SHAKEDOWN BEHAVIOUR OF COMPOSITE CYLINDERS WITH CROSS HOLE

Transcription

1 HKEDOWN BEHIOU OF COMPOITE CYLINDE WITH CO HOLE Hafeng Chen Deatment f Mechancal Engneeng Unvesty f tathclyde Glasgw, ctland, UK Wehang Chen Deatment f Mechancal Engneeng Unvesty f tathclyde Glasgw, ctland, UK Tanba L Deatment f Mechancal Engneeng Unvesty f tathclyde Glasgw, ctland, UK James Ue Deatment f Mechancal Engneeng Unvesty f tathclyde Glasgw, ctland, UK BTCT In ths study, bth the lwe and ue bund shakedwn lmts f a clsed-end cmste cylnde wth wthut a css hle subject t cnstant ntenal essue and a cyclc themal gadent ae calculated by the Lnea Matchng Methd (LMM). Cnvegence f ue and lwe bund shakedwn lmt f the cmste cylndes s sught and shakedwn lmt nteactn dagams f the numecal examles dentfyng the egns f evese lastc lmt and atchet lmt ae esented. The effects f temeatue-deendent yeld stess, mateals dscntnutes, cmste cylnde thckness and the exstence f css hle n the shakedwn lmts ae dscussed f dffeent gemety aametes. Fnally, a safety shakedwn envele s ceated by fmulatng the shakedwn lmt esults f dffeent cmste mateal and cylnde thckness ats wth dffeent css hle szes. Keywds: lwe and ue bund, shakedwn, lnea matchng methd, cmste cylnde INTODUCTION Mateals have lagely been ket esnsble f efmance mvements n many aeas f stuctues technlgy. The cntnuus develment f cmutatnal stuctues technlgy and the advanced cmste mateals has mved stuctual efmance, educed eatnal sk, and shtened ductn tme []. On the the hand ne f the mst mtant easns f usng cmste mateals s the eductn f weght []. Wth the achevements n aesace ndusty, the stengtht-weght at f engneeng cmnents has becme a vey mtant desgn cten snce a hgh stength-t-weght at esults n a bette efmance and geate shea stength. The lwe weght esults n lwe fuel cnsumtn and emssns. tength-t-weght at can be nceased by takng mateals beynd the elastc lmt and defnng lmts f allwable accumulated lastc stan. Ths aach can be aled f the desgn f cmste essue cylnde subjected t cyclc mechancal and themal lads. The nvestgatn f the elastc and elastc-lastc behavu f unfm cylnde unde cnstant ntenal essue and cyclc themal lads wth css hle ae esented by the well-knwn Bee-lke dagam n [] and [4]. The lcal stess cncentatn s edstbuted aund the mateal bundaes f cmste cylndes unde cyclc themal lads. Ths changes the fatgue lfe and elastc shakedwn lmts f the cylnde. The elastc shakedwn lmt s the hghest cyclcal lad that shakes dwn t an elastc esnse n the fst few cycles f lad. When the elastc shakedwn lmt s exceeded, the cylnde may exeence ethe lastc shakedwn atchettng. In many alcatns, t s allwable f a stuctue t be wthn the elastc shakedwn lad lmt, but lastc shakedwn altenatng lastcty, unde whch lcal lw-cycle fatgue falue mde ccus, and atchettng that ultmately leads t ncemental lastc cllase, ae nt emtted. Cnsequently the shakedwn lmt s a atculaly mtant desgn cndtn t the essue cylnde. uch desgn technques eque an undestandng f the elastclastc behavu f stuctue. The elastc-lastc esnse f a mateal s lad ath deendent and mst cmmnly smulated by ncemental Fnte Element nalyss (FE). Ths allws nvestgatn f any tye f lad cycle but als eques detaled lad hsty and nvlves sgnfcant cmute efft. T avd such dffcultes, dect methds ae ncated nt fnte element analyss n de t evaluate the shakedwn lmt. The mateal mdel s assumed t be elastc efectly lastc, and the lad dman cntanng all ssble lad aths elmnates the need t knw a ecse lad hsty. uch methds nclude mathematcal gammng methds [5-7], the Genealzed Lcal tess tan (GLO) -nde methd [8], the Elastc Cmensatn Methd (ECM) [9,], and the Lnea Matchng Methd (LMM) [-5]. mng these dect methds, the LMM s cnsdeed t be the mst adatable methd t actcal engneeng alcatns that nvlve cmlex cyclc them-mechancal lad cndtns. Othe dect methds eque secfc gams that ae nt avalable suted cmmecally, have dffcultes t effectvely analyze cmlex engneeng stuctues. The stable and accuate esults f LMM n shakedwn analyss have been cnfmed n many ndustal alcatns, f examle, the

