32, WORLDEARTHQUAKEENGINEERING Vol.32,No.4 Dec.2016 : (2016) !"#$%&' ()* 1, 2, 3 (1. (,) ;2. * (+,-.,) ; 3./ 0

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1 32, WORLDEARTHQUAKEENGINEERING Vol.32,No.4 Dec.2016 : (2016) !"#$%&' ()* 1, 2, 3 (1. (,) ;2. * (+,-.,) ; 3./ 0 1 (,) ) +,: :; <=>?,@ABCDE FGBCDE HI3,JKL23M N8 O< 83 PQR8 1~2<STU VWX8Y Z?<[P\ ]^ E_,`abc ACI STUVWX [P\ ]3 -,, Z 23,1 <STUVWX[P\ ] Z DE, X P < #,G H, X P <, X[P\ ]3 <,,G Z =>? < 3 H GR, 45 \ < 0 ^ E_<,67 STUVWX<[P\ ] -./:[P\ ]; 6;3 ; ;STUVWX 01 :TU375.4 :A CalculationofshearstrengthofRCcolumnbyBayesianprobabilitymodel DANGWei 1,SHENShifei 2,LIUXi 3 (1.Xi anaeronauticaluniversity,xi an710077,china;2.xi anthermalpowerresearchinstituteco.ltd, Xi an710054,china;3.schoolofcivilengineering,chang anuniversity,xi an710061,china) Abstract:Undertheconditionofsmalsamplespace,Bayesianprobabilitystatisticalmethodcanestimatetheun knownparametersthroughapriorinformationabouttheparameters,whichisdiferentfrom theclasicalstatistical frequencyinnature.thispapercolects83groupsoftestdatathattheshearbearingcapacityofreinforcedconcrete columnwithshearspanratiobetween1-2underlowreversedcyclicloading,andselectstheaci318-08specifi cationformulaonshearbearingcapacityofreinforcedconcretecolumnasbayesianpriormodel,andthenintegrates thetwotypesofinformationusingbayesianprobabilitystatisticaltheorytoestablishtheposteriorprobabilitymodel ofreinforcedconcretecolumnshearbearingcapacity.thenthepapersimplifiesthecolumnshearstrengthprobabili tymodelthroughadynamicupdatethateliminatesthesecondaryfactorsafectingcolumnshearstrengthbyusingthe Bayesianposteriormodelparametersrejectingtheory.Finaly,analysesverifythecorectnesofthemethodon Bayesianprobabilitysimplifiedcolumnshearbearingcapacitycalculationmodel,andatthesametime,compare thecalculationvaluesofposteriormodelsthatareobtainedundertheconditionsofdiferentpriormodels.results showthattheapproachinheritsthecompletenesofthepriormodelandtheaccuracyofagreatnumberofexperi mentaldatainformation,soitcanmoreaccuratelypredicttheshearstrengthofconcretecolumn. Keywords:shearstrength;seismicbehaviour;analyticalmodel;Bayesiantheory;reinforcedconcretecolumn : ; : : (1975-),,!,"#$% &' E mail: @qq.com

