloped rolling-type bearings (SRBs). bearings(SRBs). (SRBs).

Figure 2. Typical hysteretic behavior of FPBs and SRBs. FigureFigure 2. 2.TypicalTypical hysteretic hysteretic behavi behavioror of of FPBs FPBs and and SRBs. SRBs. Appl. Sci. 2020, 10, 7200 3 of 21

SRBs designed with a constant sloping angle and damping force [28–33], as illustrated in Figure1b, mechanically exhibit the so-called twin-flag hysteretic behavior as also shown in Figure2. The details about the unique features of SRBs and the notations presented in the figures will be further explained in Section 1.2. In the past decade, there have been some studies on mechanically modifying and upgrading SRBs to offer designers greater flexibility to optimize their performance as much as possible at different design stages. Without limiting the horizontal acceleration control performance, Wang et al. [34] numerically investigated the influences of different combinations of sloping angles and damping forces on the horizontal displacement control performance of SRBs, and accordingly, proposed some empirical formulas for practically and conservatively approximating the maximum horizontal displacement of SRBs under a given seismic demand characterized by the effective peak acceleration and corner period. In addition to SRBs designed with constant parameters, those designed with linearly variable damping forces, as well as those designed with stepwise variable sloping angles and damping forces, were analytically and experimentally studied to meet diverse control performance goals [35]. The post-earthquake self-centering capability of SRBs (or more generalized forms, the isolation bearings possessing flag-shaped hysteretic behavior) was theoretically, numerically, and experimentally studied [35,36].

1.2. Formulas and Considerations The simpliﬁed and linearized equation of motion of FPBs for a single-degree-of-freedom (SDOF) system shown in Figure1a is given in Equation (1) [3–6].

.. .. Mg . F = M u + ug = u + FFPBsgn u , (1) − RFPB where F is the inertia force; M is the mass of the protected object above the isolation bearing; g is the acceleration of gravity; RFPB is the curvature radius of the spherical sliding surface; FFPB is the friction damping force, which is usually approximated by the product of Mg and the friction coefficient µFPB . .. between two mutual-contact sliding surfaces; u, u, and u are the displacement, velocity, and acceleration of the upper bearing plate (or the protected object) relative to the lower bearing plate (the ground .. usually), respectively; ug is the horizontal acceleration excitation. As observed from Figure2 and Equation (1), once the design displacement DD, RFPB, and FFPB are determined, the effective horizontal stiffness Keff and the effective horizontal period Teff at DD can be mathematically calculated by Equations (2) and (3), respectively [3–6]. Mg FFPB 1 µFPB Ke f f = + = + Mg, (2) RFPB DD RFPB DD tv s Mg 1 T = = e f f 2π Mg 2π µ . (3) + FFPB g 1 + FPB g RFPB DD RFPB DD The unique features of FPBs can be directly observed from Equations (2) and (3) [3–6]. The effective horizontal stiffness is proportional to the mass of the protected object above, thus easily overcoming the inherent and accidental eccentricity problems, especially when the protected object is a highly irregular architectural design. In addition, the effective horizontal period is independent of the mass of the protected object above. In other words, the effective horizontal period, which is the most important design parameter when performing the linear static or dynamic analysis [37], will not be affected if the mass of the protected object after construction varies in use or is different from the initial assumed mass. The simplified and linearized equations of motion of SRBs at the slope and arc rolling stages for the SDOF system shown in Figure1b are given in Equations (4) and (5), respectively [32–34].

.. .. MgθSRB . F = M u + ug = sgn(u) + F sgn u , (4) − 2 SRB Appl. Sci. 2020, 10, 7200 4 of 21

.. .. Mg . F = M u + ug = u + FSRBsgn u , (5) − 4RSRB where θSRB is the sloping angle of the sloped rolling surface; FSRB is the friction damping force; RSRB is the curvature radius of the arc rolling surface. The unique feature of SRBs can be directly observed from Equation (4) and Figure2[ 29–33]. With a fixed sloping angle and damping force, the horizontal acceleration transmitted to the protected object above can remain constant regardless of any intensity and frequency content of external disturbance. Accordingly, SRBs do not have a fixed natural period and can offer maximum horizontal decoupling between the protected object and external disturbance.

1.3. Objectives of This Study In view of quite different features and hysteretic behavior revealed by FPBs and SRBs, their passive control performances might not be readily and objectively compared without some rules. In this study, first, some criteria are considered to objectively design FPBs with the same horizontal acceleration control performance by referring to the designed damping force and the maximum horizontal displacement response of SRBs under a given seismic demand. Then, based on the considered criteria, the passive control performances of FPBs are numerically and quantitatively compared with those of SRBs, in terms of horizontal acceleration, isolation displacement, and residual displacement, under a large number of far-field and pulse-like near-fault ground motions. It is emphasized that with a given constant horizontal acceleration control performance, numerically comparing seismic responses of FPBs with those of SRBs based on an objective design criterion, rather than seeking optimum design parameters separately for FPBs and SRBs, is the main purpose of this study.

2. Ground Motions Considered Three groups of horizontal ground motions recorded at ground surfaces with different site conditions (or soil effects) during past local and global earthquakes were adopted as horizontal acceleration inputs in this study. More precisely, the ground motion histories were mathematically .. substituted into ug in Equations (1), (4) and (5). The first group (denoted as Group 1 hereafter) contained 2553 sets of local ground motion records with peak ground acceleration (PGA) values varying from 0.082 g to 1.260 g and peak ground velocity (PGV) varying from 0.006 m/s2 to 2.800 m/s2, which were collected by the Taiwan Strong Motion Instrument Program (TSMIP) during 29 July 1992 to 31 December 2006. Among these, 16 sets of records were classified as pulse-like near-fault ground motions [38], and the remaining were classified as far-field ground motions. The second and third groups (hereafter denoted as Group 2 and Group 3, respectively) correspondingly contained 25 sets of far-field ground motion records and 41 sets of pulse-like near-fault ground motion records [38] during past global earthquakes, which were collected by the Pacific Earthquake Engineering Research Center (PEER). The PGA and PGV values of ground motion records of Group 2 varied from 0.152 g to 0.621 g and from 0.208 m/s2 to 0.704 m/s2, respectively, and the PGA and PGV values of ground motion records of Group 3 varied from 0.144 g to 1.129 g and from 0.440 m/s2 to 1.479 m/s2, respectively. The basic information of these ground motions, including the earthquake, year of occurrence, station name (as the denotation hereafter), PGA, and PGV, is summarized in Table1. Note that adopting Group 1 aims to more clearly identify the influences of different design parameters on wide-ranging seismic responses of SRBs, while adopting Groups 2 and 3, can further discuss and understand the individual potential merits of FPBs and SRBs subjected to far-field and pulse-like near-fault ground motions. Thus, it can be seen that a few ground motions classified into Groups 2 and 3 were duplicated in Group 1. Appl. Sci. 2020, 10, 7200 5 of 21

Table 1. Ground motions considered for horizontal acceleration inputs (* denotes the ground motions duplicated in Groups 1 and 2 (or 3)).

