applied sciences

Article Energy-Based Prediction of the Displacement of DCFP Bearings

Jiaxi Li 1,*, Shoichi Kishiki 2, Satoshi Yamada 3,*, Shinsuke Yamazaki 4, Atsushi Watanabe 4 and Masao Terashima 4

1 Department of Architecture and Building Engineering, School of Environment and Society, Tokyo Institute of Technology, J2-21, 4259 Nagatsuta, Midori-ku, Yokohama 226-8503, Japan 2 Laboratory for Future Interdisciplinary Research of Science and Technology, Institute of Innovative Research, Tokyo Institute of Technology, J2-21, 4259 Nagatsuta, Midori-ku, Yokohama 226-8503, Japan; [email protected] 3 Department of Architecture, Graduate School of Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 4 Nippon Steel Engineering, Osaki Center Building, 1-5-1 Osaki, Shinagawa, Tokyo 141-8604, Japan; [email protected] (S.Y.); [email protected] (A.W.); [email protected] (M.T.) * Correspondence: [email protected] (J.L.); [email protected] (S.Y.)



Received: 26 June 2020; Accepted: 28 July 2020; Published: 30 July 2020


Abstract: Isolation systems are currently being widely applied for earthquake resistance. During the design stage for such systems, the displacement response and input energy of the isolation layer are two of the main concerns. The prediction of these values is also of vital importance during the early stages of the structural design. In this study, the simple prediction method of double concave friction pendulum (DCFP) bearings is proposed, which can relate the response displacement of the isolation layer to the ground velocity through energy transfer with sufficient accuracy. Two friction models (the precise and simplified model) and a constant friction coefficient of double concave friction pendulum (DCFP) bearings are comprehensively validated by full-scale sinusoidal dynamic tests under various conditions. In addition, a response analysis, based on previous studies, was conducted using the friction models under selected unidirectional earthquake excitations, and the accuracy of using the simplified model in the response analysis was verified. Based on the response analysis data, this article verifies and optimizes the proposed prediction method by parameterizing the characteristics of earthquakes and combining the energy balance in order to gain a deeper understanding of the design of the isolation systems.

Keywords: seismic isolation; double concave friction pendulum bearing; full-scale dynamic test; friction dependencies; prediction method; response displacement; input energy; isolation system design

1. Introduction Many kinds of isolators are currently being proposed, and the isolation system is being widely applied for earthquake resistance [1–5]. A double concave friction pendulum (DCFP) bearing, which is a type of base isolation technique that detaches structures from the ground to help stabilize buildings during earthquakes, is widely used in earthquake-prone regions. Its composition is shown in Figure1.

Appl. Sci. 2020, 10, 5259; doi:10.3390/app10155259 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5259 2 of 25 Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 26

Upper Concave Plate Slider

Sliding Surface Composite PTFE Layer

Lower Concave Plate

FigureFigure 1.1. CompositionComposition ofof aa doubledouble concaveconcave frictionfriction pendulumpendulum (DCFP)(DCFP) bearing.bearing.

TheThe clear clear force–displacement force–displacement relationship relationshi ofp friction of friction pendulum pendulum bearings bearings (FPBs) makes (FPBs) numerical makes analysisnumerical one analysis of the one best of ways the best in which ways in to which study andto study predict and theirpredict performance. their performance. The main The focus main isfocus on theis on calculation the calculation of the of friction the friction force. force This. This is because is because the frictionthe friction coeffi coefficient’scient’s dependence dependence on pressure,on pressure, velocity velocity and and temperature temperature is the is mostthe most important important characteristic characteristic for the for performance the performance of a FPBof a (FrictionFPB (Friction Pendulum Pendulum Bearing). Bearing) Therefore,. Therefore, the lubricant the material lubricant used material in an FPB used and in the an characteristics FPB and the ofcharacteristics this material of are this very material important. are very Quaglini important. et al. Quaglini (2012) proposed et al. (2012 an experimental) proposed anmethodology experimental formethodology the characterization for the ofcharacterization self-lubricating of materials self-lubricating based on materials pressure, velocity, based on external pressure, temperature velocity, andexternal displacement temperature through and small-scaledisplacement specimens through [small1]. In- additionscale specimens to external [1]. temperature,In addition to particular external attentiontemperature, was particular paid to the attention temperature was paid increase to the ontemperature the sliding increase surface on caused the sliding by friction surface heating caused becauseby friction this heating increase because in temperature this increase had ain significant temperature influence had a onsignificant the behavior influence of the on FPB the during behavior an earthquakeof the FPB event.during Moreover, an earthquake the measurement event. Moreover, of this the temperature measurement increase of this is very temperature difficult toincre performase is duringvery difficult dynamic to perform excitation. during To confrontdynamic thisexcitation. difficulty, To confront Lomiento this et difficulty, al. (2013) Lomiento proposed et a al. friction (2013) modelproposed that a takes friction into model account that the takes vertical into load, account velocity the and vertical cycling load, eff ect velocity (degradation and cycling of friction effect characteristics(degradation of due friction to the characteristics repetition of cyclesdue to and the consequentrepetition of temperature cycles and consequent rise) by performing temperature 1D prototyperise) by performing dynamic tests 1D prototype on single concave dynamic friction tests on pendulum single concave (SCFP) friction bearings pendulum [2]. Quaglini (SCFP) et al.bearings(2014) proposed[2]. Quaglini a 3D et finite al. (2014) element proposed model a (FEM) 3D finite of anelement SCFP bearingmodel (FEM) to estimate of an SCFP friction bearing heating, to estimate and the estimatedfriction heating, temperature and the was estimated validated temperature with experimental was validated data measured with experimental by thermocouples data measured embedded by inthermocouples the concave sliding embedded plate [ in3]. the Following concave this sliding work, platesome studies [3]. Following proposed this simplified work, some temperature studies simulationproposed simplified methods. Kumar temperature et al. (2015) simulation proposed methods. a simplified Kumar model et al. to (2015) calculate proposed the representative a simplified temperaturemodel to calculate of the sliding the representative surface in thermal temperature calculations of the (Method sliding 2surface in the article)in thermal and calculations considered pressure,(Method velocity2 in the and article) temperature and considered dependency. pressure, The analysis velocity results and temperaturewere verified dependency. by 2D prototype The dynamicanalysis tests results of the were SCFP verified bearings by [ 4 2D]. The prototyp distributionse dynamic of the tests maximum of the displacement SCFP bearings under [4]. 30 sets The ofdistributions ground motions of the inmaximum the original displacement model (Method under 130 in sets the of article) ground and motions the simplified in the original model model were compared,(Method 1 andin the a relatively article) and small the di simplifiedfference was model found were between compared, the models; and a thus,relatively the simplified small difference model couldwas found be applied between instead the ofmodels; the original thus, model.the simplified Furthermore, model withcould the be appearance applied instead of multiple of the concaveoriginal frictionmodel. pendulumFurthermore bearings, with the (MCFP appearance bearings), of Biancomultiple et concave al. (2018) friction proposed pendulum a simplified bearings rheological (MCFP modelbearings), to simulate Bianco et the al. temperature (2018) proposed rise in a the simplified MCFP bearings rheological [5]. model to simulate the temperature rise in the MCFP bearings [5]. 2. State-of-the-Art Isolator Displacement Prediction

2. StateTwo-of of-the the- mainArt Isolator matters Displacement of concern during Prediction the design of the isolation systems were (I) whether the maximum displacement response of the isolation layer during the earthquake was within the capacity Two of the main matters of concern during the design of the isolation systems were (I) whether and (II) in the aspect of energy, whether the energy absorption capacity of the isolation system was the maximum displacement response of the isolation layer during the earthquake was within the enough in comparison to the input energy during the earthquake. In the design, nonlinear time-history capacity and (II) in the aspect of energy, whether the energy absorption capacity of the isolation analysis could be conveniently performed to obtain these values. However, in the early stages of the system was enough in comparison to the input energy during the earthquake. In the design, nonlinear structural design, the structural configurations were not well defined. Thus, simplified prediction time-history analysis could be conveniently performed to obtain these values. However, in the early methods were generally introduced in various structural codes to obtain the design displacement or stages of the structural design, the structural configurations were not well defined. Thus, simplified the energy demand of the isolation system under the specified seismic loads: prediction methods were generally introduced in various structural codes to obtain the design (I)displacementTypical Simplified or the energy Prediction demand Methods of the isolation for the system Design under Displacement the specified of the seismic Isolation loads: System (I) IntroducedTypical Simplified in ASCE P (Americanrediction M Societyethods of for Civil the Engineers),Design Displacement Euro Code of and the AIJ Isolation (Architectural System InstituteIntroduced of Japan)in ASCE [6– 8(American]. Society of Civil Engineers), Euro Code and AIJ (Architectural Institute of Japan) [6–8]. Appl. Sci. 2020, 10, 5259 3 of 25

The methods introduced in ASCE and Euro Code are similar: they can predict the response displacement and the response force based on the simplified method of equivalent linear analysis under certain conditions. However, the two methods introduced in AIJ can give the tendency of the response reduction or the relation between the response force and displacement under various parameters by using the equivalent SDOF (Single Degree of Freedom) system or using energy balance, respectively.

(1) Equivalent Lateral Force Procedure (ASCE [6])

This method can be applied for the design of a seismically isolated structure if it meets several requirements, including the structure being located at a site with S1 (mapped acceleration parameter at one-second periods) less than 0.60 g; the effective period of the isolated structure at the maximum displacement, TM, being less than or equal to 3.0 s; and so on. Based on this method, the design displacement, DD, can be predicted. g SD1 TD D = × × (1) D 2 4 π BD × × where SD1 is the spectral acceleration parameter at 1 s periods, TD is the effective period of the seismically isolated structure, BD is the numerical coefficient related to the effective damping and g is the gravity acceleration.

(2) Simplified Linear Analysis (Euro Code [7])

Similar to the equivalent lateral force procedure introduced in ASCE, this method may be applied to isolation systems with equivalent linear damped behavior if they also conform to conditions such as the distance from the site to the nearest potentially active fault, with a magnitude M 6.5, being greater ≥ than 15 km and their effective period being smaller than 3 s and larger than three times the fundamental period of the superstructure, assuming a fixed base. It assumes that the superstructure is a rigid solid, and then the displacement of the stiffness center, due to the seismic action, can be calculated in each horizontal direction, from the following expression:

M Sc(Te f f , ξe f f ) ddc = × (2) Ke f f ,min where Sc (Teff, ξeff) is the spectral acceleration defined by an elastic response spectrum that takes into account the appropriate value of the effective damping ξeff. Keff is the effective horizontal stiffness of the isolation system, and Teff is the effective period.

(3) Simplified Response Prediction Method for the Maximum Response Using an Equivalent Single-Degree-of-Freedom System (AIJ [8]

The simple prediction for the maximum response values of an equivalent SDOF system involves the evaluation of the dynamic characteristics of the structure by using viscous damping and hysteretic damping, followed by the prediction of the maximum response using an elastic response spectrum. Two variations are included. In the response prediction method 1, the maximum response is obtained through a convergence calculation that considers the changes in the equivalent period and in the equivalent damping factor due to the yielding of the SDOF system in the spectrum response. For the response prediction method 2, Method 1 is simplified based on the assumption that the response spectrum curve is constant, regardless of the period. The maximum response is also estimated by some response curves. By using these curves, it is possible to grasp visually the tendency of the response reduction in the structures without a convergence calculation.

