ELASTO-PLASTIC ANALYSIS OF STRUCTURES USING HEXAHEDRICAL ELEMENTS WITH EIGHT NODES AND ONE-POINT QUADRATURE

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1 Mcánica Compuacional Vol XXV, pp Albro Cardona, Norbro Nigro, Vicorio Sonzogni, Mario Sori. (Eds.) Sana F, Argnina, Novimbr 26 ELASTO-PLASTIC ANALYSIS OF STRUCTURES USING HEXAHEDRICAL ELEMENTS WITH EIGHT NODES AND ONE-POINT QUADRATURE Dilni Schmid a, Armando M. Awruch b and Inácio B. Morsh b a Cnro d Mcânica Aplicada Compuacional (CEMACOM), Univrsidad Fdral do Rio Grand do Sul, Av. Osvaldo Aranha, 99, 3º andar, 935-9, Poro Algr, RS, Brasil, dilni_s@homail.com, hp:// b Cnro d Mcânica Aplicada Compuacional (CEMACOM), Univrsidad Fdral do Rio Grand do Sul, Av. Osvaldo Aranha, 99, 3º andar, 935-9, Poro Algr, RS, Brasil, amawruch@ufrgs.br, hp:// Kywords: Elaso-plasic analysis, Gomrically nonlinar analysis, Hxahdrical lmn wih on-poin quadraur Absrac. Th main purposs of his work ar h formulaion and applicaion of a compuaional algorihm for h laso-plasic saic analysis of srucurs, including fini displacmns and roaions. An hxahdrical isoparamric fini lmn wih igh nods and on-poin quadraur is usd. Mchanisms o avoid hourglass mods as wll as volumric and shar locking ar inroducd. A coroaional formulaion is mployd o dal wih h gomrically nonlinar analysis, whras an xplici algorihm (basd in Eulr s schm) is implmnd for h laso-plasic analysis. Th applid consiuiv modls includ h Mohr-Coulomb as wll as h Von Miss yild cririon wih isoropic hardning. Numrical xampls wih highly nonlinar bhavior ar prsnd o dmonsra h rang of applicabiliy of h formulaion. Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

2 862 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH INTRODUCTION Low ordr hr-dimnsional fini lmns hav bn usd wih fficincy in many solid mchanics problms. Howvr, volumric locking is ncounrd for incomprssibl or nar incomprssibl marials and shar locking appars in bnding-dominad siuaions whn no spcial lmn chnology is mbddd o ovrcom his dficincy. Among h idas o limina his problm, rducd ingraion may b usd. Nvrhlss, h rsuls achivd hrough h us of hs lmns can b unsaisfacory or vn maninglss whn spurious mods ar xcid. Hnc, h us of h rducd ingraion lmns rquirs an fficin numrical sabilizaion schm o supprss h spurious mods. Low ordr fini lmns wih rducd ingraion and hourglass sabilizaion ar spcially araciv du o hir compuaional fficincy. Th numrical fficincy gaind by working wih a lowr numbr of Gauss poins is in paricular noicabl whn h numrical cos of a fini lmn analysis is srongly coupld o h numrical ffor a h lmn lvl. This is h cas whn xplici compuaions ar prformd and also compuaions basd on highly complx consiuiv modls. In a non-linar fini lmn analysis, i is ncssary o ingra h consiuiv rlaions o obain h unknown incrmn in h srsss. Ths rlaions dfin a s of ordinary diffrnial quaions and mhods for ingraing hm ar usually classifid as xplici or implici. Sinc xplici schms mploy h sandard laso-plasic consiuiv law and rquir only firs drivaivs of h yild funcion and plasic ponial, hy can b usd o dsign a gnral purpos ingraor, as h on proposd by Sloan al. (2), which can b usd for a wid rang of modls. By bing mor sraighforward o implmn han implici mhods, an xplici mhod will b usd in his work o ingra h consiuiv rlaions. Svral auhors hav bn working in h subjc of low ordr fini lmns wih hourglass sabilizaion. Th hr-dimnsional rducd ingraion concp was mainly dvlopd by Blyschko, Liu and coworkrs. Liu al. (994) dvlopd an undringrad igh-nod hxahdral lmn whr h dilaaional rm of h normal srain componns as wll as som shar srain componns ar rad in a spcial way. Basd on h muliplquadraur formulaion givn in Liu al. (994), Hu and Nagy (997) as wll as Duar Filho and Awruch (24) proposd a nw simpl on-poin quadraur hxahdral lmn. Th srain and srss vcors ar firsly xpandd in a Taylor sris a h lmn cnr up o bilinar rms. Th consan rms ar usd o compu h lmn inrnal forc vcor and h linar and bilinar rms ar usd o form h hourglass rsising forc vcor. As shown in Liu al. (994), h coroaional sysm is mployd o rmov hos mods associad wih shar locking and h dilaaional par of h gradin marix is valuad only a h cnr of h lmn o avoid volumric locking. In h formulaion of Liu al. (998), h lmn proposd by Liu al. (994) is implmnd for larg dformaion laso-plasic analysis. Th auhors work wih four poin quadraur which has h advanag ha plasic fron can b capurd mor accuraly. On h ohr hand, h advanag of compuaional fficincy is parially los. Thus, h objciv of his work is o vrify h compuaional fficincy and h robusnss of h igh-nod hxahdral lmn wih on-poin quadraur dvlopd by Hu and Nagy (997) and by Duar Filho and Awruch (24), addd wih h laso-plasic schm proposd by Sloan al. (2), in h analysis of srucural and gochnical problms. Th numrical simulaion prsnd hr was compard wih rsuls obaind by ohrs auhors and by a commrcial sofwar and i is shown ha criical problms wih physical and gomrically nonlinar analysis may b solvd wih h on-poin quadraur hr-dimnsional lmn. Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