Earthquake Year Station Name PGA (g) PGV (m/s) Earthquake Year Station Name PGA (g) PGV (m/s) Group 1 - 1992–2006 - 0.082–1.260 0.006–2.800 El Centro 1940 American Imperial Valley 0.281 0.309 Kakogwa (000) 0.240 0.208 American Imperial 1979 Delta (352) 0.350 0.330 Shin-Osaka (090) 0.233 0.218 Valley American Japan Kobe 1995 1987 Poe Road (270) 0.475 0.412 MRG (000) 0.214 0.270 Superstition Hills American Loma Gilroy Array #3 (000) 0.559 0.363 Fukushima (000) 0.185 0.314 1989 Prieta Capitola (090) 0.439 0.296 Yae (000) 0.158 0.212 Iran Manjil 1990 Abbar (L) 0.515 0.425 Turkey Kocaeli 1999 Duzce (270) 0.364 0.557 Group 2 American Landers 1992 Yermo Fire (360) 0.152 0.291 TCU074 (EW) * 0.597 0.704 Beverly Hills-12520 Mulhol (035) 0.621 0.288 TCU045 (NS) 0.522 0.460

American Canyon Country (270) 0.472 0.411 TCU088 (EW) * 0.519 0.137 1994 Taiwan Chi-Chi 1999 Northridge LA-Saturn St (020) 0.468 0.372 TCU047 (NS) 0.407 0.333 LA-Hollywood Stor FF (360) 0.358 0.274 NST (NS) 0.399 0.329 Nishi-Akashi (000) 0.483 0.468 TCU089 (EW) * 0.355 0.508 Japan Kobe 1995 Tadoka (000) 0.296 0.245 American San 1971 Pacoima Dam (upper left abut) (164) 1.129 1.144 Taiwan Chi-Chi 1999 TCU128 (EW) * 0.144 0.642 Fernanado American 1987 Parachute Test Site (225) 0.429 1.342 Yarimca (150) 0.320 0.719 Superstition Turkey Kocaeli American 1999 1989 LosGatos (090) 0.409 0.957 Yarimca (060) 0.226 0.697 Lexington Dam American Cape New Zealand 1992 Petrolia (090) 0.605 0.885 2010 HORC (S72E) 0.472 0.698 Mendocino Darﬁeld Taiwan Turkey Erzincan 1992 Erzincan (NS) 0.385 1.071 2016 CHY089 (NS) 0.288 0.577 Group 3 Meinong Appl. Sci. 2020, 10, 7200 6 of 21

Table 1. Cont.

Earthquake Year Station Name PGA (g) PGV (m/s) Earthquake Year Station Name PGA (g) PGV (m/s) Pacoima Dam (upper left) (194) 0.989 1.033 HWA019 (EW) * 0.411 1.384 Rinaldi (228) 0.869 1.479 HWA008 (NS) 0.343 0.861 Sylmar-Converter Sta East (011) 0.851 1.209 MND016 (EW) * 0.306 1.336 American 1994 Northridge Sylmar-Olive View Med FF (360) 0.798 1.293 HWA007 (EW) * 0.295 1.034 Sylmar-Converter Sta (052) 0.617 1.162 HWA012 (EW) * 0.285 0.866 Jensen Filter Plant Generator 0.569 0.761 HWA063 (NS) 0.258 0.997 Building (022) TCU067 (EW) * 0.498 0.983 HWA028 (NS) 0.258 0.517 CHY101 (NS) 0.398 1.073Taiwan Hualien 2018 HWA009 (EW) * 0.255 1.104 TCU052 (EW) * 0.357 1.746 HWA008 (EW) * 0.235 0.992 CHY101 (EW) * 0.340 0.672 HWA 062 (EW) * 0.213 0.956 TCU075 (EW) * 0.330 1.161 TRB042 (EW) * 0.208 0.656 Taiwan Chi-Chi 1999 TCU102 (EW) * 0.310 0.874 HWA062 (NS) 0.207 0.764 CHY024 (EW) * 0.282 0.529 HWA011 (NS) 0.203 0.871 TCU063 (EW) * 0.183 0.440 HWA050 (NS) 0.202 0.763 TCU102 (NS) 0.172 0.713 TRB042 (NS) 0.190 0.826 TCU128 (NS) 0.166 0.626 PGA = peak ground acceleration; PGV = peak ground velocity. Appl. Sci. 2020, 10, 7200 7 of 21

3. Eﬀects of Diﬀerent Design Parameters on the Displacement of SRBs with Identical Acceleration Control Performance

3.1. Analytical Model for SRBs The simplified analytical model for characterizing the horizontal dynamic behavior of SRBs adopted in this study, conceptually and practically, can be decomposed into two parts, a multilinear elastic model and a rigid-perfectly plastic model to correspondingly represent the backbone and energy dissipation of SRBs, as illustrated in Figure3, from which the physical meanings of Equations (4) and (5) can be understood more clearly. For simplicity and essential comparison without other more complicated factors, the influences arising from different velocities, axial pressures, temperatures, and breakaway frictions on the friction damping behavior of SRBs [5,39] were neglected. In addition, the external disturbance and dynamic responses in only one horizontal principle direction were consideredAppl. Sci. 2020, herein. 10, x FOR PEER REVIEW 7 of 20

Figure 3. IdealizedIdealized composition of analytical models for SRBs. 3.2. SRB Models 3.2. SRB Models As summarized in Table2, four SRB models were designed with sloping angles of 3, 4, 6, and 8 As summarized in Table 2, four SRB models were designed with sloping angles of 3, 4, 6, and 8 degrees, hereafter denoted as SRB-3, SRB-4, SRB-6, and SRB-8, respectively. To keep an approximately degrees, hereafter denoted as SRB-3, SRB-4, SRB-6, and SRB-8, respectively. To keep an constant horizontal acceleration response of 0.08 g for these four models at the slope rolling stage, approximately constant horizontal acceleration response of 0.08 g for these four models at the slope which is different from the comparison basis of Wang et al.’s previous study [34], the friction damping rolling stage, which is different from the comparison basis of Wang et al.’s previous study [34], the forces were accordingly designed as 550 N, 460 N, 300 N, and 120 N for SRB-3, SRB-4, SRB-6, and SRB-8, friction damping forces were accordingly designed as 550 N, 460 N, 300 N, and 120 N for SRB-3, SRB- respectively. In addition, to coordinate different sloping angles of 3, 4, 6, and 8 degrees at the slope 4, SRB-6, and SRB-8, respectively. In addition, to coordinate different sloping angles of 3, 4, 6, and 8 rolling stage with a fixed curvature radius of 100 mm at the arc rolling stage, the arc rolling ranges degrees at the slope rolling stage with a fixed curvature radius of 100 mm at the arc rolling stage, the (or horizontal displacement ranges) at this stage for SRB-3, SRB-4, SRB-6, and SRB-8 were designed to arc rolling ranges (or horizontal displacement ranges) at this stage for SRB-3, SRB-4, SRB-6, and SRB- be 10.5 mm, 14 mm, 21 mm, and 28 mm, respectively. The friction damping force designed at the arc 8 were designed to be 10.5 mm, 14 mm, 21 mm, and 28 mm, respectively. The friction damping force rolling stage, certainly, was the same as that at the slope rolling stage. The hysteresis loops of SRB-3, designed at the arc rolling stage, certainly, was the same as that at the slope rolling stage. The SRB-4, SRB-6, and SRB-8 are illustrated in Figure4 for better comparison. hysteresis loops of SRB-3, SRB-4, SRB-6, and SRB-8 are illustrated in Figure 4 for better comparison.

