ELASTOPLASTIC ANALYSIS OF PLATE WITH BOUNDARY ELEMENT METHOD

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1 Intrntionl Journl of chnicl nginring nd Tchnology (IJT) olum 9, Issu 6, Jun 18, , Articl ID: IJT_9_6_97 Avilbl onlin t htt://.im.com/ijmt/issus.s?jty=ijt&ty=9&ity=6 IN Print: nd IN Onlin: IA Publiction cous Indxd LATOPLATIC ANALYI OF PLAT WITH BOUNDARY LNT THOD Drtmnt of chnicl nginring, Fkults Tknik, Univrsits uhmmdiyh urkrt. ABTRACT This ork is th dvlomnt of boundry lmnt mthod for lstolstic lt nlysis to includ lstic-linr hrdning mtril. Th lt is subjctd to bnding, in-ln nd combind bnding nd in-ln. Th lstic zon is vlutd by using von iss critrion. Th cll discrtiztion is imlmntd to solv numriclly th domin intgrl cusd by lsticity. To vlut th nonlinr trm in th formultion of th boundry lmnt, totl incrmntl tchniqu is imlmntd. Th cbility of th dvlomnt in this ork ill b rsntd by hving numricl xmls. Kyords: Boundry lmnt mthods, lsticity, shr dformbl lt, totl incrmnt mthod. Cit this Articl:, lstolstic Anlysis of Plt ith Boundry lmnt thod, Intrntionl Journl of chnicl nginring nd Tchnology, 9(6), 18, htt://.im.com/ijt/issus.s?jty=ijt&ty=9&ity=6 1. INTRODUCTION Th y to nlyz lt ill dnd on th lods orking on it. Th lods cn b idntifid s bnding, in-ln or combind bnding nd in-ln. Whn th lod is in-ln, th lt is trtd s D roblm nd it cn b nlyzd by th ln strss thory[1]. In th cs of bnding, th lt is solvd using lt bnding thory. Thr r to kinds of lt bnding thory i.. th clssicl[] nd shr dformbl thory[]. For th combind on, th surosition of lt bnding thory nd D ln strss thory r considrd. As fr s Author knos, th lstolstic nlysis of lt by B cn b sn in [4], [5], [6], [7], [8], [9], [1] nd [11]. All th orks only considrd n lstic-rfctly lstic mtril. In this r, th ork by uriyono[9] is xtndd to includ lstic-linr hrdning mtril. Th rltionshi btn strsss nd strins is shon in Fig. 1 hn th mtril ith linr hrdning bhvior is lodd unixilly. Th totl strin is considrd s sum of th lstic nd lstic comonnts s follos: ε + = ε ε 1 htt://.im.com/ijt/indx.s 864 ditor@im.com

2 lstolstic Anlysis of Plt ith Boundry lmnt thod σ hr ε = is th lstic rt of th totl strin nd ε rrsnts th lstic rt. Figur 1 Linr hrdning mtril bhvior For initil loding, lstic bhvior is obtind for σ Y < Onc th strss hs xcdd th initil yild strss Y, on cn hv σ σ < hr σ is subsqunt yild strss nd its vlu dnds on th lstic flo rul. Th strss-strin rltionshi for monotonic loding in tnsion hs th form hich my b rittn s σ = Y + ε, if σ > Y 4 σ = Y + Tε 5 By considring qution (1), It llos qution (5) to b xrssd s σ = Y + ( ε + ε ) 6 ubstituting qution (4) into qution (6) yilds T σ Y σ = Y + Tε + T 7 vntully, th folloing qution is obtind: σ T = Y + ε = Y + H ' ε 8 T 1 hr H is th lstic modulus dfind s th slo of th hrdning curv, no xrssd in trms of strss vrsus lstic strin. For th cs of linr hrdning, H is constnt, but in gnrl, it chngs continuously long th hrdning curv, i.. htt://.im.com/ijt/indx.s 865 ditor@im.com