(4) Response Prediction of the Seismic Isolation Level Based on Energy Balance (AIJ [8]) Appl. Sci. 2020, 10, 5259 4 of 25

The energy balance method is an analysis method that evaluates seismic resistance based on the balance of the seismic input energy to buildings due to ground motions via energy spectra and the energy absorbed by the building. In this method, the relationship between the maximum shear force coefficient and the maximum displacement (response reduction effect) of the seismic isolation level can be shown in the curves that are related to the seismic isolation period and ground motion level, by assuming that the total hysteresis energy absorbed during the earthquake is equal to the energy dissipated by two repetitions of the steady state loop at the maximum displacement. In the above methods, (1) and (2) are based on equivalent linear analysis with strict application conditions. Meanwhile, (3) and (4) provide the tendency of the response reduction, but the response displacement of the isolation layer cannot be predicted under the specified seismic loads.

(II) Simplified Prediction Methods for the Energy Demand of the Isolation System Introduced in Some Design Methods Based on the Energy Concept [9–12].

Besides the maximum response of the structure, the input energy is also very important for earthquake design methods based on the energy concept [9], in which the energy demand can be taken as the input energy. This value is usually given as the energy spectrum, which can provide the input energy to structures during an earthquake [10]. However, this process is based on response analysis under a certain earthquake record, which makes it difficult to relate the input energy to the characteristics of the group of earthquakes in the construction site. Thus, some methods that can predict the total input energy based on earthquake characteristics were proposed in some earthquake design methods based on energy [11]:

(1) Approximate Inelastic Input Energy Spectra

Based on Iwan’s study [10], it was found that an inelastic input energy spectrum could be approximated by an elastic spectrum that corresponds to an equivalent viscous damping and an equivalent natural period. However, Hadjian [12] has shown that this method is not suitable for ground motions with a highly harmonic nature.

(2) Another Method Proposed by Uang [11]

This method can predict the maximum input energy of a structure with a specified ductility ratio if the strong motion duration at a given site is known. However, this maximum value is the maximum input energy that is evaluated throughout the whole period range and means that the maximum input energy of a structure within a specific period cannot be predicted. The energy balance equation used in this study is shown in Equation (3) [11]. When an isolation system consisting of DCFP bearings is subjected to earthquake ground motions, the input energy E is accumulated in the elastic strain energy of the bearing, Es, and the kinetic energy of the upper structure, Ek, and is mainly consumed by the frictional resistance of the bearing in Eh and by the damping of the system in Eξ: E = Es + Ek + Eh + Eξ (3)

where, Z E = m ag dy (4) − × × m v2 E = × (5) k 2

In these formulas, E is the total input energy (the relative input energy in [11]), Es is the elastic strain energy, Ek is the kinetic energy (the relative kinetic energy in [11]), Eh is the hysteresis energy, Eξ is the damping energy, m is the mass of the upper structure, ag is the ground acceleration, dy is the a Appl. Sci. 2020, 10, 5259 5 of 25

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 26 segment of the relative displacement between the upper structure and the ground and v is the relative velocityAkiyama between has the shown upper that structure the total and in theput ground. energy, E, based on the SDOF system, can provide a very Akiyamagood estimate has shown of the input that the energy total inputfor multi energy,-storyE ,buildings based on [9]. the Therefore, SDOF system, in this can study, provide an aSDOF very goodsystem estimate with a damping of the input ratio energy of 0% for was multi-story applied in buildings order to simplify [9]. Therefore, the computation in this study, and an to SDOFmake systemthe influence with a of damping the earthquake ratio of 0%inputs was on applied the response in order of to the simplify DCFP thebearings computation clearer. and As a to result, make the influenceinput energy of the can earthquake be expressed inputs as: on the response of the DCFP bearings clearer. As a result, the input energy can be expressed as: E  Es  Ek  Eh (6) E = Es + Ek + Eh (6) To sum up, up, in in the the early early stages stages of of the the structural structural design, design, simple simple response response prediction prediction methods methods are areof vital of vital importance. importance. Two Twocategories categories of prediction of prediction methods methods are usually are usually applied, applied, as introduced as introduced above: above:methods methods based on based equivalent on equivalent linear analysis linear analysis and methods and methods based basedon energy on energy demand demand and supply. and supply. The Thefirst firstcategory category mainly mainly focuse focusess on onthe the response response displacement displacement of ofthe the isolation isolation layer layer,, and and the the second category mainly concerns the input energy ofof thethe isolationisolation system.system. Furthermore,Furthermore, most methods in both categories categories function function according according to the to theresponse response spectrum spectrum or the or energy the energy spectrum spectrum from the from response the analysis.response analysis. However, However, this study this proposed study proposed a method a method that is that directly is directly based based on theon the characteristics characteristics of earthquakeof earthquake records records and and can relatecan relate both both the response the response displacement displacement and the and input the energyinput energy to the ground to the velocityground velocity history withhistory suffi withcient sufficient accuracy. accuracy Therefore,. There a deeperfore, a understanding deeper understanding can be gained can be in gained order toin provideorder to anotherprovide perspectiveanother perspective for the design for the methods design methods for isolation for isolation systems. systems.

3. Performed Experiments

3.1. Testing Setup The experiments in this study were conducted at the University of California, San Diego,Diego, in the Caltrans Seismic Response Modification Modification Device (SRMD) Test Facility [[13].13]. An overview of the test setup is shown in Figure 2 2.. AfterAfter thethe specimenspecimen waswas placedplaced insideinside thethe facility,facility, anan axialaxial forceforce waswas addedadded to control the contact pressure between thethe slider and the concave plate. Additionally, Additionally, four horizontal actuators made the loading table move horizontally in order to control the velocity velocity and and loading loading path. path.

Figure 2. Figure 2. Test setup. 3.2. Dependency Tests 3.2. Dependency Tests Pressure, velocity and temperature dependency equations were introduced in previous studiesPressure, [14–16 ]velocity. The applicability and temperature of the pressuredependency and equations velocity dependency were introduced equations in previous for specimens studies [14–16]. The applicability of the pressure and velocity dependency equations for specimens with with slider diameters of 300 and 400 mm and spherical radii of the concave plates (Rs) of 4500 mm wereslider verified diameters under of 300 diff erentand 400 pressures mm and through spherical the radii dependency of the concave test introduced plates (R ins) Tableof 45001. Themm input were wavesverified were under unidirectional different pressures sinusoidal through displacement the dependency variations, test and introduced the amplitude, in Table maximum 1. The velocity input (Vmax),waves were period unidirectional and number sinusoidal of cycles weredisplacement determined. variations, and the amplitude, maximum velocity (Vmax), period and number of cycles were determined.

Table 1. Test procedure of the dependency test. Appl. Sci. 2020, 10, 5259 6 of 25

Table 1. Test procedure of the dependency test.

Spec. Test Pressure Amplitude Vmax Period Cycle num. num. N/mm2 mm mm/s s num. ± (1) Velocity dependency test T01 60 200 400 3.14 4 ϕ ϕ 300 400 (2) Pressure dependency test T02 40 20 62.83 4 T03 60200 20 62.83 4 T04 80 20 62.83 4

3.3. ASCE Tests The dynamic tests shown in Table2 were conducted to validate the friction models under various situations. The test procedure was designed based on ASCE/SEI 7-16. Thus, it was named the ASCE test.

Table 2. Test procedure of the ASCE test.

Spec. Test Pressure Amplitude Vmax Period Cycle num. num. N mm2 mm mm/s s num. \ ± T01 268 392 4.26 3 T02 10 14.646 4.26 20 T03 100 146.369 4.26 3 T0460 200 292.738 4.26 3 T05 268 392.269 4.26 3 T06 400 585.476 4.26 3 T07 400 585.476 4.26 3 ϕ300 ϕ400 T08 40 400 585.476 4.26 3 T09 80 400 585.476 4.26 3 T10 30 440 644.024 4.26 1 T11 90 440 644.024 4.26 1 T12a 7 T12b300 439.107 4.26 7 60 T12c 6 T13 268 392.269 4.26 3

4. Numerical Simulation of the Experiments

4.1. Friction Dependency The pressure dependency of the friction coefficient could be considered by a pressure dependency factor γ, which is related to the bearing stress σ at the contact area of the concave plate and the slider. The pressure dependency equation was obtained by a previous experimental study [14]:

0.19 γ = 2.03 σ− + 0.068 (7) × The dependency of the friction coefficient on velocity was considered by a velocity dependency factor α. This factor was related to the velocity of the upper concave plate relative to the lower concave plate v, which could be described by the following equation [14]:

0.019v α = 1 0.55 e− (8) − × Figure3 shows the applicability verification of the pressure and velocity dependency equations for ϕ300 and ϕ400 specimens, based on the dependency test introduced in Table1. In Figure3, the experimental friction coefficients calculated from the dependency test were normalized by the Appl. Sci. 2020, 10, 5259 7 of 25 values at 60 N/mm2 and 400 mm/s, respectively, in order to minimize the influence of product variation. Appl.AsAppl. a Sci. result,Sci. 20 202020, ,Equations10 10, ,x x FOR FOR PEER PEER (7) REVIEW andREVIEW (8) are both highly consistent with the experimental results, and77 thus, of of 26 26 the previously proposed pressure and velocity dependency equations are effective for DCFP bearings previouslywithpreviouslyϕ300 and proposedproposedϕ400 slider pressurepressure diameters. andand velocityvelocity dependencydependency equationsequations areare effectiveeffective forfor DCFPDCFP bearingsbearings withwith φ300 φ300 and and φ400 φ400 slider slider diameters. diameters. ExperimentalExperimental datadata of of φ φ330000 ExperimentalExperimental datadata of of φ φ400400 DDependencyependency equation equation

1100.55.55exp(exp(00.019.019vv)) 22.03.030.019.1900.068.068

VelocityVelocity (mm/s) (mm/s) Pressure,Pressure,σσ(MPa)(MPa) (a)(a) (b)(b)

