International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Probabilistic hazard model of inelastic oscillator based on semi-theoretical solutions of first passage… Mori, Yasuhiro; Takashima, Masato; Kojima, Sayo; Ozaki, Fuminobu Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Probabilistic Hazard Model of Inelastic Oscillator based onSemi-theoretical Solutions of First Passage ProblemYasuhiro MoriProfessor, Grad. School of Environmental Studies, Nagoya Univ., Nagoya, JapanMasato TakashimaGrad. Student, Grad. School of Environmental Studies, Nagoya Univ., Nagoya, JapanSayo Kojima,Kozo Keikaku Engineering, Inc., Tokyo JapanFuminobu OzakiAssoc. Professor, Grad. School of Environmental Studies, Nagoya Univ., Nagoya, JapanABSTRACT: In this study, an approximate method of estimating the exceedance probability of themaximum ductility factor of an inelastic oscillator is proposed based on the equivalent linearizationtechnique using the capacity spectrum (CS) method. The CS method seeks to determine the first point atwhich the demand spectrum crosses the CS. Because the auto-correlation function of the responsespectrum is only a function of the difference between the logarithms of two natural periods, the CSmethod can be interpreted as the first passage problem of a stationary standard normal stochasticprocess. Because the auto-correlation function is not continuous when the difference is equal to zero,and because the crossing event cannot be modeled as a Poisson process when the CS is close to thehorizontal axis, the mean crossing rate is semi-theoretically estimated based on Monte Carlo simulation.The results are further modified to create a general model. The accuracy of the proposed method isdemonstrated by using numerical examples.1. INTRODUCTIONPredictors of seismic structural demands (such asinterstory drift ratios) that are faster than nonlin-ear dynamic analysis (NDA) are useful for struc-tural performance assessment and design. Severaltechniques for realizing such predictors have beenproposed by using the results of a nonlinear staticpushover analysis (e.g., Luco 2002; Chopra & Goel2002; Yamanaka et al. 2003; Mori et al. 2006).These techniques often use the maximum responseof an inelastic oscillator (computed via NDA) thatis equivalent to the original frame.In reliability-based seismic design of a structure,it is necessary to probabilistically express the maxi-mum response of the inelastic oscillator. This infor-mation can be obtained via NDA, but requires thou-sands of samples. In practice, the use of simplermethods, such as an equivalent linearization tech-nique (EqLT), using an elastic response spectrumseems more reasonable; design spectra are beingdeveloped on the basis of probabilistic approachessuch as uniform hazard spectrum (UHS) and con-ditional mean spectrum (CMS, Baker & Jayaram2008).A UHS is obtained by plotting the response withthe same (i.e., uniform) exceedance probability fora suite of elastic oscillators with different naturalperiods; hence, a UHS represent no specific groundmotion (Abrahamson 2006). Although there existscertain correlation among the spectral responses ofelastic oscillators to a ground motion (e.g., Baker112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015& Jayaram 2008), perfect correlation is implicitlyassumed in the use of a UHS. In such a scenario,the response may be overestimated via EqLT whena very rare event is considered.The correlation among spectral responses maybe considered by using a CMS, which is the meanspectrum conditional to the event that the spectraldisplacement of an elastic oscillator with a certainperiod equals the displacement with, for