3 Mcánica Compuacional Vol XXV, pp (26) PRINCIPLE OF VIRTUAL WORK In a fini lmn rprsnaion, h principl of virual work is givn by: δ W = δ u b dv + δ u p ds, () in V whr h suprscrip dsignas h ranspos; δu is h virual displacmn vcor in h lmn ; b is h body forc vcor applid in h lmn domain V ; p is h racion vcor applid on h lmn boundary S ; W is h lmn inrnal virual work givn by: δ W in = V in S δ ε dv, (2) whr is h srss vcor in h lmn and δε is h virual srain vcor du o δu. If h srain in h lmn is inrpolad in rms of nodal displacmn by: hn, quaion 2 can b rwrin as: whr B is h gradin marix. ε = B U, (3) in δ W = δ U B dv, (4) V 3 ONE-POINT QUADRATURE EIGHT-NODE HEXAHEDRAL ELEMENT WITH HOURGLASS CONTROL For an igh-nod hxahdral lmn, h spaial coordinas, x i, and h displacmn componns, u i, in h lmn ar approximad in rms of nodal valus, x ia and u ia, by: whr h rilinar shap funcions ar xprssd as: x 8 i = Naxia, (5) a = 8 i = Nauia, (6) a = u Na ( ξ, η, ζ ) = ( + ξaξ ) ( + ηaη ) ( + ζ aζ ), (7) 8 and h subscrip i dnos coordina componns (x, y, z) ranging from on o hr and a dnos h lmn nodal numbrs ranging from on o igh. Th rfrnial coordinas ξ, η ζ of nod a ar dnod by ξ a, η a ζ a, rspcivly. If h following column vcors ar dfind for nodal coordinas in h spaial sysm and h naural sysm: [ x, x, x, x, x, x, x, x ] [ y, y, y, y, y, y, y, y ] [ z, z, z, z, z, z, z, z ] [,,,,,,, ] x = x =, (8) x = y =, (9) x = z =, () ξ = , () Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

4 864 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH [,,,,,,, ] [,,,,,,, ] η = , (2) ζ = , (3) h Jacobian marix a h cnr of h lmn (ξ = η = ζ = ) can b valuad as: ξ x ξ y ξ z J ( ) = 8 η x η y η z, (4) ζ x ζ y ζ z and is drminan j o can b wrin as: j o ξ x ξ y ξ z = d J ( ) = η x η y η z = V, (5) 52 8 ζ x ζ y ζ z whr V is h volum of h 8-nod hxahdral lmn. To idnify h dformaion mods of h lmn, as i can b sn in Liu al. (994), h gradin submarics B a () a h cnr of h lmn ar dfind as follows: a ( ) ( ) N a x b ( ) B b, (a =,2,...,8). (6) Na = = 2 y b 3 Na ( ) z If h invrs marix of J() is dnod by D, hn h gradin vcors b, b 2 and b 3 in quaion 6, according o Liu al. (998), can b shown o b: b = { b a } = [ D ξ + D2 η + D3 ζ 8 ], (7) b2 = { b2 a } = [ D2 ξ + D22 η + D23 ζ 8 ], (8) b3 = { b3 a } = [ D3 ξ + D32 η + D33 ζ 8 ]. (9) To allvia volumric locking, h ida undrlying rducd-slciv ingraion is usd. Th gradin marix is dcomposd ino wo pars: ( ξ, η, ζ ) = ( ) + ˆ ( ξ, η, ζ ) B B B, (2) whr B ( ) is h gradin marix corrsponding o h dilaaional par of h srain vcor, valuad a h lmn cnr only, and B ˆ ( ξ, η, ζ ) is h gradin marix corrsponding o h dviaoric par of h srain vcor. Thn, quaion 4 can b rwrin as: Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

