Acta Scientiarum. Technology ISSN: Universidade Estadual de Maringá Brasil

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1 Aca cieniarum. Technlgy IN: Universidade Esadual de Maringá Brasil hang, Hsu Yang A mehdlgy fr analysis f defecive pipeline by inrducing sress cncenrain facr in beam-pipe finie elemen frmulain Aca cieniarum. Technlgy, vl. 38, núm. 3, juli-sepiembre, 26, pp Universidade Esadual de Maringá Maringá, Brasil Available in: hp:// Hw cie Cmplee issue Mre infrmain abu his aricle Jurnal's hmepage in redalyc.rg cienific Infrmain ysem Newrk f cienific Jurnals frm Lain America, he Caribbean, pain and Prugal Nn-prfi academic prjec, develped under he pen access iniiaive

2 Aca cieniarum hp:// IN prined: IN n-line: Di:.425/acasciechnl.v38i A mehdlgy fr analysis f defecive pipeline by inrducing sress cncenrain facr in beam-pipe finie elemen frmulain Hsu Yang hang Pnifícia Universidade Caólica d Paraná, Rua Imaculada Cnceiçã, 55, Prad Velh, 825-9, Curiiba, Paraná, Brazil. hsu.shang@pucpr.br ABTRACT. This wrk prpses a mehdlgy fr defecive pipe elasplasic analysis using he Euler Bernulli beam-pipe elemen frmulain. The virual wrk equain is mdified incrprae he sress cncenrain facr in beam-pipe elemen frmulain. The sress cncenrain facr is evaluaed a priri by a 2D r 3D finie elemen mdel accrding he defec prfile. In his wrk, a semicircular defec and a recangular defec are cnsidered. The sress cncenrain facr is insered in he beampipe elemen elasplasic frmulain, and several applicains are presened shw he applicabiliy f he prpsed mehd. Keywrds: Euler-Bernulli beam elemen, sress cncenrain facr, virual wrk equain, elasplasic analysis. Uma medlgia de análise de du cm defei aravés da inrduçã de far de cncenraçã de ensões na frmulaçã de elemens finis de viga-du REUMO. Ese rabalh prpôs uma medlgia de análise elasplásica em du cm defei, uilizand frmulaçã d elemen viga-du Euler-Bernulli. A equaçã d rabalh virual é mdificada para incrprar na frmulaçã d elemen viga-du far de cncenraçã de ensões, que é deerminada previamene pel mdel de elemens finis 2D u 3D cnfrme perfil d defei. Nese rabalh, defei semicircular e defei reangular sã cnsiderads. O far de cncenraçã de ensões é inserid na frmulaçã elasplásica d elemen viga-du e diversas aplicações sã apresenadas nese rabalh para msrar a aplicabilidade d méd prps. Palavras-chaves: Elemen da viga Euler-Bernulli, far de cncenraçã de ensões, equaçã d rabalh virual, análise elasplásica. Inrducin eel pipelines are widely used fr cnveying naural gas and crude il, and heir derivaives. eel is used due is high capaciy f ransprain and lw manufacuring cs cmpared her maerials. The pipeline is ypically cnsruced frm carbn seel, because f is high mechanical srengh, and als because i is cheaper han her maerials. Neverheless, depending n he derivaive cnveyed r he cndiin f he sil surrunding he pipeline, carbn seel can be subjeced crrsin ha generaes surface defecs and reduces pipe wall hickness. Cnsequenly, he mechanical srengh f he pipe is cmprmised. Addiinally, her defecs in he pipe culd be creaed during insallain. In such a siuain, he cnveying pressure shuld be reduced in rder avid rupure f he pipeline in he regin f he defecs; hwever, his culd affec he cnveying capaciy. Anher sluin culd be he inerrupin f perain repair he defecive par. Bh cases invlve financial implicains. Thus, i is necessary develp a mehdlgy assess he effec f limiing he peraing pressure in defecive pipes wih mre accuracy, wihu being ime cnsuming. An efficien way fr such an assessmen is he use f semi-empirical mehds such as B3G (American ciey f Mechanical Engineers [AME], 99), r numerical mehds such as finie elemen mehd. The defec creaes irregulariies n he surface and inrduces, a he viciniy f defec, he effec f sress cncenrain, which increases he sress surrunding he defec regin a higher sress level. A 3D finie elemen mdel can be used analyze he sress-cncenrain effec wih accuracy when a prper mesh refinemen is emplyed (Kim, him, Huh, & Kim, 22; Chi, G, Kim, Kim, & Kim, 23; Kim & n, 24). Hwever, such a mdel nly cnsiders lcal effecs, and cann evaluae he effec f lcal sress cncenrains in he glbal behavir f he pipeline. The sluin f he 3D finie elemen mdels f he whle pipeline sysem Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26

3 34 hang needs lng prcessing ime and high cmpuainal effr. Therefre, he presen wrk prpses a mehdlgy fr he analysis f defecive pipelines. This mehdlgy includes a finie elemen glbal saic elasplasic analysis in a pipeline wih he Vn Mises mdel fr isrpic maerial hardening. The equilibrium equain fr elasplasic analysis is slved by he al Lagrangian frmulain. imulaneusly, i incrpraes a sress cncenrain facr in a hree-nde beam-pipe elemen frmulain. This mehdlgy unifies he glbal analysis f he pipeline, simulaed by he Euler Bernulli beam-pipe mdel by means f beam elemens, wih he lcal analysis f he defecive pipe. This is achieved by inrducing a sress cncenrain facr, evaluaed a priri by 2D r 3D finie elemen mdels. A cmpuainal cde named APC3D was develped in Frran by he auhr f his wrk implemen he prpsed mehdlgy. A brief lieraure review is presened cncerning a sress cncenrain sudy in defecive pipes wih a finie elemen mehd develped fr pipeline elasplasic analysis. everal sudies have been cnduced fr finie elemen mdeling f defecive pipes. The defec culd have differen gemeries, such as hse frmed by crrsin pis, lcaed in he lngiudinal direcin. The effecs f he ineracin f defecs n he pipe-wall srengh were invesigaed (Chuchaui & Pick, 996; Kim e al., 22). me sudies have sudied a numerical mehd based n an equilibrium equain (Crnin & Pick, 22) deermine he burs pressure. The mehd called weighed deph difference deals wih irregular prfiles n he crrsin surface. I furher invesigaes he effecs f pi ineracin frm adjacen crrsin. The effec f sress cncenrain in he defec regin als plays an impran rle in he evaluain f he residual srengh f he defecive pipe. everal finie elemen mdels were develped calculae he sress cncenrain facr wih differen defec gemeries (Kim & n, 24). me sudies fcused n pipe and pressure vessel glbal-lcal analysis (aa, 25). Crack-ip pening displacemen has been adped as a failure crierin. The resuls f he analysis were cmpared wih he repred resuls. A beam elemen fr elasplasic analysis in a pipe wihu defecs has als been develped (Nwzarash & Mhareb, 24). This elemen was based n beam elemen frmulain wih w ndes, each wih si degrees f freedm. This elemen is capable f calculaing lads such as aial frce, shear lad, rsin, bending, and inernal and eernal pressures. The elasicperfec-plasic mdel was included in he finie elemen beam frmulain. A mehdlgy was presened fr a lcal failure analysis f a defecive pipe, wih he defec represened by a semi-ellipic gemery (Adib, Jalluf, chmi, Carmasl, & Pluvinage, 27). The mehdlgy includes he sress cncenrain effec a he ip f he defec, and a furher prbabilisic analysis was presened. Anher analysis has presened a lcal fracure crierin f a pipe wih nch-ype defecs based n 3D finie elemen frmulain (Oh, Kim, Baek, Kim, & Kim, 27). Recenly, a number f enriched finie elemen frmulains such as he eended finie elemen mehd have