example,10% exceedance probability in 50 years. However,the authors have shown that the EqLT using CMScan provide fairly optimistic estimates (Mori et al2011).A new approximation method is proposed usingsemi-theoretical solutions of the first passage prob-lem by converting a probabilistic elastic displace-ment response spectrum into a stationary standardnormal random process. The accuracy and applica-bility of the method are discussed by using numer-ical examples.2. EQUIVALENT LINEARIZATION TECHNIQUES2.1. Equivalent Linearization TechniqueIn an EqLT, the maximum displacement of an in-elastic oscillator with the elastic natural period, T1,and the damping factor, h1, is approximated by us-ing the maximum displacement of an elastic oscil-lator with the equivalent natural period, Teq, and theequivalent damping factor, heq, asSID(T1;h1) ≈ SED(Teq;heq) (1)where SD(T ;h) is the spectral displacement of anoscillator with the natural period, T , and the damp-ing factor, h; the superscripts E and I represent theelastic and inelastic responses, respectively. OFten,Teq and heq are expressed as functions of the max-imum ductility factor of the inelastic oscillator, µ ,which is defined asµ = SID(T1;h1)/δy (2)where δy is the yield displacement of the oscilla-tor. Several linearization techniques have been pro-posed (e.g., Iwan 1980, Shimazaki 1999); amongthem, the following formulae proposed by Shi-mazaki for an oscillator with a bilinear backbonecurve are used in this study.Teq = T1 ·√µ1− k2(1−µ)(3)heq = 0.25 · (1−1/√µ)+h1 (4)where k2 is the second stiffness ratio of the back-bone curve.2.2. Capacity Spectrum MethodThe capacity spectrum (CS) method (Freeman1978) can be used to graphically estimate the in-elastic displacement as the intersection of the ca-pacity spectrum and the demand spectrum (DS). Totake into account the effect of heq, the demand spec-trum must be adjusted by multiplying it with thedamping reduction factor, Fh(heq), defined as theratio of the spectral response of an elastic oscilla-tor with a damping factor to that with the dampingfactor, h1. Because heq is a function of the unknownvalue µ , an iterative procedure is generally requiredfor its determination.In contrast, the response can be estimated di-rectly by considering the demand and capacityspectra in an ordinal T -SD coordinate rather thanan SD-SA coordinate, as shown in Fig.1 (Mori &Maruyama 2007). The SD axis can be transformedlinearly into the axis of the maximum ductility fac-tor, µ , by dividing the SD axis by the yield displace-ment of the inelastic oscillator. The T axis can alsobe expressed in terms of µ because Teq is a functionof µ , as expressed by Eq.(3). Then, the capacityspectrum can be obtained by connecting the corre-sponding values in the linear (vertical) and nonlin-ear (horizontal) µ coordinates.On the basis of Eq.(3), the capacity spectrum,CS(T ), of an inelastic oscillator with a bilinearbackbone curve and mass equal to unity can be ex-pressed as,CS(T ) = δy ·µ=9.8 ·Cyk1· (1− k2) ·T2T 21 − k2 ·T 2=9.8 ·Cy ·T 214pi2· (1− k2) ·T2T 21 − k2 ·T 2; T ≥ T1(5)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015δ ADemand Spectrum (elastic)   076543210Capacity SpectrumA’g(T) = Capacity Spectrum ×Demand Spectrum (inelastic)          = ⋅µ( ) 	 µ1 4 90EqLTµµ T  T  2T  T3T EqLTeg.   