5 Mcánica Compuacional Vol XXV, pp (26) 865 in Expanding ˆ ( ξ, η, ζ ) ( ) ˆ ( ) ( ) δ W = δ U [ B + B ξ, η, ζ ] ξ, η, ζ dv. V B in a Taylor sris a h lmn cnr up o bilinar rms, quaion 2 can b rwrin as: ( ξ, η, ζ ) ( ) ˆ, ( ) ˆ, ( ) ˆ ξ ξ η η, ζ ( ) ζ 2 ˆ ( ) + 2 ˆ ( ) + 2 ˆ ( ) B = B + B + B + B + B ξη B ηζ B ξζ,, ξη, ηζ, ξζ whr B() is h on-poin quadraur gradin marix conribud from boh h dilaaional and dviaoric pars: ( ) = ( ) + ˆ ( ) (2) (22) B B B. (23) Th ohr rms on h righ-hand sid of quaion 22 ar h gradin marics corrsponding o non-consan dviaoric srain. Th firs and scond drivaivs of B ar obaind afr som dious algbra and can b found in Liu al. (998). Th srss vcor is also xpandd in a Taylor sris abou h lmn cnr up o bilinar rms: (,, ) ( ) ˆ, ( ) ˆ, ( ) ˆ ξ η, ζ ( ) 2 ˆ ( ) + 2 ˆ ( ) + 2 ˆ ( ) ξ η ζ = + ξ + η + ζ + ξη ηζ ξζ., ξη, ηζ, ξζ By subsiuing quaion 22 and quaion 24 ino quaion 2, w can ingra and obain h inrnal virual work of h lmn as: ( ) ( ) ( ) ( ) ( ) ( ) in ˆ ˆ ˆ δ W ˆ ˆ,,,,, ( ) ˆ = δ U + ξ ξ + η η + ζ, ζ ( ) + B 3 B 3 B 3 B Bˆ, ξη ( ) ˆ ( ) + Bˆ, ξη, ηζ ( ) ˆ ηζ ( ) + Bˆ,, ξζ ( ) ˆ, ξζ ( ) V 9 9 9, whr h firs rm on h righ-hand sid of quaion 25 is h on-poin quadraur inrnal virual work. Th ohr rms ar also valuad a h lmn cnr o provid h sabilizaion of h lmn. By assuming ha h Jacobian is a consan, /8 of h lmn volum, h on-poin quadraur lmn inrnal forc vcor wihou hourglass conrol can b xprssd by: ( ) ( ) f = B V (24) (25). (26) Th lmn siffnss marix for h undringrad lmn can b obaind by using h srss-srain law = C ε in conjuncion wih h srain displacmn rlaion givn by quaion 3: f = K U, (27) whr K is h lmn siffnss marix valuad a h lmn cnr wihou hourglass conrol and i is givn by: ( ) ( ) K = B C B V, (28) Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

6 866 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH which is rank insufficin and may xhibi spurious singular mods. To limina hs hg spurious singular mods, i is ncssary o add h hourglass-rsising forc, f, o h lmn inrnal forc vcor as: By obsrving quaion 25, 26 and 29, f hg in hg f = f + f. (29) hg f may b dfind as: = Bˆ ( ) ( ) + Bˆ ( ) ( ) + Bˆ, ξ ˆ, ξ, η ˆ, η, ζ ( ) ˆ, ζ ( ) Bˆ, ξη ˆ + Bˆ, ξη, ηζ ˆ, ηζ + Bˆ, ξζ ˆ, ξζ ( ) ( ) ( ) ( ) ( ) ( ) V If h firs and scond drivaivs of h srss vcor can b drivd from h marial consiuiv quaions, h lmn sabilizaion siffnss marix K sab may b also dfind as:. (3) hg sab f = K U. (3) This marix is addd o h lmn siffnss marix, K, so h lmn siffnss marix K is rank sufficin and is givn by: sab K = K + K. (32) To avoid h drivaion of h rlaionships bwn h firs and scond drivaiv of h srss vcor and h nodal displacmn vcor in quaion 3, Hu and Nagy (997) proposd a sabilizaion marix, E, o saisfy following consiuiv rlaions: ˆ ˆ, ξ = E ε, ξ, ˆ ˆ, η = E ε, η, ˆ ˆ, ζ = E ε, ζ, ˆ ˆ, ξη = E ε, ξη, ˆ ˆ, ηζ = E ε, ηζ, ˆ ˆ, ξζ = E ε, ξζ. whr E is h lasic marial modulus marix, which is h simpls form for compuaion and is givn as: whr, E 3x3 6x6 = 3x3 (33), (34) 2µ = 2 µ, (35) 2µ and µ is h Lamé consan. Elaso-plasic marial bhavior or damag is characrizd by a suddn sofning of h marial if a crain srss limi is rachd. To prvn xcssiv siffnss and nhanc h bhavior of laso-plasic marials, Rs (25) proposd h us of an opimal paramr µ op in h sabilizaion marix. Th facor µ op can b sn as h smalls paramr which yilds an hourglass-fr dformaion parn and i can b obaind as follows: op µ = µ H, (36) E + H Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

