CHAPTER 4 FAILURE OF UNFLAWED PRESSURE VESSEL

Transcription

1 50 CHAPTER 4 FAILURE OF UNFLAWED PRESSURE VESSEL It s ssntal t avd falus n pplns and cylndcal pssu vssls n pw plants f safty as wll as cnmc lablty. Knwldg n th maxmum pssu, th pssu vssl/pp ln can wthstand and th pdctn mthds f bust pssu a utmst mptant t th dsgns and th pats. Nnlna Fnt lmnt analyss (FEA) f cylndcal pps hav bn cad ut wth th us f ANSYS sftwa t fnd th clatn f stss-stan bhavu masud n unaxal tnsl tst spcmns wth pfmanc f th matal n actual cylndcal pssu vssls. Als, fw cass f wll-knwn pdctv quatns a xamnd f pdctablty. 4. THEORETICAL BACKGROUND OF FAILURE ESTIMATES OF UNFLAWED PIPES OR PRESSURE VESSEL Th dsgn f an unflawd pssu vssl pp und ntnal pssu qus th study f tw mds f falu. Th fst ccus whn th dfmatn bcms xcssv and th s a pssblty f pmannt dfmatn. Th scnd ccus at a hgh pssu and taks th fm f bustng th pssu vssl pp.

2 5 F a clsd nd cylndcal vssl subjctd t ntnal pssu p, th lastc stsss a (Chstph t al 00): p R (4.) p R (4.) z p (4.) wh s th adal stss ;, th hp stss;, th axal stss; R,th nn adus; R, th ut adus ; s any adus btwn R and R and = R /R. Fm Equatns (4.) t (4.), th man stss s qual t th axal stss. z Th sha stss, = - z = z -, s R p (4.4) ( ) Yld ccus at th nn sufac whn th maxmum sha stss achs th sha yld stss and th pssu at fst yld (p y ), assumng th vn Mss yld ctn, can b xpssd n tms f th tnsl yld stss ys as ys p y (4.5) If th pssu s asd bynd th yld pssu, th yld spads adally thugh th wall. F an lastc-pfctly plastc matal, yld ccus wthn th vstand zn (R s )(say, R R s ) f an ntnal pssu:

3 5 p ys Rs Rs ln R R (4.6) whn R s =R, th ntnal pssu s th cllaps pssu (p c ): pc ys ln (4.7) If th cylnd s t man lastc n unladng, th sdual sha stss at th nn sufac yldng. ys p c shuld nt xcd th sha yld n vs If th cylnd s cmpltly vstand t th ut sufac, th mnmum damt at that wll gv vs yldng n th mval f th pssu s gvn by ys pc ys. Fm Equatn (4.7), ths gvs ln (4.8) Slvng Equatn (4.8) gvs =.8. Abv a valu f =.8, a cylnd cannt b vstand t th ut damt wthut causng vsd yldng. Th maxmum pssu that can b cntand lastcally wthut vsd yldng s gvn by th pssu t gv a chang n sha stss fm ys t - ys at th nn sufac. ys p aut (4.9)

4 5 Cylnds wth lss than.8 can b cmpltly vstand wthut vs yldng ccung n unladng; thf, th maxmum pssu that can b cntand lastcally aft autfttag s th cllaps pssu p c. Th vaus mthds usd t stmat maxmum pssu p m, may b classfd nt fu gups. In th fst gup, a th mpcal fmula basd n sm tst ss, xampl Faupl s fmula and Wllng & Ubng fmula. Th scnd gup uss ths f falu basd upn th th mat tnsl stngth f th matal. Th thd gup s basd n th maxmum pssus fm plastc stss-stan latns n smpl tnsn and assumng small nfntsmal stans, whas n th futh gup, ths a btand by assumng lag fnt stans (Chstph t al 00). 4.. Empcal Fmula Basd n Sm Tst Rss Faupl (956) ppsd an xpssn f bust pssu: ys p ln (4.0) c ys Equatn (4.0) has nt bn dvd usng plastcty thy t pdct th cllaps. It stms fm th bsvatn that cllaps ccus at a pssu btwn th valu pdctd by Equatn (4.7) and th valu pdctd whn was usd n th plac f ys Th at f ys / s usd as th wghtng fact t ntplat btwn th tw lmts. Basd n th pssu-tst ss f sval stl cylnds, Ubng & Wllng (960) ppsd th fmula: p m. u ln (4.).4 u

