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27420078JOURNALOFEARTHQUAKEENGINEERINGANDENGINEERINGVIBRATIONVol.27No.4Aug.2007:2007-02-02;:2007-04-21:(10572107):(1974-),,,,.E2mail:gzeying@126.com:100021301(2007)0420030205Newmark1,2,1(1.,710055;2.,041004):Newmark,,,1,;2,1,:;Newmark2;;:TU311.3:AIntegralschemeofdirectNewmarkpreciseintegrationmethodanditsapplicationtoseismicresponseanalysisGUOZeying1,2,LIQingning1(1.SchoolofCivilEngineering,XianUniversityofArchitectureandTechnology,Xian710055,China;2.SchoolofEngineering,ShanxiNormalUniversity,Linfen041004,China)Abstract:BasedonthedirectNewmarkpreciseintegrationmethod,twostep2by2stepintegralschemesarepresentedbyapplyingtheGaussintegrationalgorithmandthepreciseexponentialmatrixcalculation.Theaccuracyandstabil2ityofthesetwointegralschemesareanalyzed.Bycomparingthetwointegralschemes,itcanbeseenthatthepre2cisionofthefirstschemeishigherthanthatofthesecondone.Thestabilityofthefirstschemeapparentlysatisfiestheconditionsofalgorithmicstabilityanalysistheory,whilethesecondschemeisunstable.Thereforethefirstschemeisappliedtotheseismicresponseanalysisofstructures.Exampleshowstheadaptabilityofthefirstintegralschemetotheearthquakeresponseanalysis.Keywords:seismicresponse;directNewmarkpreciseintegrationmethod;step2by2stepintegralscheme;stabilityanalysis,:Mx+CÛx+Kx=-MIxg=F(1):x,Ûx,x;I;xg;M,C,K(1),:Newmark[1],,[2,3],,,,,[3][4]Newmark2,,,Newmark2[4],,,,11NewmarkNewmark2,=1/2,=1/4,,(),,[1],:Ûxi+1=Ûxi+(t/2)xi+(t/2)xi+1(2)xi+1=xi+tÛxi+(t2/4)(xi+1+xi)(3):xi,Ûxi,xiti;xi+1,Ûxi+1,xi+1ti+1;ti+1=ti+t,t,i=0,1,2,,n(2):xi+1=(2/t)(Ûxi+1-Ûxi)-xi(4)(4)ti+1(1),:Mxi+1+CÛxi+1+Kxi+1=Fi+1(5):Ûxi+1=-[(2/t)M+C]-1Kxi+1+[(2/t)M+C]-1[(2/t)MÛxi+MÛxi+Fi+1](6)(6):Ûx=Hx+f(7):H=-[(2t)M+C]-1K,f=[(2/t)M+C]-1[(2/t)MÛxi+Mxi+F](8),(7)ti+1:xi+1=Txi+ti+1tiexp[H(ti+1-s)]f(s)ds(9)T=exp(Ht):T()=eH=I+Ta,Ta=H+(H)2/2!++(H)l/l!(l)(10)Tk=2Tk-1+Tk=1Tk-1(k=1,2,3,N),T=I+TN(11)(9)ti+1:xi+1=Txi+(t/2)(8/9T1f(s1)+(5/9)T2f(s2)+(5/9)T3f(s3))(12):y1=0,y2=-0.6,y3=0.6,sj=ti+t2(1+yj),Tj=exp(H(t/2)(1-yj))(j=1,2,3Tj),21:(12)xi+1;(7)Ûxi+1;(5)xi+1;2:(12)xi+1;(3)xi+1;(2)134:NewmarkÛxi+12[5],1,2.11:xi+1=Txi+(t/2)[(8/9)T1f(s1)]+[(5/9)T2f(s2)]+[(5/9)T3f(s3)](13)Ûxi+1=Hxi+1+f(14)xi+1=-(c/m)xi+1-(k/m)xi+1+(1/m)Fi+1(15):f(sj)={1/[(2/t)m+c]}[(2/t)mxi+mxi+F(sj))T=exp(Ht),Tj=exp