3 dσ H ' = 9 dε In this r, shr dformbl lt thory is considrd instd of clssicl lt thory. Th totl incrmntl tchniqu is usd to vlut th nonlinr systm of qution. Th cll discrtiztion mthod using 9-nods qudriltrl cll is lid to nlyz th domin intgrls hich mrg in th formultion. In this r, th nottion of Crtsin tnsor is mloyd, ith Grk indics chnging from 1 to nd th Ltin indics chnging from 1 to.. B FORULATION.1. Govrning qution Gnrl formul for lstolstic lt nlysis is dtrmind by considring lstic strins tht r du to bnding nd mmbrn lodings, thrfor th totl strin rts cn b xrssd s χ ε = χ + χ ; 1 = ε + ε ; 11 And γ γ = 1 Whr, χ r th rts of th totl bnding strins, ε r th rts of th totl in-ln strins, nd γ th rts of th shr strin rts. Th rts of th totl bnding nd in-ln strins comris linr nd nonlinr comonnts. Th nonlinr comonnts of qutions (1) nd (11) r du to lsticity () nd thy cn b rittn s nd χ = χ 1 ε = ε 14 On th othr hnd, th rts of th rsultnts of th strss cn lso b rittn s = ; 15 for momnt rsultnts = Q ; 16 Q for shr rsultnts nd N = N N 17 for mmbrn strss rsultnts. Th rltionshis btn th strsss nd dislcmnts cn b sttd s htt://.im.com/ijt/indx.s 866 ditor@im.com

4 lstolstic Anlysis of Plt ith Boundry lmnt thod υ = D (, β+ β, γ, γ δ) ; 18 nd = (, Q C + ); 19 N υ = B( u, β + u β, uγ, γ δ ) N ; hr, D = h, B = h nd xrssd s: C = kh. Th quilibrium qution cn b (1 + υ) Q = ; 1 nd Q, q = ; N, β =.. Dislcmnt nd trss Intgrl qution As rsntd in [9] th dislcmnt intgrl qution for lstolstic nlysis cn b xrssd in qution (4) for rottion nd dflction nd qution (5) for in-ln dislcmnt. Th qutions r th xrssion of dislcmnt rts of ny oints X in th domin to th vlus of th rts of dislcmnt nd th rts of th trction on th boundry. nd = ij j &( X') W ( X', & ( d P ( X', ( d + W X ', X ) q ( X ) d + i i ( X ', X ) ( X ) ij j i ( & χ & d 4 = u ( X') U ( X', t ( d T ( X', u j ( d+ θ ( ', X ) N ε X ( X ) d 5 Whr, W ij (X,, P ij (X,, χ ij (X,X), U ij (X,, T ij (X,, nd ε ij (X,X) r nmd fundmntl solutions. Thy cn b found nd xlind in [7]. qution (4) nd (5) r th solution for ny oint in th domin, to hv solutions on th boundry oints, th limiting rocss s X x Є is conductd to hv: = ij j C & ( x') W ( x', & ( d P ( x', ( d+ W x', X ) q ( X ) d + nd ij i i ( x ', X ) ( X ) ij j i ( & χ & d 6 = C u ( x') U ( x', t ( d T ( x', u j( d+ θ ( ', X ) N ε x ( X ) d 7 htt://.im.com/ijt/indx.s 867 ditor@im.com

