. Ton Tat HOAG LA A SIMPLE WAY TO ESTABLISH THE EQUATIO OF SHELLS s YIELD SURFACE. FACULTY OF CIVIL EGIEERIG, UIVERSITY OF ARCHITECTURE, HCM CITY, VIETAM ABSTRACT: A yield suface is a five dimensional suface in te six dimensional space of stesses. Te yield suface is usually convex and te state of stess of inside te yield suface is elastic. Wen te stess state lies on te suface te mateial is said to ave eaced its yield point and te mateial is said to ave become plastic. Fute defomation of te mateial causes te stess state to emain on te yield suface, even toug te suface itself may cange sape and sie. Tis is because stess states tat lie outside te yield suface ae non pemissible in ate independent plasticity, toug not in some models of viscoplasticity. Tis pape deals wit te yield suface of sells. KEYWORDS: yield suface, sells, elastic, plastic, viscoplasticity ITRODUCTIO Yield suface is usually expessed and visualied in tee dimensional space of pincipal stesses (,, 3 ), a two o tee dimensional space extended by te constant stess (I,J,J 3 ) o a vesion of tee dimensional space tat is Haig Westegaad stess. Yield suface is a fequent pesence on Tesca yield suface, Hube Von Mises yield suface, Mo Coulomb yield suface, Ducke Page yield suface, Besle Piste yield suface, Wanke William yield suface,... (Figue H,H&H3) Figue H. Tesca yield suface, 3D& D Figue H. Hube Von Mises yield suface Figue H3. Mo Coulomb yield suface, 3D & D copyigt FACULTY of EGIEERIG HUEDOARA, ROMAIA 37
AALS OF FACULTY EGIEERIG HUEDOARA Intenational Jounal Of Engineeing RELATIOSHIP BETWEE STRESS, DEFORMATIO AD DISPLACEMET Te intenal foces in te sell tat known as: θ, ϕ, M θ, M ϕ and Q. Assumptions: (i) coss section emains plane duing defomation, (ii) te impact of negligible sea is ignoed. Equations: YIELD CODITIO Hube Mises s condition: θ θd ϕ ϕd Mθ θ d Mϕ ϕd θ θ θ ( vcotgϕ w) θ ( v w ) ϕ ϕ ϕ ϕ ( v w) θ cotgϕ ( v w ) ( ) ϕ θ cosϕ Q ( ) sin ( Q) ϕ θ ϕ ( M ) M cosϕ ϕ θ ϕ P P Q ϕ ( ) ( ) ( ) 3 3 Coulomb Tesca s condition: max min One pesented it in 3D and D (plane stess state, 3 ) as sown in Figue H4 and Figue H5 Figue H4. Stess state in 3D Figue H5. Plane stess state, 3, D Figue H6. Isotopic mateial Figue H7. Ototopic mateial Easily find te Hube Mises condition mentioned in te law nonlinea but Coulomb Tesca condition mentioned simple linea law in te pocess of analysis fo te sell. If one conside te sell mateial wit yield stess diffeence between tension and compession, we will sow in Figue H6 & H7. To simplify fute, we can assume tat te ototopic popety of mateial consideed only in tension (, > ). Ten we esow on Figue H8. Fo efeence, we eplace (ϕ, θ) by (.). Relationsip between i and sown in Figue H & H. 38 Tome X (Yea ). Fascicule 3. ISS 584 673
AALS OF FACULTY EGIEERIG HUEDOARA Intenational Jounal Of Engineeing Figue H8. Simple model fo ototopic mateial Figue H9. Simple analysis Figue H. Relationsip between and Figue H. Relationsip between and ESTABLISHMET EQUATIO OF YIELD SURFACE Relationsip between i, i, I and : Wen te stess state is expessed as te edges of te yield suface, te legal condition is equied pependicula between (, ) and tese edges. At te cone, (, ) will old between two lines pependicula to te two neigboing sides. One conside Figue H9, if te ate of defomation at a point depends on te fist quadant, one find te coesponding state of stess at tis point is cone B on te yield suface. Moe specifically as follows ( ) : stess state at cone B, ( ) : stess state at cone C, : stess state at cone D, One easily ave : egime B, : egime B, : copyigt FACULTY of EGIEERIG HUEDOARA, ROMAIA 39
AALS OF FACULTY EGIEERIG HUEDOARA Intenational Jounal Of Engineeing Tome X (Yea ). Fascicule 3. ISS 584 673 4 : egime A, : egime F, : egime E, Similaly one ave, : egime B, if : egime B, if : egime C, if : egime D, : egime E, By using tis equation: i i i espect to espect to espect to espect to espect to espect to espect to espect to In addition one ave, d d d M d M d d d d
AALS OF FACULTY EGIEERIG HUEDOARA Intenational Jounal Of Engineeing copyigt FACULTY of EGIEERIG HUEDOARA, ROMAIA 4 Similaly, 4 M 4 M ILLUSTRATIO Fo case (*) Pesentation wit dimensionless: n n 4M m 4M m f g p q t k Results ae sown in Table (*). Te ote cases ae similaly done. Table (*).Results
AALS OF FACULTY EGIEERIG HUEDOARA Intenational Jounal Of Engineeing COCLUSIOS Wen eseacing plastics fo te sell we need to conside conditions fo yield and yield suface of te sell in te case of ototopic mateials. Tis pape sows te simple way to ave equation of yield suface of te sell fo limit analysis metod. REFERECES [.] Roge Fosdisk, Eic Volkmann. omality and convexity of te yield suface in nonlinea plasticity. IMA, 789, Mac 99. [.] Yuan Gao Zang. Limit Analysis Using te Ellipsoid Yield Suface. Poc. R. Soc. Lond. A 994 445, 59 57. [3.] Fan Rausce. Tesca s o Von Mises yield condition. Austia [4.] ULg, Remaks on plastic mateial beavio. Belgium [5.] Stefan Soae, Doel Banabic. A fou paamete in plane isotopic yield function. 4 Cluj apoca, Romania AALS OF FACULTY EGIEERIG HUEDOARA ITERATIOAL JOURAL OF EGIEERIG copyigt UIVERSITY POLITEHICA TIMISOARA, FACULTY OF EGIEERIG HUEDOARA, 5, REVOLUTIEI, 338, HUEDOARA, ROMAIA ttp://annals.fi.upt.o 4 Tome X (Yea ). Fascicule 3. ISS 584 673