[H(t/2)(1-yj)],j=1,2,3:xi+1Ûxi+1=AxiÛxi+Lf(t)(16)A=a11a12a21a22=T+t2HdH0md-H0t2cdHT+t2H2d+HHH0md+2tH0m-t2HH0cd-H0c(17)H0=t2m+ct=c(1+),H=-2+2t,d=8/9T1+(5/9)T2+(5/9)T3,L(A[5],L)Tn,=2Tn;[6]=t;=(C/2)M(10)(11)T,[7]l=4,N=15T1,T2,T3,T,(12),A(A):(A)=maxi1(iAi),,(A),MATLAB,[8],(A)10.02(01)1(A)-Fig.1Spectralradiusoftransformmatrixversusdampingratio1,,t5,,(A)1,12.222,:2327xi+1Ûxi+1xi+1=Axixixi+Lf2(t),A=a11a12a13a21a22a23a31a32a33a11=T,a12=H0md,a13=t2H0md,a21=2t(T-1),a22=2tH0md-1(18)a23=H0md,a31=4t2(T-1),a32=2t(2tH0md-2),a33=2tH0md-1(19)1,L,[8],(A)22(A)-Fig.2Spectralradiusoftransformmatrixversusdampingratio2,,t5,,(A)1,,231:2001x1x2+1.2-0.4-0.40.6Ûx1Ûx2+6-2-24x1x2=010f(t)x1x2t=0=00,Ûx1Ûx2t=0=00,f(t)=sin(t),0t1.00,0t=0.2,21Table1Displacementresponseofthesystem120.2x10.00046970.00044500.00045470.00045470.0014729x20.03954840.03774560.01836230.01836230.0535150.4x10.0078850.0075850.0139630.00756820.011514x20.2823570.2723020.3379290.2051000.2874200.6x10.0395050.0381210.0630670.0533480.045149x20.8006470.7736040.9524270.9155330.7696700.8x10.1169760.1130010.1169360.2305430.120352x21.4898921.440521.667252.451541.4081701.0x10.2534390.2449570.1673020.7078330.24737x22.1062762.037112.184814.586811.98257/%x1-5.25963.133179.3213.583x2-4.55853.57117.76935.322:(1)1,2;(2)1,,334:Newmark,,;(3)()121:12,13Fig.3Sectiondimensionofwall2:13T,,6m,3m,3-,:12900kg,24400kgRayleighElCentro(NS)Taft7,35GalMATLAB,()145234(ElCentro)Fig.4Timehistorycurvesofdisplacements5(Taft)Fig.5Timehistorycurvesofdisplacements2(ElCentro)Table2Maximumtopdisplacementsofthestructure/mt/s-0.023858.221-0.023578.24-0.023928.223(Taft)Table3Maximumtopdisplacementsofthestructure/mt/s-0.00348719.951-0.00345019.95-0.00354019.954Newmark2[4],,,,,2,;1,,Newmark21,,:[1]NewmarkNM.Amethodofcomputationforstructuraldynamics[J].JournalofEngineeringMechanicsDivision,1959,85(3):67-94.[2].[J].,1994,34(2):131-136.[3].[J].,1995,12(3):253-260.[4]GuoZeying,LiQingning,ZhangShoujun.DirectNewmark2precisionintegrationmethodofseismicanalysisforshort2legshearwallstructure[C]//Advancesinstructuralengineering.Beijing:SciencePress,2006:180-184.[5],,.[M].:,1992.[6],.[J].,2001,21(3):22-28.[7]WangMengfu,AuFTK.AssessmentandimprovementofprecisetimestePintegrationmethod[J].ComputersandStructures,2006,84(12):779-786.[8],,.[J].,2002,22(3):1-8.4327