32, 4 2016 12 WORLDEARTHQUAKEENGINEERING Vol.32,No.4 Dec.2016 :1007-6069(2016)04-0223-08!"#$%&' ()* 1, 2, 3 (1. (,) 710077;2. * (+,-.,) 710054; 3./ 0 1 (,) 710061) +,: 23456789:; <=>?,@ABCDE FGBCDE HI3,JKL23M N8 O< 83 PQR8 1~2<STU VWX8Y Z?<[P\ ]^ E_,`abc ACI318-08 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 :2015-08-16; :2016-05-23 : (1975-),,!,"#$% &' E mail:759314520@qq.com
224 32 23, 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 ACI318-08 P- H &,' Z DE 5, 3, <3 -, FGX[P\ ] H # [6]()cUVW&'*3 (GB50010-2010) 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 ACI318-08 P- ;γ(x,θ) Gbc ACI318-08 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 >,'(- (1)? @ 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=1 1.2 56789:; 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
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 13 + +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)
226 32 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-2D1 205 199 26.9 0.2 2 1.54 0.2 84.38 40.148 91.423 2.102 0.923 SC-2D2 202 199 29.8 0.4 2 1.54 0.2 109.07 40.867 94.160 2.669 1.158 SC-2D3 203 197 29.8 0.55 2 1.54 0.2 98.59 40.597 93.538 2.428 1.054 SC-2D4 203 200 25.2 0.65 2 1.54 0.2 92.51 39.182 88.560 2.361 1.045 SC-2D5 206 197 25.2 0.97 2 1.54 0.2 81.34 39.08 88.330 2.081 0.921 SC-2D6 203 196 24.54 0.85 2 1.54 0.2 75.85 37.974 85.568 1.997 0.886 SC-2D7 203 197 19.65 1 2 1.54 0.2 61.74 35.72 78.467 1.728 0.787 SC-3D1 202 198 27.16 0.2 2 1.54 0.565 85.06 66.238 89.542 1.284 0.950 SC-3D2 203 199 16.3 0.4 2 1.54 0.565 79.58 61.308 78.172 1.298 1.018 [8] SC-3D3 203 199 23.5 0.55 2 1.54 0.565 94.77 65.204 86.696 1.453 1.093 SC-3D4 203 200 28.58 0.65 2 1.54 0.565 104.41 67.989 92.447 1.536 1.129 SC-3D5 203 199 28.58 0.75 2 1.54 0.565 93 67.601 91.920 1.376 1.012 SC-3D6 204 199 23.5 0.85 2 1.54 0.565 72.52 65.526 87.124 1.107 0.832 SC-3D7 203 200 19.2 1 2 1.54 0.565 66.25 63.326 82.273 1.046 0.805 SC-4D1 203 198 19.2 0.2 2 1.54 1 74.38 94.682 92.294 0.786 0.806 SC-4D2 203 197 28.6 0.4 2 1.54 1 111.13 98.728 100.731 1.126 1.103 SC-4D3 203 196 28.6 0.55 2 1.54 1 106.76 98.154 100.145 1.088 1.066 SC-4D4 203 199 19.97 0.65 2 1.54 1 101.43 95.649 93.657 1.06 1.083 SC-4D5 204 198 19.97 0.75 2 1.54 1 86.04 95.568 93.578 0.9 0.919 SC-4D6 204 199 21.88 0.85 2 1.54 1 85.46 97.129 96.106 0.88 0.889 DZ-1 148 148 25.25 0.13 1 0.85 0.54 60.37 34.431 65.607 1.753 0.920 DZ-2 148 148 25.25 0.24 1 0.85 0.54 55.76 34.432 65.609 1.619 0.850 DZ-4 144 138 23.75 0.61 1 0.95 0.54 68.5 30.535 60.330 2.243 1.135 DZ-5 133 138 23.75 0.64 1 1.02 0.54 63.01 28.202 57.282 2.234 1.100,! [9] DZ-8 145 140 26.25 0.86 1 0.92 0.54 68.7 31.86 62.884 2.156 1.092 DZ-9 145 135 26.25 0.87 1 0.96 0.54 67.72 30.533 61.270 2.218 1.105 DZ21-3 203 202 40 0.31 1.5 1.48 0.215 143.77 47.145 134.576 3.049 1.068 DZ21-4 204 203 40 0.52 1.5 1.46 0.215 162.19 47.645 135.285 3.404 1.199 DZ22-2 204 206 37.375 0.62 1.5 1.44 0.371 159.45 59.483 126.675 2.681 1.259 DZ11-2 200 199 31.625 0.32 1.5 1.15 0.22 123.97 42.705 106.343 2.903 1.166 DZ11-4 200 201 23.125 0.74 1.5 1.15 0.22 95.94 39.291 94.397 2.442 1.016 12A 250 250 20.625 0.25 2 0.34 0.71 89 113.347 