5 54 Wh th tu stss and th tu stan at th mat lad a ( ) (4.) u u ln( ) (4.) wh, s th nmnal stan at th mat lad. 4.. Fmula Basd upn Ultmat Tnsl Stngth Sctn VIII f th ASME Bl cd (96) gvs th fllwng fmula f allwabl maxmum pssu f unfd pssu vssls: p m f f.5 (4.4) Sdbg (94) ppsd th fmula, basd upn assumptn f unfm stss-dstbutn thughut th wall by cnsdng th avag stss valu and falu as a functn f th sgnfcant stss ctahdal sha stss. 4 p m (4.5) Maxmum stss ctn s gvn by: p ( ) (4.6) m

6 55 Maxmum sha stss ctn s gvn by: p m (4.7) Th latns (4.6) and (4.7) as wll as th n ppsd by Sdbg n th Equatn (4.5) can b cnsdd as appxmat, du t th assumptn f unfm stss dstbutn thughut th wall. In d t pvd th actual stss dstbutn, th plastc ang must b cnsdd. Tun (90) has dvlpd a fmula, cnsdng th matal t b pfctly plastc wth n stan-hadnng. Assumng small stans and maxmum sha stss thy f falu, th maxmum pssu s xpssd by: p ln (4.8) m Assumng pfctly plastc matal wth ctahdal sha stss n plac f th maxmum sha stss, Nada (9) btand pm ln (4.9) n Usng th tsnal stss-stan n-pw latn (, wh and n a matal cnstants) n plac f th tnsl stss-stan latn, as suggstd by Baly (90), Nada (9) dtmnd a thtcal quatn f th maxmum pssu, basd upn th maxmum sha stss plastcty latn as n pm n (4.0) Equatn (4.0) has bn fd t as Baly-Nada quatn.

7 56 Basd upn th tnsl stss-stan latn, Nada (950) btand th maxmum pssu latn as n n pm (4.) Man & Rmtt (958) btand an xpssn, dfnng th maxmum pssu usng th fnt stan thy and th nstablty cndtn dp 0. Basd n ths xpssn, thy dvlpd an appxmat xpssn f maxmum pssu as p m ln (4.) ( ) Svnssn (958) btand pssu-xpansn latn. H dvd a smplfd quatn f fully plastc yldng f a gd plastc matal by applyng a fact t Tun s quatn f th ffct f stan hadnng. Th pssu xpansn latn suggstd by Svnssn s n th fm f ss, whl Man & Rmtt (958) fund that, bcaus f apd cnvgnc f th ss, th fst tw tms gv suffcntly accuat ss. Smplfd fmula ppsd by Svnssn s, p m u 0.5 ln u 0.7 (4.) u 4.. Pssu - Expansn Rlatn Th pssu xpansn latn f an stpc cylndcal vssl und ntnal pssu s dvd h basd n fnt stan cnsdatns. Th quatn f qulbum f an lmnt n a thck-walld cylnd ntally at a adus and nstantanusly at a adus ( + u), can b psntd by th quatn f qulbum (MacGg t al 948):

8 57 d u) (4.4) d( u) ( Wh u s th adal dsplacmnt. Snc lag dsplacmnts a nvlvd, th adal and tangntal (ccumfntal) stans a dfnd as u ln (4.5a) u ln (4.5b) If th dsplacmnt u s lmnatd btwn Equatns (4.5a) and (4.5b), th fllwng stan cmpatblty quatn may b dvd d ( u) (4.6) d( u) F cnvnnc, ffctv stss and stan a mad pptnal t th ctahdal sha valus whch a dfnd as [( z ) ( z ) ( ) ] (4.7) [( z ) ( z ) ( ) ] (4.8) It wll b bsvd that and a als tu stss and natual stan as dtmnd n a smpl tnsl tst. In ths pblm, th pncpal axs f stss and stan man fxd and cncdnt. Thus, a dfmatn law f plastc flw can b mplyd and s xpssd as a functn f. On fm f quatn whch has bn mplyd t psnt th stss-stan law f many matals s

9 58 n (4.9) Wh and n a matal cnstants, n bng fd t as th stanhadnng xpnnt. F ductl matals, th nst f nstablty s dfnd by th fllwng cndtn (Jnas t al 976). at u (4.0) Usng Equatn (4.9) n Equatn (4.0), th matal cnstant s n u (4.) Th th matal cnstant n Equatn (4.9) can b xpssd n tms f th mat tnsl stngth f th matal (Svnssn 958) as n (4.) n In a ss f bust tsts n a numb f clsd nd cylndcal vssls (Cssland & Bns 955, Cstantn 965, Magtsn 978), t was fund that th axal stan ( z ) was small and culd b nglctd. In th plastc ang, th matal s ncmpssbl.(.. th Pssn s at psnt cas /). F th (4.) z ( ) (4.4) Substtutng z = 0 and n Equatn (4.8), th ffctv stan s

