Concrete damaged plasticity model

Transcription

1 Conree damaged asiiy model Conree damaged asiiy model is a maerial model for he analysis of onree sruures mainly under dynami loads suh as earhquakes(only aes an be analyzed under he dynami loads like ime hisory analysis,). This model is also apiable o brile maerials suh as rok, morar and eramis. Conree is suiable for refleing he brile haraerisis in a low onfining pressure and is no suiable for he duiliy behavior in he behavior under high onfining pressure. Therefore, he behavior of onree under high hydrosai pressure is no inluded in he analysis. This onsrain model refles he irreversible damage effes assoiaed wih he fraure mehanism of he onree, whih is relaively under low onfining pressure. These effes an refle he following behavioral haraerisis. -. Differen yield srenghs in ension and ompression -. Reduion of differen elasi srenghs in ension and ompression -. Resoraion effe of siffness under yli loading Conree damaged asiiy models in Midas were developed based on he model proposed by Lubliner (989) and Lee&Fenves (998). Addiive srain rae deomposiion el : Toal Srain, : Plasi sran, el : Elasi Srain Sress-Srain Relaionship D ( d) D : : D : The Elasi siffness wihou damage D ( d) D : The dereased Elasi siffness by damage d : damage faor (from (no damage) o (damage degree is %)) The damage assoiaed wih he fraure mehanism of he onree is represened

2 Civil Chaper 8 Conree damaged asiiy model by a reduion in he elasi siffness, whih is denoed by he value of he damage faor ( d ) in his model. The effeive sress used Modulus of Elasiiy and srain rae in oninuum mehanis onep is expressed as follows, and he relaionship beween Cauhy sress ( ) and effeive sress ( ) an be expressed using he damage faor. Here, Cauhy sress refers o aual sress. D : d In Conree Damaged Plasiiy, he oeffiien value ( d ) of he arbirary seion of he maerial an represen he effeive bearing area for an apied fore, and if d, he effeive sress and he Cauhy sress an be regarded as he same. he damage degree is alulaed by he hardening parameer ( ) and he effeive sress as follows. d d, Hardening Variables Damage sae in ension and ompression an be expressed independenly using equivalen asi srain, a ension and ompression. Miroraking and rushing in onree an be refleed by inreasing or dereasing he hardening oeffiien( h ). These oeffiiens adjus he hange of yield surfae and he derease of elasi siffness., h,

3 Yield Funion The yield funion ( F ) is expressed as he effeive sress area and deermines he yield or damage sae. The yield funion ( F ) is using he following equaion proposed by Lubliner (989) and modified by Lee and Fenves (998). In he yield funion, and are he non-dimensional maerial onsans, and he values of and are he ensile and ompressive srenghs by. 2 max max F, I 3J ˆ ˆ, 2 b b Maauley brake : x x x 2 Generally, b of onree is.~.6, is.8~.2 (Lubliner 989) I : s invarian of effeive sress ensor J 2 : 2 nd invarian of effeive deviaori sress ensor : Maximum prinipal sress of effeive sresses max In Plane sress, one of he prinipal sresses of 3 axes is unondiionally, so he value of ˆ max beomes zero and an be expressed more simy by he following expression. 2 max F, 3 ˆ I J

4 Civil Chaper 8 Conree damaged asiiy model I 3J ˆ 2 2 Uiaxial ension ˆ 2 Biaxial ension ˆ Uniaxial ompression I 3J ˆ 2 b, b Biaxial ompression I 3J 2 Figure. Yield Fuion of Plane sress Flow Rule The flow poenial funion whih defines he asi deformaion is onsised on he basis of he Druker-Prager and is expressed as effeive sress. Asymmeri siffness ours beause i does no use he same funion as he yield funion. The flow poenial funion used in he Conree Damaged Plasiiy model is shown below. 2 G an 3J2 I an 3 : Dilaany angle : Used o adjus he rae of hange of asympoe by eenriiy : Tensile failure sress in uniaxial