2 , 458K Z, 45 \ < 0 ^ <,J 223 R67 < G [1-2] 45 Z STUVWX<[P \ ], + ^ E_, STUVWX[P\ ]<"#, 23 <X[P\ ], X< P 6 < 45, c G STUVW'> 6<, G P, [< 3 - c < P- [3-5] K ^ E_< <, ^ E_ +,, GX P a <!! ", # X [P\ ]< $%45, Z 45 +^ E_Gbc ACI P- H &,' Z DE 5, 3, <3 -, FGX[P\ ] H # [6]()cUVW&'*3 (GB ) X[P\ ]3 45 +,, 23 <X[P\ ]3 # <3 J# [6] 3 H GR, F- 45<./ 1!"#4 1.1!"# Z 451 X P,0`1K - 'G H&, *Z- (1) 3 P] [7] : V(X,Θ)=V d (X)+γ(X,θ)+σε (1) :X STUVWX P < ;Θ(θ,σ) BC< DE, Z?# < 45 ^ E_G HI3;V d *2 N8<STUVWXP]3 -, # Z bc ACI P- ;γ(x,θ) Gbc ACI P- <& 3,θ=[θ 1,θ 2,,θ p ] T G X<& 4E;ε567, ε~(0,1);σ H& N8<! 8#9:<- (1) ; (? *:(1) 4 σ 2 X (2)ε $ & 3 γ (X,θ)<< BC<, # Z p =E >,'(- AE, B- (2): ln[v(x,θ)]=ln[v d (X)]+ p θ i h i (x)+σε (2) Z DEI345 ^ E_J- (2)& C, (D DE< I3 * p(θ) B CDE Θ <E =E,f(Θ), a : f(θ) =κl(θ)p(θ) (3) L(Θ) ^ E_< =E;κ E F, κ=[ L(Θ)P(Θ)d(Θ)] -1 (4) GH+ ade<,)i Z < J_ *C,DE< : p(θ σ) =1,p(σ) 1/σ (5) i= :; 8; 1.1 8=><?K P\ ] =E: { } L(Θ)= Π 1 LM σ [V i-v d (x i )-γ(x i,θ) ] Π Φ[- V i-v d (x i )-γ(x i,θ) ] σ? σ ΠΦ[ V i-v d (x i )-γ(x i,θ) ] (6) K σ

3 4, : 3 STUVWX[P\ ] 225,φ( ) =E,Φ( ) =E LM^ :V i =V d (x i )+γ(x i,θ)+ σε;?lm :V i <V d (x i )+γ(x i,θ)+σε;klm :V i >V d (x i )+γ(x i,θ)+σε #NOGLM <^ E_ H 1.3 " 9:; G- (2) H7 - (7), P\ >, Z? < 45G DE HI3, ( #` - (7)G^ E_ H3 : ln(v/v d )=γ(x,θ)+σε (7) 2 < =8>$?@ 2.1 "#A89 >$ *567 y 7 x 1,x 2,,x m P N8> Q4 [1-2] : y i =θ 1 +θ 2 x 12 +θ 3 θ θ m x 1m +σε i (8) ε i N(0,1), ZRS :Y=Xθ+σε,"T θ=[θ 1,θ 2,,θ m ] T σbc U,Y~N n (Xθ,σ 2 I n ), " =E : L(θ,σ)=( 1 2)n/2 exp{- 1 2πσ 2σ (Y-Xθ)T (Y-Xθ)}=( 1 2)n/2 exp{- 1 2πσ 2σ 2[S 2 n +(θ-^θ) T X(θ-^θ)]} (9) ^θ=(x T X) -1 X T Y,S 2 n =(Y-X^θ) T (Y-X^θ) (10) VWQ DE(θ,σ)< C<, 1DE< :P(θ,σ) 1/σ, - (3)C," DE (θ,σ)< : f(θ,σ) L(θ,σ)P(θ,σ) (θ,σ) 1 σ n+1exp{- 1 2σ 2[θ-^θT X T X(θ-^θ)]} (11) f(θ,σ)g σ8 R + K HX,Y DE θ< =E: f(θ)= R+ f(θ,σ)dσ 1/[S 2 n +(θ-^θ) T X T X(θ-^θ)] n/2 (12) - (12) 3 υ=n-m-1, DE θ, S Q 0 =X T X/S 2 n< m t =E<, "DE θ< t,0 θ~t m (υ,θ,q 0 ) J_ t < O DE θ< Z [4 2.2 BC σ 2 9DE >$ - (11) G θ8 R m KX, 2 < [1] : f(σ)= f(θ,σ)dθ 1 n R m σ υ+1exp(υs2 2 2) (13) - (13) 3 \DE υ,] DE υs 2 n <^ Gamma =E<, "DE σ 2 < ^ Gamma,0 σ 2 ~Gamma(υ+1,υS 2 n ) J_^ Gamma < O DE σ 2 < Z 3 8 F G? 3.1 HE8 I? KA_`ab,$# [8],[9] [10] c 83 X P^ E_, PQR 1~2 Y STUVWX[P\ ]<,J_^ '>8 ] UVW R P QR ST T [ 4 < ` ^ E_,' H2 (B 1)