Figure 3. Verificationof (a) pressure and (b) velocity dependency equations through the dependency test. FigureFigure 3. 3. Verification Verification of of ( a(a) )pressure pressure and and ( (bb) )v velocityelocity dependency dependency equations equations through through the the dependency dependency test. test. The temperature dependency equation is as follows: TheThe temperature temperature dependency dependency equation equation is is as as follows: follows: β =1.131.13 expexp(( 0.0070.007TT)) (9)   1.13× exp(−0.007T) ((99)) wherewhere β β is isis the thethe temperature temperaturetemperature dependency dependencydependency factor factorfactor and andandT TisT is theis thethe temperature temperaturetemperature of theofof thethe contact contactcontact area areare betweenaa betweenbetween the thesliderthe sliderslider and and theand concave thethe concaveconcave plate plate inplate Celsius inin CelsiusCelsius [15,16 ]. [15,16[15,16 As the].]. AsAs temperature, thethe temperature,temperature,T, is di ffi TT,cult , isis difficult todifficult accurately toto accuratelyaccurately measure measureduringmeasure testing, duringduring a numericaltesting,testing, aa numerical methodnumerical proposed methodmethod by proposedproposed Constantinou, byby Constantinou,Constantinou, M.C. et al. [4 M.C.,M.C.17] will etet al. beal. [4,17 introduced[4,17]] willwill be tobe introducedsimulateintroduced the to to temperature simulate simulate the the in temperature temperature the next section. in in the the n nextext section. section. TTheThehe friction friction coefficient coecoefficientfficient atat eacheach timetime duringduring excitationexcitation cancan bebe calculatedcalculated by: by:   (10) µ = µ 00 γ α β (10(10)) 0 × × × wherewhere µµ, , γγ, , αα andand ββ areare thethe frictionfriction coefficient,coefficient, pressurepressure dependencydependency factor,factor, velocity velocity dependencydependency where µ, γ, α and β are the friction coefficient, pressure dependency factor, velocity dependency factor factorfactor and and temperature temperature dependency dependency factor factor at at each each time, time, respectively, respectively, and and µ µo ois is the the friction friction coefficient coefficient atandat 60 60 temperatureN/mm N/mm22 ( (γγ = = 1), dependency1), 400 400 mm/s mm/s ( factor(αα = = 1) 1) atandand each anan atmosphere time,atmosphere respectively, temper temperature andatureµ ofoofis aroundaround the friction 20˚C 20˚C ( coe(ββ = = ffi1), 1),cient which which at 60 N/mm2 (γ = 1), 400 mm/s (α = 1) and an atmosphere temperature of around 20 C(β = 1), which was waswas selected selected asas 0.075 0.075 for for allall the the specimens specimens in in this this study.study. µ µo owas was calculated calculated from from◦ the the trend trend lineline of of thethe frictionselectedfriction coefficient coefficient as 0.075 for versus versus all the the the specimens time time from from in the thisthe dependency dependency study. µo was test test calculated T01 T01 in in Table Table from 1, 1, the which which trend was was line when when of the the the friction time time wascoewasffi approachinapproachincient versusgg zero, thezero, time asas shown fromshown the inin dependencyFigureFigure 4.4. TheThe test experimentalexperimental T01 in Table valuesvalues1, which inin this wasthis figure whenfigure therepresentrepresent time was thethe frictionapproachingfriction coefficientscoefficients zero, as nearnear shown zerozero in displacementdisplacement Figure4. The where experimentalwhere thethe velocityvelocity values waswas in this400400 mm/s figuremm/s and representand thethe pressurepressure the friction waswas 2 60coe60 N/mm ffiN/mmcients22. . near zero displacement where the velocity was 400 mm/s and the pressure was 60 N/mm .

0.10.1

µ Experimental value µ Experimental value 0.080.08 TrendlineTrendline 0.0750.075 0.060.06 0.040.04 yy = = - 0.007ln(x)-0.007ln(x) ++ 0.0556 0.0556 0.020.02 Friction Coefficient Friction Coefficient Friction Coefficient Friction Coefficient 00 00 55 1010 1515 TimeTime (s)(s) ..

Figure 4. Determination of µo. FigureFigure 4. 4. Determination Determination of of µ µo.o. 4.2. Friction Models 4.2. Friction Models 4.2. FrictionTwo friction Models models are proposed as precise and simplified models, and both considered the pressure,TwoTwo velocityfrictifrictionon models andmodels temperature areare proposedproposed dependency asas preciseprecise introduced andand simplifiedsimplified in Section modelsmodels 4.1, ,. andand Meanwhile, bothboth consideredconsidered the precise thethe pressure,pressure, velocityvelocity andand temperaturetemperature dependencydependency introducedintroduced inin SectionSection 44.1..1. MeanwMeanwhilehile, , thethe preciseprecise modelmodel applies applies the the precise precise temperature temperature simulation simulation model model shown shown in in Figure Figure 5 5aa andand t thehe simplified simplified modelmodel appliesapplies thethe simplifiedsimplified temperaturetemperature simulationsimulation modelmodel shownshown inin FigureFigure 5b,5b, wherewhere rrcontactcontact isis thethe radiusradius ofof thethe bearingbearing andand thethe monitormonitor pointspoints areare virtualvirtual pointspoints inin thethe modelmodel usedused forfor temperaturetemperature Appl. Sci. 2020, 10, 5259 8 of 25 model applies the precise temperature simulation model shown in Figure5a and the simplified model applies the simplified temperature simulation model shown in Figure5b, where rcontact is the radius of Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 26 the bearing and the monitor points are virtual points in the model used for temperature simulation, whichsimulation, is proposed which is by proposed Constantinou, by Constantino M.C. et al.u, [ 4M.C.,17]. et As al. the [4,17 precise]. As modelthe precise has a model lot of monitoringhas a lot of pointsmonitoring along points the sliding along direction, the sliding the direction, simulation the results simulation are more results precise. are more However, precise. for theHowever, simplified for model,the simplified which model, only applies which oneonly monitoring applies one point, monitoring the calculation point, the speedcalculation is much speed faster. is much Therefore, faster. theTherefore, validation the ofvalidation the simplified of the modelsimplified is very model important is very inimportant judging in whether judging it whether can be used it can instead be used of theinstead precise of the model. precise model.

Sliding Direction 2 πrcontact

2 rcontact

Monitoring Points Monitoring Point

(a) (b) Figure 5. TemperatureTemperature simulation models used in the (a) precise modelmodel andand ((b)) simplifiedsimplified model.model.

In both the precise modelmodel and the simplifiedsimplified model,model, the temperature increase of the monitoring points can bebe calculated byby [[17]:17]:

Z t √DD t q(qt(t )τd)dτ ∆T(Tt)(t =)  0 − (11(11)) kk√π 0 √τ where qq(t()t) is is the the heat heat flux flux at the at surface the surface of the of solid; the ∆ solidT(t); isΔ theT(t) temperature is the temperature increase at increase time t, compared at time t, tocompared the temperature to the temperature at t = 0; τ is aat time t = parameter0; τ is a time that parameter varies between that 0varies and t; betweenD is the thermal 0 and t di; Dffusivity is the ofthermal the solid; diffusivity and k is of the the thermal solid; and conductivity k is the thermal of the conductivity solid [17]. In of this the study, solid [17]. the values In this ofstudy,k and theD werevalues adopted of k and from D the were “JSME adopted Data Book: from Heat the “JSME Transfer”, Data which Book: are Heat 0.016 Transfer”, W/(mm C) which and 4.07 are mm 0.0162/s, ·◦ respectivelyW/(mm∙°C) and [18]. 4.07 Then mm the2/s, temperature respectively at [18]. the Then contact the surface temperature of the at slider the contact and the surface concave of plates the slider can beand calculated the concave by taking plates average can be ofcalculated the temperature by taking of theaverage monitoring of the pointstempe withinrature the of the contact monitoring surface. pointsFor within DCFP the Bearings, contact thesurface. instantaneous heat flux, q(t), can be defined as follows [4]: For DCFP Bearings, the instantaneous heat flux, q(t), can be defined as follows [4]:  p  v(t) 2  µ(t)p(t) if δ π rcontact /2 q(t) =  v(t)2 2 (12) µ(t)p(t) if ≤ · r / 2 q(t)  02 otherwisecontact (12)  0 otherwise where µ(t), p(t) and v(t) are the coefficient of friction, the pressure at the contact area and the relative velocitywhere µ(t) between, p(t) and the v(t) upper are the and coefficient lower concave of friction, plates the at timepressuret, respectively; at the contactδ is area the and lateral the distance relative fromvelocity the between center of the the upper slider toand the lower monitor concave point plates of interest at time and t, rrespectively;contact is the contact δ is the radius. lateral distance from the center of the slider to the monitor point of interest and rcontact is the contact radius. 4.3. Numerical Results Using Friction Models 4.3. Numerical Results Using Friction Models The comparison of the first hysteresis curve between the friction models and the experimental resultsThe of comparison the DCFP bearings of the first with hysteresis 400 mm-diameter curve between sliders the under friction both models strong and the weak experimental excitations areresults shown of the in DCFP Figure bearings6, and show with high 400 mm accuracy-diameter for bothsliders friction under models. both strong The and input weak excitations excitations in Figureare shown6a,b in are Fig theure first 6, and cycles show in T01high and accuracy T02 of for the both ASCE friction test, respectively.models. The input Meanwhile, excitations as an in errorFigure criterion, 6a,b are thethe valuesfirst cycles of the in post-yield T01 and T02 stiff ofness, the eASCEffective test, sti ffrespectively.ness, effective Meanwhile, damping andas an energy error dissipatedcriterion, the per value cycles were of the also post calculated-yield stiffness, to quantify effective the validity stiffness, of the effective models [damping6]. In all of and the energy ASCE tests,dissipated the simplified per cycle modelwere also shows calculated a similar to accuracyquantify tothe the validity precise of model, the models with both[6]. In models all of the showing ASCE hightests, accuracythe simplified in the model simulation shows of a the similar behaviours accuracy of to the the DCFP precise bearings model, [19 with]. both models showing high accuracy in the simulation of the behaviours of the DCFP bearings [19]. Appl. Sci. 2020, 10, 5259 9 of 25 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 26

Experimental Precise Simplified

φ400-T01-Cyc1 φ400-T02-Cyc1 Vmax=392mm/s Vmax=15mm/s

(a) (b) FigureFigure 6. 6. AccuracyAccuracy ofof the precise and and simplified simplified models models under under (a ()a )strong strong excitation excitation (seismic) (seismic) and and (b) (bweak) weak excitation excitation (wind (wind-like).-like). 4.4. Numerical Results Using a Constant Friction Coefficient 4.4. Numerical Results Using a Constant Friction Coefficient In order to find the constant friction coefficient that was suitable for the most amount of time In order to find the constant friction coefficient that was suitable for the most amount of time during most seismic situations, a nominal friction coefficient, µn, was defined. The value of µn was during most seismic situations, a nominal friction coefficient, µn, was defined. The value of µn was 0.043 for both sizes of the sliders based on experimental results, and it represented the nominal friction 0.043 for both sizes of the sliders based on experimental results, and it represented the nominal coefficient under seismic load conditions. This value was determined by taking the average of the friction coefficient under seismic load conditions. This value was determined by taking the average friction coefficient of the DCFPB at zero displacement points in the second cycle (points c and d in of the friction coefficient of the DCFPB at zero displacement points in the second cycle (points c and Figure7) of the input sine wave with a constant pressure of 60 N /mm2 and a maximum velocity of d in Figure 7) of the input sine wave with a constant pressure of 60 N/mm2 and a maximum velocity 400 mm/s (T01 of the dependency test in Table1). of 400 mm/s (T01 of the dependency test in Table 1). ) ) m m m m ( ( t t n n e e m m e e c c a a l l p p s s a b c d e f g i i d d

l l a a t t n n o o z z

i Cycle 1 Cycle 2 Cycle 3 Cycle 4

i Cycle 1 Cycle 2 Cycle 3 Cycle 4 r r zontal displacement (mm) zontal displacement zontal displacement (mm) zontal displacement o o H H Hori Hori Time (s)

FigureFigure 7. 7.Experimental Experimental data data used used to to obtain obtainµ n.µn.