5 Deformaion under Uniaxial behavior of Tension and Compression The inrease or derease of he equivalen asi srain (, ) used as he hardening oeffiien is alulaed by he inegral equaion of he following srain., d d in uniaxial ension in uniaxial ompression The elasi siffness of he maerial for unloading in he srain sofening zone of he sress-srain urves shown in he below figure may be damaged and degraded. The degree of siffness degradaion is quie differen in ension and ompression. And The degree of siffness degradaion is hanged as an independen aording o asi deformaion and is refleed as an independen variable. The damage faor ( d, d ) for ensile and ompression deermined by asi deformaion and has a value beween and if here is no damage., d d d d d d

6 Civil Chaper 8 Conree damaged asiiy model E d E el Figure 2. (a) Response of onree o uniaxial loading in ension u E d d E el Figure 2. (b) Response of onree o uniaxial loading in ompression

7 Deformaion under Uniaxial yli ondiions The mehanism of siffness reduion under uniaxial yli loading is muh more omiaed han he uniaxial loading, beause i involves he opening and losing of miro raks reaed in he previous sep. Under uniaxial yli loading, he elasi siffness is resored o some exen aording o he hange of load, whih ays an imporan role in he behavior of onree under yli loading. When ension is onvered ino ompression, he siffness in ompression inreases beause he ensile raks is losed. The modulus of elasiiy for a onree damaged asiiy model an be expressed using he damage faor d whih is alulaed by a funion of he sress and d, d (damage faor for ension and ompression) for he uniaxial, he funion is as follows: d s d s d s, s The oeffiien, s, s are used o express he resoraion effe of siffness by he onversion of sress and are alulaed aording o he following equaions. s w r w * ( ) s w r w * ( ) * r ( ) H if if The maerial oeffiien, w, w are a resoraion faor ha adjuss he degree of resoraion for he siffness in ension and ompression when a onversion of sress

8 Civil Chaper 8 Conree damaged asiiy model ours. Midas progams assume w = and w =. The below figure shows he resored shape of ompressive siffness under he srain sofening sae. E d E w w Figure 3. Elasi siffness by resoraion faor( w ) in ompression

9 E w w d E ( d ) E ( d )( d ) E w w E Figure 4. Resoraiion of siffness aording o Tension - Compression Tension load ( w, w ) Muli-axial Sress Sae Lee and Fenves (998) an be summarized as follows so ha various variables defined in he uniaxial sress sae an be apied in he muli-axis sae. max r ˆ ˆ r r ˆ ˆ ˆ i i r 3 ˆ i min 3 ˆ ˆ Maauley brake : i x x x 2

10 Civil Chaper 8 Conree damaged asiiy model ˆ max ˆ, min are maximum and minmum haraerisi value of asi srain ensor, ˆ h ˆ, ˆ r ˆ ˆ h ˆ, r ˆ ˆ ˆ ˆ 2 ˆ 3 Visoasi regularizaion The problems for onvergene an our in he ase of Imii analysis of maerial models ha exhibi sofening behavior and siffness reduion, and his diffiuly an be overome hrough visoasi regularizaion. Conree damaged asiiy models an be regularizeded by using visous and onsequenly, he alulaeion of sresses o be ouside he yield surfae are possible. The visosiy model for regularizaion is based on he Duvau-Lions (976) model. The inelasi srain, in his model is expressed as he follows. is he projeion of on he yielding surfae, and is he relaxaion as he visosiy oeffiien in he visous sysem. for ime inremen. is a value

11 D : D : d Relaionship of Sress-srain: v Inelasi srain : D :, D ( d ) D p n n n d d d Viso asiiy reduion faor :,, Sress : d D n n n n Algorihmi Tangenial Siffness : v n n v n d d d D d d d n n n vn, n n Referenes. Lubliner J, Oliver J, Oller S, e al. A Plasi-Damage Model for Conree. In J Solids Sru, 989, 25: Lee J, Fenves G V. Plasi-damage model for yli loading of onree sruures. J Eng Meh-ASCE, 998, 24:892-9