4 J 1 HE8 $%KL Table1 Experimentaldataandcalculationresults D # ^> b/mm T u R PQ h/mm T /(N mm -2 ) n R λ ρ v /% ρ l /% V test/kn V d /kn V/kN V test /V d V test /V SC-2D SC-2D SC-2D SC-2D SC-2D SC-2D SC-2D SC-3D SC-3D [8] SC-3D SC-3D SC-3D SC-3D SC-3D SC-4D SC-4D SC-4D SC-4D SC-4D SC-4D DZ DZ DZ DZ ,! [9] DZ DZ DZ DZ DZ DZ DZ A B A B A B A B A B

5 4, : 3 STUVWX[P\ ] 227 MJ 1 HE8 $%KL u R D # ^> b/mm h/mm /(N mm -2 ) n PQ R λ T T ρ l /% ρ v /% V test/kn V d /kn V/kN V test /V d V test /V 21B A B A B A B A B A B A B A B A B A B A [ ] [10] 34B B A A A B B A B B A B A B A B A B A B A B

6 MJ 1 HE8 $%KL u R D # ^> b/mm h/mm /(N mm -2 ) n PQ R λ T T ρ l /% ρ v /% V test/kn V d /kn V/kN V test /V d V test /V J 2 =8NO Table2 Parametersrejectingproces "#9 P (- (7) X< P, `1bc ACI STUVW P- -,0: V d = 1 6 (1+ N 14bh f 槡 cb w d+f y A sv s h 0 (14) J_ K a h i (x) `1 h 1 (x)=ln2 & E3,h 2 (x)=ln(h/2h),h3(x)=ln(ρ l f y / ), h 4 (x)=ln(ρ s f yv / ),h 5 (x)=ln(n/ bh),h 6 (x)=ln(b/h),h 7 (x)=ln(a/h),h 8 (x)=ln(f y / ) 8 1 ^ ; E_<?K,`Z *,`Z- (7) H DEI3 (?X P - : V=V d 0 59( H 2h )-0 77 (ρ l f y )0 28(ρ s f yv ) ( N bh )0 11 ( b h )-1 49 ( a h )-0 07 ( f y ) 0 1 (15) 3.3 "#Q - <, B εσ 3, < σ G P < U <, ( ( U < G (15) H J_DE θ=[θ 1,θ 2,,θ 8 ] T < 3 θ i <7 4E(coeficientofvariation): cov(θ i )= μ i σ i (16) μ i σ i θ i <Z θ i < 7 4E 0, G < h i (x)g P < U,G < DE(θ,σ) Z 45 HI3 K <F, DE σ< Z U 0 <DE A B 2 $ 2 ( G DE h 7 (x),h 6 (x),h 8 (x),h 1 (x),h 5 (x),de σ 2 H+U 7, G D E h 2 (x),de σ , B h 2 (x),h 3 (x),h 4 (x)gx P U,, <- : V=V d ( H 2h )-0 8 (ρ l f y ) (ρ s f yv ) -0 5 (17) G- (17) F, A l =ρ l f y,/,a s =ρ s f yv /,λ=h/2h < 7 - (18) - (19) # [6] <3,J- (18) R& 3N8 O,"# ) c 3 - Jbc < N8,G < # V=λ -0 8 A 0 4 l A -0 5 s V d (18)