2 blem f the defectve elne [] and a sue heate utlet enetatn tube late []. In ths ae, the lnea matchng methd s aled f shakedwn analyss f a cmste cylnde wth css hles subjected t cnstant ntenal essue and cyclc themal lads. The Bee-lke shakedwn lmt dagams f cmste cylnde ae ltted f dffeent cmste mateal and thckness ats wth and wthut css hles. Thee css-hle szes ae cnsdeed, all elatvely small n cmasn wth the the cylnde dmensns. The bjectve f the nvestgatn s t fmulate a safety shakedwn lmt egn f ndustal uses usng the calculated shakedwn lmt esults f dffeent cmste mateal at and cylnde thckness at wth dffeent css hle szes. NUMEICL POCEDUE The basc assumtn f the analyss f shakedwn s that the mateal s cnsdeed t be stc, elastc-efectly lastc and satsfyng the vn Mses yeld cndtn. In de t slve blems n actce wth hgh temeatue effects, the yeld stess f the mateal s cnsdeed t be temeatuedeendent. Ths deendence s mlemented at Gauss nts and elated t evey ladng vetex f ladng dman. Let a bdy subjected t cyclc hsty f vayng temeatue θ ( x, t) wthn the vlume f the stuctue and suface lads P ( x, t) actng ve at f the stuctue s suface T be cnsdeed. The vaatn s cnsdeed t be ve a tycal cycle t t. Hee dentes a lad aamete, allwng a whle class f ladng hstes t be taken nt accunt. On the emande f the suface, dented u, the dslacement s u. Cesndng t these ladng hstes, a lnea elastc slutn hsty s btaned; θ P ˆ j ˆ j + ˆ () j whee θ ˆ j and P ˆ ae the elastc slutns cesndng t j θ ( x, t) and P ( x, t), esectvely. F shakedwn cyclc blems the cyclc stess hsty, dung a tycal cycle t t, esectve f mateal etes s gven by ˆ j( x, t) j( x, t) + ρj( x ) () whee ρ dentes a cnstant esdual stess feld n equlbum j wth ze suface tactns n T, whch cesnds t the esdual state f stess at the begnnng and end f the cycle. Based un the knematc theem f Kte [6] and Melan's lwe bund shakedwn theem [7] the LMM cedue has ved t duce vey accuate ue and lwe bund shakedwn lmts [] [7].. Ue Bund Pcedue. Kte's theem states: F all Knematcally dmssble (K) stan ate hstes T T c c c c ˆ j ( x, t) & ε j ddt D( & ε j ) ddt j & ε jddt () T c whee j dentes a state asscated wth ε& c j ( all stan ate hstes that accumulate ve a cycle) at yeld. Then s whee s s the shakedwn lmt. Kte's theem s als called ue bund shakedwn theem. f They [] shws the fm ( ) f the ue bund theem that allws the LMM t be dslayed as a gammng methd. [8] shws that the yeld cndtn and the lnea mateal vde the same stess f stan ate hsty at an ntal K ε&. s a esult the matchng cndtn s: j L j j (4) whee j s the asscated stess at yeld. F the vn Mses yeld cndtn, matchng cndtn (4) becmes: y μ (5) & ε Whee ε& dentes the vn Mses effectve stan ate and μ dentes shea mdulus. The ue bund multle can be btaned by a sngle teatn that begns wth the evaluatn f a vayng shea mdulus μ by matchng the stess due t the lnea mdel and the yeld cndtn at the stan ate ε& j yelded by the evus teatn. Each teatn vdes bth a knematcally admssble stan ate hsty and an equlbum dstbutn f esdual stess, whle ue bunds ae geneated such that they cnvege t the mnmum ue bund.. Lwe Bund Pcedue. Melan's theem states: If a tme cnstant esdual stess feld ρ exsts such that suestn wth nduced elastc stesses ˆ j ( x, t) fms a safe state f stess eveywhee n the stuctue,.e. f ( ˆ j ( x, t) + ρj ( x )) (6a) then (6b) s Melan's theem can