73.573 0.785 1.2101 12B 250 250 20.625 0.25 2 0.34 0.36 94 73.841 67.457 1.273 1.393 13A 250 250 20.625 0.125 1 0.34 0.92 144 137.047136.677 1.051 1.054 13B 250 250 20.625 0.125 1 0.34 0.45 143 83.996 120.051 1.702 1.191 14A 250 250 20.625 0.125 2 0.34 0.18 72 53.52 69.299 1.345 1.039 14B 250 250 20.625 0.125 2 0.34 0.09 74 43.361 79.575 1.707 0.930 15A 250 250 20.625 0.125 1 0.61 2.33 179 296.201232.279 0.604 0.771 15B 250 250 20.625 0.125 1 0.61 1.12 167 159.622180.968 1.046 0.923 16A 250 250 20.625 0.125 2 0.61 0.51 98 90.768 87.341 1.08 1.122 16B 250 250 20.625 0.125 2 0.61 0.27 101 63.678 84.384 1.586 1.197
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 250 250 30.625 0.25 1 0.34 1.53 132 213.166172.204 0.619 0.767 22A 250 250 30.625 0.25 2 0.34 0.71 102 120.609 81.912 0.846 1.245 22B 250 250 30.625 0.25 2 0.34 0.36 101 81.103 77.522 1.245 1.303 23A 250 250 30.625 0.125 1 0.34 0.92 154 144.306150.581 1.067 1.023 23B 250 250 30.625 0.125 1 0.34 0.45 151 91.254 136.465 1.655 1.107 24A 250 250 30.625 0.125 2 0.34 0.18 70 60.778 82.340 1.152 0.850 24B 250 250 30.625 0.125 2 0.34 0.09 74 50.619 97.200 1.462 0.761 25A 250 250 30.625 0.125 1 0.61 2.33 205 303.459248.989 0.676 0.823 25B 250 250 30.625 0.125 1 0.61 1.12 195 166.881197.959 1.169 0.985 26A 250 250 30.625 0.125 2 0.61 0.51 98 98.027 98.694 1 0.993 26B 250 250 30.625 0.125 2 0.61 0.27 94 70.937 98.356 1.325 0.956 27A 250 250 30.625 0.25 2 0.95 2.44 145 315.883171.857 0.459 0.844 27B 250 250 30.625 0.25 2 0.95 1.22 137 178.175137.394 0.769 0.997 28A 250 250 30.625 0.125 2 0.95 1.27 121 183.812138.904 0.658 0.871 28B 250 250 30.625 0.125 2 0.95 0.61 126 109.314119.474 1.153 1.055 32A 500 500 29.75 0.25 2 0.34 0.85 431.7 543.315335.928 0.795 1.285 32B 500 500 29.75 0.25 2 0.34 0.42 411.7 349.17 307.818 1.179 1.337 33A 500 500 29.75 0.125 1 0.34 1.02 587.5 620.043612.229 0.948 0.960 33B 500 500 29.75 0.125 1 0.34 0.48 614.2 376.233542.845 1.633 1.131 34A 500 500 29.75 0.125 2 0.34 0.19 275 245.298322.329 1.121 0.853 [ ] [10] 34B 500 500 29.75 0.125 2 0.34 0.1 270.8 204.663371.454 1.323 0.729 35B 500 500 29.75 0.125 1 0.61 1.36 695.8 773.553829.442 0.899 0.839 36A 500 500 29.75 0.125 2 0.61 0.61 355.8 434.928398.830 0.818 0.892 36A-1 500 500 29.75 0.125 2 0.61 0.51 385.8 389.778391.128 0.99 0.986 36A-2 500 500 29.75 0.125 2 0.61 0.44 365.8 358.173387.129 1.021 0.945 36B 500 500 29.75 0.25 2 0.61 0.27 366.7 281.445388.936 1.303 0.943 37B 500 500 29.75 0.25 2 0.95 1.42 595.8 800.67 570.106 0.744 1.045 38A 500 500 29.75 0.125 2 0.95 1.48 531.7 827.733577.230 0.642 0.921 38B 500 500 29.75 0.125 2 0.95 0.74 546.7 493.623487.901 1.108 1.121 41B 250 250 56.75 0.25 1 0.34 1.53 240 227.803197.496 1.054 1.215 42A 250 250 56.75 0.25 2 0.34 0.71 117.5 135.245 98.574 0.869 1.192 42B 250 250 56.75 0.25 2 0.34 0.36 125 95.739 98.209 1.306 1.273 43A 250 250 56.75 0.125 1 0.34 0.92 168 158.931177.978 1.057 0.944 43B 250 250 56.75 0.125 1 0.34 0.45 163.5 105.88 169.923 1.544 0.962 45A 250 250 56.75 0.125 1 0.61 2.33 213.5 318.085280.089 0.671 0.762 45B 250 250 56.75 0.125 1 0.61 1.12 216 181.506231.063 1.19 0.935 46A 250 250 56.75 0.125 2 0.61 0.51 111 112.652121.718 0.985 0.912 46B 250 250 56.75 0.125 2 0.61 0.27 113.5 85.562 127.316 1.327 0.891 47A 250 250 56.75 0.25 2 0.95 2.44 168 330.519192.979 0.508 0.871 47B 250 250 56.75 0.25 2 0.95 1.22 169 192.811159.559 0.877 1.059 48A 250 250 56.75 0.125 2 0.95 1.27 136 198.437160.930 0.685 0.845 48B 250 250 56.75 0.125 2 0.95 0.61 132.5 123.94 145.372 1.069 0.911
