混凝土的破坏准则

2003-10-22BD2013-410:00-11:45a.m..§5.112342§5.2IiJitheprincipalstressinvarants,theinvarantsofprincipalstressdeviatortensor.()()σ1σ2σ3IiJiI1=σ1+σ2+σ3J1=0I2=-(σ1σ2+σ2σ3+σ1σ3)J2=[σ12+σ22+σ32-σ1σ2-σ2σ3-σ1σ3]/3I3=σ1σ2σ3J3=(S13+S23+S33)/32theoctahedralnormalstress&theoctahedralshearstressσoctτoctσoctτoctσoctτoct()()()()221323222113213231331JIoctmoct=-+-+-===++=sssssstssssshydrostaticaxis,deviatoricplane,&meridianπOPrONξONρπONπ3Lodeρπσ1πθξρθ(ξ,ρ,θ)Haigh-Westergaard2/32322333cos2JJJm===qrsx⎟⎠⎞⎜⎝⎛≤≤30pqξρθf(I1,J2,J3)σ1,σ2,σ3ξθθθθ=0°θ=60°θ=30°§5.31σ1σ2σ3()0,,321=sssf42I1I2I33ξρθ4σoctτoctθ5σmτmθ1230°≤θ≤60°1234πξξ11θ=0°ft′f′bcσrσz132sssss===zr2θ=60°fc′f′bt321sssss===zr3θ=30°0.5(σ1-σ3,0,σ1-σ3)σm=0.5(σ1+σ3)[σ1,0.5(σ1+σ3),σ3]251I1ξ2ρt/ρc≤1tc34ρt/ρcπξ0.5ξ=-7fc′0.85ChinnZimmerman(1965)I1=-79fc′§5.4Rankine(1876)fttttfff′=′=′=321sssI1,J2,θρ,ξ,θTresca61864Trescakk=⎟⎠⎞⎜⎝⎛---13322121,21,21maxsssssskξTrescaσ3=0Hognestad(1951)Mohr-CoulombDrucker-PragerMohr-Coulombσfsttan-=ccϕMohr-CoulombDrucker-PragerMohr-Coulomb()0,2121=-+=kJIJIfaI1J20.5ρt=ρc7Drucker-Prager(theoctahedralshearstresstheory)VonMisesσ2kJoct32322==tξσ3=0vonMisesVonMisesTresca8§5.5OttensonWilliam-WarnkeOttenson19771OttosenI1J2cos3θ()013cos,,122221=-′+′+′=cccfIbfJfJaJIflqHaigh-Westergaard()01322,,2=-++=xrlrqrxbafλ=λ(cos3θ)0abλOttensonPossion()⎥⎦⎤⎢⎣⎡=-ql3coscos31cos211KK03cos≥q()⎥⎦⎤⎢⎣⎡--=-qpl3coscos313cos211KK03cos≤q2abK1K21f′cθ=60°2f′t=0.1f′cθ=0°3θ=0°σ1=σ2=-1.16f′cσ3=04θ=60°ξ/f′c,ρ/f′c=-5,4a=1.2759b=3.1962K1=11.7365K2=0.98013a0b09a=b==0λvonMisesa=0λDrucker-PragerλWilliam-Warnke1974William-Warnke19740≤θ≤60°0≤θ≤60°10≤θ≤60°WilliamWarnke0≤θ≤60°ρt=ρcvonMisesDrucker-Prager()01,2222=-+=byaxyxf1/4P1-P-P2-P3P1-P-P2θ=0°θ=60°rtrcP1(0,b)P2(m,n)yrtP1P2n⎟⎟⎠⎞⎜⎜⎝⎛=⎥⎦⎤⎢⎣⎡+⎟⎠⎞⎜⎝⎛=⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=21,23,,2/14242222/122bnambnamyfxfyfxfn10ab223bnma=P2(m,n)12222=+bnam*P2(m,n)ρt,ρcbcmr23=()ctbnrr5.0--=*ρtρcab()tcctcarrrrr45222--=ctccttbrrrrrr5425222-+-=**Oρ,θρθqrsin=x()byt--=rqrcos()[]1cossin22222=--+bbatrqrqrρθ0≤θ≤60°()()[]qqqqrqrqrqr22222/1222222sincoscossinsin2cosbaababbattt++-+-=**ρtρcρ(θ)()()()()[]()()22222/12222222cos445cos42cos2tctcctttcctctccrrqrrrrrqrrrrrqrrrqr-+--+--+-=()()()[]2/121323222132122cossssssssssq-+-+---=ρt/ρc=1vonMisesDrucker-Pragerρt/ρc0.5ρt/ρc=0.5121≤ctrr112σmθ=0°θ=60°22105⎟⎟⎠⎞⎜⎜⎝⎛′+⎟⎟⎠⎞⎜⎜⎝⎛′+=′=′cmcmctcmtfafaaffssrtθ=0°22105⎟⎟⎠⎞⎜⎜⎝⎛′+⎟⎟⎠⎞⎜⎜⎝⎛′+=′=′cmcmcccmcfbfbbffssrtθ=60°τmtτmcθ=0°θ=60°2252Jm=t22J=rσm5θ=0°θ=60°5/rt=m()()01,51,,=-=qsrtrrqtsmmmmf312f(I1,J2,θ)f(σm,τm,θ)3ρt/ρc0.54120°60°5670°≤θ≤60°8000210≤≤aaa000210≤≤bbb()()mcmtsrsr5.098a2≤0b2≤01011a0=b0a1=b1=a2=b2=0vonMisesa0=b0a1=b1a2=b2=0Drucker-Prager4121fc′θ=60°f′c02ft′θ=0°f′t/fc′3fbc′θ=0°fbc′0fbc′/fc′4(θ=0°,01x)(σm/fc′,τm/fc′)=(-1x,1r)5(θ=60°,02x)(σm/fc′,τm/fc′)=(-2x,2r)σm0()000000==⎟⎟⎠⎞⎜⎜⎝⎛′=xrsxrccmtf1bcbcbcfafafa′+′-′=15294322210()tbcbcttbcffffaffa′+′′-′+′-′=2562312113()()()⎟⎠⎞⎜⎝⎛′′-′+′-′+′′+′+′′-′-′=bcttbctbctbcbctbctffffffffffffa9231322256561121112xxxrx0202010=++xxaaa220211024aaaaa---=x220100bbbxx--=1335/63122221--+⎟⎠⎞⎜⎝⎛+=xrxbb()()⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛-++-⎟⎠⎞⎜⎝⎛+=313115/231022020022xxxxxxxrb5LaunayGachon94.15.167.38.115.02121=====′=′rrxxbctff14William-Warnke(σxy,σyz,σzx,τxy,τyz,τzx)§5.6Bresler-Pister(1958)1BreslerPister2000sstcba+-=τ0=τoct/fc′σ0=σoct/fc′a,bcBresler-Pister212τot/τoc=1/Hsich-Ting-Chen(1982)1HsichOttensonλλ=bcosθ+cσ1I115J2σ10111222=-′+′+′+′ccccfIdfcfJbfJasHsich-Ting-Chen21vonMisesDrucker-PragerRankineMillsZimmerman2Hsieh-Ting-ChenOttosenθ=60°33σ2/σ3=1~1.3Kotsovos(1979)1Kotsovos(1979)William-WarnkeWilliam-Warnke857.005.0633.0⎟⎟⎠⎞⎜⎜⎝⎛+′=′coctcocttffst(θ=0°)724.005.0944.0⎟⎟⎠⎞⎜⎜⎝⎛+′=′coctcoctcffst(θ=60°)()()()()()22222222222cos445cos4cos2octtoctcocttoctcocttoctcocttocttoctcoctcocttoctcocttoctcoctcoctttqtttttqtttttqtttt-+--+--+-=τocttτoctcτoctθ=0°θ=60°1621KotsovosWilliam-Warnke2σ00.05fcτot/τoc0.5(τot/τoc0.5)σ0-20.14fcτot/τoc1.0σ0-5.0fcPodgorski(1985)1PodgorskiOttosen02210=++-octoctoctcpcctts()⎥⎦⎤⎢⎣⎡-=-bqa3coscos31cos1pc1c2c0αβ5θ=30°0:-0.5:-1fbc=1.25fc212Ottosen3σ0-5.0fcfbc=-1.1fc(1996)()()baqqsst5.1sin5.1cos000ctdccccba+=⎟⎟⎠⎞⎜⎜⎝⎛--=()()()()213232221032103131/ffffffccoctfffccoctffffssssssttsssss-+-+-==++==(σ1fσ2fσ3f)17abccctdbτ0=0σ0=bbT/T/Tσ1f=σ2f=σ3f=ftttb=fttt/fcd1.0d0σ0=baσ0=-∞τ0cθθ=0°c=ctθ=60°c=ccθ=0°~60°ct≥c≥ccαβ1.5~2.0123445(13)