7 Mcánica Compuacional Vol XXV, pp (26) 867 whr E rprsn h Young s modulus and H is h hardning modulus a h ons of plasificaion. Thn, by subsiuing quaion 33 in quaion 3, h lmn sabilizaion siffnss marix is obaind in h following form: K sab ˆ ( ) ˆ,, ( ) ˆ, ( ) ˆ, ( ) ˆ =, ( ) ˆ ξ ξ + η η + ζ, ζ ( ) + 3 B E B 3 B E B 3 B E B ˆ ( ) ˆ ( ) ˆ,,, ( ) ˆ, ( ) ˆ, ( ) ˆ B ξη E B ξη + B ηζ E B ηζ + B ξζ E B, ξζ ( ) V Th lmn dvlopd so far is fr of volumric locking and has no spurious singular mods. Howvr, i is no suiabl o pla/shll analysis owing o h shar and mmbran locking in hin srucurs and i canno pass h pach s if h msh is irrgular. To rmov shar locking, i is shown by Hu and Nagy (997) ha h gradin submarics corrsponding o h assumd shar srain is wrin in an orhogonal coroaional coordina sysm roaing wih h lmn and ach shar-srain componn is linarly inrpolad in on rfrnial coordina dircion only: which implis: whr B ˆ xy, Bˆ yz (,, ) = ( ) + ˆ, ( ) (,, ) = ( ) + ˆ, ( ) (,, ) = ( ) + ˆ ( ) xy xy xy ζ (37) ε ξ η ζ ε ε ζ, (38) ε ξ η ζ ε ε ξ, (39) yz yz yz ξ ε ξ η ζ ε ε η, (4) xz xz xz, η ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Bˆ = Bˆ = Bˆ = Bˆ = Bˆ =, (4) xy, ξ xy, η xy, ξη xy, ηζ xy, ξζ Bˆ = Bˆ = Bˆ = Bˆ = Bˆ =, (42) yz, η yz, ζ yz, ξη yz, ηζ yz, ξζ Bˆ = Bˆ = Bˆ = Bˆ = Bˆ =, (43) xz, ξ xz, ζ xz, ξη xz, ηζ xz, ξζ Bˆ xz ar h gradin marics corrsponding o h dviaoric srain componns ε ˆ xy, ε ˆ yz ε ˆ xz, rspcivly. Th on-poin quadraur lmn will pass in h pach s whn i is skwd if h gradin marix B a () is rplacd by h uniform gradin B a () dfind by Flanagan and Blyschko (983) as: B' a = a ( ξ, η, ζ ) dv V B. (44) V 4 CORROTATIONAL APPROACH FOR PHYSICAL AND GEOMETRICALLY NONLINEAR ANALYSIS I has bn shown ha h liminaion of h shar locking dpnds on h propr ramn of h shar srain. I is ncssary o aach a local coordina sysm o h lmn so ha h srain nsor in his local sysm is rlvan for h ramn. Th coroaional sysm dscribd in Liu al. (998) is mployd for his purpos. By using h coroaional sysm, h ingraion of h laso-plasic consiuiv quaions bcoms asir. Thorically, h moion of a coninuous mdium can always b dcomposd ino a rigid body moion followd by a pur dformaion. If h fini lmn discrizaion is fin nough o provid a valid approximaion of h coninuum, his dcomposiion can b Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