been applied analyze crack prpagain in pipes (Zhang, Ye, Liang, Zhang, & Zhi, 24). A review f he lieraure shws ha he majriy f sudies cncenraed n lcal failure analysis f defecive pipes, while a few sudies fcused n develping a mehdlgy fr he glbal-lcal analysis f defecive pipes. aemen f prblem In his secin, a brief presenain will be made n he equilibrium equain f a pipe and n he finie elemen mdel emplyed simulae he glbal behavir f a pipeline. The frmulain presened in his secin will be mdified incrprae sress cncenrain facrs as a mehdlgy f glbal-lcal analysis. The elemen used simulae pipeline in his wrk is a hree-nde isparameric elemen, wih a Hermiian shape funcin (Dha, Tuz, & Canin, 985; Bahe, 996). Each nde has si degrees f freedm. The siffness mari is evaluaed by numerical inegrain. There are seven Gauss inegrain pins alng he elemen. The elemen mdel can be seen in Figure. Figure. paial hree-nde beam elemen. Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26

4 A mehdlgy fr analysis f defecive pipeline 35 Accrding Bahe (996), he equain f virual displacemen in he Tal Lagrangian Frmulain is given by Equain : () The erm n he lef represens inernal virual wrk, and he erm n he righ represens eernal virual wrk. The eernal virual wrk is given by Equain 2: (2) B f : Cmpnens f frce applied eernally per i uni vlume a ime ; f : Cmpnens f racin applied n he i eernal surface per uni surface a ime ; O f : urface a ime, where he eernal surface was subjeced eernal racin; O δ u i δ u i calculaed n surface f. In Equain, he sress and srain incremens are given, respecively, by Equains 3 and 4: ε ε ε (3) (4) The virual wrk can be calculaed cnsidering he cmpnens f sress and srain f a pipe-beam elemen, accrding Equain 5 given by: (5) In Equain 5, he firs w erms represen he virual wrk f sress n he pipe, aial and angenial, respecively. The erm n he righ hand side represens he eernal virual wrk generaed by he lad applicain. The equains fr sress and srain incremens fr a pipe in he lngiudinal and angenial direcins are given by Equains 6 a 9: L NL δ ε δ ε δ ε δ Δ ε δ ε Cδ ε (6) (7) (8) (9) The srain in he lngiudinal direcin is calculaed by he sum f he linear and nn-linear srain. In he Vn Mises mdel f isrpic maerial hardening, he vlumeric plasic srain is zer. Therefre, i is apprpriae epress he general relain f sress-srain a ime as Equain (Bahe, 996): E ν Δ " Equain : p ( e Δe ) () Δ p e " Δ e ' e () In he indicial frm, Equain is Equain 2: mδ (2) is he mean sress r vlumeric sress, m epressed as Equain 3: ii m (3) 3 Frm he Vn Mises mdel, he yield cndiin a ime is epressed as Equain 4: ( ) vm f 2 esc (4) 2 3 esc is he yield sress a ime. The sluin f Equain 4 is deermined by using he Newn-Raphsn algrihm (Bahe, 996), cnsidering ha he ieraive prcedure is carried u a each numerical inegrain pin. The energy nrm is adped as he crierin f cnvergence. A mehdlgy fr sress cmpensain in virual wrk equain In rder incrprae a sress cncenrain facr (CF) in beam-pipe elemen frmulain, his wrk prpses a mehdlgy inrduce an CF in he virual wrk equain. Cnsider ha he CF is bained frm a pipe in a 2D r 3D finie elemen mdel, subjec ensin r inernal pressure lading. Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26