Table 2. Design parameters of SRBs and FPBs. Table 2. Design parameters of SRBs and FPBs. Arc Rolling Stage Slope Rolling Stage Arc Rolling Stage Slope Rolling Stage Range / Design Friction Sloping Angle (◦) Friction Model Range / Design Curvature Sloping Angle (°) Displacement Curvature DampingFriction Upper Bearing Lower Bearing DampingFriction Model Displacement Radius (mm) Upper Lower (DD)i (mm) Radius ForceDamping (N) Plate Plate ForceDamping (N) (DD)i Bearing Bearing SRB-3 10.5 100(mm) Force 550 (N) 0 3Force 550 (N) SRB-4(mm) 14 100 460 0Plate 4Plate 460 SRB-3SRB-6 2110.5 100 100 300550 0 0 6 3 300550 SRB-4SRB-8 2814 100 100 120460 00 8 4 120460 FPB-3-2M 63.78 2550 550 - - - FPB-3-3MSRB-6 494.7021 19,800 100 550300 -0 - 6 -300 FPB-6-2MSRB-8 90.6428 1810 100 300120 -0 - 8 -120 FPB-3-2MFPB-6-3M 704.3063.78 14,100 2550 300 550 ------FPB-3-3M 494.70 19,800 550 - - - FPB-6-2M 90.64 1810 300 - - - FPB-6-3M 704.30 14,100 300 - - -

Figure 4. Hysteresis loops of four different SRB models with the same horizontal acceleration control performance.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 20

Figure 3. Idealized composition of analytical models for SRBs.

3.2. SRB Models As summarized in Table 2, four SRB models were designed with sloping angles of 3, 4, 6, and 8 degrees, hereafter denoted as SRB-3, SRB-4, SRB-6, and SRB-8, respectively. To keep an approximately constant horizontal acceleration response of 0.08 g for these four models at the slope rolling stage, which is different from the comparison basis of Wang et al.’s previous study [34], the friction damping forces were accordingly designed as 550 N, 460 N, 300 N, and 120 N for SRB-3, SRB- 4, SRB-6, and SRB-8, respectively. In addition, to coordinate different sloping angles of 3, 4, 6, and 8 degrees at the slope rolling stage with a fixed curvature radius of 100 mm at the arc rolling stage, the arc rolling ranges (or horizontal displacement ranges) at this stage for SRB-3, SRB-4, SRB-6, and SRB- 8 were designed to be 10.5 mm, 14 mm, 21 mm, and 28 mm, respectively. The friction damping force designed at the arc rolling stage, certainly, was the same as that at the slope rolling stage. The hysteresis loops of SRB-3, SRB-4, SRB-6, and SRB-8 are illustrated in Figure 4 for better comparison.

Table 2. Design parameters of SRBs and FPBs.

Arc Rolling Stage Slope Rolling Stage Range / Design Sloping Angle (°) Curvature Friction Friction Model Displacement Upper Lower Radius Damping Damping (DD)i Bearing Bearing (mm) Force (N) Force (N) (mm) Plate Plate SRB-3 10.5 100 550 0 3 550 SRB-4 14 100 460 0 4 460 SRB-6 21 100 300 0 6 300 SRB-8 28 100 120 0 8 120 FPB-3-2M 63.78 2550 550 - - - FPB-3-3M 494.70 19,800 550 - - - Appl.FPB-6-2M Sci. 2020, 10 , 7200 90.64 1810 300 - - - 8 of 21 FPB-6-3M 704.30 14,100 300 - - -

FigureFigure 4. HysteresisHysteresis loops loops of offour four different diﬀerent SRB SRB models models with withthe same the horizontal same horizontal acceleration acceleration control controlperformance. performance.

3.3. Numerical Simulations and Results A to-be-protected object above the SRB models was assumed to be important equipment with a mass of 1000 kg located on the first floor of a building structure, e.g., a storage system for wafers used in high-tech industries. For simplicity, the SDOF assumption still held in the numerical simulation, i.e., the isolation mode predominated the overall dynamic behavior of the isolated equipment. At each time step of ground motions of Groups 1, 2, and 3, the Newmark-β method with linear acceleration was used for solving Equations (4) and (5) correspondingly when at the slope and arc rolling stages. During two or more consecutive time steps in numerical integration, there could be no relative motion between the upper and lower bearing plates, as shown in Figure1b. Therefore, the sticking phase [40], which will significantly affect the post-earthquake re-centering performance, was taken into .. consideration. By setting u in Equations (4) and (5) equal to zero, the critical characteristic strengths at the slope and arc rolling stages, which are the criteria to identify if the slipping or sticking phase occurs [40], could be defined and calculated by Equations (6) and (7), respectively. More precisely, C if F is calculated to be smaller than the designed damping force FSRB at a specific time step (e.g., SRB .. . at ti), the sticking phase will occur. In this case, at the time step ti, u and u should be set to zero, C and u should remain the same as that at the previous time step ti-1. Note that FSRB at the slope rolling .. C stage is only related to ug once θ is determined, while F at the arc rolling stage is also related to u, .. SRB in addition to ug, once RSRB is determined.

C .. MgθSRB F = Mug + sgn(u) , (6) SRB 2

C .. Mg FSRB = Mug + u . (7) 4RSRB With sloping angles of 3, 4, 6, and 8 degrees at the slope rolling stage, the magnitudes of FC after .. SRB excitations, i.e., ug = 0, were correspondingly 256.83 N, 342.43 N, 513.65 N, and 684.87 N, of which the ﬁrst two and the last two are smaller than and larger than their designed damping forces, respectively. .. It also means that for the designs with sloping angles of 3 and 4 degrees, if the magnitudes of ug during external disturbance are correspondingly larger than 0.030 g and 0.012 g with the same direction as u or larger than 0.082 g with the opposite direction to u, the motion between the superior and inferior bearing plates can still be in the slipping phase. The horizontal maximum and residual displacement responses of SRB-3, SRB-4, SRB-6, and SRB-8 subjected to ground motions of Groups 1, 2, and 3 are presented in Figures5–7, respectively. As for the maximum horizontal displacement responses subjected to ground motions of Group 1, another presentation, i.e., the maximum horizontal displacement responses versus PGV in a logarithm scale, is also provided for better observation. The maximum horizontal displacement responses of SRBs have been numerically demonstrated to be almost linearly proportional to PGV on a logarithm scale [41]. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 20

3.3. Numerical Simulations and Results A to-be-protected object above the SRB models was assumed to be important equipment with a mass of 1000 kg located on the first floor of a building structure, e.g., a storage system for wafers used in high-tech industries. For simplicity, the SDOF assumption still held in the numerical simulation, i.e., the isolation mode predominated the overall dynamic behavior of the isolated equipment. At each time step of ground motions of Groups 1, 2, and 3, the Newmark-β method with linear acceleration was used for solving Equations (4) and (5) correspondingly when at the slope and arc rolling stages. During two or more consecutive time steps in numerical integration, there could be no relative motion between the upper and lower bearing plates, as shown in Figure 1b. Therefore, the sticking phase [40], which will significantly affect the post-earthquake re-centering performance, was taken into consideration. By setting 𝑢 in Equations (4) and (5) equal to zero, the critical characteristic strengths at the slope and arc rolling stages, which are the criteria to identify if the slipping or sticking phase occurs [40], could be defined and calculated by Equations (6) and (7), respectively. More precisely, if 𝐹 is calculated to be smaller than the designed damping force FSRB at a specific time step (e.g., at ti), the sticking phase will occur. In this case, at the time step ti, 𝑢 and 𝑢 should be set to zero, and u should remain the same as that at the previous time step ti-1. Note that 𝐹 at the slope rolling stage is only related to 𝑢 once θ is determined, while 𝐹 at the arc rolling stage is also related to u, in addition to 𝑢 , once RSRB is determined.