µ=fffiflflffi   !" # $%& '()*+ ,* *-. /-. /Figure 1: Capacity-Spectrum Method in T -SD Coordi-nate)where k1 and Cy are the elastic stiffness and yieldbase shear coefficient of the oscillator, respectively;the acceleration attributable to gravity is 9.8 (m/s2).The following damping reduction factor (Kasaiet al 2003) is used in this research:Fh(h) =(D(h)−1) · (5 ·T )+1 ; 0≤ T ≤ 0.2D(h) ; 0.2≤ T ≤ 2D(h) ·[√h/h1 · (T −2)/40+1]; 2≤ T ≤ 8(6)whereD(h) =√1+25 ·h11+25 ·h(7)When the probability distribution and auto-correlation function of the n-year maximumSED(T ;h) are available, the exceedance probabilityof the n-year maximum displacement response ofan inelastic oscillator can be estimated by MonteCarlo simulation. This simulation finds the inter-section of each sample of SED(T ;h), which is a de-mand spectrum and CS(T )/Fh(heq), which is here-after designated a factored CS, g(T ), (see Fig.1).However, such a procedure requires extensive com-putational effort, and thus, a more practical methodis investigated in the next section.Figure 2: Schematic Illustration of CS and DS in Stan-dard Normal Stochastic Process3. CS METHOD AS FIRST PASSAGE PROBLEMIN STANDARD NORMAL STOCHASTIC PRO-CESSIn the CS method, the event that the equivalent nat-ural period, Teq, is longer than teq corresponds tothe event that SED(T ;h) is always above the factoredCS in the range of (T1, teq) (the hatched area inFig.2(a)).According to previous studies such as those byBaker & Jayaram (2008), the auto-correlation func-tion of SED(T ;h) is dependent only on the differencebetween the logarithms of two natural periods asshown in Eq.(8) and Fig.3 (Baker & Jayaram 2008).KSD(ζ ) = 1− cos(pi2−0.366 · ln(10) · |ζ |)(8)(ζ = log(T )− log(T1), T1 ≥ 0.109(s))Therefore, SED(T ;h) can easily be transformedinto a stationary standard normal stochastic pro-cess, X(τ) where τ = log(T ) + 1. Then, the CSmethod can be interpreted as the first passage prob-lem of a stationary standard normal stochastic pro-cess crossing a factored CS, which is accordinglytransformed, downward as shown in Fig.2(b)(Moriet al 2011). By this transformation, the DS is nor-malized and the information regarding structural312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Natural period (s)0.1 1 1000. T1 = 1.01.0Correlation coefficientT1 = 0.3Figure 3: Auto-correlation Model of SED(T ;h)characteristics and seismic hazard, except for theauto-correlation function of SED(T ;h), is reflected tothe shape and position of the factored CS.When SED(T,h) is lognormally distributed, it canbe transformed into a stationary standard normalstochastic process byX(τ) =ln(SED(τ,h))−µln(SED(τ))σln(SED(τ))(9)in which µln(SED(τ)) and σln(SED(τ)) are the mean andstandard deviation of ln(SED(τ)), respectively.4. FIRST PASSAGE PROBLEM OF STATIONARYSTANDARD STOCHASTIC PROCESS4.1. Theoretical Solution of Mean Crossing RateConsider an event that a stationary standard normalstochastic process, X(τ), crosses a constant thresh-old a downward. The mean crossing rate, νa, canbe expressed byνa =σV√2piφ(a) (10)in which φ(•) is the standard normal probabilitydensity function, and σV is the standard deviationof V (τ) = dX(τ)/dτ , which can be expressed asσV =√−d2KX(ζ )dζ 2∣∣∣∣ζ=0(11)in which KX(ζ ) is the auto-correlation function ofX(τ), which depends only on the time difference,ζ = τ1− τ2.If the events of crossing a threshold a are rare,the occurrence of the events can be modeled by aPoisson process, and the probability, P0(tL), that nocrossing occurs within time interval (0, tL) can beapproximately expressed as,P0(τL) = P[X(τ) > a;0 < τ < τL]= exp(−νa · τL) (12)If the threshold varies with time, the mean cross-ing rate can be expressed as,νa(τ) =[A− a˙{1−Φ(a˙σV)}]·φ{a(τ)} (13)in which a˙ = da(τ)/dτ,Φ(•) is the standard nor-mal probability distribution function, andA = σV√2piexp{−12(a˙σV)2}(14)Similar to Eq.