228 32 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 1.3507 1.0068 4 0.3898 0.0230 J 2 =8NO Table2 Parametersrejectingproces 0.0208-0.7727-0.7652 0.2775-0.5077 0.1118-1.4912-0.0731 0.1009 0.0205-0.5412-0.7649 0.2816-0.5088 0.1130-1.5189 0.0984 0.0206-0.6152-0.7615 0.2719-0.5073 0.0998 0.0988 0.0210-0.1409-0.7693 0.3050-0.5047 0.0915 0.0211-0.7907 0.3209-0.4906 0.0933 0.0234-0.8077 0.3887 0.5032 0.0798 0.3249-0.3293 3.2 "#9 P (- (7) X< P, `1bc ACI318-081 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 ) -0 51 ( N bh )0 11 ( b h )-1 49 ( a h )-0 07 ( f y ) 0 1 (15) 3.3 "#Q - <, B εσ 3, < σ 2 0.0208 8 + 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 2 0 0 0798, B h 2 (x),h 3 (x),h 4 (x)gx P U,, <- : V=V d ( H 2h )-0 8 (ρ l f y ) -0 39 (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)
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 < 4 1.3507 0.3898, V d K &, < V test /V< 4 1.0068 0.0230 1, 4U,- & <P] ^, U 1 ^ Jbc ACI318-08 <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 $%URS Fig.1 ComparisonbetweensimplifiedmodelcalculationsandcalculationsfromSpecificationACI318-08 J 3 E"# DE"# 9RS Table3 Comparisonbetweenpriormodelandposteriormodel c (GB50010-2010) bc ACI318-08 0.79 1.05 1.3507 1.0068 4 0.083 0.085 0.390 0.023 1 2 Q"#$%UV XY $%URS Fig.2 ComparisonbetweensimplifiedmodelcalculationsandcalculationsfromthecodeofChina
230 32 D # [6]()cUVW&'*3 (GB50010-2010) P- K -,V test /V d < 4 0.79 0.083, V d KA &, < V test /V< 4 1.05 0.085, GRB 3 3 B,)c P3 - bc ACI318-08 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 K @ (1) Z 45& STUVWX[P\ ],bc ACI318-081 - 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] GB50010-2010UVW&'*3 [S]. : c1,2010 GB50010-2010CodeforDesignofConcreteStructures[S].Beijing:ChinaArchitectural&BuildingPres,2010. [4] GB50011-20011 *3 [S]. : c1,2001. GB50011-2001CodeforSeismicDesignofBuildings[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):195-201. WUTao,LIUXi,XINGGuohua.StudyontheshearcapacityofreinforcedconcretecolumnbasedonBayesiantheory[J].EngineeringMechan ics,2013,30(5):195-201.(inchinese) [7] SongJ,KangW H,KimKS,etal.Probabilisticshearstrengthmodelsforreinforcedconcretebeamswithoutshearreinforcement[J].Structural Engineering&Mechanics,2010,11(1):15-38. [8],.STUVW X8Y Z?< [ [J]. @0,1987(1):1-10. LUZhanqin,CHENJiakui.Thebendingstrengthandductilityinreinforcedconcretecolumnsunderlowreversedcyclicloading[J].Journalof SouthwestJiaotongUniversity,1987(1):1-10. [9],!.STUVW X P [J]. 1* (,1995,12(4):32-38. ZHANGXianjin,HUXingsheng.Shearstrengthanalysisofreinforcedconcreteframecolumn[J].JournalofWuhanCityColege,1995,12(4): 32-38. [10],,,,. X 6<^ [M]. :,1979:6-10. HirosawaMasaya,EndoRiho,NakayamaShinsuck,etal.Experimentalstudyonseismicperformanceofreinforcedconcrete[M].Beijing:Seis mologicalpres,1979:6-10.