als be called lwe bund shakedwn theem statc shakedwn theem. On the bass f Melan's lwe bund shakedwn theem, a lwe bund f shakedwn lmt can be cnstucted n the same cedue by maxmsng the lwe bund lad aamete unde the cndtn whee f any tentally actve lad/temeatue ath, the stesses esultng fm the suestn f ths cnstant esdual stess feld ρ wth the themal-mechancal elastc stess j ˆ j nwhee vlate the temeatue-deendent yeld cndtn. Hence, as the abve ue bund teatve cess vdes a sequence f esdual stess felds t s ssble t evaluate a lwe bund at each teatn by scalng the elastc slutn s that ˆ + ρ eveywhee satsfes yeld. The lwe bund f j j s shakedwn lad multle can be wtten as: max. Iteatn tes f LMM hakedwn nalyss. vey sgnfcant advantage f the methd cmes fm the

3 ablty t use standad cmmecal fnte element cdes whch have the faclty t allw the use t defne the mateal behavu. Ths has been dne n the cde BQU wth use subutne UMT. Essentally, BQU caes ut a cnventnal ste by ste analyss and, thugh the use f use utne, each ncement s enteeted n tems f an teatn f the methd. t each ncement, the use utne UMT allws a dynamc esctn f the Jacban whch defnes the elatnsh between ncements f stess and stan. Fg. esents a flw chat shwng the + teatn stes n BQU f estmatng the shakedwn lmt usng the ue and lwe bund theem. detaled teatn f lwe bund and ue bund shakedwn lmt s gven n [7]. Calculate cnstant esdual stess Defne stan ate asscated wth n vetces at the lad hsty Calculate the shakedwn lmt multle: (Kte's ue bund theem) (Melan's lwe bund theem) + μ k ε y k laye f steel and an ute laye f alumnum., m, ae the nne adus, mddle adus, and ute adus f the cmste cylnde, esectvely. The aea suundng the hle, whch can be an nstumentatn tang a t f the flud enty ext, s exected t be the mst ctcal egn snce ths s a stuctue dscntnuty causng the se f the lcal stess cncentatn. T mve the mechancal efmance f ths ctcal egn, the mateal suundng the hle s selected t be hgh efmance steel. The thckness f ths steel nset s equal t. The shakedwn esults ae btaned f thee dffeent adus ats:.5,.75,..thee css-hle adus ats ae mdelled:.,.,.. The sze f the steel nset n Fg. emans cnstant f all ats. The maxmum adus ats defned n ths ae meet the equement f ME B&P Cde ectn III Dvsn, n whch the lmtatn f shuld be less equal t / f efated cylndcal shells [9]. The analyss s efmed f thee cmste s mateal ats:,, Whee s and stands f the vlume f steel and alumnum, esectvely. F bette esult cmasn, n all the cases the nne adus s chsen t be mm whle length s L 9mm. Cnvegence cndtn: + e e N MPa, θ C teel nset aea Yes m L ( t) θ + θ ( t) MPa, θ Fg. LMM flw dagam f + teatn ste COMPOITE CYLINDE GEOMETY The gemetcal shae and the mateal etes f the cmste cylnde wth a css hle ae as shwn n Fg. and Table, esectvely. The cmste thck cylnde has an nne Fg. Gemetcal shae f cmste cylnde

4 Table Mateal ety aametes f the steel and alumnum Tye Yung s mdulus E (GPa) Ceffcent f Pssn s themal exansn α Yeld stess Themal Cnductvty Densty at ν ( C ) y (MPa) κ - (W/mK) (Kg/mm ) teel..4x E-6 lumnum 7..6x 5 5.7E-6 4 FINITE ELEMENT MODELLING The cmste cylndes ae analysed usng BQU tye CD nde quadatc bck elements wth educed ntegatn scheme. The cmste cylndes wth css-hles have thee lanes f symmety. Hence, t mnmze the sze f the mdel, these symmety bunday cndtns ae aled t a quate mdel. clse D vew f a cmste cylnde wth css-hle s shwn n Fg.. The cnstant ntenal essue s aled t the man cylnde be and hle be. The cut end f the cylnde s cnstaned such that the lane sectn emans lane dung ladng. The clsed-end bunday cndtn s eesented by alyng a unfm axal thust t the end f the cylnde. The hles ae assumed t have en-ended bunday cndtn. The aled cyclc themal