𝑀𝑔𝜃 𝐹 =𝑀𝑢 + sgn𝑢, (6) 2

𝑀𝑔 𝐹 =𝑀𝑢 + 𝑢. (7) 4𝑅 With sloping angles of 3, 4, 6, and 8 degrees at the slope rolling stage, the magnitudes of 𝐹 after excitations, i.e., 𝑢 =0, were correspondingly 256.83 N, 342.43 N, 513.65 N, and 684.87 N, of which the first two and the last two are smaller than and larger than their designed damping forces, respectively. It also means that for the designs with sloping angles of 3 and 4 degrees, if the magnitudes of 𝑢 during external disturbance are correspondingly larger than 0.030 g and 0.012 g with the same direction as u or larger than 0.082 g with the opposite direction to u, the motion between the superior and inferior bearing plates can still be in the slipping phase. The horizontal maximum and residual displacement responses of SRB-3, SRB-4, SRB-6, and SRB- 8 subjected to ground motions of Groups 1, 2, and 3 are presented in Figures 5–7, respectively. As for the maximum horizontal displacement responses subjected to ground motions of Group 1, another presentation, i.e., the maximum horizontal displacement responses versus PGV in a logarithm scale, Appl. Sci. 2020, 10, 7200 9 of 21 is also provided for better observation. The maximum horizontal displacement responses of SRBs have been numerically demonstrated to be almost linearly proportional to PGV on a logarithm scale [41].In these In these ﬁgures, figures, in addition in addition to mean to responses,mean response the 25ths, the and 25th 75th and percentile 75th percentile responses responses are also provided are also providedfor further for statistical further statistical comparison. comparison. Since all theSince SRB all models the SRB were model designeds were withdesigned the same with curvaturethe same curvatureradius at theradius arc at rolling the arc stage, rolling the stage, numerical the numerical results atresults the arc at the rolling arc rolling stage (including stage (including still at still the atequilibrium the equilibrium position position if there if arethere any) are are any) covered are covered in the ﬁguresin the figures and the and following the following discussions. discussions.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20 (a) Maximum displacement responses.

(b) Residual displacement responses. (b) Residual displacement responses. (b) Residual displacement responses. Figure 5. Comparisons of displacement control performances of different SRB models subjected to Figure 5. Comparisons of displacement control performances of different SRB models subjected to groundFigure 5.motions Comparisons of Group of of 1. displacement control control performances performances of of different diﬀerent SRB models subjected to ground motions of Group 1. ground motions of Group 1.

(a) Maximum displacement responses. (b) Residual displacement responses. (a) Maximum displacement responses. (b) Residual displacement responses. (a) Maximum displacement responses. (b) Residual displacement responses. Figure 6. Comparisons of displacement control performances of different SRB models subjected to FigureFigure 6. ComparisonsComparisons of of displacement displacement control control performances performances of of different diﬀerent SRB SRB models subjected to to groundFigure 6.motions Comparisons of Group of 2. displacement control performances of different SRB models subjected to groundground motions motions of Group 2. ground motions of Group 2.

(a) Maximum displacement responses. (b) Residual displacement responses. (a) Maximum displacement responses. (b) Residual displacement responses. (a) Maximum displacement responses. (b) Residual displacement responses. FigureFigure 7. 7. ComparisonsComparisons of of displacement displacement control control performances performances of of different diﬀerent SRB SRB models models subjected subjected to to Figure 7. Comparisons of displacement control performances of different SRB models subjected to groundFigureground 7.motions motions Comparisons of of Group Group of 3. 3.displacement control performances of different SRB models subjected to ground motions of Group 3. ground motions of Group 3. As observed from these figures, based on retaining the same horizontal acceleration control performance,As observed adopting from athese constant figures, sloping based angle on retaining and damping the same force horizontal for SRBs accelerationmight not entirelycontrol controlperformance, their horizontal adopting maximuma constant and slopingsloping residual angleangle displacement andand dampingdamping responses forceforce forforsimultaneously. SRBsSRBs mightmight notnotOn averageentirelyentirely andcontrol in terms their horizontalof distribution, maximum the increase and residual in damp diingsplacement forces (or responses energy dissipation simultaneously. capability) On average or the decreaseand in terms in sloping of distribution, angles will the lead increase to a decrease in damp ininging horizontal forcesforces (or(or isolation energyenergy displacementdissipationdissipation capability)capability) but an increase oror thethe indecrease residual in displacement,sloping angles evenwill lead though to a mostdecrease of th ine horizontal horizontalresidual displacement isolationisolation displacementdisplacement responses butbutof all anan theincreaseincrease SRB modelsinin residualresidual subjected displacement,displacement, to ground eveneven motions thoughthough of Group mostmost of of2, thi.e.the, residualfar-field grounddisplacement motions, responses were very of alllimited the SRB (an models subjected to ground motions of Group 2, i.e., far-field ground motions, were very limited (an ordermodels of subjected 10−4 to 10 to− 3ground m) and motions practically of Group acceptable. 2, i.e., far-field The satisfactory ground motions, self-centering were very performances limited (an −4 −3 order of 10 −4 toto 1010−3 m)m) andand practicallypractically acceptable.acceptable. TheThe satisfactorysatisfactory self-centeringself-centering𝐹 performancesperformances revealed by SRB-6 and SRB-8 is no surprise since their calculated values of after excitations as revealed by SRB-6 and SRB-8 is no surprise since their calculated values of 𝐹 after excitations as perrevealed Equation by SRB-6 (6) were and SRB-8larger isthan no surpritheir designse sinceed their damping calculated forces. values Their of higher𝐹 after potential excitations energy as comparedper Equation with (6) SRB-3 were and larger SRB-4 than when their at design the sameed damping horizontal forces. displacement Their higher response potential can alsoenergy be physicallycompared understood.with SRB-3 andTaking SRB-4 SRB-8, when under at the the same 1987 horizontal American displacement Superstition, response Parachute can Test also Site be (225)physically as an exampleunderstood. for an Taking insight SRB-8, into the under details, the as 1987 shown American in Figure Superstition, 8, the displacement Parachute response Test Site at the(225)(225) end asas ofanan the exampleexample ground forfor motion anan insightinsight and into intothe finalthethe details,details, residual asas displacementshownshown inin FigureFigure response 8,8, thethe displacement displacementafter free vibration responseresponse were atat 573.20thethe endend mm ofof thetheand groundground 0.54 mm, motionmotion respectively. andand thethe finalForfinal SRB-3 residualresidual and displacementdisplacement SRB-4, in some responseresponse cases, whenafterafter free freethe magnitudesvibrationvibration werewere of 573.20 mm and 0.54 mm, respectively. For SRB-3 and SRB-4, in some cases, when the magnitudes of Appl. Sci. 2020, 10, 7200 10 of 21

As observed from these figures, based on retaining the same horizontal acceleration control performance, adopting a constant sloping angle and damping force for SRBs might not entirely control their horizontal maximum and residual displacement responses simultaneously. On average and in terms of distribution, the increase in damping forces (or energy dissipation capability) or the decrease in sloping angles will lead to a decrease in horizontal isolation displacement but an increase in residual displacement, even though most of the residual displacement responses of all the SRB models subjected 4 to ground motions of Group 2, i.e., far-field ground motions, were very limited (an order of 10− to 3 10− m) and practically acceptable. The satisfactory self-centering performances revealed by SRB-6 C and SRB-8 is no surprise since their calculated values of FSRB after excitations as per Equation (6) were larger than their designed damping forces. Their higher potential energy compared with SRB-3 and SRB-4 when at the same horizontal displacement response can also be physically understood. Taking SRB-8, under the 1987 American Superstition, Parachute Test Site (225) as an example for an insight into the details, as shown in Figure8, the displacement response at the end of the ground Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 20 motion and the final residual displacement response after free vibration were 573.20 mm and 0.54 .. mm, at respectively. the end of ground For SRB-3 motions andwere SRB-4, sufficiently in somelarge cases, to when overcome the magnitudes their designed of u dampingg at the end forces of, groundtheir self motions-centerin wereg performances sufficiently large were tostill overcome acceptable their but designed not as dampinggood as the forces, performances their self-centering of SRB-6 𝑢𝑢̈𝑔𝑔 performancesand SRB-8. Inwere brief, stillan excessive acceptable damping but not force as good for asSRBs the will performances cause the sticking of SRB-6 phase and SRB-8.occurring In brief,more anfrequently excessive during damping motion force and for thus SRBs generally will cause has the an sticking adverse phase effect occurring on the self more-centering frequently capability during of motionSRBs. and thus generally has an adverse effect on the self-centering capability of SRBs.