TheThe accuracy accuracy ofof usingusing µn underunder strong strong excitation excitation and and weak weak excitation excitation is show is shownn in Figure in Figure 8. The8. Theresults results showed showed that that µn µwasn was suitable suitable for for strong strong excitations excitations but but could could not be be used used under under weak weak excitations.excitations. In the the other other tests tests in in Table Table 2,2, which which can can all all be be considered considered to to be be strong strong excitations, excitations, the thesimulation simulationss using using µn allµn showedall showed that that the theforce force–deflection–deflection relation relationss of the of DCFP the DCFP bearings bearings had hadno large no largedifferences differences [19]. [ 19 Therefore,]. Therefore, the the constant constant friction friction coefficient, coefficient, µµn, n, could could be be applied applied for for the the rough rough behaviorbehavior estimation estimation of of DCFP DCFP bearings bearings under under seismic seismic loads. loads. For For example, example, it it could could be be applied applied for for the the responseresponse force force simulation simulation when when the the response response displacement displacement of of the the bearing bearing is is known. known. In the numerical simulation of DCFP bearings, pressure, velocity and temperature dependencies all have influence on the behavior of the bearings. However, for DCFP bearings under stable pressure, the temperature variation caused by friction heating will dominate the influence. This influence is more obvious when the temperature is relatively low (e.g., 20–100 ◦C), meaning that it is under weak excitations, as shown in Figures6b and8b. However, when the temperature is higher, the influence will ease up, as shown in Figures6a and8a, and this is the reason why the constant friction coe fficient can be applied for rough estimation under strong excitations such as seismic loads. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 26

Experimental Constant

Appl. Sci. 2020, 10, 5259 φ400-T01-Cyc1 φ400-T02-Cyc1 10 of 25 Vmax=392mm/s Vmax=15mm/s Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 26

Experimental Constant

(a) (b) φ400-T02-Cyc1 Figure 8. Accuracy of usingφ400 µn -underT01-Cyc1 (a) strong excitation (seismic) and (b) weak excitation (wind-like). Vmax=392mm/s Vmax=15mm/s In the numerical simulation of DCFP bearings, pressure, velocity and temperature dependencies all have influence on the behavior of the bearings. However, for DCFP bearings under stable pressure, the temperature variation caused by friction heating will dominate the influence. This influence is more obvious when the temperature is relatively low (e.g., 20–100 °C), meaning that it is under weak excitations, as shown in Figures 6b(a) and 8b. However, when the temperature(b) is higher, the influence will ease up, as shown in Figures 6a and 8a, and this is the reason why the constant friction coefficient can beFigure Figureapplied 8. 8.Accuracy Accuracy for rough ofof usingusingestimation µµnn under under ( a)) strong strong strong excitation excitation excitations (seismic) (seismic) such and and as (b (bseismic) )weak weak excitation excitationloads. (wind (wind-like).-like). 5. Numerical Analyses of the Dynamic Response of Isolated Structures 5. NumericalIn the numerical Analyses simulation of the Dynamic of DCFP Response bearings, of pressure, Isolated Structuresvelocity and temperature dependencies 5.1.all have Mechanical influence Model on the behavior of the bearings. However, for DCFP bearings under stable pressure, 5.1.the Mechanical temperature Model variation caused by friction heating will dominate the influence. This influence is moreThe obvious mechanical when modelthe tem usedperature in this is relatively study is shownlow (e.g., in Figure20–1009 °C), and meaning considered that both it is under static andweak The mechanical model used in this study is shown in Figure 9 and considered both static and dynamicexcitations, friction as shown [20]. in To Figure demonstrates 6b and the8b. relationHowever, of when the response the temperature of the isolation is higher, layer the influence and the dynamic friction [20]. To demonstrate the relation of the response of the isolation layer and the earthquakewill ease up, records as shown more in clearly, Figures the 6a upperand 8a, structure and this is setthe asreason a rigid why body. the constant friction coefficient earthquake records more clearly, the upper structure is set as a rigid body. can be applied for rough estimation under strong excitations such as seismic loads. Fr 5. Numerical Analyses of the Dynamic Response of Isolated Structures kra=N/2Rs 5.1. MechanicalUpper ConcaveModel Plate y Center of Upper Concave Plate The mechanical model used in this study is shown in Figure 9 and considered both static and Yield Surface Spring (a) Restoring force of pendulum movement dynamic friction [20]. To demonstrate the relation of the response of the isolation layer and the Center of Yield Surface Ff Ff earthquakeCenter of Lower records Concave Plate more clearly, the upper structure is set as a rigid body. (c) & Original Neutral position rf N (b) kf Fr Lower Concave Plate OPY OPY kra=N/2Rs Spring (b) Elastic stiffness Spring (c) Friction force Upper Concave Plate y (a) y of the entire bearing Center of Upper Concave Plate Yield Surface y Spring (a) Restoring force of pendulum movement Ff Center of Yield Surface Ff Ff Center of Lower Concave Plate OY N & Original Neutral position (c) rf rf N (b) kf OPY Lower Concave Plate OPY OPY Spring (b) & spring (c) work as an entirety

Spring (b) Elastic stiffness Spring (c) Friction force Figure 9. Unidirectional(a) mechanical model ofy a rigidof the structure entire bearing with DCFP bearings. Figure 9. Unidirectional mechanical model of a rigid structure with DCFP bearings. y In Figure9, the center of the lower concave plate is taken as the original neutral position. In Figure 9, the center of the lower concave plate is taken as the originalFf neutral position. The The stiffness of the upper structure is considered to be infinite in order to make the behavior of the stiffness of the upper structureOY is considered to be infinite in order to makeN the behavior of the DCFP DCFP bearing clearer. Whether sliding between the slider and the concave plate will occur or not rf OPY is considered by a yield surface with a radius rf [21,22]. OY is the center of the yield surface, y is the displacement of the upper structure relative to the center ofSpring the (b) lower & spring concave (c) work as plate an entirety and OPY is the distance from OY to y. The internal forces of the DCFP bearing are considered in terms of three springs: spring (a), spring (b) and spring (c). Spring (a) represents the restoring force of the pendulum Figure 9. Unidirectional mechanical model of a rigid structure with DCFP bearings. movement. kra is the stiffness, N is the vertical force that acts on the DCFP bearing, and Rs is the In Figure 9, the center of the lower concave plate is taken as the original neutral position. The stiffness of the upper structure is considered to be infinite in order to make the behavior of the DCFP Appl. Sci. 2020, 10, 5259 11 of 25 spherical radius of the upper and lower concave plates. In this study, the DCFP bearings with slider diameters equal to 400 mm and Rs values equal to 4500 mm are set, and a vertical load that can provide 60 MPa of pressure on the slider surface is applied. Therefore, kra equals 0.838 kN/mm. Spring (b) represents the elastic stiffness of the entire DCFP bearing with a value of kf = 1900 kN/mm, based on experimental results. Spring (c) represents the friction force between the slider and two concave plates. Spring (b) and spring (c) work entirely to simulate the friction force. The friction force is represented by spring (b) when there is no sliding and by the friction models, introduced in Section2, during sliding.

5.2. Input Ground Motions In order to see whether the simplified model could be used instead of the precise model in the response analysis and later as the analysis database for the prediction method, the earthquake inputs shown in Table3 were selected. These inputs include earthquake records with various intensities, various durations and both near and far fields. Additionally, in order to study the behavior of the isolation system under earthquakes with similar intensities, each earthquake record was amplified to four input waves with PGVs (Peak Ground Velocities) equal to 0.25, 0.50, 0.75 and 1.00 m/s. In Table3, the first column shows the abbreviation of the earthquake record, which is consist of the name and the measuring station of the earthquake.

Table 3. Input earthquake motion.

Abv. Earthquake Station PGV (m/s) Duration (s) Field JKB Kobe JMA Kobe 0.893 30 Far KNA Kobe Nishi-Akashi 0.373 41 Near TC1 Chi-Chi TCU129 0.554 90 Near NCC Northridge Canyon Country-WLC 0.449 20 Far LPG Loma Prieta Gilroy Array 0.447 40 Near IVD Imperial Valley Delta 0.330 100 Near TSD Tohoku JMA Sendai 0.545 180 Near TIM Tohoku JMA Ishinomaki 0.376 300 Near

5.3. Accuracy of the Simplified Model As shown in Figure 10, the response analysis results using the simplified model were as good as those using the precise model under both strong and weak earthquake excitations. This was because, in both cases, the slider was on the monitoring point of the simplified model for most of time. In the response analysis of all the other input waves introduced in Table3, the response analysis results using the simplified model could always provide results with similar accuracy to those obtained using the precise model. Thus, the simplified model was used in the response analysis instead of the precise modelAppl. in order Sci. 2020 to, 10 save, x FOR time. PEER REVIEW 12 of 26

Precise Simplified

(a) (b) FigureFigure 10. Accuracy10. Accuracy of of the the forceforce displacement displacement curve curve using using the simplified the simplified model under model JKB under (Kobe JKB (Kobe earthquakeearthquake measured measured by by JMA JMA Kobe Kobe station) station):: (a) PGV (a) PGV= 0.25m/s;= 0.25 (b) m PGV/s; ( b= )1.0m/ PGVs.= 1.0 m/s.

5.4. Relation between Maximum Response Displacement and PGV Based on the simulation results, the relationship between the maximum response displacement of the isolation layer (ymax, which is the maximum relative displacement between the upper concave plate and the lower concave plate) and the peak ground velocity (PGV) of the input earthquake is shown in Figure 11. It can be seen that, even though they have the same PGV, different earthquake inputs can still cause different ymax. The difference may be caused by other earthquake characteristics such as the ground acceleration and the bearing’s friction coefficient during excitation. These factors are also related to the transfer of the input energy in the isolation layer. The relationship between them will be studied in the next chapter.

1250 PGV=250 1000 PGV=500 PGV=750 750 PGV=1000 (mm)

max 500 y

250

0 0 250 500 750 1000 1250 PGV(mm/s) Figure 11. Relationship between the maximum response displacement of the isolation layer and the peak ground velocity under unidirectional earthquake inputs.

6. Proposed Prediction Method A simple method is proposed in this chapter to predict the response of an isolation system under a certain unidirectional earthquake input and the energy transfer during a process based solely on the earthquake record. Furthermore, the method was simplified and optimized based on the response analysis results. Finally, how this method could be applied to general design factors was also studied.

6.1. Mechanical Model

6.1.1. Relationship between Ground Velocity, the Response Displacement and the Input Energy of the Isolation Layer. By using the model in Figure 9, the relative movement and the energy balance in a base-isolated building become the relative movement and the energy balance in the base-isolated layer. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 26

Precise Simplified

(a) (b)

Appl. Sci.Figure2020 ,10.10, Accuracy 5259 of the force displacement curve using the simplified model under JKB (Kobe12 of 25 earthquake measured by JMA Kobe station): (a) PGV = 0.25m/s; (b) PGV = 1.0m/s.

5.4. Relation between Maximum Response Displacement and PGV Based on the simulationsimulation results,results, the the relationship relationship between between the the maximum maximum response response displacement displacement of ofthe the isolation isolation layer layer (ymax (y,max which, which is the is the maximum maximum relative relative displacement displacement between between the upper the upper concave concave plate plandate the and lower the lower concave concave plate) andplate) the and peak the ground peak ground velocity velocity (PGV) of (PGV) the input of the earthquake input earthquake is shown inis shownFigure 11in .Figure It can be11. seen It can that, be seen even that, though even they though have they the same have PGV, the same different PGV, earthquake different earthquake inputs can inputsstill cause can distillfferent causey maxdifferent. The diymaxfference. The difference may be caused may be by ca otherused by earthquake other earthquake characteristics characteristics such as suchthe ground as the ground acceleration acceleration and the and bearing’s the bearing’s friction coefrictionfficient coefficient during excitation. during excitation. These factors These are factors also arerelated also to related the transfer to the of transfer the input of energythe input in theenergy isolation in the layer. isolation The relationship layer. The relationship between them between will be themstudied will in be the studied next chapter. in the next chapter.