10 59 (4.5) Usng Equatns (4.7) and (4.4), th ffctv stss s ( ) (4.6) Dfnng x ( u) R and usng Equatn (4.6), w can wt Equatns (4.4) and (4.6) n th fm d x dx (4.7) d x dx (4.8) Usng Equatn (4.) n Equatn (4.8), th cmpatblty quatn bcms d x dx (4.9) Fm Equatn (4.5), at th nn sufac can b wttn as ln x, at x x (4.40) Th slutn f Equatns (4.9) and (4.40) can b wttn n th fm x ln (4.4) x x Usng th abv slutn (4.4), th ffctv stan fm Equatn (4.5) s x ln (4.4) x x

11 60 At th ntatn f th nstablty, th matal s ncmpssbl and th s n chang n th vlum f th matal bf and aft dfmatn. As th mdnal stan s nglgbl, th gwth f th cylndcal shll n th axal dctn s nglgbl. Ths ndcats that th aa f th css-sctn f th matal bf and aft dfmatn mans th sam that s [( R u ) ( R u ) ] ( R R ) (4.4) Dvdng Equatn (4.4) by R x (4.44) 0 x Usng Equatns (4.5) and (4.44) n Equatn (4.4), th ffctv stans at th nn sufac R ) and at th ut sufac R ) a ( ( 0 ln x (4.45) 0 ln - (4.46) Nw, dvdng Equatn (4.7) by Equatn (4.9) and usng Equatn (4.5) d (4.47) d At th nn sufac, p and and at th ut sufac, 0 and. Intgatng Equatn (4.47), and usng th pw-law xpssn f stss n Equatn (4.9) f th matal, th ntnal pssu can b xpssd n th fm (Svnssn 958): p n d (4.48)

12 6 Th appld pssu n Equatn (4.48) s wttn n tms f th ffctv stans. Falu s assumd t tak plac whn th dvatv f th appld pssu wth spct t th ffctv stan vanshs. Ths s an nstablty ctn fllwd f th psnt cas. dfnd n Equatn * (4.46) s a functn f whch lads t a nn-lna quatn f. Th dvatn f ths nn-lna quatn s psntd blw: At th pnt f nstablty, p s maxmum and dp d * 0 (4.49) wh, * nstablty. s th ffctv stan f th nn sufac at th ntatn f Assumng f n (4.50) and dffntatng Equatn (4.48) wth spct t, n gts dp d f f (4.5) It s knwn fm Equatn (4.46) that (4.5)

13 6 Equatn (4.5) mpls that (4.5) Usng Equatns (4.50) and (4.5), n can wt n n n f f (4.54) Usng Equatn (4.54) n Equatn (4.5), t s fund that n f d dp (4.55) Dffntatng Equatn (4.5) wth spct t, n can gt d d, whch mpls d d (4.56) Usng Equatns (4.55) and (4.56) n Equatn (4.49), n gts a nn-lna quatn: 0 n (4.57) It shuld b ntd that dfnd n Equatn (4.46) s a functn f. Hnc, Equatn (4.57) s a nn-lna quatn n tms f, whch s slvd by usng Nwtn-Raphsn mthd. Th scnd dvatv f p wth spct t at * s fund t b ngatv. Hnc, th valu f p at * bcms a maxmum whch s dntd by p m. F valuatn f p m fm Equatn (4.48), th lmts f th ntgatn namly, and a

14 6 * btand by substtutng n Equatn (4.46). Substtutng ths valus n Equatn (4.48) and ntgatng, gvs th maxmum pssu, p m. A tn-pnt Gauss ul s adptd f valuatng th ntgal n Equatn (4.48). F a lng cls-ndd cylndcal vssls, th axal stan ( z ) at falu s ntcd t b small cmpad t th hp stan ( ) and th cmpssv adal stan ( ) s appxmatly qual t th tnsl hp stan ( ) (.. 0 and )(Chstph t al 00). z 4. FAILURE PRESSURE ESTIMATES OF UNFLAWED PIPES Gnally, th pdctn mthds gv latnshps amng bust pssu, matal ppts and gmtcal dmnsn f th pp cncnd. Ths pdctns a basd n sval matal mdls and falu ths (Zhu & Ban 007). Numb f nvstgats has ppsd thtcal, numcal and mpcal mthds f th pdctn f bust pssu f ntnally pssusd pps (Sdbg 94, Nada 950, Faupl 956, Svnssn 958). Sval nvstgats hav cad ut xpmnts f th abv (Faupl 956, Magtsn 978, Zhng & L 006). Gnally, th pdctn mthds gv latnshps amng bust pssu, matal ppts and gmtcal dmnsn f th pp pssu vssl cncnd. Ths pdctns a basd n sval matal mdls and falu ths (Zhu & Ban 007). Chstph t al (00) xamnd falu data n pssu vssls f vaus matals and mad a cmpasn n fquntly usd ths f valdatn and futh us n th dsgn. Cnductng bust tsts by a sngl agncy s dffc du t lack f faclty and cst nvlvd. Faupl (950) s t b cmmndd f pvdng data n th pfmanc f cylnds / pps