0.6 originalDuring excitation ground motion original excitation 1 During ground motion originalDuring excitation and after plus durationground of motion zero input originalDuring excitation and after plus durationground of motion zero input 0.4 )

2 0.5 0.2

0 0

-0.2 -0.5 Displacement (m) Acceleraction (m/sec -0.4 -1 0 10 20 30 40 0 10 20 30 40 Time (sec) Time (sec)

FigureFigure 8.8.DisplacementDisplacement response response histories histories and residualand residual displacement displacement of SRB-8 of under SRB- the8 under 1987 Americanthe 1987 Superstition,American Superstition, Parachute TestParachute Site (225). Test Site (225). 4. Numerical Comparisons between FPBs and SRBs 4. Numerical Comparisons between FPBs and SRBs 4.1. Analytical Model for FPBs 4.1. Analytical Model for FPBs The simplified analytical model for describing the horizontal dynamic behavior of FPBs adopted in thisThe study, simplified conceptually analytical and model practically, for describing can be decomposed the horizontal into dynamic two parts, behavior a linear of elasticFPBs adopted model andin this a rigid-perfectly study, conceptually plastic andmodel practically, to correspondingly can be decomposed represent the into backbone two parts, and a linear energy elastic dissipation model ofand FPBs, a rigid as illustrated-perfectly plastic in Figure model9, from to correspondingly which the physical repr meaningesent the of backbone Equation and (1) can energy be understood dissipation moreof FPBs, clearly. as illustrated Similarly, in for Figure simplicity 9, from and which essential the physical comparison, meaning the of influences Equation of (1) di canfferent be understood velocities, axialmore pressures, clearly. Similarly, temperatures, for simplicity and breakaway and essential frictions comparison, on the friction the dampinginfluences behavior of different of FPBs velocities, [5,39] wereaxial neglected.pressures, Intemperatures, addition, the and external breakaway disturbance frictions and on dynamic the friction responses damping in only behavior one horizontal of FPBs principle[5,39] were directionneglected. were In considered addition, herein. the external disturbance and dynamic responses in only one horizontal principle direction were considered herein.

Figure 9. Idealized composition of analytical models for FPBs.

4.2. 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 (DD)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., Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 20

𝑢 at the end of ground motions were sufficiently large to overcome their designed damping forces, their self-centering performances were still acceptable but not as good as the performances of SRB-6 and SRB-8. In brief, an excessive damping force for SRBs will cause the sticking phase occurring more frequently during motion and thus generally has an adverse effect on the self-centering capability of SRBs.

Figure 8. Displacement response histories and residual displacement of SRB-8 under the 1987 American Superstition, Parachute Test Site (225).

4. Numerical Comparisons between FPBs and SRBs

4.1. Analytical Model for FPBs The simplified analytical model for describing the horizontal dynamic behavior of FPBs adopted in this study, conceptually and practically, can be decomposed into two parts, a linear elastic model and a rigid-perfectly plastic model to correspondingly represent the backbone and energy dissipation of FPBs, as illustrated in Figure 9, from which the physical meaning of Equation (1) can be understood more clearly. Similarly, for simplicity and essential comparison, the influences of different velocities, axial pressures, temperatures, and breakaway frictions on the friction damping behavior of FPBs [5,39]Appl. Sci. were2020 , neglected.10, 7200 In addition, the external disturbance and dynamic responses in only11 ofone 21 horizontal principle direction were considered herein.

Figure 9. IdealizedIdealized composition composition of of analytical analytical models models for for FPBs. FPBs. 4.2. FPB Models 4.2. FPB Models To quantitatively and objectively discuss the potential advantages of FPBs and SRBs in diverse 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 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. design criteria for the FPB models first and then were compared with those of the FPB models Ground motions of Groups 2 and 3, as detailed in Section2 and listed in Table1, were also employed designed. Ground motions of Groups 2 and 3, as detailed in Section 2 and listed in Table 1, were also herein. More precisely, first, the average of maximum horizontal displacement responses of each SRB employed herein. More precisely, first, the average of maximum horizontal displacement responses model, respectively, subjected to ground motions of Groups 2 and 3 was considered as the design of each SRB model, respectively, subjected to ground motions of Groups 2 and 3 was considered as displacement (DD)i (i = 2 and 3) for the FPB model, as given in Equation (8). At the design displacement, the design displacement (DD)i (i = 2 and 3) for the FPB model, as given in Equation (8). At the design the FPB model had the same horizontal acceleration response as the SRB model, i.e., 0.08 g. Through displacement, the FPB model had the same horizontal acceleration response as the SRB ·model, i.e., 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 Table2. 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.

Pn max j=1 Di,j (D ) = , i = 2 and 3, (8) D i n where i represents the ground motions of Group i (i = 2 and 3); j represents the jth 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 = max 25 and 41, respectively); Di, j is the maximum horizontal displacement response of SRB-3 or SRB-6 subjected the jth ground motion of Group i. Appl.Appl. Sci. Sci. 2020 2020, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 1111 ofof 2020