1250 PGV=250 1000 PGV=500 PGV=750 750 PGV=1000 (mm)

max 500 y

250

0 0 250 500 750 1000 1250 PGV(mm/s) Figure 11. Relationship between the maximum response displacement of the isolation layer and the peak ground velocity under unidirectional earthquakeearthquake inputs.inputs.

6. Proposed Prediction Method 6. Proposed Prediction Method A simple method is proposed in this chapter to predict the response of an isolation system under A simple method is proposed in this chapter to predict the response of an isolation system under a certain unidirectional earthquake input and the energy transfer during a process based solely on the a certain unidirectional earthquake input and the energy transfer during a process based solely on earthquake record. Furthermore, the method was simplified and optimized based on the response the earthquake record. Furthermore, the method was simplified and optimized based on the response analysis results. Finally, how this method could be applied to general design factors was also studied. analysis results. Finally, how this method could be applied to general design factors was also studied. 6.1. Mechanical Model 6.1. Mechanical Model 6.1.1. Relationship between Ground Velocity, the Response Displacement and the Input Energy of the 6.1.1.Isolation Relationship Layer. between Ground Velocity, the Response Displacement and the Input Energy of the Isolation Layer. By using the model in Figure9, the relative movement and the energy balance in a base-isolated buildingBy using become the themodel relative in Figure movement 9, the relative and the movement energy balance and the in the energy base-isolated balance in layer. a base-isolated buildingAs onebecome matter the ofrelative concern movement during theand designthe energy stage, balance the prediction in the base of-isolated the maximum layer. relative displacement between the upper and lower concave plates of the bearing during earthquakes is the focus of this study. Figure 12 shows the relative movement of the base-isolated layer during strong earthquake pulses, where vu is the absolute velocity of the upper structure, vg is the absolute velocity of the ground, and y shows the relative displacement between the upper and lower concave plates, which can also be called the response displacement of the isolation layer. Then, the response velocity of the isolation layer (the relative velocity between the upper and lower concave plates) can be expressed as: . v = y = vu vg (13) − During a strong earthquake pulse, the history of vg is known and the value of vu is mainly determined by the dynamic friction coefficient of the bearing. Therefore, y can be predicted by integral v, which is the difference between vu and vg. Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 26

As one matter of concern during the design stage, the prediction of the maximum relative displacement between the upper and lower concave plates of the bearing during earthquakes is the focus of this study. Figure 12 shows the relative movement of the base-isolated layer during strong earthquake pulses, where vu is the absolute velocity of the upper structure, vg is the absolute velocity of the ground, and y shows the relative displacement between the upper and lower concave plates, which can also be called the response displacement of the isolation layer. Then, the response velocity of the isolation layer (the relative velocity between the upper and lower concave plates) can be expressed as:

v  y  vu  vg (13)

During a strong earthquake pulse, the history of vg is known and the value of vu is mainly determinedAppl. Sci. 2020, 10 by, 5259 the dynamic friction coefficient of the bearing. Therefore, y can be predicted13 of by 25 integral v, which is the difference between vu and vg.

vu Upper Concave Plate Slider vg Lower Concave Plate

Ground y

FigureFigure 12. 12. RelativeRelative movement movement of of the the base base-isolated-isolated layer. layer.

Based on on the the response response analysis analysis results, results, t thehe ground ground velocity record of the earthquake was found found toto be greatly related to, and work simultaneoussimultaneouslyly with, the the transferred energy and the response displacement ofof the the isolation isolation layer layer during during earthquakes. earthquakes As shown. As shownin Figure in 13 Figure, during 13, one during earthquake, one earthquake,several large seve velocityral large pulses velocity (from pulses one ground (from one peak ground velocity, peakvgi velocity,1, to the vg nexti-1, to peak, the nextvgi, and peak, to vg thei, Appl. Sci. 2020, 10, x FOR PEER REVIEW − 14 of 26 next peak, vgi+1, is onei+1 velocity pulse) exist. The difference between vgi 1 and vgi-1 i was definedi as the and to the next peak, vg , is one velocity pulse) exist. The difference between− vg and vg was defined asvelocity the velocity diffSinceerence difference the of elastic this velocityofstrain this energy velocity pulse, was ∆pulse, alsovgi, whichrelatively Δvgi, couldwhi smallch show couldin this the study,show intensity the valueintensity of theof it velocity couldof the be velocity pulse: pulse: ignored under large velocity pulses. Then, ΔEri in this time interval can be expressed as: ∆vgi = vgi vgi 1 (14) Eri  −Ehri − (16) vgi  vgi  vgi1 (14)

It was found that, when a large Δvgi exists in the earthquake, it will lead to large input energy in 300 ) vgi s the structural/ system, ΔEi, and a large response displacement of the isolation layer (a large relative

m 200

m i displacement( between the upper concave plate and the lower concave plate), Δy , simultaneously.

y 100 ∆vgi= vgi – vgi-1 t i

Since the upperc structure is considered to bevg a rigid body in this study, the total input energy, E, is o 0 i+1 l

e h consumed v by the hysteresis energy E . However, during a velocity pulse, some of the input energy -100 d vgi-1 n k s

will first beu temporarily stored in the kinetic energy, E , and the elastic strain energy, E . In order to

o -200

r beginning of the velocity pulse

eliminate theG interference of Ek, the change in energy from the beginning of the velocity pulse to the -300 time kinetic energy0 recovers2 to4 zero, 6which is8 approximately10 12 the time14 interval16 from18 yi−1 to20 yi and is Time (s) i considered to be10 very4 useful as it is related to the peak response displacement change, Δy . In Figure 10 13, yi and yi−1 are two neighboring peak response displacements of the isolation layer, and every time 8 a peak response displacement is achieved,∆E Ek becomes zero; ymax is the maximum amplitude of the ) i ∆Eri = ∆Ehri + ∆Esri (∆Ekri ≈ 0) J

( 6 i response displacementy of the isolation layer; Δy is one of the peak response displacement changes of g r

e 4 kinetic energy recovers to zero

the isolationn layer, and the one shown in the figure is the maximum. Then, in this time interval, the E 2 i i total input energy, ΔEr , can be expressed as the sum of the consumed energy.E ΔEhr and the change in the elastic strain0 energy, ΔEsri, with the change in kinetic energy, ΔEkri, Ehwere also considered to be zero: 0 2 4 6 8 10 12 14 16 18 20 Time (s)

Eri  Ehri  Esri (Ekri  0) (15) ) y

m i-1 m (

t n e

m ∆yi = yi – yi-1 e c a l p s i y D yi ymax

Figure 13.FigureTime 13. history Time history of ground of ground velocity, velocity, input input energy energy and and response response displacement displacement of of the the base-isolated base- layer duringisolated NCC layer (Northridge during NCC ( earthquakeNorthridge earthquake measured measured by Canyon by Canyon Country-WLC Country station)-WLC station) with its PGV equals towith 250 its mm PGV/s. equals to 250 mm/s.

To see exactly how the ground velocity record determines the response displacement of the isolation layer, the velocity history of the upper structure, vu, and the ground, vg, based on the response analysis, is shown in Figure 14, where, based on Equation (13), the area between the velocity history of the upper structure and the ground is the response displacement of the isolation layer. It can be seen from the time history of vu that the acceleration of the upper structure during large velocity pulses is roughly constant. Therefore, in the prediction, it is reasonable to consider the friction coefficient during large velocity pulses to be constant.

Appl. Sci. 2020, 10, 5259 14 of 25

It was found that, when a large ∆vgi exists in the earthquake, it will lead to large input energy in the structural system, ∆Ei, and a large response displacement of the isolation layer (a large relative displacement between the upper concave plate and the lower concave plate), ∆yi, simultaneously. Since the upper structure is considered to be a rigid body in this study, the total input energy, E, is consumed by the hysteresis energy Eh. However, during a velocity pulse, some of the input energy will first be temporarily stored in the kinetic energy, Ek, and the elastic strain energy, Es. In order to eliminate the interference of Ek, the change in energy from the beginning of the velocity pulse to the time kinetic energy recovers to zero, which is approximately the time interval from yi 1 to yi and is − considered to be very useful as it is related to the peak response displacement change, ∆yi. In Figure 13, yi and yi 1 are two neighboring peak response displacements of the isolation layer, and every time − a peak response displacement is achieved, Ek becomes zero; ymax is the maximum amplitude of the response displacement of the isolation layer; ∆yi is one of the peak response displacement changes of the isolation layer, and the one shown in the figure is the maximum. Then, in this time interval, the total input energy, ∆Eri, can be expressed as the sum of the consumed energy. ∆Ehri and the change in the elastic strain energy, ∆Esri, with the change in kinetic energy, ∆Ekri, were also considered to be zero: ∆Eri = ∆Ehri + ∆Esri (∆Ekri = 0) (15) Since the elastic strain energy was also relatively small in this study, the value of it could be ignored under large velocity pulses. Then, ∆Eri in this time interval can be expressed as:

∆Eri = ∆Ehri (16)

To see exactly how the ground velocity record determines the response displacement of the isolation layer, the velocity history of the upper structure, vu, and the ground, vg, based on the response analysis, is shown in Figure 14, where, based on Equation (13), the area between the velocity history of the upper structure and the ground is the response displacement of the isolation layer. It can be seen from the time history of vu that the acceleration of the upper structure during large velocity pulses is roughly constant. Therefore, in the prediction, it is reasonable to consider the friction coefficient during large velocity pulses to be constant. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 26

300 vu 200 Peak response displacement change vg

100 of the isolation layer Δyi 0 0 2 4 6 8 10 12 14 16 18 20 -100 Time (s) -200 Absolute Velocity (mm/s) (mm/s) Velocity Absolute -300

Figure 14. TimeTime history history of of the the ground ground velocity velocity and and upper upper structure structure velocity velocity during during earthquakes where the NCC and PGV are equalequal toto 250250 mmmm/s./s.

In Figure Figure 15 15,, the the same same response response displacement displacement is shown is shown in the in time the history time history of the response of the response velocity ofvelocity the isolation of the isolation layer, v. layer, v.

300 v 200 m/s) m/s) 100 0 0 2 4 6 8 10 12 14 16 18 20 -100 Peak response displacement change Time (s) -200 of the isolation layer Δyi Relative Velocity (m Velocity Relative -300 Figure 15. Time history of the relative velocity between the upper and lower concave plates during earthquakes where the NCC with PGV are equal to 25 mm/s.