7 4, : 3 STUVWX[P\ ] 229 V=0 006( a h )-2 ( f y ) 0 3 V d (19) 4 RS T $ 1 (,V test /V d < , V d K &, < V test /V< , 4U,- & <P] ^, U 1 ^ Jbc ACI <GR([^ J& <GR,$ 1 (, #1 P]<3 J^ LM, bc 1-3 <\ ] ^, 3 J <3,- 45 \ [DE <, ^ E_< G <A0,,- 45 Z <.,G H, < &W ^ 1 1 Q"#$%UVWX ACI318-08Y $%URS Fig.1 ComparisonbetweensimplifiedmodelcalculationsandcalculationsfromSpecificationACI J 3 E"# DE"# 9RS Table3 Comparisonbetweenpriormodelandposteriormodel c (GB ) bc ACI Q"#$%UV XY $%URS Fig.2 ComparisonbetweensimplifiedmodelcalculationsandcalculationsfromthecodeofChina

8 D # [6]()cUVW&'*3 (GB ) P- K -,V test /V d < , V d KA &, < V test /V< , GRB 3 3 B,)c P3 - bc ACI P3 - R, 1, 0 1,- bc 3 X[P\ ] ; < 0,-! 0,K & < <3 &W23 B, 4, 1 - Z < G <3 &W 2 # [6] J^ [)cuvw&'*3 X[P\ ]3 <GR $ 1 2 (,KA 1, bc ACI318-08,- K& < ^,, F- 45<./ 5 (1) Z 45& STUVWX[P\ ],bc ACI K X[P\ ],VPQR R (2) #1 <P]3 &WJ^ LM&W, KA& X<[P\ ] 1-3 <\ ] ^,- STUVWX[P\ ]3 < (3) Z 45,6789:; <=>?,I3, + < ", Z #1 - (18),(19)6GSTUVWX[P\ ] H3 = [1],. 23$% [M]. :,2006. ZHUHuiming,HANYuqi.Bayesianmultivariatestatisticalinferencetheory[M].Beijing:SciencePres,2006:1-11. [2]. 23$%[M]. :,1991. ZHANGYaoting.Bayesianstatisticalinference[M].Beijing:SciencePres,1991. [3] GB UVW&'*3 [S]. : c1,2010 GB CodeforDesignofConcreteStructures[S].Beijing:ChinaArchitectural&BuildingPres,2010. [4] GB *3 [S]. : c1,2001. GB CodeforSeismicDesignofBuildings[S].Beijing:ChinaArchitectural&BuildingPres,2001. [5] ACICommitee318.BuildingCodeRequirementsforStructuralConcrete(ACI318-08)andCommentary[S].AmericanConcreteInstitute, FarmingtonHils,MI,2008. [6],,c. <STUVWX[P\ ]3 [J]. ],2013,30(5): WUTao,LIUXi,XINGGuohua.StudyontheshearcapacityofreinforcedconcretecolumnbasedonBayesiantheory[J].EngineeringMechan ics,2013,30(5): (inchinese) [7] SongJ,KangW H,KimKS,etal.Probabilisticshearstrengthmodelsforreinforcedconcretebeamswithoutshearreinforcement[J].Structural Engineering&Mechanics,2010,11(1): [8],.STUVW X8Y Z?< [ LUZhanqin,CHENJiakui.Thebendingstrengthandductilityinreinforcedconcretecolumnsunderlowreversedcyclicloading[J].Journalof SouthwestJiaotongUniversity,1987(1):1-10. [9],!.STUVW X P [J]. 1* (,1995,12(4): ZHANGXianjin,HUXingsheng.Shearstrengthanalysisofreinforcedconcreteframecolumn[J].JournalofWuhanCityColege,1995,12(4): [10],,,,. X 6<^ [M]. :,1979:6-10. HirosawaMasaya,EndoRiho,NakayamaShinsuck,etal.Experimentalstudyonseismicperformanceofreinforcedconcrete[M].Beijing:Seis mologicalpres,1979:6-10.