0.08·0.08·gg.. Through Through referring referring to to SRB-3 SRB-3 subjected subjected to to ground ground motions motions of of Group Group 2, 2, the the friction friction damping damping force force designeddesigned forfor thethe FPBFPB modelmodel (denoted(denoted asas FPB-3-2MFPB-3-2M hehereafter)reafter) waswas thethe samesame asas thatthat forfor SRB-3,SRB-3, asas illustratedillustrated inin FigureFigure 10a.10a. InIn otherother words,words, atat anyany horizontalhorizontal displacement,displacement, FPB-3-2MFPB-3-2M andand SRB-3SRB-3 hadhad thethe samesame energyenergy dissipationdissipation areaarea causedcaused byby thethe frictionfriction dampingdamping force.force. Similarly,Similarly, throughthrough referringreferring toto SRB-3SRB-3 subjectedsubjected toto groundground motionsmotions ofof GroupGroup 3,3, SRB-6SRB-6 subjectedsubjected toto groundground motionsmotions ofof GroupGroup 2,2, andand SRB-6SRB-6 subjectedsubjected to to groundground motionsmotions of of GroupGroup 3, 3, the the frictionfriction dampingdamping forcesforces designed designed for for thethe FPBFPB modelsmodels (correspondingly(correspondingly denoteddenoted asas FPB-3-3M,FPB-3-3M, FPB-FPB-6-2M,6-2M, andand FPB-6-3MFPB-6-3M hereafter)hereafter) werewere identicalidentical toto thosethose forfor SRB-3,SRB-3, SRB-6,SRB-6, andand SRB-6,SRB-6, respectivelyrespectively,, asas illustratedillustrated inin FigureFigure 10.10. TheThe curvedcurved slidingsliding surfacessurfaces designeddesigned forfor FPB-3-2M,FPB-3-2M, FPB-3-3M,FPB-3-3M, FPB-6-2M,FPB-6-2M, andand FPB-6-3M,FPB-6-3M, asas wellwell asas thethe slopedsloped rollingrolling surfacessurfaces designeddesigned forfor SRB-3SRB-3 andand SRB-6,SRB-6, areare concepconceptuallytually illustratedillustrated inin FigureFigure 1111 forfor aa betterbetter comparisoncomparison ofof theirtheir differentdifferent geometricalgeometrical dimensiodimensions.ns. BasedBased onon thethe designdesign criteriacriteria aforementioned,aforementioned, thethe designdesign parametersparameters ofof FPB-3-2M,FPB-3-2M, FPB-3-3M,FPB-3-3M, FPFPB-6-2M,B-6-2M, andand FPB-6-3MFPB-6-3M cancan bebe determined,determined, asas alsoalso listedlisted inin TableTable 2.2. Note:Note: basedbased onon thethe designdesign criteria,criteria, whenwhen subjectedsubjected toto groundground motionsmotions ofof GroupGroup 3,3, i.e.,i.e., near-faultnear-fault pulse-likepulse-like groundground motions,motions, ththee FPBFPB modelsmodels werewere designeddesigned withwith aa considerableconsiderable curvaturecurvature radius,radius, whichwhich mightmight notnot oftenoften bebe seenseen forfor practicalpractical applicationapplication whenwhen simplysimply consideringconsidering far-fieldfar-field groundground motions.motions. ∑ 𝐷 ∑ 𝐷,, 𝐷𝐷 == , , 𝑖 𝑖 = = 2 2 andand 3, 3, (8)(8) 𝑛𝑛 wherewhere ii representsrepresents thethe groundground motionsmotions ofof GroupGroup ii ((ii == 22 andand 3);3); jj representsrepresents thethe jjthth groundground motionmotion ofof GroupGroup 22 oror 3;3; nn representsrepresents thethe numbernumber ofof groundground motionmotionss ofof GroupGroup 22 oror 33 (for(for GroupsGroups 22 andand 3,3, nn == 2525 andand 41,41, respectively);respectively); 𝐷𝐷, isis thethe maximummaximum horizontalhorizontal displacedisplacementment responseresponse ofof SRB-3SRB-3 oror SRB-6SRB-6 Appl. Sci. 2020, 10, 7200 , 12 of 21 subjectedsubjected thethe jjthth groundground motionmotion ofof GroupGroup ii..

SRB-3SRB-3 SRB-6SRB-6 FPB-3-2M,FPB-3-2M, FPB-3-3M FPB-3-3M FPB-6-2M,FPB-6-2M, FPB-6-3M FPB-6-3M ion (g) t ion (g) t a a r r Accele Accele

(DD)i (DD)i (DD)i (DD)i Displacement Displacement Displacement Displacement ((aa)) SRB-3, SRB-3, FPB-3-2M, FPB-3-2M, and and FPB-3-3M. FPB-3-3M. ( (bb)) SRB-6, SRB-6, FPB-6-2M, FPB-6-2M, and and FPB-6-3M. FPB-6-3M.

FigureFigure 10. 10.10. Illustrations Illustrations of of design design criteria criteria for forfor FPB FPBFPB models. models.models.

FigureFigure 11. 11. IllustrationsIllustrations of ofof curved curvedcurved sliding slidingsliding and andand sloped slopedsloped rolling rollingrolling surfaces surfacessurfaces correspondingly correspondinglycorrespondingly designed designeddesigned for FPB forfor FPBandFPB and SRBand SRB models.SRB models. models.

TheThe designdesigndesign criteria criteriacriteria for forfor the thethe FPB FPBFPB models modelsmodels aforementioned aforemaforementionedentioned were basedwerewere onbasedbased the meanonon thethe response meanmean response subjectedresponse subjectedtosubjected ground toto motions groundground of motionsmotions Group 2 of orof Group 3,Group which 22 isoror called 3,3, whichwhich Benchmark isis calledcalled I hereafter. BenchmarkBenchmark For II more hereafter.hereafter. comprehensive ForFor moremore comprehensivecomparisoncomprehensive and comparisoncomparison considering andand the consideringconsidering eﬀects caused thethe effeeffe bycts diversects causedcaused ground byby diversediverse motions, grougrou dindndﬀ motions,erentmotions, FPB differentdifferent models FPBdesignedFPB modelsmodels based designeddesigned on the basedbased maximum onon thethe horizontalmaximummaximum displacementhorizonthorizontalal displacementdisplacement response of responseresponse each SRB ofof modeleacheach SRBSRB subjected modelmodel to each ground motion, i.e., (DD)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). Among these, the FPB models designed based on the responses of SRB-3 and SRB-6 subjected to any ground motion of Group 2 were denoted as generalized forms of FPB-3-2 and FPB-6-2, respectively, and those subjected to any ground motion of Group 3 were denoted as generalized forms of FPB-3-3 and FPB-6-3, respectively.

max (DD) = D , i = 2 and 3; j = 1 n. (9) i,j i, j ∼ 4.3. Comparison Results and Discussions The protected object above the FPB models and the associated conditions were the same as those introduced in Section 3.3. For numerically simulating the FPB models, at each time step of ground motions of Groups 2 and 3, the Newmark-β method with linear acceleration was used for solving Equation (1). The same method as that described in Section 3.3 but a diﬀerent formula, as given in Equation (10), was employed herein to consider the sticking phase [40] for the FPB models.

C .. Mg FFPB = Mug + u , (10) RFPB Appl. Sci. 2020, 10, 7200 13 of 21

The maximum horizontal acceleration and displacement responses, as well as the residual displacement responses of SRB-3, FPB-3-2M, FPB-3-3M, FPB-3-2, FPB-3-3, SRB-6, FPB-6-2M, FPB-6-3M, FPB-6-2, and FPB-6-3 subjected to ground motions of Groups 2 and 3, are presented in Figures 12–14, respectively. For better comparison, the response ratios of FPB-3-2M to SRB-3, FPB-3-3M to SRB-3, FPB-3-2 to SRB-3, FPB-3-3 to SRB-3, FPB-6-2M to SRB-6, FPB-6-3M to SRB-6, FPB-6-2 to SRB-6, and FPB-6-3 to SRB-6 are also presented. The numbers indicated in these figures represent the amounts of ground motions under which the control performances of the FPB models were better than those of the corresponding SRB models. In addition to mean responses, the 25th and 75th percentile responses are also provided for further statistical comparison. The numerical results of SRB-3 and SRB-6 at the arc rolling stage, i.e., the maximum horizontal acceleration responses less than 0.08 g shown in Figure 12, · are covered in the figures and the following discussions. As observed from Figure 13, evidently, since all the maximum horizontal displacement responses were larger than zero, the situation that the SRB or FPB models were not activated (i.e., were not functional) under the ground motions considered did not occur. Figures 15 and 16 present the displacement and acceleration response histories as well as the hysteresis loops of SRB-3, FPB-3-2, SRB-6, and FPB-6-2 under 1999 Taiwan Chi-Chi, TCU074 (EW) (one of the ground motions of Group 2) and those of SRB-3, FPB-3-3, SRB-6, and FPB-6-3 under 1999 Taiwan Chi-Chi, TCU052 (EW) (one of the ground motions of Group 3), respectively. Note again that this numerical study aimed to discuss the potential merits of FPBs and SRBs quantitatively based on the design criteria described in Section 4.2, rather than to design practically acceptable horizontal isolation displacement as small (or satisfactory) as possible for FPBs and SRBs against the ground motion considered. As observed from Figures 12, 15 and 16, the distribution of maximum horizontal acceleration responses of the FPB models was much more scattered than that of the SRB models, even considering the responses of the SRB models at the arc rolling stage (only SRB-6 under one far-field ground motion). It is mainly because of the unique feature of SRBs—constant acceleration control. Under 15 (13) and 20 (21) sets (i.e., the majority) of far-field ground motions, FPB-3-2M (FPB-3-2) and FPB-6-2M (FPB-6-2) had better horizontal acceleration control performances than SRB-3 and SRB-6, respectively, in particular, the latter. Under 28 (28) sets and 37 (36) sets (i.e., the majority) of pulse-like near-fault ground motions, FPB-3-3M (FPB-3-3) and FPB-6-3M (FPB-6-3) had better horizontal acceleration control performances than SRB-3 and SRB-6, respectively, in particular, the latter. Without considering the extremes, the response ratios on average, regardless of being subjected to far-field or pulse-like near-fault ground motions, were less than unity. To sum up, under the ground motions considered, if simply observing the amount or the average of response ratios, the FPB models might potentially have an advantage over the SRB models in terms of horizontal acceleration control. In addition, by comparing FPB-6-2M (FPB-6-2) with FPB-3-2M (FPB-3-2) and FPB-6-3M (FPB-6-3) with FPB-3-3M (FPB-3-3) in terms of maximum horizontal acceleration, with the same acceleration control performance, the FPB models designed with a larger design displacement, a smaller curvature radius, and a smaller damping force might potentially have a better horizontal acceleration control performance than those designed with a smaller design displacement, a larger curvature radius, and a larger damping force. Appl. Sci. 2020, 10, 7200 14 of 21 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 20 ) 2 Max acceleraction (m/sec acceleraction Max Ratio