6.1.2. Prediction of the Response Displacement of the Isolation Layer

In this section, the peak response displacement changes in the isolation layer, Δyi, were primarily predicted by simplifying Figure 14. It should be mentioned that Δyi was predicted instead of ymax. This was because the value of ymax was highly related to former response displacement, which made it difficult to be derived theoretically. However, the derivation of Δyi was more related to the corresponding velocity pulse. Furthermore, since the maximum Δyi was usually larger than ymax, the maximum Δyi could be considered to be a conservative value of ymax. As shown in the left figure of Figure 16, the solid line shows a ground velocity pulse from vgi−1 to vgi to vgi+1 and the two dashed lines show the slope from vgi−1 to vgi and the slope from vgi to vgi+1, respectively, which can be understood as the average ground acceleration. The dotted line is the absolute velocity of the upper structure, and its slope can be roughly considered to be constant, as introduced in Figure 14. Furthermore, since the points where vu and vg intersect are determined by the former velocity history, the intersection in the left figure of Figure 16 was randomly picked between vgi−1 and vgi, referred to as vui−1. At the intersection points, vu was equal to vg, which meant that v was equal to zero, and this meant that the direction of the response displacement of the isolation layer, y, would change from the next second. Then, until the next intersection, y would be in the same direction; reach the peak displacement change, Δyi; and reverse direction again. Then, how the calculation of Δyi can be simplified under a large velocity pulse is shown in the right figure of Figure 16. In the simplification, vui−1 and vgi−1 are assumed to be equal, which is a conservative assumption for predicting Δyi:

vui1  vgi1 (17) The acceleration between two adjacent peak ground velocities is assumed to be the average value, kgi and kgi+1: Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 26

300 vu 200 Peak response displacement change vg

100 of the isolation layer Δyi 0 0 2 4 6 8 10 12 14 16 18 20 -100 Time (s) -200 Absolute Velocity (mm/s) (mm/s) Velocity Absolute -300

Figure 14. Time history of the ground velocity and upper structure velocity during earthquakes where the NCC and PGV are equal to 250 mm/s.

Appl. Sci.In Figure2020, 10, 5259 15, the same response displacement is shown in the time history of the response15 of 25 velocity of the isolation layer, v.

300 v 200 m/s) m/s) 100 0 0 2 4 6 8 10 12 14 16 18 20 -100 Peak response displacement change Time (s) -200 of the isolation layer Δyi Relative Velocity (m Velocity Relative -300 Figure 15. Time history of the relative velocity between the upper and lower concave plates during earthquakes wh whereere the NCC with PGV are equal to 25 mmmm/s./s.

6.1.2. Prediction Prediction of the Response Displacement of the Isolation Layer

In this section, the peak response displacement changes in the isolation layer, ∆yi, were primarily In this section, the peak response displacement changes in the isolation layer, Δyi, were primarily predicted by simplifying Figure 14. It should be mentioned that ∆yi was predicted instead of ymax. predicted by simplifying Figure 14. It should be mentioned that Δyi was predicted instead of ymax. This was because the value of ymax was highly related to former response displacement, which made This was because the value of ymax was highly related to former response displacement, which made it difficult to be derived theoretically. However, the derivation of ∆yi was more related to the it difficult to be derived theoretically. However, the derivation of Δyi was more related to the corresponding velocity pulse. Furthermore, since the maximum ∆yi was usually larger than ymax, corresponding velocity pulse. Furthermore, since the maximum Δyi was usually larger than ymax, the the maximum ∆yi could be considered to be a conservative value of ymax. maximum Δyi could be considered to be a conservative value of ymax. As shown in the left figure of Figure 16, the solid line shows a ground velocity pulse from vgi 1 As shown in the left figure of Figure 16, the solid line shows a ground velocity pulse from vgi−1 Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of− 26 to vgi to vgi+1 and the two dashed lines show the slope from vgi 1 to vgi and the slope from vgi to to vgi to vgi+1 and the two dashed lines show the slope from vgi−1 to− vgi and the slope from vgi to vgi+1, vg respectively,i+1, respectively, which which can be can unde berstood understood as the as average the average ground ground acceleration. acceleration. The Thedotted dotted line lineis the is the absolute velocity of the upper structure,vg  vg and its slope canvg be roughlyvg considered to be constant, absolute velocity of the upper structure,kg  i and iits1 ;slopekg can bei roughly1 i considered to be constant, as i  i1  vu vg (18) introducedas introduced in Figure in Figure 14. 14 Further. Furthermore,more, since sinceti the thepoints points where where vuti 1andand vg intersectintersect are aredete determinedrmined by theby the former former velocity velocity history, history, the the intersection intersection in in the the left left figure figure of of Figure Figure 1616 waswas randomly picked where Δti is the time interval between vgi−1 and vgi; Δti+1 is the time interval between vgi and vgi+1. between vgi 1 and vgi, referred to as vui 1. At the intersection points, vu was equal to vg, which between vgi−1 and vgi, referred to as vui−1. At the intersection points, vu was equal to vg, which meant The friction− coefficient was assumed− to be the nominal friction coefficient, µn, which was equal meant that v was equal to zero, and this meant that the direction of the response displacement of the thatto 0.043 v was and equal could to zero be, and used this to meant roughly that simulate the direction the behaviorof the response of the displacement bearing under of the earthquake isolation isolation layer, y, would change from the next second. Then, until the next intersection, y would be layer,excitations. y, would Therefore, change from the next second. Then, until the next intersection, y would be in the same in the same direction; reach the peak displacement change, ∆yi; and reverse direction again. Then, direction; reach the peak displacement change, Δyi; and reverse direction again. Then, how the how the calculation of ∆yi can be simplifiedku  under  g a largekg / kg velocity pulse is shown in the right figure calculation of Δyi can be simplified under ai largen velocityi pulsei is shown in the right figure of Figure(19) of Figure 16. In the simplification, vui 1 and vgi 1 are assumed to be equal, which is a conservative 16. In the simplification, vui−1 and vgi−1 −are assumed2 − to be equal, which is a conservative assumption assumptionwhere g is the for gravity predicting acceleration∆y : in mm/s . for predicting Δyi: i Then, the predicted value will be Δyi-pre, as shown in Figure 16, which is the predicted peak vui 1 = vgi 1 (17) − − response displacement of the isolation layer.vu i1  vgi1 (17) The acceleration between two adjacent peak ground velocities is assumed to be the average Vel. (mm/s) vgi vgi value, kgi and kgi+1: Time (s) vg

Δyi vu -pre

kgi ku kgi+1 kg i i kui kgi+1

vu vgi+1 vgi-1 i-1 vu =vg i-1 i-1

FigureFigure 16. 16.Method Method toto predictpredict thethe responseresponse displacement displacement of of the the isolation isolation layer layer (0 (0<

It can be seen from Figure 16 that, when the value of kui is close to kgi, Δyi-pre is close to zero. However, based on the response analysis results, Δyi can also be relatively large in this situation. Therefore, a special prediction method was proposed for these cases, as shown in Figure 17, in which the case of kui/kgi being equal to 1.0 is shown. Based on the response analysis results, the cases of kui/kgi being between 0.8 and 1 should use this method and the cases of kui/kgi being between 0 and 0.8 should use the method in Figure 16. Appl. Sci. 2020, 10, 5259 16 of 25

The acceleration between two adjacent peak ground velocities is assumed to be the average value, kgi and kgi+1: vgi vgi 1 vgi+1 vgi kg = − − ; kg + = − (18) i ∆ti i 1 ∆ti+1

where ∆ti is the time interval between vgi 1 and vgi; ∆ti+1 is the time interval between vgi and vgi+1. − The friction coefficient was assumed to be the nominal friction coefficient, µn, which was equal to 0.043 and could be used to roughly simulate the behavior of the bearing under earthquake excitations. Therefore,

kui = µn g kgi/ kgi (19) × × where g is the gravity acceleration in mm/s2. Then, the predicted value will be ∆yi-pre, as shown in Figure 16, which is the predicted peak response displacement of the isolation layer. It can be seen from Figure 16 that, when the value of kui is close to kgi, ∆yi-pre is close to zero. However, based on the response analysis results, ∆yi can also be relatively large in this situation. Therefore, a special prediction method was proposed for these cases, as shown in Figure 17, in which the case of kui/kgi being equal to 1.0 is shown. Based on the response analysis results, the cases of kui/kgi being between 0.8 and 1 should use this method and the cases of kui/kgi being between 0 and 0.8 should use the method in Figure 16. Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 26

Vel. (mm/s) 1.5kgi+1 Time (s) vgi

ku vg 1.5 vu kg i kui=kgi kgi+1 vg vg i-1 vui-1 i+1

kgi 0.5

FigureFigure 17. 17. MethodMethod to to predict predict the the response response displacement displacement of of the the isolation isolation layer layer (0.8 (0.8 << kukui/kg/kgi ≤ 1.0).1.0). i i ≤

Therefore,Therefore, as as shown shown in in the the right right half half of of Figure Figure 16, 16 ,for for the the cases cases of of kukui/ikg/kgi beingi being between between 0 0and and 0.8, 0.8, Δ∆yyi-ipre-pre (the(the predicted predicted value value of of the the peak peak response response displacement displacement change change of of the the isolation isolation layer) layer) can can be be predictedpredicted by by vgvgi-1i-1, vg, vgi, ikg, kgi, ikg, kgi+1i +and1 and kukui, whichi, which are are given given by by the the time time history history data data of of the the earthquake earthquake recordrecord and and Equation Equation (19). (19).

1 vg  vg  kg  ku 1 i vgi ivg1i 12 2 i kgi kui i ∆yyi  prepre = (kgi(kgkui )ku() ( − −) )(1 (1 + − )) i 2 i i kg ku kg + − 2− × − × kgi i × kui ikg− ii 11 (kg ku ) (kg kg + ) 2 (20) = i− i × i− i 1 (vg vg ) (20) (kg 2 kukg2 )(ku(kgkg kg)  ) i i 1   −i ii i i i+1 i ×1  −  − 2 × 2 × − (vgi vgi1 ) 2  kgi (kui  kgi1 )

As for the cases of kui/kgi being between 0.8 and 1, the value of vgi−1, vgi, kgi and kgi+1 should be amplified by 1.5 before using Equation (20). The accuracy was between 1 and 2.1, as shown in Figure 18, which is a conservative prediction with an error caused by the assumptions shown in Equations (17)–(19).

1500 PGV=250 PGV=500 1000 PGV=750

(mm) PGV=1000 500 pre - i y

Δ 0 -1500 -500 500 1500

-500

-1000

-1500 Δyi (mm)

Figure 18. Accuracy of the prediction method for the maximum response displacement change of the isolation layer under selected earthquake inputs. Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 26

Vel. (mm/s) 1.5kgi+1 Time (s) vgi

ku vg 1.5 vu kg i kui=kgi kgi+1 vg vg i-1 vui-1 i+1

kgi 0.5

Figure 17. Method to predict the response displacement of the isolation layer (0.8 < kui/kgi ≤ 1.0).

Therefore, as shown in the right half of Figure 16, for the cases of kui/kgi being between 0 and 0.8, Δyi-pre (the predicted value of the peak response displacement change of the isolation layer) can be predicted by vgi-1, vgi, kgi, kgi+1 and kui, which are given by the time history data of the earthquake record and Equation (19).