5 hr, C ij (x ) r clld fr trm. In th cs of smooth boundry, th vlu of th fr trm is.5. qution (4), (5), (6), nd (7) r dislcmnt intgrl qutions. On th othr hnd, th intgrl strss qutions r & ( X ') = W ( X ', & ( d P ( X ', & ( d + + k k k k k l χ ( X ', X ) & γθ γθ ( X ) d 1 l [ (1 + υ ) (1 υ ) & θθ δ ] for momnt rsultnts, Q X ') = W ( X ', ( d P ( X ', ( d + β ( k k k + l βγθ ( X ', X ) γθ ( X ) 8 W ( X ', X ) q& ( X ) d 8 k W ( X ) d βk ( X ', ) q χ & d 9 for shr rsultnts, nd N ( X') = U ( X', t ( d T ( X', u ( d + γ γ γ γ 1 8 [ (1 + υ ) N (1 υ )N δ ] θθ ε γθ ( ', X ) N γθ X ( X ) d for mmbrn rsultnts... Discrtiztion nd ystm of qution Numricl mthod is usd to solv qutions of (6), (7), (8), (9) nd (). Discrtiztion is conductd on th boundry nd rt of th domin hich is xctd to yild. Qudrtic is ormtric lmnts r lid to discrtiz th boundry nd 9 nods qudriltrl clls r imlmntd in th domin. Aftr discrtiztion, th mtrix form of th qutions (6) nd (7) cn b obtind s H G = u H u + b T + u G t u T N 1 hr [H] nd [G] r clld th influnc mtrics of boundry lmnt, [T] is th influnc mtrix of lsticity. Th surscrit nd u indict th mod of lt nd D rsctivly. { }, { u }, { } nd { t } r th rt vctors of th dislcmnt nd th trction on th boundry. { b } is th rt vctors of th lod on th domin nd { } nd { N } r th lsticity trms. Onc boundry conditions r imosd, qutions (1) cn b rsntd s T & = + u T N [ A]{ x} { f } hr, [A] is th cofficint mtrix, { x } is th column mtrix of th unknon nd {f } is th column mtrix of rscribd boundry vlus. In th sm y, th strss intgrl qutions of qution (8), (9) nd () cn b xrssd in mtrix form s htt://.im.com/ijt/indx.s 868 ditor@im.com

6 lstolstic Anlysis of Plt ith Boundry lmnt thod htt://.im.com/ijt/indx.s = u u u N T T T b b u H H H t G G G N Q Th totl incrmntl tchniqu is usd to solv th nonlinr systm of qutions of () nd (). Th dtil lgorithm of th tchniqu cn b sn in [9] s this ork is xtndd from tht ork. Hovr hn vluting th lstic zon, lstic-rfctly lstic or lsticlinr hrdning mtril r considrd.. NURICAL XAPL To sho th cbility of this ork to tk into ccount lstic-linr hrdning mtril, xmls r rsntd. Th first xml is squr lt of simly suortd lt (s Fig. ). Ths xmls hv bn rortd in [5], [9] nd [11] for lstic-rfctly lstic mtril. Figur imly suortd squr lt In this r th sm lt is rsntd ith linr hrdning mtrils of H =.7 nd H =.15. Th rortis of th lt r = 1.9 GP, ν =., σ Y = 16 P nd h/ =.1. Th nondimnsionl rmtrs r introducd s follos. Q q = nd 1 D W = hr = σ y h /4, is th cntr lt dflction nd D=h /1(1-ν ). Th rsult of th hrdning mtrils for squr lt cn b sn in Fig.. As it cn b sn tht th biggr vlu of H mk th mtril hrdns during th lstic flo, it mns t th sm lod ftr lstic, mtril ith biggr vlu of H hs lss dflction. Figur Dflction t cntr lt for vrious H. h q q imly uortd

7 Th scond xml is rctngulr lt ith cntr hol subjctd to unixil tnsion of σ. Th gomtry nd loding condition is shon in Fig. 4(). This is tyicl ln strss roblm nd hs lso bn studid using th B by Alibdi [1]. Th mtril is ssumd to hrdn linrly nd hs rortis of = 7GP, ν =., σ Y = 4 P nd H =.. Du to th symmtry conditions nd to sss th snsitivity solution to th discrtiztion, only qurtr of th lt is discrtizd s follos: odl A, 6 lmnts on th boundry ith 1 domin clls (s Fig. 4(b)) odl B, 6 lmnts on th boundry ith domin clls (s Fig. 4(c)) odl C, 8 lmnts on th boundry ith 41 domin clls (s Fig. 4(d)) Fig. 5 shos th comrison of th dislcmnt in x-dirction of oint A btn th currnt nlysis of th thr modls nd th on obtind in [1]. Th currnt nlysis of th thr modls r crrid out in 6 lod sts to chiv th finl lod hich is found to giv convrgd solution. In this figur th nondimnsionl rmtrs of U x nd Q r dfind s U x = u x /r 1 nd Q=σ/σ yild. It cn b sn tht odl B nd C giv convrgd rsults. trss comonnt σ xx t th nt sction of th lt is lottd in Fig. 6. Th rmtr of Dist is dfind s Dist = y/r. Th rsults r comrd ith th on obtind in [1]. Onc gin, convrgd solutions r givn by odl B nd C. At lst, th dvlomnt of th lstic zon is rsntd in Fig. 7. Figur 4 Gomtry nd discrtiztion of th rctngulr lt. Figur 5 Dislcmnt t osition of oint A htt://.im.com/ijt/indx.s 87 ditor@im.com