5 36 hang T inrduce he effecs f an CF in he virual wrk equain, cnsider ha he muliplicain f his facr wih incremenal sress is shwn by Equains 5 and 6. Observe ha when here is n effec f defecs in he pipe, he muliplier will assume uniy as is value. If his ccurs, Equains 5 and 6 reurn Equain 6 and 7, which are he incremenal sress equains presened in he previus secin, accrding Equain 5 and 6. k ( k ) k k ( ) (5) (6) Term k represens he sress cncenrain facr when nly lngiudinal sress is generaed, and k when nly angenial sress is generaed. Inrducing Equains 5 and 6 in Equain 5 resuls in he fllwing Equain 7: ( ( k ) ) V ( k ) V ε d V ( ) δ ε d V δ We δ (7) Neverheless, when he CF is inrduced in Equain 5, he glbal cndiin f equilibrium will be affeced and he ieraive prcedure f Newn-Raphsn algrihm will n cnverge. Cnsider he cndiin shwn in Figure 2. The equilibrium sae is achieved when he sum f he inernal frces is zer. In his wrk, he energy cnvergence crierin is adped, and he inernal energy f a slid mus be he same as he eernal energy prduced by he eernal frce, accrding Equain 8 and 9. Figure 2. ae f equilibrium f a beam wihu defec. * F ( kδ ) da ( k ) Δ da A A ( ( ) ) k δ ε d V * ( ( k ) ) δ ε d V δ W in (8) V V e (9) When inrducing an CF he virual wrk equain by muliplying he sress incremens, he cndiin f equilibrium f Equain 5 will n be saisfied. The reasn fr his is ha he CF inrduces an verbalance in he equain equilibrium evaluain. This verbalance is evaluaed by muliplying he sress incremen by he facr (k-), as shwn in Figure 3. In rder aain he equilibrium cndiin in he virual wrk equain, and simulaneusly, aain he cnvergence crierin, he verbalance shuld be eliminaed. Thus, he inernal frce is calculaed by Equain 8, and he equain f virual wrk (Equain 5) cann be simply evaluaed by inrducing Equains 5 and 6. Hwever, i is necessary sum he eernal energy par, he verbalance energy prduced by CF, as shwn by Equain 9. I is simply summing an verbalance value n he lef hand side f Equain 5, and simulaneusly, summing he same verbalance value n he righ hand side. In her wrds, mahemaically, here are n addiins Equain 5. Thus, he crierin f equilibrium cninues be aained. Figure 3. ae f equilibrium n saisfied in he defecive beam. Applicains In his secin, a number f applicains f he sress cmpensain mehd are presened. A cmpuainal cde, named APC3D, is develped in Frran carry u he numerical applicains. In rder validae he prpsed mehdlgy, he firs case presens a sudy n pipe wih a defec in a semicircular prfile, subjeced inernal pressure and ensin in he free end. The secnd case shws an analysis in a pipe wih defec in a semicircular prfile subjeced bending. The las case is a pipe wih a recangular defec subjeced inernal pressure. The resuls bained by APC3D f hese hree cases are cmpared by cmmercial sfware. The maerial adped fr all cases in his sudy is cld-wrked seel. As he maerial in his analysis is cnsidered ehibi bilinear isrpic hardening behavir, he Newn-Raphsn algrihm is Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26

6 A mehdlgy fr analysis f defecive pipeline 37 adped. The yield surface epansin is cnrlled by he Vn Mises equain. The mechanical prperies f he maerial are E 25, E T 75GPa and y 42MPa. Pipe wih semicircular defec subjeced inernal pressure and ensin Cnsider he pipe mdel shwn in Figure 4. A defec wih a semicircular prfile is lcaed in he middle f pipe and has a 2.5 mm as radius; φ e 2; φ i 9; L mm; F 5 kn. The analysis is divided in w cases. In he firs case, nly ensin is applied a he end f mdel. In he secnd case, nly inernal pressure is applied. Figure 5. Mesh wih aisymmeric elemen using cmmercial sfware. Figure 4. chemaic represenain f defecive pipe, and he mesh adped by APC3D. Pipe subjeced ensin and inernal pressure. The curve f CF is bained frm he cmpuainal mdel cnsiued by aisymmeric elemens via ANY, Figure 5, and i is deermined in he lngiudinal direcin in he firs case, and in he angenial direcin