1 vgi  vgi1 2 kgi  kui yi  pre   (kgi  kui )( ) (1  ) Appl. Sci. 2020, 10, 5259 2 kgi kui  kgi1 17 of 25 (20) (kg  ku )(kg  kg )   i i i i1   2 2 (vgi vgi1 ) 2  kgi (kui  kgi1 ) As for the cases of kui/kgi being between 0.8 and 1, the value of vgi 1, vgi, kgi and kgi+1 should be − amplifiedAs by for 1.5 the before casesusing of kui/ Equationkgi being between (20). 0.8 and 1, the value of vgi−1, vgi, kgi and kgi+1 should be amplified by 1.5 before using Equation (20). The accuracy was between 1 and 2.1, as shown in Figure 18, which is a conservative prediction The accuracy was between 1 and 2.1, as shown in Figure 18, which is a conservative prediction with an error caused by the assumptions shown in Equations (17)–(19). with an error caused by the assumptions shown in Equations (17)–(19).

1500 PGV=250 PGV=500 1000 PGV=750

(mm) PGV=1000 500 pre - i y

Δ 0 -1500 -500 500 1500

-500

-1000

-1500 Δyi (mm)

FigureFigure 18. Accuracy 18. Accuracy of theof the prediction prediction method method for the m maximumaximum response response displacement displacement change change of the of the isolationisolation layer layer under under selected selected earthquake earthquake inputs. inputs.

6.1.3. Prediction of Energy Transfer The input energy of the isolation layer was defined by Equation (4) in the introduction. Since the response displacement of the isolation layer, y, can be predicted, the total input energy to the isolation layer between vgi 1 and vgi can also be predicted by vgi-1, vgi, kgi, kgi+1 and kui, which were given by − time history data of the earthquake records and Equation (19):

1 vgi vgi 1 2 ∆Ei pre = m kgi (kgi kui) ( − − ) − × × 2 × − × kgi (21) 1 kui 2 = 2 m (1 ) (vgi vgi 1) × × − kgi × − − The equivalent velocity [9] of the total input energy is: s kui V∆Ei pre = 1 vgi vgi 1 (22) − − kgi × − −

The amount of ∆Ei consumed by the hysteresis energy, ∆Ei, can be predicted by vgi 1, vgi, kgi, − kgi+1 and kui, which were given by the time history data of the earthquake records and Equation (19):

∆Ehri pre = m kui ∆yi pre − × × ku−(kg kg + ) (23) = ∆E pre i× i− i 1 i kg (ku kg + ) − × i× i− i 1 The equivalent velocity of the consumed energy is: s s ku ku (kg kg ) = i i i i+1 V∆Ehri pre 1 vgi vgi 1 × − (24) − − kgi × − − × kg (ku kg + ) i × i − i 1

The accuracy of V∆Ei-pre was between 0.8 and 1.9, and the accuracy of V∆Ehi-pre was between 0.8 and 1.6. It can be seen that this prediction method is relatively conservative. Appl. Sci. 2020, 10, 5259 18 of 25

6.2. Simplification and Optimization of the Prediction Method Based on the Analysis Results

6.2.1. Simplification of kgi+1 By combining Equations (21) and (23), Equation (20) can be expressed in the form of energy as:

1 kui 2 kui (kgi kgi+1) m kui ∆yi pre = m (1 ) (vgi vgi 1) × − (25) × × − 2 × × − kgi × − − × kg (ku kg + ) i × i − i 1

kgi+1 was only contained in the rightmost fraction of Equation (25), which can be considered as the percentage of energy absorption, and it was attempted to simplify this fraction in order to not only simplify the equation but also make the method easier to apply in the design stage. Therefore, it is generalized with a generalization parameter, ε:

kg + = ε kg (26) i 1 − × i Then, the predicted percentage of the absorption can be expressed, in the form of equivalent velocity of energy, as:

s uv tu (1 + ε) kui V∆Ehr pre kui (kgi kgi+1) kgi i − = × − = × (27) ( ) kui V∆Ei pre kgi kui kgi+1 ε + − × − kgi

By comparing the prediction value of Equation (27) to the analysis result points, as shown in FigureAppl. 19 Sci., 0.25 2020 is, 10 considered, x FOR PEER REVIEW to be the most suitable value of ε for the general calculation. It19 should of 26 be mentioned that, since the percentage of absorption will not exceed one, the value of V∆Ehri-pre/V∆Ei-pre be mentioned that, since the percentage of absorption will not exceed one, the value of VΔEhri –pre/VΔEi can be taken as one when ku /kg is larger than one. –pre can be taken as one wheni i kui/kgi is larger than one.

1.5

ε = 0.1 1 Ei Δ

/V 500 1500

Ehri ε = 0.25 PGV=250 Δ V 0.5 ε = 1 PGV=500 PGV=750 PGV=1000 Prediction 0 0 0.5 1 1.5 kui/ kgi FigureFigure 19. Simplified19. Simplified prediction prediction of of the the percentage percentage of absorption absorption by by assuming assuming different different values values of ε. of ε.

6.2.2.6.2.2. Simplification Simplification of theof the Prediction Prediction Equation Equation ofof VΔEi∆Ei –-prepre

EquationEquation (22) (22) gives gives the the prediction prediction equation equation ofof VΔ∆EiEi –-prepre. .It It shows shows a linear a linear relationship relationship between between VΔEi –pre and the absolute value of the peak ground velocity difference, |Δvgi|, which equals to |vgi- V∆Ei-pre and the absolute value of the peak ground velocity difference, |∆vgi|, which equals to |vgi- vgi-1|. vgi-1|. By observing the relationship between the analysis results for VΔEi and |Δvgi|, as shown in By observing the relationship between the analysis results for V∆Ei and |∆vgi|, as shown in Figure 20, Figure 20, it was found that the value of kgi could be taken as a constant without causing large it was found that the value of kg could be taken as a constant without causing large differences. differences. i Therefore, the slope of the prediction line, based on Equation (22), is determined as: 500 1500 2000 s s PGV=250 r V∆Ei pre kui kui kui − = 1PGV=500 1 = 1 | | (28) − kgi ≈ − kgave − 930 ∆vgi PGV=750 1500 PGV=1000 Prediction 1000 (mm/s) Ei Δ V

500

0 0 500 1000 1500 2000 |Δvg | (mm/s) i

Figure 20. Simplified prediction of VΔEi.

Therefore, the slope of the prediction line, based on Equation (22), is determined as:

VE  pre ku ku kui i  1  i  1  i  1  (28) vgi kgi kgave 930

where the kgave = average value of kgi (the kgi corresponding to the maximum response displacement of the isolation layer) of the eight selected earthquakes with the PGV amplified as 250 mm/s2, which was 930 mm/s2. The value 930 mm/s2 can be considered to be a constant value that can be used in most cases where the input earthquakes have PGVs larger than 250 mm/s2, because the eight earthquakes selected in this study were considered to be representative. Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 26

be mentioned that, since the percentage of absorption will not exceed one, the value of VΔEhri –pre/VΔEi –pre can be taken as one when kui/kgi is larger than one.

1.5

ε = 0.1 1 Ei Δ

/V 500 1500

Ehri ε = 0.25 PGV=250 Δ V 0.5 ε = 1 PGV=500 PGV=750 PGV=1000 Prediction 0 0 0.5 1 1.5 kui/ kgi Appl. Sci. 2020, 10, 5259 19 of 25 Figure 19. Simplified prediction of the percentage of absorption by assuming different values of ε.

6.2.2. Simplification of the Prediction Equation of VΔEi –pre where the kgave = average value of kgi (the kgi corresponding to the maximum response displacement of the isolationEquation layer) of (22) the gives eight the selected prediction earthquakes equation of withVΔEi –pre the. It PGV shows amplified a linear relationship as 250 mm between/s2, which was VΔEi –pre and the absolute value of the peak ground velocity difference, |Δvgi|, which equals to |vgi- 930 mm/s2. The value 930 mm/s2 can be considered to be a constant value that can be used in most vgi-1|. By observing the relationship between the analysis results for VΔEi and |Δvgi|, as shown in 2 cases whereFigure the 20, input it was earthquakes found that the have value PGVs of kgi largercould be than taken 250 as mm a constant/s , because without the causing eight earthquakes large selecteddifferences. in this study were considered to be representative.

500 1500 2000 PGV=250 PGV=500 PGV=750 1500 PGV=1000 Prediction 1000 (mm/s) Ei Δ V

500

0 0 500 1000 1500 2000 |Δvg | (mm/s) i

FigureFigure 20. 20.Simplified Simplified predictionprediction of of VΔVEi.∆ Ei. Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 of 26 6.2.3. SimplifiedTherefore, and the Optimized slope of the Prediction prediction Methodline, based on Equation (22), is determined as: 6.2.3. Simplified and Optimized Prediction Method By applying Equations (27)V andE  (28),pre the proposedku predictionku methodku can be simplified, as shown By applying Equations (27)i and  (28), the i proposed  predictioni   methodi can be simplified, as in Figure 21. 1 1 1 (28) shown in Figure 21. vgi kgi kgave 930

where the300 kgave = average value of kgi (the kgi corresponding to the maximum response displacement 500 1500 (1) 2000 2 of the isolation250 layer) of the eight selected earthquakes with the PGVPGV=250 amplified as 250 mm/s , which 2 2 was 930 200mm/s . The value 930 mm/s can be considered to be a constantPGV=500 value that can be used in 2 most cases150 where the input earthquakes have PGVs 1500 larger thanPGV=750 250 mm/s , because the eight earthquakes100 selected in this study were considered to be representative.PGV=1000 ∆vgi 50 Prediction 1000 0 (mm/s) Ei Δ

-50 6.8 7 7.2 7.4 7.6 7.8 V -100 Ground Ground velocity (mm/s) (1) 500 -150   g Time (s) V  pre  1 n  vg 90 Ei 930 i 0 80 0 500 1000 1500 2000 (2) |Δvgi| (mm/s) ∆E 70 i ∆Ehr y y (kJ) i 1.5 (2) ku 60 (1  )  i V  pre Ehri kg Energ  i 50 V  pre kui Ei   kg 40 1 i Ei Ei 6.8 7 7.2 7.4 7.6 7.8 Δ /V 500 1500

Time (s) (3) Ehri ε = 0.25 PGV=250 Δ

30 V 0.5 PGV=500 20 PGV=750 PGV=1000 10 Prediction ∆yi 0 0 0 0.5 1 1.5 6.8 7 7.2 7.4 7.6 7.8 ku / kg -10 i i -20 (3) Equation of Energy

Response Response Disp. y (mm) 1 2 n  m  g  yi  pre   m V -30 2 Ehri Time (s) Figure 21. Simplified prediction method. Figure 21. Simplified prediction method.

The accuracies of VΔEi –pre, VΔEhri –pre and Δyi-pre when using this simplified prediction method are shown in Figures 22–24, respectively. The accuracy of VΔEi –pre was between 0.8 and 1.5, the accuracy of VΔEhri –pre was between 0.7 and 1.6, and the accuracy of Δyi-pre was between 0.7 and 1.8. Appl. Sci. 2020, 10, 5259 20 of 25

The accuracies of V∆Ei-pre, V∆Ehri-pre and ∆yi-pre when using this simplified prediction method are shown in Figures 22–24, respectively. The accuracy of V∆Ei-pre was between 0.8 and 1.5, the accuracy of V∆Appl.Appl.Ehri -pre Sci.Sci. 2020was2020,, 1010 between,, xx FORFOR PEERPEER 0.7 REVIEWREVIEW and1.6, and the accuracy of ∆yi-pre was between 0.7 and 1.8.2121 ofof 2626

20002000

15001500 (mm/s) (mm/s) (mm/s) 10001000 pre pre pre pre pre pre – – – Ei Ei Ei Ei Ei Ei

Δ PGV=250 Δ PGV=250 Δ PGV=250 V V V 500500 PGV=500PGV=500 PGV=750PGV=750 PGV=1000PGV=1000 00 00 500500 10001000 15001500 20002000 VVΔEi (mm/s)(mm/s) ΔEi

Figure 22. Accuracy of the simplified prediction method for VΔEi. FigureFigure 22. 22.Accuracy Accuracy of of the the simplified simplified prediction method method for for VΔEiV. ∆Ei.