15 64 that w tstd und ntnal pssu. Tabl 4. psnts fw bust data n pps f dffnt matals. Th matal cnstants n Equatn (4.9) f dffnt matals a psntd n Tabl 4.. F th abv pps, bust pssu calculatns hav bn cad ut usng Faupl s fmula, Sdbg s fmula and Svnssn s smplfd fmula and psntd n Tabl 4.. Tabl 4. Gmtcal dmnsns and matal ppts f pps Pp n. Matal Cylndcal pp dmnsns (mm) Out damt, D Thcknss, t Tnsl stngth ppts (MPa) Yld stss, ys Ultmat stngth, Bust pssu (MPa) Tst Rfnc AISI Faupl 956 AISI Faupl N lnd Mn- M-V tubng Gad 5 Sup duplx Stanlss stl Faupl Lasbkan & Aksanya 04 Tabl 4. Matal cnstants f pps Pp n. Pw law Invs Rambg Osgd latn n Yung's mdulus, E (GPa) n R

16 65 4. FINITE ELEMENT METHOD Zhu & Ban (007) hav statd that th applcatn f FEA f th bust pssu pdctn f pplns ptntally ffs gat accuacy but wth apppat falu ctn. At psnt, us f th Fnt Elmnt Mthd t pdct th bust pssu f cylndcal shlls has bcm a tnd snc cnfguatns that can b xamnd a nt lmtd and th fft and xpns qud a latvly mnmal (Xu t al 008). T undstand th cmbnd stss ths f falu and th us n dsgn, fnt lmnt analyss (FEA) has bn cad ut f assssng th falu f unflawd stl cylndcal pssu vssls fm th masud ppts f unaxal tnsl spcmns by As t al (009). Thy fund that th falu pssu stmats fm fnt lmnt analyss (FEA) (basd n th glbal plastc dfmatn) w n gd agmnt wth tst ss f thn as wll as thckwalld cylndcal vssls mad f ductl stl matals. Hwv thy hav nt pvdd an nsght f Glbal Plastc Dfmatn (GPD) n latn wth th ffctv stss lvls wthn th wall f th cylnd. Fm an ngnng standpnt, damag and falu f stuctus f t a glbal mchansm, whch nvlvs lag szs, wth cmplx gmts and ladng cndtns. Stuctual falu s chaactzd by falu cta, dscbng ladng stss/stan cndtn whch, whn xcdd, tgg catastphc glbal falu,.., lss f functnalty f th stuctu. Fnt lmnt analyss (FEA) has bn cad ut h t fnd th pfmanc f th matal n actual cylndcal pssu vssls. Ths nds an analyss whch taks matal nn-lnaty nt accunt. In d t pfm th nn-lna analyss, stss-stan data f th matal s t b suppld n addtn t th Yung's mdulus (E) and Pssn's at ( ). A cnsttutv latnshp that xpsss th stss as an xplct functn f stan s wdly utlzd n th xpmntal stss analyss. Th latnshp s psntd as (Bna t al 995):

17 66 n R nr E (4.58) 0 wh 0 =, s th mat tnsl stngth f th matal and n R s E th paamt dfnng th shap f th nn-lna stss-stan latnshp. Th sngl valud xpssn (4.58) psnts ssntally th nvs f Rambg-Osgd quatn. Th matal cnstants n Equatn (4.58) f dffnt matals a psntd n Tabl 4.. Th lastc-plastc pcss qus cntnuus assssmnt f stss and plastc stan at all pnts f th stuctu, as th appld lad ncass. Hnc th lad s appld n a squnc f latvly small ncmnts and wthn ach stp a chck n qulbum and stss s mad. As ladng stats, th pgam stats t tat th stss abv th yld stss t cnsd th plastc ffcts. It shuld b ntd that th Scant mdulus (E s ) and th Scant Pssn s at ( s ) n th pptnal lmt a qual t th Yung s mdulus (E) and th Pssn s at ( ) f th matal spctvly. Th whl nn-lna cuv s cnsdd t cnsst f a numb f staght lns, ach bng dsgnatd as a lad stp. An ax-symmtc fu nd quadlatal fnt lmnt (Elmnt typ: D PLANE 8) avalabl n ANSYS sftwa packag s utlzd t mdl th cylndcal pp. Axal dsplacmnt s suppssd at bth nds f th cylndcal shll t hav n axal gwth und ntnal pssu (s Fgu 4.a). Fgu 4.b shws a typcal hyd bust tstd cylnd.