8 868 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH prformd a h lmn lvl. If h rigid body moion is liminad from h oal displacmn fild which corrsponds o larg displacmns and roaion bu small srains, h pur dformaion par is always a small quaniy rlaiv o h lmn dimnsions. 4. Coroaional srss updas For srss and srain updas, w assum ha all variabls a h prvious load sp n ar known. Thn, i is only ncssary o calcula h srain incrmn from h displacmn fild wihin h load incrmn [ n, n+ ], and h procdur dscribd by Liu al. (998) o df calcula h dformaion par ( u ˆ ) of h displacmn incrmn in a coroaional sysm df is usd. In his work, u ˆ is rfrrd o h mid-poin configuraion ( n+/2 ). Dnoing h spaial coordinas of h prvious load sp configuraion, Ω n, and h currn configuraion, Ω n +, as x n x n+ in h fixd global Carsian coordina sysm Ox, h coordinas in h corrsponding coroaional Carsian coordina sysm, Ox ˆn and Ox ˆn+, can b obaind by h following ransformaion ruls: x ˆ = R x, (45) n n n ˆ n+ = R n + xn + x, (46) whr R n and R n + ar h orhogonal ransformaion marics which roas h global coordina sysm o h corrsponding coroaional coordina sysm, rspcivly (dfind in Liu al. (998) and in Duar Filho and Awruch (24)). Sinc h srain incrmn is rfrrd o h configuraion a = n+/ 2, assuming h vlociis wihin h incrmn [ n, n+ ] ar consan, i is obaind: x n+ / 2 = ( xn + xn+ ), (47) 2 x R x. (48) ˆ n+ / 2 = n+ / 2 n+ / 2 Similar o polar dcomposiion, an incrmnal dformaion can b sparad ino h summaion of h pur dformaion and h pur roaion. Ling u indica h displacmn incrmn wihin h load incrmn [ n, n+ ], i may b wrin: df ro u = u + u, (49) df ro whr u u ar, rspcivly, h dformaion par and h pur roaion par of h displacmn incrmn in h global coordina sysm. In ordr o obain h dformaion par of h displacmn incrmn rfrrd o h configuraion a = n+/2, i is ncssary o find h rigid roaion from Ω n o Ω n +. I can b shown ha h oal roaion displacmn incrmn can b xprssd as: ( x xˆ ) = u R ( xˆ xˆ ) ro u = xn + xn Rn + / 2 ˆ n + n n + / 2 n + n. (5) Thn, h dformaion par of h displacmn incrmn rfrrd o Ω n+ / 2 is: ( x xˆ ) df ro u = u u = R ˆ n + / 2 n + n. (5) Thrfor, h dformaion displacmn incrmn in h coroaional sysm O x n 2 is obaind as: ˆ +/ Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

9 Mcánica Compuacional Vol XXV, pp (26) 869 df df uˆ = R n + / 2 u = xˆ n+ xˆ n. (52) Sinc h coroaional coordina sysm roas wih h configuraion, i is usd h coroaional Cauchy srss, which is objciv, as srss masur. Th ra of dformaion (or vlociy srain vcor), ε, also dfind in h coroaional coordina sysm, is usd as h masur of h srain ra: df df vˆ vˆ ε = dˆ = +, (53) 2 xˆ xˆ df whr ˆv is h dformaion par of h vlociy in h coroaional sysm ˆx. Thn, h srain incrmn is givn by h mid-poin ingraion of h vlociy srain nsor, n+ df df ˆ uˆ uˆ ε ˆ = d dτ = +. (54) 2 ˆ ˆ x n n + / 2 x n + / 2 Onc h srain incrmn is obaind by quaion 54, h srss incrmn can b calculad wih h laso-plasic schm dscribd in Scion 5, and h oal srain and srss can hn b updad as: ε ˆ ˆ ˆ n+ n (55) ˆ ˆ ˆ n + n (56) 4.2 Consiuiv quaions and soluion of h incrmnal sysm of quaions As h marial ra for h Cauchy srss nsor is no a fram-invarian ra, i is mployd h Grn-Naghdi objciv ra, which, according o Liu al. (998) and Duar Filho and Awruch (24), givs h following marial angn marix: T C 6x6 6x3 ( ) = + Tˆ ( ) 3x6 3x3, (57) whr m n dnos h m n zro marix, C is a 6 6 srss-srain marix and Tˆ ( ), h iniialsrss marix, is dfind blow: Tˆ ( ) = symm (58) Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

10 87 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH Th T() marix is arrangd o b compaibl wih h following ordring of srain and roaion componns: [ ε ε ε ε ε ε ω ω ] ε =. (59) ω 3 Thn, h quilibrium quaion a h j h iraion can b wrin in h coroaional coordina sysm as: whr K ˆ Uˆ = Pˆ fˆ, (6) j j Û is h displacmn incrmn vcor; Pˆ j is h xrnally applid nodal poin ˆ forcs and h angn siffnss marix K j and h inrnal nodal forc vcor f j ar: j sab ( C + Tˆ ) B'( K K ˆ j = B' ( ) j ) +, (6) hg f ˆ = f + f. (62) j Th angn siffnss and nodal forcs ar ransformd ino h global coordina sysm as: K j = R Kˆ R, (63) j ( ˆ ˆ ) j j r = R P f = R r ˆ, (64) j j j j j j whr R is h ransformaion marix of h coroaional sysm dfind by Liu al. (998) and by Duar Filho and Awruch (24). 5 ELASTO-PLASTIC CONSTITUTIVE EQUATION During a ypical sp of an laso-plasic fini lmn analysis, h forcs ar applid in incrmns and h corrsponding nodal displacmn incrmns ar found from h global quilibrium quaions. Onc hs displacmns ar known, h srain incrmns a h ingraion poins wihin ach lmn ar drmind using h srain displacmn rlaions. If h srsss associad wih an imposd srain incrmn caus plasic yilding, h lasoplasic consiuiv quaion may b wrin in h following incrmnal form: = C ε. (65) p According o Own and Hinon (98), h laso-plasic srss-srain marix, C p, for h paricular cas of an associad flow rul is givn by: C p Caa C = C H + ˆ a Ca, (66) whr C is h lasic srss-srain marix, a is h yild surfac gradin and H is h hardning modulus, ha can b xprssd, in h cas of linar hardning, as a funcion of h lasic modulus E and h plasic angn modulus E T as: ET H = E E. (67) T Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