ladng s duced by assumng that the utsde suface f the cylnde s at ambent temeatue whle the ntenal suface temeatue s fluctuatng fm ambent t hghe values. Thee themal stess extemes ae used f ths cyclc lad hsty: - Fstly, accdng t dffeent themal cnductvtes f the steel and alumnum, a themal stess s duced by the mst sgnfcant nnlnea themal gadent alng the thckness. Ths mst sgnfcant themal lad s calculated by a steady-state themal analyss, - ecndly, a themal stess ccung at the hghest unfm temeatue s aled due t the mateal msmatch. Ths themal stess s adted knwng that themal exansns between the steel and alumnum ae sgnfcantly dffeent, - Fnally, a ze themal stess feld s selected t smulate a unfm ambent temeatue f the whle cylnde. When the ambent temeatue θ emans at C, the magntudes f the maxmum vn Mses effectve them elastc stesses f the abve themal ladng extemes can be detemned by the maxmum temeatue dffeence θ between the nne suface and ute suface f the cmste cylnde. Hence these themal and mechancal lad ath extemes can be chaactesed by the ntenal essue and the maxmum temeatue dffeence θ. The efeence cnstant elastc mechancal stess can be calculated by the alumnum ntenal essue y MPa whle the efeence temeatue dffeence θ θ C detemnes the efeence cyclc themal elastc stesses. When the temeatue-deendent yeld stess Y (T ) s adted, the actual lad fact s udated n an teatve way dung the calculatn. The adted temeatue-deendent yeld stess s gven n Equatn (7) f steel and esented n Table f alumnum; ( T ).4 ( MPa / C ) T (7) Table Temeatue-deendent yeld stess f alumnum Temeatue ( ) ( ) T y Y Y (MPa) (a) (b) (c) Fg. Quate fnte element mdels f dffeent mateal ats s s s a) b) c) 5 EULT ND DICUION 5. Ue and Lwe Bund esults wth Temeatue Deendent and Indeendent Yeld tess. Based un the knematc theem f Kte [6], the LMM cedue has ved t duce hghly accuate ue bund [] and lwe bund shakedwn lmts [7]. The cnveged values f bth ue and lwe bunds shakedwn lmts f the cmste s cylnde ae shwn n Fg. 4 whee mateal ad,.75,.. n nteactn dagam cmsed f the shakedwn lmt f dffeent ats f vayng themal lad and cnstant mechancal lad s als esented. Ths lmt s dvded nt tw egns; evese lastcty lmt B *B*, and atchet lmt BC B*C. hakedwn wll nt ccu when the aled lad s beynd the evese lastcty lmt B/ *B*, whee the emanent stans settle nt a clsed cycle, a stuatn als knwn as cyclc altenatng lastcty. These emanent lastc stans wll ncease ndefntely f the aled lad s beynd the atchet lmt BC/ B*C, a cndtn knwn as atchettng. The nt C cesnds t the lmt lad f the aled mechancal lad. Thee ae sgnfcant dffeences between the evese lastcty lmt *B* adtng temeatue-deendent yeld stess and the evese lastcty 4

5 lmt B cnsdeng temeatue-ndeendent yeld stess. It s mtant t adt temeatue-deendent yeld stess f a stuctue assessment unde hgh temeatue vaatns. In de t smlfy the calculatns, the temeatue-ndeendent yeld stess s adted when the vaatn f eatng temeatue aaches t ze the temeatue vaes wthn a lmted ange. The temeatue effects n the yeld stess may be gned n such cndtns. Fg. 4b shws tycal ue and lwe bund sequences cnvegng afte 8 teatns f lad nt (Fg. 4a) θ.6.4. θ * * B B.5 cnsdeng temeatue-ndeendent yeld stess, and f lad nt *(Fg. 4a) cnsdeng temeatue-deendent yeld stess. It can be bseved that bth the ue bund and lwe bund cnvege t the exact shakedwn lmt vng that LMM duces hghly accuate ue bund and lwe bund shakedwn lmt esults. F bette vew, the esults btaned n the cmng sectn ae ue bund shakedwn lmt wth temeatue-ndeendent yeld stess. hakedwn lmt multle Ue bund wth cnstant yeld stess (MPa) Ue bund wth temeatue deendent yeld stess Ue bund wth cnstant yeld stess (MPa) Lwe bund wth cnstant yeld stess (MPa). Ue bund wth temeatue deendent yeld stess Lwe bund wth temeatue deendent yeld stess C (a) (b) Fg. 4 a) Ue and lwe bunds shakedwn lmt nteactn cuves f the cmste cylnde b) the cnvegence cndtn f teatve s cesses f shakedwn analyss (nt and *, subjected t changng themal lads nly) (,. 