13 21 15 20

(a) Group 2 (25 sets of far-field ground motions).

28 36 28 37

(b) Group 3 (41 sets of pulse-like near-fault ground motions).

FigureFigure 12. Comparisons 12. Comparisons of ofmaximum maximum horizontal horizontal accelerationacceleration responses responses of SRBs of SRBs and FPBsand subjectedFPBs subjected to to differentdiﬀerent ground ground motions. motions. Appl. Sci. 2020, 10, 7200 15 of 21 Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20

13 21 14 18

(a) Group 2 (25 sets of far-field ground motions).

28 36 29 37

(b) Group 3 (41 sets of pulse-like near-fault ground motions).

Figure 13.Figure Comparisons 13. Comparisons of maximum of maximum horizontal horizontal displaceme displacementnt responses responses of SRBs of SRBs and FPBs and subjectedFPBs subjected to diﬀerent ground motions. to different ground motions. Appl. Sci.Appl. 2020 Sci., 102020, x FOR, 10, 7200PEER REVIEW 16 of 21 16 of 20 Residualdisplacement (m) Ratio

11 10 9 5

(a) Group 2 (25 sets of far-field ground motions). Residual displacement (m)

13 1 16 2

(b) Group 3 (41 sets of pulse-like near-fault ground motions).

Figure 14. Comparisons of residual displacement responses of SRBs and FPBs subjected to diﬀerent Figure 14. Comparisons of residual displacement responses of SRBs and FPBs subjected to different ground motions. ground motions. Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 20

Appl.Appl. Sci. 2020,, 10,, 7200x FOR PEER REVIEW 1717 of of 2120 ) ) 2 2 ) ) 2 2 Displacement (m) Displacement Acceleration (m/sec Acceleration (m/sec Acceleration Displacement (m) Displacement

(m/sec Acceleration (m/sec Acceleration ) )

2 2 ) ) 2 2 Displacement (m) Displacement Acceleration (m/sec Acceleration (m/sec Acceleration Displacement (m) Displacement (m/sec Acceleration (m/sec Acceleration

(a) Displacement response (b) Acceleration response histories. (c) Hysteresis loops. histories. (a) Displacement response (b) Acceleration response histories. (c) Hysteresis loops. Figure 15.histories. Numerical results of SRB-3, FPB-3-2, SRB-6, and FPB-6-2 under 1999 Taiwan Chi-Chi, FigureTCU074 15. (EW)Numerical (Group 2). results of SRB-3, FPB-3-2, SRB-6, and FPB-6-2 under 1999 Taiwan Chi-Chi, Figure 15. Numerical results of SRB-3, FPB-3-2, SRB-6, and FPB-6-2 under 1999 Taiwan Chi-Chi, TCU074 (EW) (Group 2). TCU074 (EW) (Group 2). TCU052(EW) TCU052(EW) TCU052(EW) 3 1 1 SRB-3 SRB-3 SRB-3 FPB-3-3 FPB-3-3 FPB-3-3 TCU052(EW) TCU052(EW) TCU052(EW) 1.53 0.51 0.51 SRB-3 SRB-3 SRB-3 FPB-3-3 FPB-3-3 FPB-3-3 1.50 0.50 0.50

-1.50 -0.50 -0.50

-1.5-3 -0.5-1 -0.5-1 0 50 100 0 50 100 -3 -1.5 0 1.5 3 Time (sec) Time (sec) Displacement (m) -3 TCU052(EW) -1 TCU052(EW) -1 TCU052(EW) 30 50 100 10 50 100 1-3 -1.5 0 1.5 3 SRB-6 Time (sec) SRB-6 Time (sec) SRB-6Displacement (m) FPB-6-3 FPB-6-3 FPB-6-3 TCU052(EW) TCU052(EW) TCU052(EW) 1.53 0.51 0.51 SRB-6 SRB-6 SRB-6 FPB-6-3 FPB-6-3 FPB-6-3 1.50 0.50 0.50