in he secnd case. The Vn Mises sress is calculaed in pin A, Figures 4 and 5, which is lcaed in he r f he defec. The curve f CF is bained as a rai f Vn Mises sress, via ANY, and nminal sress. The mesh adped by APC3D wih a beam-pipe elemen is shwn in Figure 4, and i is he same fr he ensin and inernal pressure cases. In he firs case, he resuls bained by APC3D and by ANY are cmpared, as can be seen in Figure 6. Observe ha in he linear regime, he mehd f sress cmpensain presens he same resuls as ha bained by he aisymmeric elemen mdel. The sress evaluaed by APC3D is he same, r alms cinciden, wih he sress bained by ANY in he linear regime. Hwever, when he maerial yielding begins, here are differences beween he resuls shwn by APC3D and ANY. These differences are epeced due he differences in elemen frmulain adped in his analysis, cnsidering ha he presen mehdlgy uses a hree-nde beam pipe elemen, and ANY uses an aisymmeric elemen. Figure 6. Calculaed sress curves bained frm APC3D and ANY. In addiin he analysis f Vn Mises sress, he displacemen is calculaed in Pin A. In cnras he sress, he displacemen calculain cnduced by APC3D des n invlve an CF, even in he linear and plasic regimes. The resuls bained by APC3D and ANY are shwn in Figure 7. The magniude f displacemen is similar fr APC3D and ANY, which shws he applicabiliy f he prpsed mehdlgy. The same mehdlgy f analysis is applied he same defecive pipe subjeced inernal pressure in he secnd case f analysis. In rder shw he applicabiliy f he prpsed mehd, he resuls bained frm ANY are cmpared resuls bained frm APC3D, cnsidering he effecs f a angenial CF, and wihu he effecs f a angenial CF. Observe ha he resuls prvided by he APC3D wihu angenial CF Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26

7 38 hang curve are significanly differen hse frm ANY, shwn in Figure 8. On he her hand, he APC3D wih angenial CF curve prvides resuls similar hse frm ANY, wih an aisymmeric elemen mdel. Figure 7. Cmparisn f displacemen curve bained frm APC3D and ANY. elemen mdel. Therefre, he mesh refinemen f hese mdels plays an impran rle in he CF calculain. Furhermre, he curve f he CF is bained by inerplain f he Vn Mises sress versus he lading sep, which is anher surce f numerical errr. The manner in which he inerplain is cnduced can adversely affec he accuracy f he analysis. These w aspecs cnsiue he surce f numerical errr, and his errr culd be accumulaed during an ieraive prcedure, such as Newn-Raphsn. The sluin f he virual wrk equain has been carried u based n he displacemen mehd. Once he displacemen is bained, he srain can be evaluaed, fllwed by he sress incremen. The prpsed mehdlgy culd deermine wih reasnable accuracy he sress wih a cncenrain facr in he beam-pipe elemen mdel. Hwever, he srain and displacemen are n calculaed wih he same accuracy. uch difficuly in baining accurae resuls is based n he difficuly in deermining he srain and displacemen frm he sress incremened by he CF, in her wrds, he inverse prcess. This limiain is inheren he finie elemen frmulain adped in his wrk. Figure 8. Vn Mises sress cmparisn; he values are bained a he r f defec by APC3D wihu CF and wih CF, and ANY; pipe subjeced inernal pressure. Frm he bservain f resuls, i is pssible inser an CF in he beam-pipe elemen frmulain wihu cmprmising he applicabiliy f he riginal elemen frmulain when using he prpsed mehd. This prvides cndiins reduce he elemen mdel cnsiued by a 2D r 3D elemen in a mdel cnsiued by a unidirecinal elemen. Cnsequenly, his reduces he cmpuainal ime prcessing. Neverheless, he prpsed mehdlgy presens several