18 67 p Axal Radal Fgu 4. a Ax-symmtc mdl f a cylndcal pssu vssl subjctd ntnal pssu Fgu 4.b Phtgaph f hyd-bust pssu tstd stl cylnd Th stss-stan data f th matal fm th un-axal tnsl spcmn s fttd n th latn [Equatn (4.58)] and th numcal data s gnatd s suppld t th ANSYS sftwa. Nn-lna analyss pcds

19 68 wth spcfd numb f sub-stps and uns up t a phnmnn knwn as Glbal Plastc Dfmatn (GPD). Th ffctv stss valus can b pst pcssd f th ndd tm stps and f th qud lcatns. Effctv stsss can b plttd as a functn f appld ntnal pssu at dffnt lcatns namly nn sufac, mddl thcknss and ut sufac. It can b bsvd that th ffctv stsss ncas at dffnt ats at dffnt lcatns but mt at a pnt knwn as th pnt f Glbal Plastc Dfmatn (GPD). Th cspndng ntnal pssu s th bust pssu. 4.4 RESULTS AND DISCUSSION Tabl 4. psnts fw bust data n pps f dffnt matals whch a avalabl n ltatu. Rfncs a pvdd n th Tabl 4.. Th matal cnstants n Equatn (4.9) as wll as (4.58) f dffnt matals s psntd n Tabl 4.. F th abv pps, bust pssu calculatns hav bn cad ut usng Faupl s fmula, Sdbg s fmula and Svnssn s smplfd fmula and psntd n Tabl 4.. Th stss stan cuv gnatd f th pp s shwn n Fgu 4.. Fm th pst pcssng f fnt lmnt analyss th ffctv stss vaatn btand f pp at dffnt lcatns s shwn n Fgu 4.. It can b bsvd that th ntnal pssu s 089 MPa at th vnt whn ffctv stss valus at nn sufac, mddl thcknss and ut sufacs cncd. Smlaly falu pssus f all th cylnds hav bn valuatd and ncludd n Tabl 4.. It s bsvd that th thugh th thtcal fmula stmat falu pssus satsfactly wll wth many pps, th accuacy s nt cnsstnt. Hwv, th pdctn by fnt lmnt analyss s cnsstntly dpndabl.

20 69 A cmpasn f bust pssu stmatns usng dffnt fmula and Fnt Elmnt Analyss n latn t th xpmntal valus a psntd n Fgu 4.4. It s bsvd that thugh th thtcal fmula stmat falu pssus satsfactly wll wth analysd pps, th accuacy s t b vfd f th pps f vaus dmnsns and matals. Hwv, th pdctn by fnt lmnt analyss s lkly t b cnsstntly dpndabl du t th cls smulatn f gmty and matal bhavu n th analyss. Tabl 4. Bust pssu cmputd by thtcal fmula and Fnt Elmnt Analyss Pp n. Tst Faupl's fmula (4.0) Falu pssu (MPa) Sdbg's Equatn (4.5) Svnssn's fmula (4.) Fnt Elmnt Analyss (FEA)

21 Stss, MPa Stan Fgu 4. Stss-stan cuv f AISI 0 stl Effctv Stss, MPa Intnal Pssu, MPa Inn sufac Mdddl lay Out sufac Fgu 4. Effctv stss plt f a cylnd

22 7 Pssu n MPa Faupl's fmula Sdbg s quatn Svnssn' smplfd fmula Fnt Elmnt Analyss (FEA) Tst Pssu n MPa Fgu 4.4 Cmpasn f analyss wth tst valus 4.5 CONCLUDING REMARKS Easy t us and handy quatn such as Faupl s fmula can b usd n fndng bust pssu f cylndcal pps/pssu vssls aft valdatng f th spcfc matal. Nn-lna fnt lmnt analyss mthdlgy smulats th al matal bhavu and hnc t s m dpndabl. Mv, t may b xtndd t th gmts such as cncal and llpsdal tc.