11 Mcánica Compuacional Vol XXV, pp (26) Ingraion of h consiuiv quaions According o Liu al. (998), wih h us of a coroaional sysm, h ingraion of h fini dformaion laso-plasic consiuiv quaions aks a simpl form, as h small dformaion hory. Th ingraion of h consiuiv quaions is prformd a lmn lvl in h coroaional sysm. Th ingraion schm, proposd by Sloan al. (2) is usd in his work o dal wih h physical nonlinariy. Th ingraion schm can b summarizd in compuing an lasic rial srss sa, finding h yild surfac inrscion poin (considring also h cas of laso-plasic unloading), updaing h srsss and rsoring h srsss o h yild surfac. Wih dnoing h iniial srss sa, h lasic rial srss sa can b asily compud according o = +, whr = C ε. Th problm of finding h srsss a h yild surfac inrscion poin in is quivaln o finding h scalar quaniy α which saisfis h following non-linar quaion: f ( + αc ε ) = f ( ) =, (68) in whr f is h yild funcion. A valu of α = indicas ha ε causs purly plasic dformaion, whil a valu of α = indicas purly lasic dformaion. Thus, for an lasic o plasic ransiion, w hav < α < and h lasic par of h srss incrmn is givn by αc ε. If h iniial srss sa is lying on h yild surfac and h angl θ bwn h yild surfac gradin a and h angnial lasic srss incrmn is largr han 9º, h srss incrmn may cross h yild surfac wic. This possibiliy is causd by h us of a olranc o h valu of h yild funcion which prmis h srsss o ly jus ousid h yild surfac, and i can b chckd wih: cos θ = a a 2 2, (69) whr is h norm in h spac L 2. Sloan al. (2) givs a daild algorihmic 2 dscripion of h mhod o find α, wih sufficin dails in ordr o b implmnd in a fini lmn cod. Onc α is found, h porion of h angnial lasic srss incrmn o plasic dformaion is compud wih ha corrsponds ( α) and usd o compu h srss incrmn: = λc a, (7) whr h plasic srain-ra muliplir, λ, which for h paricular cas of h von Miss yild cririon wih isoropic hardning is qual o h quivaln plasic srain, can b found as follows: 5.2 Corrcion of srsss o h yild surfac a λ =. (7) Η + a C a A h nd of ach incrmn in h xplici ingraion procss, h srsss may divrg from h yild condiion. Th xn of his violaion, which is commonly known as yild surfac drif, dpnds on h accuracy of h ingraion schm and h nonlinariy of h Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

12 872 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH consiuiv rlaions. Sloan (987) suggsd ha, onc h ingraion is prformd accuraly, h amoun of yild surfac drif will b small and any rmdial acion is opional. Ohr auhors, including Crisfild (99), srongly advoca som form of iraiv srss corrcion as h ffcs of violaing h yild condiion ar cumulaiv. Th radial rurn algorihm, usd o corrc h srsss in his work, consiss in finding h closs-poin-projcion of h srss sa ono h yild surfac, if h srss sa lis ousid h yild surfac. A small corrcion o h plasic srain-ra muliplir is compud wih h following quaion: f δλ = H + a C a, (72) whr f is h valu of h yild funcion for h uncorrcd srsss. Th srsss and h quivaln plasic srain ar hn updad according o h following xprssions: = δλc a, (73) pl pl ε = ε δλ, (74) whr h subscrip in quaion 73 and quaion 74, as wll as in quaion 69 rfrs o uncorrcd variabls. 6 NUMERICAL EXAMPLES Numrical applicaions involving h von Miss as wll as h Mohr-Coulomb yild cririon xhibiing highly nonlinar bhavior, ar prsnd o s and o vrify h lmn prformanc for physical and nonlinar gomrically analysis. Rsuls ar compard wih hos rpord by ohr auhors and wih h commrcial cod ABAQUS (24). 6. Elasic-plasic canilvr bam This simulaion prsns h laso-plasic rspons of a canilvr bam subjcd o a ransvrs shar load of 8 kn a on nd. Th hr displacmns dgrs of frdom of all h nods a h fixd nd ar prscribd. Th lngh of h bam is L = 24. m, whil h widh is B =. m and h high is H = 4. m. Th marial has isoropic hardning wih bilinar uniaxial srss-srain rlaion and h marial paramrs ar: lasic modulus E =. 4 kn/m 2, Poisson s raio ν =.3, iniial yild srss = 3. 2 kn/m 2, and h angnial modulus afr yilding, E T =. 3 kn/m 2. An schmaic rprsnaion of h xampl can b found on Figur. Figur : Elasic-plasic canilvr bam Th rsuls achivd wih h prsn lmn ar compard wih hos givn by h 8 nod brick lmn wih incompaibl mod (calld C3D8I) in ABAQUS. Th msh usd in h calculaions wih h prsn lmn and C3D8I has 48 8 (lngh high widh) Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