(12), if the events of crossingthreshold a(τ) are rare, the occurrence of theseevents can also be modeled by a nonstationary Pois-son process, and the probability, P0(τL), that nocrossing occurs within time interval (0,τL) can beapproximately expressed asP0(τL) = exp(−∫τL0νa(τ)dτ)(15)4.2. Mean Crossing Rate Based on SimulationTo estimate a mean crossing rate by using Eqs.(12)or (15), the second-order derivative of the auto-correlation function of X(τ) must be available atζ = 0, as shown in Eq.(11). However, as shownin Eq.(8) and Fig.3, the auto-correlation function ofSED(T ;h) is not continuous at ζ = 0; accordingly, itis not differentiable.Even if a mean crossing rate can be estimated ac-curately, the exceedance probability cannot be ac-curately estimated by using Eqs.(12) or (15) if thethreshold is close to the horizontal axis. Under suchcircumstances, the crossing event would not be rareand cannot be modeled by a Poisson process.In this research, attempts are made to estimateσV in Eq.(10) based on Monte Carlo simulation sothat the exceedance probability can be estimated byusing Eqs.(12) or (15).The auto-correlation function expressed byEq.(8) is affected by the nonlinear transformationof SED(T ;h) into a standard normal stochastic pro-cess. If SED(T ;h) is lognormally distributed, the412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20151 1000.	 Correlation coefficientNatural period (ζ)Figure 4: Auto-correlation Model of SED(T ;h)auto-correlation function of X(τ) can be expressedas (Der Kiureghian and Liu 1985)KX(ζ ) =ln(1+KS(ζ ) ·VSD(T1) ·VSD(T ))√ln(1+V 2SD(T ))·√ln(1+V 2SD(T ))(16)in which VSD(T ) and KS(ζ ) are the coefficient ofvariation (c.o.v.) and the auto-correlation functionof SED(T ;h), respectively. Fig.4 illustrates examplesof KX(ζ ) when SED(T ;h) is lognormally distributedwith constant c.o.v. equal to 0.5, 1.0, 1.5, or 2.0along with that expressed by Eq.(8).4.2.1. σV for Crossing Constant ThresholdσV is determined iteratively for each constantthreshold within the range of a = −3 to 2.5 byMonte Carlo simulation with 40,000 samples sothat the exceedance probability estimated by thesimulation agrees with that estimated by Eqs.(10)-(12). Fig.5 shows examples of agreement and non-agreement. The increment of τ in the simulation isset to be 0.01.During the course of research, it was found thata constant value of σV cannot provide an accurateestimate of the exceedance probability, particularlywhen a is large (see Fig.5(b)). Thus, σV is modeledhere as a function of τ , expressed by Eqs.(17)-(19).σV (a,τ,dK) = σV0(a) ·{1− f (τ) ·g(a,dK)} (17)in whichf (τ) = 0.68 ·Φ(ln(τ)− ln(0.06)0.5)(18)0 0.5 0.5 result 	Figure 5: Example of Satisfactory Agreement and Non-agreement of Exceedance Probability between Simula-tion and Semi-theoretical Result (a = 1)0.9− 0.8− 0.7− 0.6− 0.5−	0.9− 0.8− 0.7− 0.6− 0.5− Figure 6: S(dK)andR(dK) Estimated by Simulationg(a,dK) =1−Φ(√−a−R(dK)S(dK))(a < 0)1 (a ≥ 0)(19)f (τ) in Eq.(17) considers that σV depends on τ ,and g(a,dK) in the equation considers that the de-pendency of σV on τ depends on threshold level a.In Eq.(19), R(dK) and S(dK) are determined itera-tively, as shown in Fig.6 and modelled by a linearfunction of dK = (KX(0.01)−KX(0))/0.01 asR(dK) = 0.503 ·dK+2.54 (20)S(dK) = 0.162 ·dK+0.815 (21)Then, σV0(a) in Eq.(17) is determined for eachthreshold based on the simulation.By expressingthe horizontal axis with log{1−Φ(a)} and the ver-tical axis with log{σV0(a)}, σV0(a) is intriguinglyplotted on a straight line as shown in Fig.7. Accord-ingly, σV0(a) is modeled asσV0(a) = 10{k(dK)−1.034·log(1−Φ(a))} (22)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015-2.5 -2 -1.5 -1 -0.5 001230123012301230123	        SimulationRegressed line)}(log{ 0 aVσ)}(1log{ aΦ−Figure 7: σV0 Estimated by Simulationin whichk(dK) = −0.42 ·dK+0.686 (23)4.2.2. σV for Crossing Time-varying ThresholdWhen the threshold varies with time, further mod-ification on the mean crossing rate is necessary.Based on the iterative analysis, the mean cross-ing rate is determined considering the instantaneousslope of the threshold, a˙, asνa(τ, a˙) = νa(τ) ·(1+ a˙ · 1h(dK,τ))(24)in which νa(τ is estimated by using Eq.