20002000

15001500 (mm/s) (mm/s) (mm/s) 10001000 pre pre pre pre pre pre – – – Ehri Ehri Ehri

Δ PGV=250 Δ PGV=250 Δ PGV=250 V V V 500 500 PGV=500PGV=500 PGV=750PGV=750 PGV=1000PGV=1000 00 00 500500 10001000 15001500 20002000 V (mm/s)(mm/s) VΔΔEhriEhri (mm/s)

ΔEhri FigureFigureFigure 23. 23.23.Accuracy AccuracyAccuracy of ofof the thethe simplified simplifiedsimplified prediction prediction methodmethod method forfor for VVΔVEhri∆.. Ehri.

20002000

15001500 (mm) (mm) (mm) 10001000 pre| pre| pre| pre| pre| pre| - - i - i i y y y PGV=250

|Δ PGV=250

|Δ PGV=250 |Δ 500500 PGV=500PGV=500 PGV=750PGV=750 PGV=1000PGV=1000 00 00 500500 10001000 15001500 20002000 |Δ|Δy || (mm)(mm) |Δyii| (mm)

Figure 24. Accuracy of the simplified prediction method for |i∆y |. FigureFigure 24.24. AccuracyAccuracy ofof thethe simplifiedsimplified predictionprediction methodmethod forfor |Δ|Δyyii|.|. i

Appl. Sci. 2020, 10, 5259 21 of 25 Appl. Sci. 2020, 10, x FOR PEER REVIEW 22 of 26

6.3.6.3. Application of thethe PredictionPrediction MethodMethod inin DesignDesign InIn orderorder toto givegive aa generalgeneral predictionprediction ofof thethe maximummaximum responseresponse displacementdisplacement changechange andand inputinput energyenergy of of an an isolation isolation layer layer under under a group a group of earthquakes of earthquakes with a certain with aintensity, certain the intensity, specific the earthquake specific characteristics,earthquake characteristics, such as ∆vgi and suchkg i as, should Δvgi and be replaced. kgi, should Therefore, be replaced. Equation Therefore, (29) is proposed, Equation where (29) isζ isproposed, the generalization where ζ is parameter the generalization of the ground parameter velocity. of As the shown ground in Figure velocity. 25, basedAs shown on the in earthquake Figure 25, records,based on|∆ thevgi earthquake| was assumed records, to be |Δ 1.5vg timesi| was the assumed peak ground to be 1.5 velocity times (PGV)the peak in ground this study. velocity (PGV) in this study.

∆vgi = ς PGV (29) vg i   × PGV (29)

2000

ζ = 2.0 ζ = 1.5 1500 ) ζ = 1.0

(mm/s 1000 PGV=250 vg|

|Δ PGV=500 500 PGV=750 PGV=1000

0 0 500 1000 1500 2000 PGV (mm/s) FigureFigure 25. 25. RelationshipRelationship between between |Δ|∆vgvg|| and and PGV. PGV.

kgkgii was assumed basedbased onon thethe averageaverage valuevalue ofof kgkgii ofof thethe eighteight selectedselected earthquakesearthquakes withwith PGVsPGVs amplifiedamplified toto 250250 mmmm/s,/s, whichwhich waswas 930930 mmmm/s:/s:

PGV kgkgi=930930  ((30)30) i × 250 Therefore,Therefore, thethe predictionprediction methodmethod cancan bebe generalizedgeneralized as:as:

r n  g V  pre  1  µn g  PGV (31) V Ei pre = 1 930× ς PGV (31) ∆Ei − − 930 × ×

s (1   ) n  g V  pre  V  pre  Ehri Ei (1 + ε) µn PGVg (32) V pre = V pre      ×  × (32) ∆Ehri ∆Ei n g 930 PGV − − × µn g + ε 930 250 × × × 250 VV  prepre2 ∆Ehrii ∆yyi i prepre= − ((33)33) − 22 µn  g × n × where = 1.5 and = 0.25. where ζζ = 1.5 and ε = 0.25. TheThe accuraciesaccuracies areare shownshown inin FiguresFigures 26 26 and and 27 27:: Appl. Sci. 20202020, 10, 5259x FOR PEER REVIEW 2223 of 2526 Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 26

Figure 26. Prediction of VΔEi and VΔEhri by only using PGV. FigureFigure 26. 26.Prediction Prediction of ofV V∆ΔEiEi and V∆ΔEhriEhri byby only only using using PGV PGV..

1500 500 1500 1500 500 PGV=250 1500 PGV=250 PGV=500 PGV=500 PGV=750 PGV=750 1000 PGV=1000 1000 PGV=1000 Prediction

(mm) Prediction | i (mm) y | i |Δ y 500 |Δ 500

0 0 0 500 1000 1500 0 500PGV (mm/s)1000 1500 PGV (mm/s) FigureFigure 27. 27. PredictionPrediction of of |Δ|∆yi| byby onlyonly usingusing PGV.PGV. Figure 27. Prediction of |Δyi| by only using PGV. In Figures 26 and 27, the data points show the analysis results for the maximum V , V and ∆y In Figures 26 and 27, the data points show the analysis results for the maximum∆Ei VΔ∆EiEhri, VΔEhri andi underIn various Figures earthquake 26 and 27, inputs the data with points their show PGVs the amplified analysis to results 250, 500, for 750 the and maximum 1000 mm V/s;ΔEi the, VΔ dashedEhri and Δyi under various earthquake inputs with their PGVs amplified to 250, 500, 750 and 1000 mm/s; the linedashedΔyi under shows line various the shows prediction theearthquake prediction result inputs under result with earthquake under their earthquake PGV groupss amplified groups with di ffwithtoerent 250, different PGVs.500, 750 ThePGVs. and prediction 1000 The predictionmm/s; results the areresultsdashed a little are line abovea shows little the above the average prediction the av becauseerage result because of under the selected ofearthquake the selected value groups of value the generalizationwith of the different generalization PGVs. parameter The parameter predictionζ and ε. ζ andresults εIt. shouldare a little be mentioned above the thataverage the prediction because of results the selected would bevalue made of morethe generalization conservative byparameter enlarging ζ theand generalizationεIt. should be mentionedparameter ζ thatin Equation the prediction (31) and results by decreasing would be the made generalization more conservative parameter byε inenlarging EquationIt should the (32). generalization be By mentioned doing this, parameter that the the maximum predictionζ in Equation response results (31) displacement would and by be decreasingmade (the more maximum the conservative generalizatio∆yi can bebyn consideredenlarging the as thegeneralization conservative parameter value of ζy in , Equation as shown (31) in Figures and by 11 decreasing and 27) and the the generalizatio maximumn parameter ε in Equation (32). By doing this,max the maximum response displacement (the maximum Δyi inputparameter energy ε in of Equation the isolation (32). layer By doing under this, all the maximum earthquake response inputs with displacement same intensity (the maximum could also Δ beyi can be considered as the conservative value of ymax, as shown in Figures 11 and 27) and the maximum roughlyinputcan be energy considered predicted. of the as isolation the conservative layer under value all ofthe y maxearthqua, as shownke inputs in Figure withs 11 same and intensity 27) and the could maximum also be roughlyinput energy predicted. of the isolation layer under all the earthquake inputs with same intensity could also be 7.roughly Conclusions predicted.

Full-scale dynamic tests were conducted under various conditions (the difference in surface pressure depending on the scale of the superstructure, relative speed between the upper and lower Appl. Sci. 2020, 10, 5259 23 of 25 concave plate, amplitude, etc.) to evaluate the effects of surface pressure, velocity and temperature rise due to friction heat on the dynamic behavior of DCFP bearings for seismic isolation structures. The combination of these effects with two temperature simulation models and two friction models (the precise model and the simplified model) was proposed. After verifying the friction models by full-scale dynamic tests, it was found that the simplified model could simulate the DCFP bearing to the same standard as the precise model with much less calculation time. Furthermore, after comparing the two models by response analysis, the simplified model was further proved to be able to replace the precise model. Even though temperature and velocity have large influences on the value of the friction coefficient, the friction coefficient can be taken as constant, regardless of them, with sufficient accuracy in estimating the response force of the isolation layer under certain response displacements and in predicting the response displacement of the isolation layer (relative displacement between the upper structure and ground) under earthquakes. Furthermore, in this study, the constant friction coefficient was defined as the nominal friction coefficient, µn. By observing the response analysis results, it was found that large ground velocity pulses will lead to large input energies and large response displacements of the isolation layer, simultaneously. Then, by theoretical derivation, it was found that the response displacement and the energy transfer could be predicted directly from ground velocity using a constant friction coefficient. After verifying and optimizing the prediction method by analyzing the data, a simplified prediction method for energy transfer and response displacement was proposed. Using this method, the energy transfer and response displacement could be predicted for DCFP bearings and any friction coefficient during any earthquake with sufficient accuracy. In order to generalize the prediction method so that it could be easily used in design, specific earthquake characteristics in the method were replaced by general characteristics that could be used to describe a group of earthquakes in a construction site, such as PGV. Additionally, the generalized method can provide conservative predictions, and the conservativeness degree can be adjusted by adjusting the generalization parameters, ζ and ε. After these fundamental studies on the relationship between the response and the input energy of the DCFP bearing and the input earthquake records under unidirectional excitation, this relationship under bidirectional earthquake excitation will also be studied, with the aim of proposing a more direct and simple method for predicting the response of the isolation layer from ground motions and providing more perspectives in relation to the existing design codes.

Author Contributions: Conceptualization, S.Y. (Shinsuke Yamazaki), A.W. and M.T.; Methodology, S.K. and S.Y. (Satoshi Yamada); Software, J.L.; Validation, J.L.; Formal Analysis, J.L.; Investigation, S.Y. (Shinsuke Yamazaki), A.W. and M.T.; Resources, S.Y. (Shinsuke Yamazaki), S.K. and S.Y. (Satoshi Yamada); Data Curation, J.L.; Writing—Original Draft Preparation, J.L.; Writing—Review & Editing, S.K., S.Y. (Satoshi Yamada), S.Y. (Shinsuke Yamazaki), A.W. and M.T.; Visualization, J.L.; Supervision, S.K., S.Y. (Satoshi Yamada) and S.Y. (Shinsuke Yamazaki); Project Administration, S.K., S.Y. (Satoshi Yamada) and S.Y. (Shinsuke Yamazaki); Funding Acquisition, S.K., S.Y. (Satoshi Yamada) and S.Y. (Shinsuke Yamazaki). All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The assistance of Cameron J. Black, Ian D. Aiken and Gianmario Benzoni during the experiment at the University of California, San Diego, is highly appreciated. Conflicts of Interest: The authors declare no conflicts of interest.

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