13 Mcánica Compuacional Vol XXV, pp (26) 873 lmns. Th iniial gomry configuraion and h final dformd msh can b sn in Figur 2. Figur 2: Iniial gomry and dformd msh Th dflcion on h mid-surfac is shown in Figur 3. I may b obsrvd ha h rsul is vry clos o hos givn by h C3D8I lmn of ABAQUS. 2 Prsn soluion: msh 48 8 C3D8I Vrical displacmn (m) X (m) Figur 3: Dflcion on h mid-surfac of lasic-plasic bam 6.2 Squar pla undr concnrad load A concnrad load of 7 kn is applid a h cnr of a squar pla wih lngh L = 4 m and hicknss =.4 m, as shown on Figur 4. Th marial is an lasic-plasic modl wih isoropic hardning and Young s modulus E = 3. 7 kn/m 2, angnial modulus afr yilding, E T = 3. 6 kn/m 2, Poisson s raio ν =.3 and iniial yild srss = 4. 4 kn/m 2. Th pla has simply suppord dgs and du o symmry, only a quarr of h pla Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

14 874 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH is modld wih hr-dimnsional lmns (6 lmns wr usd in h hicknss dircion). L y x L = 4 P Figur 4: Squar pla undr concnrad load Th prsn soluion is compard wih rsuls obaind by h lmn C3D8I, in ABAQUS. Th plos of cnral dflcion vs concnrad load is givn in Figur 5. Analyzing h rsuls i is possibl o s ha h prsn soluion agrs vry wll wih h soluion achivd wih ABAQUS Prsn soluion: msh C3D8I Vrical load (kn) Cnral dflcion (m) 6.3 Mohr-Coulomb slop sabiliy analysis Figur 5: Vrical load vs cnral dflcion A slop sabiliy analysis wih h Mohr-Coulomb yild cririon is prformd in his xampl. Th slop gomry and boundary condiions ar prsnd in Figur 6. Figur 6: Slop gomry and boundary condiions Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

15 Mcánica Compuacional Vol XXV, pp (26) 875 Th uni wigh of h marial is γ = 2. kn/m 3, h cohsion c =. kn/m 2, h fricion angl φ = 4º and h dilaaional angl ψ is qual o zro. Th marial has h Young s modulus E =.5 3 kn/m 2 and h Poisson s raio ν =.3. Th ou-of-plan displacmn dgr of frdom is prscribd in all nods allowing only displacmns in h plan. Th slop is discrizd wih a msh conaining lmns ( lmn was usd in h lngh dircion). In ordr o find h facor of safy of h slop, h analysis is prformd for svral rial facors of safy of h soil paramrs ranging form. o 2.65 and h maximum displacmn vs facor of safy of ach analysis is plod in Figur 7. Th plod rsuls indica ha h facor of safy of h slop lis around 2.6. I can b sn ha h rsuls obaind wih h us of h currn formulaion ar clos o hos achivd by Smih and Griffihs (997). Bishop and Morgnsrn (96) producd chars for slop sabiliy analysis using slip circl chniqus, and hs giv a facor of safy of 2.55 for h slop considrd on his xampl. Displacmn máx -5 (m) Facor of safy Bishop and Morgnsrn (96) Smih and Griffihs (997) Prsn soluion: msh Figur 7: Maximum displacmn vs facor of safy Th normalizd displacmns ar plod in Figur 8. Th dformd msh and h naur of h failur mchanism of h slop ar also indicad on Figur 8. Figur 8: Normalizd displacmns Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

16 876 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH 6.4 Cylindrical shll wih fr dgs A cylindrical shll is submid o a pair of concnrad forcs, inducing larg displacmns and roaions. Th gomry of h cylindr is characrizd by a lngh L =.35 m, radius R = m and a consan hicknss =.94 m. Th marial has isoropic hardning and is propris ar: Young s modulus E =.5 3 kn/m 2, Poisson s raio ν =.325, iniial yild srss =.5 2 kn/m 2, and plasic angn modulus E T =.5 2 kn/m 2. No boundary condiions ar applid o h fr dgs of h shll, bing h load pair rsponsibl for h quilibrium of h cylindr. Du o symmry rasons, only on ighh of h cylindr is discrizd wih a msh conaining lmns (circumfrnc lngh hicknss). A schmaic rprsnaion of h xampl is prsnd in Figur 9. B A P z y x Figur 9: Srching of a cylindr In ordr o obain h pos-buckling rspons, h Gnralizd Displacmn Conrol Mhod, proposd by Yang and Shih (99) was implmnd. Th soluion obaind for h displacmns wih h prsn lmn, in boh poins A and B, vs h applid load ar compard wih h rsuls givn by Masud and Tham (2) and Valn al. (24). Th final dformd configuraion is plod in Figur. P Figur : Dformd msh du o h 4 kn load In Figur rsuls obaind wih h us of h currn formulaion compard wih hos publishd in h liraur. Th prsn soluion agr vry wll wih h rsuls obaind by Valn al. (24) and is vry clos o h soluion publishd by Masud and Tham (2). Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