(13) withσV estimated by Eqs.(10)-(12), andh(dK,τ) = f1(dK,τ)·(1 + f2(dK,τ) ·g1(τ) ·g2(dK,τ))(25)whereg1(τ) =−0.68 ·Φ(τ0.5); a˙ ≤ 0−0.68 ·Φ(τ−0.1 · a˙0.5); a˙ > 0(26)g2(dK,τ) =1−Φ(√a(τ)−(0.503·dK+2.54)0.162·dK+0.815);a(τ) ≤ 01 ;a(τ) > 0(27)f1(dK,τ) ={f11−1.5 · f12 ; a˙ ≤−1.5f11 + a˙ · f12 ; a˙ >−1.5(28)f11 = 0.56−0.125 ·dK+a(0) · (−0.139−0.054 ·dK) (29)f12 = 0.493+0.384 ·dK+a(0) · (−0.096−0.078 ·dK) (30)f2(dK,τ) ={f21 + a˙ · f22 ; a˙ ≤ 0.5f21 +0.5 · f22 ; a˙ > 0.5(31)f21 = 0.832+0.018 ·dK+a(0) · (−0.06−0.009 ·dK) (32)f22 = −0.489−0.276 ·dK+a(0) · (−0.008−0.032 ·dK) (33)5. NUMERICAL EXAMPLESFig.8 shows the maximum ductility factor of inelas-tic oscillators with elasto-plastic backbone curves.It is assumed that the oscillator is subjected to aseismic hazard with the following characteristics:- SED(T ;h) is lognormally distributed, with auto-correlation function given by Eq.(8). The casewhen SED(T ;h) is normally distributed is alsoconsidered, simply for reference.- The 90% non-exceedance probability of theresponse spectrum is given by the design spec-trum prescribed in the Japanese seismic pro-visions as shown in Fig.9. The characteristicperiod of the response spectrum, Tg, definedas the boundary between the acceleration con-stant domain and the velocity constant range,is equal to 0.86 s.- The c.o.v. of SED(T ;h), which is lognormallydistributed, is constant for any value of thenatural period and equal to 0.5, 1.0, 1.5, or2.0. When SED(T ;h) is normally distributed,the c.o.v. is constant and equal to 0.5.The following characteristics of the oscillator areconsidered:612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 10 20 30 40 500.010.11	0 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.110 10 20 30 40 500.010.11Figure 8: Exceedance probability of maximum ductility factor- Yield base shear,Cy =0.3 or 0.5.- Natural period, T1 = α ·Tg with α =0.25, 0.5,or 1.0.- Damping factor, h=0.05The CS transformed into the standard normalstochastic process is illustrated in Fig.10.In Fig.8, the maximum ductility factors esti-mated by Monte Carlo simulation as rigorous esti-mates, and those by EqLT using UHS, are presentedby red and yellow lines, respectively. As shown inthe figure that EqLT using UHS provides an erro-neous estimate, particularly when the c.o.v. of theCy is large and α is small. On the contrary, the pro-posed method provides a fairly accurate estimate inmost cases.6. CONCLUSIONSIn this study, an approximate method of estimatingthe exceedance probability of the maximum duc-tility factor of an inelastic oscillator is proposedbased on the equivalent linearization technique us-ing a capacity spectrum (CS) method. The CSmethod seeks to determine the first point at whichdemand spectrum (DS) crosses the CS. Because theauto-correlation function of response spectrum isonly a function of the difference between the log-arithms of two natural periods, the response spec-trum, which is DS in the CS method, can easily 0.5 1 1.5 2 2.5 3050015001000Figure 9: Design Spectrum in Japanese Seismic Provi-sion0 0.5 1 1.5 2-2024t= 0.5 1 1.5 2-2024tk2=0.000.030 0.5 1 1.5 2-2024tCS(t) VSD= 0.5 1 1.5 2-2024tCy=0.30.5CS(t)CS(t)CS(t)(i)VSD=0.5 ,Cy=0.3, k2=0.00 (ii)VSD=0.5 ,Cy=0.3,=0.25(iii) =0.25,Cy=0.3,k2=0.00 (iv)VSD=0.5 ,=0.25,k2=0.00Figure 10: CS in Standard Normal Stochastic Processbe transformed