8 lstolstic Anlysis of Plt ith Boundry lmnt thod Figur 6 trss vrition on th nt sction of th lt. Figur 7 Dvlomnt of th lstic zon for vrious lod lvl. Thirdly, th sm lt s th scond on is lodd by combind tnsion nd bnding, hr th tnsion is σ = 5 P nd th bnding is q =.184 P. Th rsult of von iss strss is rsntd in Fig.8 in comrison ith F rsult. Figur 8 on iss trss contour t finl lod 4. CONCLUION From th xmls nd rsults rsntd in this ork, it cn b concludd tht th dvlomnt in this ork hs cbility to simult both lstic-rfctly lstic nd lsticlinr hrdning mtril. ACKNOWLDGNT Th uthors ould lik to thnk to th Dirctort Gnrl of Rsrch, Tchnology nd Highr duction, inistry of Rsrch, Tchnology nd Highr duction, Rublic of Indonsi for th finncil suort. This rsrch is finncilly suortd by Pnlitin Ungguln Prgurun Tinggi ith th contrct of 11.71/A.-III/LPP//17. htt://.im.com/ijt/indx.s 871 ditor@im.com

9 RFRNC [1]. Timoshnko nd J. N. Goodir, Thory of lsticity, Third. ingor: cgr-hill Intrntionl ditions, 197. [] G. Kirchhoff, Ubr ds glichgicht und di bgung inr lstischn schib, J. Rin Ang th, no. 4, , 185. []. Rissnr, On vritionl thorm in lsticity, J. th. Phys., no. 9,. 9 95, 195. [4]. J. Krm nd J. C. F. Tlls, On boundry lmnts for Rissnr s lt thory, ng. Anl., vol. 5, no. 1,. 1 7, [5]. J. Krm nd J. C. F. Tils, Th B lid to lt bnding lstolstic nlysis using Rissnr s thory, ng. Anl., vol. 9, , 199. [6]. J. Krm nd J. C. F. Tlls, Nonlinr mtril nlysis of Rissnr s lts, in Plt Bnding Anlysis ith Boundry lmnt, 1998, [7] uriyono nd. H. Alibdi, Boundry lmnt thod for hr Dformbl Plt ith Combind Gomtric nd tril Nonlinritis, ng. Anl. Bound. lm., vol.,. 1 4, 6. [8] uriyono nd. H. Alibdi, Anlysis of shr dformbl lts ith combind gomtric nd mtril nonlinritis by boundry lmnt mthod, Int. J. olids truct., vol. 44, , 7. [9] uriyono, Boundry lmnt thod for hr Dformbl Plt ith tril Nonlinrity, ARPN J. ng. Al. ci., vol. 11, no. 8, , 16. [1]. R. C. Olivir nd. J. Krm, lstolstic nlysis of Rissnr s lts by th boundry lmnt mthod, ng. Anl. Bound. lm., vol. 64, , 16. [11] uriyono,. ffndy, nd Wijinto, tril nonlinrity lt bnding nlysis ith boundry lmnt mthod, in AIP Confrnc Procdings, 17, vol [1]. H. Alibdi, Th Boundry lmnt thod, liction to solids nd structurs. Chichstr: Wily, 1. htt://.im.com/ijt/indx.s 87 ditor@im.com