limiains, because an CF is bained frm he resuls f a 2D r 3D finie Analysis f pipe wih semicircular defec subjeced bending This eample presens an analysis f a pipe wih a semicircular defec subjeced a lading applied a he end, simulaing he siuain f a clamped beam. Unil nw, he prpsed mehd has been applied nly cases wih ensin and inernal pressure. Hwever, i has n been applied he analysis when he pipe is subjeced mmen. The disribuin f he sress is aisymmeric he neural line. Cnsider a pipe wih a semicircular defec, and he gemerical prperies shwn in Figure 9; φ e 2; φ i 9; L 5mm. The semicircular defec has a radius f 2.5 mm and is siuaed in he middle f pipe. A lad wih magniude 38 kn is applied a he end f pipe in he verical direcin, wih he her end ally clamped, as a clamped beam. In ANY, nly half a mdel has been cnsidered, cnsiued by 3D elemens, in rder reduce he cmpuainal effr, as shwn in Figure. The resrains are applied accrding symmeric cnsiderains. The same siuain was simulaed wih beam-pipe elemens in APC3D, in which he semicircular defec is represened by ne elemen. Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26

8 A mehdlgy fr analysis f defecive pipeline 39 Figure 9. chemaic represenain f defecive pipe, and he mesh adped by APC3D. Pipe subjeced mmen. Analysis f pipe wih recangular defec subjeced inernal pressure In he hird eample, a pipe wih a recangular defec subjeced inernal pressure is analyzed, cnsidering he gemeric prperies shwn in Figure 2. Due he limiain f he prpsed mehd, he beam-pipe elemen frmulain, nly pin 4 is analyzed, as shwn in Figure 3. I is pssible realize a mapping ha invlves mre pins f analysis and bain curves f he sress cncenrain facrs a hese pins. These curves can be insered in he prpsed mehd, and resuls can be bained ha invlve pins neighbrhds. Figure. Mesh wih brick elemen using cmmercial sfware. The Vn Mises curve versus lad seps is bained a he r f defec. The CF is deermined in each lad sep and he funcin ha represens he CF variain is bained. The funcin is insered in APC3D and he resuls f analysis in erms f sress are represened in Figure, frm which i can be bserved ha he APC3D wih CF presens similar resuls ha f ANY in he linear regime. Hwever, in he nnlinear regime, here are differences beween he values in cmparisn. On he her hand, he ime prcessing f APC3D wih CF is cnsiderably smaller han ha f ANY. Figure 2. chemaic represenain f a pipe wih recangular defec (Chi e al., 23). L: 23, D: 762, : 7.5, c: 5, l: 3, a: 3, mm (75% de ). Figure 3. Discreizain f a pipe wih recangular defec and principal pins. Figure. Vn Mises sress bained a he r f defec by APC3D wihu CF and wih CF, and ANY; pipe subjeced mmen. The curve ha shws he Vn Mises sress versus lad seps a pin 4 is deermined frm a 3D elemen mdel frm ANY. Frm his curve, a funcin f inerplain f he sress cncenrain facr is calculaed and is insered in APC3D. The numerical mdel f he beam-pipe elemen is cnsiued by 23 elemens and he defec is represened by an elemen wih an equivalen lengh ha f he defec. The resuls f Vn Mises sress, wih 63 lad seps, a pin 4 are presened in Figure 4, wih an inernal pressure 63 MPa. This eample shws he versailiy f he prpsed mehdlgy in analyzing differen prfiles f defecs. Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26