-1.50 -0.50 -0.50

-1.5-3 -0.5-1 -0.5-1 0 50 100 0 50 100 -3 -1.5 0 1.5 3 Time (sec) Time (sec) Displacement (m) -3 -1 -1 0 50 100 0 50 100 -3 -1.5 0 1.5 3 (a) DisplacementTime (sec response) (b) AccelerationTime response (sec) histories. (c) HysteresisDisplacement loops. (m) histories. (a) Displacement response (b) Acceleration response histories. (c) Hysteresis loops. Figure 16. Numerical results of SRB-3, FPB-3-3, SRB-6, and FPB-6-3 under 1999 Taiwan Chi-Chi, Figure 16.histories. Numerical results of SRB-3, FPB-3-3, SRB-6, and FPB-6-3 under 1999 Taiwan Chi-Chi, TCU052 (EW) (Group 3). TCU052 (EW) (Group 3). Figure 16. Numerical results of SRB-3, FPB-3-3, SRB-6, and FPB-6-3 under 1999 Taiwan Chi-Chi, As observed from Figures 13, 15 and 16, under 14 (13) and 18 (21) sets (i.e., the majority) of 5. ConclusionsTCU052 (EW) (Group 3). far-field ground motions, FPB-3-2M (FPB-3-2) and FPB-6-2M (FPB-6-2) had smaller horizontal isolation displacement5. ConclusionsBased on than the SRB-3well-developed and SRB-6, simplified respectively, analyt inical particular, models the for latter. FPBs Underand SRBs, 29 (28) this sets study and 37attempts (36) sets to (i.e.,numerically the majority) discuss of pulse-liketheir passive near-fault control ground performances, motions, FPB-3-3Min terms (FPB-3-3)of horizontal and Based on the well-developed simplified analytical models for FPBs and SRBs, this study FPB-6-3Mtransmitted (FPB-6-3) acceleration, had smaller isolation horizontal displacement, isolation and displacement post-earthquake than SRB-3residual and displacement, SRB-6, respectively, which attempts to numerically discuss their passive control performances, in terms of horizontal inhave particular, been rarely the latter. studied Note before. that theTo amountshave objective for FPB-3-2 comparison and FPB-6-2 results, shown the first in Figure criterion 13 are adopted the same in transmitted acceleration, isolation displacement, and post-earthquake residual displacement, which asthis those study shown for designing in Figure the12. FPB It is reasonablemodel was becausethat at the FPB-3-2 horizontal and FPB-6-2 isolation were displacement designed based identical on the to have been rarely studied before. To have objective comparison results, the first criterion adopted in maximumthe maximum horizontal horizontal displacement displacement responses response of SRB-3 of the and SRB SRB-6 model subjected under to a eachgiven ground seismic motion, demand, i.e., this study for designing the FPB model was that at the horizontal isolation displacement identical to Benchmarkthe two different II. If FPB-3-2 models (FPB-6-2) have the had same a larger horizontal horizontal acceleration acceleration control response performance. than SRB-3 The (or second SRB-6) the maximum horizontal displacement response of the SRB model under a given seismic demand, the two different models have the same horizontal acceleration control performance. The second Appl. Sci. 2020, 10, 7200 18 of 21 under a given ground motion, it also meant that FPB-3-2 (FPB-6-2) had a larger horizontal displacement response than SRB-3 (SRB-6), and vice versa. Without considering the extremes, the response ratios on average, regardless of being subjected to far-field or pulse-like near-fault ground motions, were less than unity. To sum up, under the ground motions considered, if simply observing the amount or the average of the response ratios, the FPB models might be potentially more effective than the SRB models in reducing maximum horizontal displacement responses. By comparing FPB-6-2M (FPB-6-2) with FPB-3-2M (FPB-3-2) and FPB-6-3M (FPB-6-3) with FPB-3-3M (FPB-3-3) in terms of maximum horizontal displacement, with the same acceleration control performance, the horizontal displacement control performances of the different FPB models seemed to be not very different. As observed from Figures 14–16, only under 9 (11) and 5 (10) sets (i.e., the minority) of far-field ground motions, FPB-3-2M (FPB-3-2) and FPB-6-2M (FPB-6-2) had superior re-centering performances to SRB-3 and SRB-6, respectively. Only under 16 (13) sets (i.e., the minority) and 2 (1) sets (i.e., the minority and even almost none) of pulse-like near-fault ground motions, FPB-3-3M (FPB-3-3) and FPB-6-3M (FPB-6-3) had superior re-centering performances to SRB-3 and SRB-6, respectively. Without considering the extremes, the response ratios on average, regardless of being subjected to far-field or pulse-like near-fault ground motions, were larger than unity. To sum up, under the ground motions considered, if simply observing the amount or the average of the response ratios, the SRB models, SRB-6 (i.e., the design with a larger sloping angle and a smaller damping force) in particular, might be potentially more effective than the FPB models in controlling residual displacement responses. It is because of its larger sloping angle (or higher potential energy, as explained in Section 3.3) at the design displacement shown in Figure 11. Another reason might be its smaller static residual displacement to which the theoretical or statistical permanent residual displacement is proportional [24,36]. However, it is emphasized again that with the same horizontal acceleration control performance, as clearly observed from Figure 13, SRB-6 almost had the largest, and even practically unacceptable, horizontal isolation displacement among all the SRB and FPB models, regardless of being subjected to far-field or pulse-like near-fault ground motions. In addition, by comparing FPB-6-2M (FPB-6-2) with FPB-3-2M (FPB-3-2) and FPB-6-3M (FPB-6-3) with FPB-3-3M (FPB-3-3) in terms of residual displacement, with the same acceleration control performance, the FPB models designed with a larger design displacement, a smaller curvature radius, and a smaller damping force might potentially have a better self-centering performance than those designed with a smaller design displacement, a larger curvature radius, and a larger damping force.

5. Conclusions Based on the well-developed simplified analytical models for FPBs and SRBs, this study attempts to numerically discuss their passive control performances, in terms of horizontal transmitted acceleration, isolation displacement, and post-earthquake residual displacement, which have been rarely studied before. To have objective comparison results, the first criterion adopted in this study for designing the FPB model was that at the horizontal isolation displacement identical to the maximum horizontal displacement response of the SRB model under a given seismic demand, the two different models have the same horizontal acceleration control performance. The second criterion was that at any horizontal isolation displacement, the two different models have the same energy dissipation capability. Some conclusions obtained from the numerical results and comparisons are made as follows.

1. This study briefly introduced and compared the features, equations of motion, and simplified analytical models of FPBs and SRBs. A large number of local and global ground motion records, including the so-called far-field and pulse-like near-fault ones, were adopted as the horizontal acceleration inputs for the numerical analysis on different SRB and FPB models designed based on the criteria considered. It is emphasized again that the purpose of the diverse ground motions considered and the models designed in this study was to obtain wide-ranging seismic responses of the models, rather than to passively reduce their seismic responses as small (or satisfactory) as possible or to propose optimum design approaches for FPBs and SRBs. Appl. Sci. 2020, 10, 7200 19 of 21

2. Based on retaining the same horizontal acceleration control performance, the numerical results of four SRB models designed with different sloping angles and damping forces under the ground motions considered indicated that suppressing horizontal isolation displacement and reducing residual displacement were always two antithetic functions for SRBs simply designed with a constant sloping angle and damping force. In other words, under the same excitation, it is evident that the larger the damping force or, the smaller the sloping angle, the smaller the horizontal isolation displacement presented, but the more significant the residual displacement obtained. 3. With the same horizontal acceleration control performance, the numerical results of hundreds of FPB models under the ground motions considered showed that the FPB models designed with a larger design displacement, a smaller curvature radius, and a smaller damping force might potentially have a better horizontal acceleration control and self-centering performances than those designed with a smaller design displacement, a larger curvature radius, and a larger damping force. 4. On the premise of retaining the same horizontal acceleration control performance, the numerical results of two SRB models and hundreds of FPB models under the ground motions considered indicated that although the SRB models possess a unique feature—constant acceleration responses—the FPB models, on average, might potentially have a slight advantage over the SRB models in terms of horizontal acceleration control. Besides, the FPB models might also be potentially more effective than the SRB models in suppressing horizontal isolation displacement. As for the re-centering capability, the SRB model designed with a larger sloping angle is potentially better than that designed with a smaller sloping angle as well as the FPB models.

In this research, the numerical study only considered two SRB models as the benchmarks for comparison with the FPB models might not be generalized enough. Nevertheless, some important quantitative results consistent with the theoretical bases and features of FPBs and SRBs, can still be obtained. These results are valuable for the whole picture of comparison of the two kinds of isolation bearings (i.e., FPBs and SRBs) or even the two types of hysteretic behavior (bilinear and twin-flag hysteretic behavior), as detailed in the fourth conclusion. In the future, SRB models designed with more different acceleration performance goals as well as designed with more diverse combinations of constant sloping angles and damping forces might be further considered as benchmarks for a more comprehensive comparison. In addition, further comparison of FPBs designed with different curvature radii and friction coefficients and SRBs designed with variable sloping angles and damping forces might also be made.

Author Contributions: Conceptualization, resources, methodology, writing—original draft preparation, supervision, project administration, and funding acquisition, S.-J.W.; software, validation, formal analysis, investigation, and data curation, Y.-L.S.; methodology, investigation, and writing—review and editing, C.-Y.Y., W.-C.L., and C.-H.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Ministry of Science and Technology (MOST), Taiwan, grant number MOST 105-2221-E-011-169-MY2. The APC was funded by the Ministry of Science and Technology (MOST), Taiwan. Acknowledgments: The study was financially aided by the Ministry of Science and Technology (MOST), Taiwan, grant number MOST 105-2221-E-011-169-MY2. The support is greatly acknowledged. Conflicts of Interest: The authors declare no conflict of interest.

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