into a stationary standard normalstochastic process. Thus, the CS method can be in-712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015terpreted as a first passage problem of a stationarystandard normal stochastic process. Because theauto-correlation function is not continuous whenthe difference is equal to zero, and because thecrossing event cannot be modeled as a Poisson pro-cess when CS is close to the horizontal axis, themean crossing rate is estimated semi-theoreticallybased on Monte Carlo simulation. The results arefurther modified to create a general model. The ac-curacy of the proposed method is demonstrated byusing numerical examples.In this study, only a single spectral displacementis considered. Because the preceding conclusionsmay be dependent on the shape of the displacementspectrum, further investigation is necessary, alongwith possible improvement and simplification ofthe proposed method.ACKNOWLEDGMENTSThe authors gratefully acknowledge the financialsupport provided through a Grant-in-Aid for Scien-tific Research (B) (No.25282100) by the Ministryof Education, Science, Sports, and Culture and theJapan Society for the Promotion of Science.7. REFERENCESAbrahamson, N., (2006). "Seismic hazard assessment:problems with current practice and future develop-ments," Proc. 1st European Conf. on Earthquake En-gineering & Seismology, Geneva, Switzerland, 1-17.Baker, J.W. and Jayaram, N., (2008). ”Correlationof Spectral Acceleration Values from NGA GroundMotion Models." EarthquakeSpectra, Vol.24, No.1,pp.299-317.Chopra, A.K. and R.K. Goel, (2002). "Amodal pushoveranalysis procedure for estimating seismic demandsfor buildings," Earthquake Engineering & StructuralDynamics, 31(3), 561-582.Freeman, S.A., (1978). "Prediction of response of con-crete buildings to severe earthquake motion,"DouglasMcHenry Int. Symp. on Concrete and Concrete Struc-tures, SP-55, ACI, 589-605.Der Kiureghian, A. and P-L. Liu, (1985). "Structural re-liability under incomplete probability information," J.Engineering Mechanics, ASCE, 112, 85-104.Iwan, W.D., (1980). "Estimating inelastic response spec-tra from elastic spectra," Earthquake Engineering &Structural Dynamics, 8, 375-388.Kasai, K., H. Ito, and A. Watanabe, (2003). "Peak re-sponse prediction rule for a SDOF elasto-plastic sys-tem based on equivalent linearization technique," J.Structural & Construction Engineering, 571, 53-62.(in Japanese)Luco, N., (2002). Probabilistic Seismic Demand Anal-ysis, SMRF Connection Fractures, and Near-sourceEffects. Ph.D. Dissertation; Stanford University, Stan-ford, California.Mori, Y., T. Yamanaka, N. Luco, and C.A. Cor-nell, (2006). "A static predictor of seismic demandon frames based on a post-elastic deflected shape,"Earthquake Engineering & Structural Dynamics, 35,1295-1318.Mori, Y. and M. Maruyama, (2007). "Seismic structuraldemands taking accuracy of response estimation intoaccount," Earthquake Engineering & Structural Dy-namics, 36, 1999-2020.Mori, Y., Kojima, S., and Ibuki, K., (2011). "Probabilis-tic Hazard Modelof Inelastic Response of SDOF Sys-tem Based on EquivalentLinearization Technique,"Application of Probability & Statisticsin Civil Engrg,8pp. (CD-ROM).Shimazaki, K., (1999). "Evaluation of structural coeffi-cient by displacement response estimation using theequivalent linear method," J. Structural & Construc-tion Engineering, 516, 51-57. (in Japanese)Yamanaka, T., Y. Mori, and M. Nakashima, (2003)."Estimation of displacement response of multi-storyframe considering modal shape after yielding," Sum-maries of Technical Papers of Annual Meeting, AIJ,B-1, 563-564. (in Japanese)8


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