9 32 hang Figure 4. Vn Mises sress bained by APC3D wihu CF and wih CF, and ANY; eample 3. Cnclusin In his wrk, an effr has been made develp a defecive pipe analysis mehdlgy reduce he cmpuainal effr fr evaluaing he perain limi pressure wih reasnable accuracy. I is well knwn ha i is pssible assess he sress cncenrain effec using a 2D r 3D finie elemen mdel. Hwever, he nnlinear sluin f such a numerical mdel demands prcessing ime. T vercme his prblem, he mehdlgy prpsed in his wrk aims inrduce a sress cncenrain facr Equain, evaluaed a priri by 2D r 3D elemens, in beam finie elemen virual wrk Equain. uch an inrducin causes inaccuracy in he sress evaluain by beam elemens, in cmparisn wih 3D r 2D elemens. In addiin, i causes an verbalance in he virual wrk Equain. This prblem was slved by using he mehdlgy f sress cmpensain, develped by he auhr f his wrk. Frm he analysis f several applicains, he resuls bained by he sress cmpensain mehd demnsrae he penial and applicabiliy f he prpsed mehd. The resuling accuracy can be imprved by using a smaller lerance fr he cnvergence crierin and increasing he rder f he plynmial Equain fr he sress cncenrain facr apprimain. On he her hand, he prpsed mehd has he capaciy handle an analysis f a pipeline cnaining a recangular defec, and simulaneusly, subjeced a cmbinain f lading in a glbal pipeline behavir analysis. uch analysis carried u by APC3D will ake less cmpuainal effr in cmparisn a numerical mdel cnsruced by 3D elemens frm cmmercial sfware. References Adib, H., Jalluf,., chmi, C., Carmasl, A., & Pluvinage, G. (27). Evaluain f he effec f crrsin defecs n he srucural inegriy f X52 gas pipelines using he INTAP prcedure and nch hery. Inernainal Jurnal f Pressure Vessels and Piping, 84(3) American ciey f Mechanical Engineers. (99). B3G-99. Manual fr deermining he remaining srengh f crrded pipelines. New Yrk Ciy, NY: The American ciey f Mechanical Engineers. Bahe, K. J. (996). Finie elemen prcedures. Upper addle River, New Jersey: Prenice Hall. Chi, J. B., G, B. K., Kim, J. C., Kim, Y. J., & Kim, W.. (23). Develpmen f limi lad sluins fr crrded gas pipelines. Inernainal Jurnal f Pressure Vessels and Piping, 8(2), Chuchaui, B. A., & Pick, R. J. (996). Behaviur f lngiudinally aligned crrsin pis. Inernainal Jurnal f Pressure Vessels and Piping, 67(), Crnin, D.., & Pick, R. J. (22). Predicin f he failure pressure fr cmple crrsin defecs. Inernainal Jurnal f Pressure Vessels and Piping, 79(4), Dha, G., Tuz, G., & Canin, G. (985). The finie elemen mehd displayed. Nrwich: Jhn Wiley & ns. Kim, Y. J., & n, B. G. (24). Finie elemen based sress cncenrain facrs fr pipes wih lcal wall hinning. Inernainal Jurnal f Pressure Vessels and Piping, 8(2), Kim, Y. J., him, D. J., Huh, N.., & Kim, Y. J. (22). Plasic limi pressures fr cracked pipes using finie elemen limi analyses. Inernainal Jurnal f Pressure Vessels and Piping, 79(5), Nwzarash, F., & Mhareb, M. (24). An elas-plasic finie elemen fr seel pipelines. Inernainal Jurnal f Pressure Vessels and Piping, 8(2), Oh, C. K., Kim, Y. J., Baek, J. H., Kim, Y. P., & Kim, W.. (27). Ducile failure analysis f API X65 pipes wih nch-ype defecs using a lcal fracure crierin. Inernainal Jurnal f Pressure Vessels and Piping, 84(8), aa, M. (25). Lcal and glbal cllapse pressure lngiudinally flawed pipes and cylindrical vessels. Inernainal Jurnal f Pressure Vessels and Piping, 82(3), Zhang, B., Ye, C., Liang, B., Zhang, Z., & Zhi, Y. (24). Ducile failure analysis and crack behavir f X65 buried pipes using eended finie elemen mehd. Engineering Failure Analysis, 45, Received n May 29, 25. Acceped n Nvember 9, 25. License infrmain: This is an pen-access aricle disribued under he erms f he Creaive Cmmns Aribuin License, which permis unresriced use, disribuin, and reprducin in any medium, prvided he riginal wrk is prperly cied. Aca cieniarum. Technlgy Maringá, v. 38, n. 3, p , July-ep., 26