17 Mcánica Compuacional Vol XXV, pp (26) Applid load (kn) Prsn soluion: p A Masud and Tham (2): p A Valn al. (24): p A Prsn soluion: p B Masud and Tham (2): p B Valn al. (24): p B Displacmn (m) Figur : Applid load vs displacmn 7 CONCLUSIONS An isoparamric hxahdrical fini lmn wih igh nods and on-poin quadraur was formulad and i was applid o analyz physical and gomrically nonlinar problms involving plas and shlls. Th Mohr-Coulomb as wll as h Von Miss yild cririon wih isoropic hardning, in h conx of srucurs wih fini displacmns and roaions, was usd. A coroaional formulaion was mployd o dal wih h gomrical nonlinar analysis whil an xplici algorihm was implmnd for h laso-plasic analysis. Good rsuls wr obaind whn compard o hos prsnd by ohr auhors and hos obaind using a commrcial sofwar, and no volumric and/or shar locking wr dcd in any cas. Fuur works will xnd applicaions of his lmn o gochnical problms, using mor complx consiuiv quaions, and kinmaic, as wll as mixd hardning, will b implmnd. REFERENCES A. Masud and C. L. Tham. Thr-dimnsional coroaional framwork for laso-plasic analysis of mulilayrd composi shlls. AIAA Journal, 38: , 2. A. W. Bishop and N. Morgnsrn. Sabiliy cofficins for arh slops. Géochniqu, :29-5, 96. ABAQUS v ABAQUS Analysis Usr's Manual. ABAQUS, Inc, 24 D. P. Flanagan and T. Blyschko. A uniform srain hxahdron and quadrilaral wih orhogonal hourglass conrol. Inrnaional Journal for Numrical Mhods in Enginring, 7:679-76, 983. D. R. J. Own and E. Hinon. Fini Elmns in Plasiciy: Thory and Pracic. Pinridg Prss Limid, 98. I. M. Smih and D. V. Griffihs. Programming h fini lmn mhod. John Wily & Sons, 997 Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://

18 878 D. SCHMIDT, A.M. AWRUCH, I.B. MORSH L. A. Duar Filho and A. M. Awruch. Gomrically nonlinar saic and dynamic analysis of shlls and plas using h igh-nod hxahdral lmn wih on-poin quadraur. Fini Elmns in Analysis and Dsign, 4:297-35, 24. M.A. Crisfild, 99. Non-linar Fini Elmn Analysis of Solids and Srucurs, volum. John Wily & Sons, 99. R. A. F. Valn, R. J. A. Souza and R. M. N. Jorg. An nhancd srain 3D lmn for larg dformaion lasoplasic hin-shll applicaions. Compuaional Mchanics, 34:38-52, 2. S. Rs. On a physically sabilizd on poin fini lmn formulaion for hrdimnsional fini laso-plasiciy. Compur Mhods in Applid Mchanics and Enginring, 94: , 25. S. W. Sloan, A. J. Abbo, and D. Shng. Rfind xplici ingraion of lasoplasic modls wih auomaic rror conrol. Enginring Compuaions, 8:2-54, 2. S. W. Sloan. Subspping schms for h numrical ingraion of lasoplasic srss-srain rlaions. Inrnaional Journal for Numrical Mhods in Enginring, 24:893-9, 987. Y.-B. Yang and M. S. Shih. Soluion mhod for nonlinar problms wih mulipl criical poins. AIAA Journal, 28:2-26, 99. Y.-K. Hu and L. I. Nagy. A on-poin quadraur igh-nod brick lmn wih hourglass conrol. Compurs & Srucurs, 65:893-92, 997. W. K. Liu, Y. Guo, S. Tang and T. Blyschko. A mulipl quadraur igh-nod hxahdral fini lmn for larg dformaion lasoplasic analysis. Compur Mhods in Applid Mchanics and Enginring, 54:69-32, 998. W. K. Liu, Y. -K. Hu, and T. Blyschko. Mulipl quadraur undringrad fini lmns. Inrnaional Journal for Numrical Mhods in Enginring, 37: , 994. Copyrigh 26 Asociación Argnina d Mcánica Compuacional hp://