75,. ) 5. Effect f the cmste mateal at. The shakedwn nteactn cuves f a cmste cylnde wth vayng mateal at cnfguatns (Fg. ) ae esented n Fg. 5. The aled essue n X-axs s nmalzed wth esect t the efeence ntenal essue and the themal stess n Y-axs s nmalzed by usng the efeence temeatue dffeence θ θ C. Fg. 5 shws that the lmt lad f the cmste cylnde educes when deceasng the vlume f steel mateal wheeas s the evese lastcty lmt s nceased wth smalle. The eductn n lmt lad s axmately n tn t the lss f steel mateal. The nceasng evese lastcty lmt s due t the dffeence n themal cnductvtes f the steel and alumnum. s the vlume f alumnum nceases, me tn f the cylnde has lage themal cnductvty, whch leads t a lwe themal elastc stess ange. Hence, when the vlume f alumnum nceases the evese lastcty lmt nceases. hakedwn lmt nteactn cuves f the cmste cylnde (. 5 ) wth css hle f dffeent cmste mateal ats and dffeent css hle ats ae.5 Lwe bund wth cnstant yeld stess (MPa) Lwe bund wth temeatue deendent yeld stess Iteatns esented n Fg. 6 whch shws that wth the addtn f a css hle, the geneal tend f the shakedwn cuves s smla t Fg. 5 - bth shw an deceasng lmt lad and nceasng evese lastcty lmt f deceasng vlume f steel. The addtn f a hle gves se t a lcal stess cncentatn. Ths s shwn t have lttle effect n the lmt lad f any mateal cnfguatn when hle damete s smalle. detaled dscussn f the effects f the hle damete s gven n sectn θ θ Fg. 5 hakedwn lmt nteactn cuves f the cmste cylnde f dffeent cmste mateal at wthut css hle 5

6 θ θ Fully steel Fully alumnum. θ θ Fully steel Fully alumnum. θ θ Fully steel Fully alumnum (a) (b) (c) Fg. 6 hakedwn lmt nteactn cuves f the cmste cylnde (. 5 ) f dffeent cmste mateal at wth dffeent css hle at: a). b). c). 5. Effect f the Hle Damete. Css hles n cmste cylndes ae stuctual dscntnutes whch ncease elastc stess due t lcal stess cncentatn. The nfluence f css hle sze,.,.,. n the shakedwn lmt nteactn cuve s shwn n Fg. 7 wth dffeent mateal at cnfguatns. Fg. 7a esents that f a mateal at f, the addtn f a hle has a lage mact n the evese lastcty lmt, whch shws the dmnance f ths stess ase t the mechansm. The addtn f a hle s shwn t have neglgble s effect n the lmt lad. When the mateal at, the lmt lad s detemned by the lage tn f the alumnum mateal due t ts lw yeld stess. The ntductn f hle has much less effect n the lmt lad than ths small mateal at. Fg. 7b shws that f a mateal at f, the addtn f a hle has a szable effect n the evese lastcty lmt, but macts the lmt lad less sgnfcantly than Fg. 7c f a mateal at f. Ths s because when the mateal at educes t, the stess cncentatn fm the hle becmes cmaable wth the stess cncentatn due t the mateal msmatch. When the sze f hle nceases, bth the lmt lad and evese lastcty lmt deceases. Fg. 7c shws that f a mateal at f, the addtn f a hle has lttle effect n the value f evese lastcty lmt, but causes a eductn n the lmt lad. The eductn n mateal caused by nceasng hle damete s the cause f the eductn n lmt lad. Lttle effect f the hle sze n the evese lastcty lmt s due t the dmnance f the mateal bunday stess ase, whch has lttle nteactn wth the stess cncentatn caused by the hle. 5.4 Effect f the Cmste Cylnde Thckness. Fg. 8 shws the effects f the adus at n the shakedwn nteactn cuve. Thee dffeent elatve thckness.5,.75,. f cmste cylnde wth a fxed mateal at f wee analyzed. Inceasng the adus at geatly nceases the lmt lad and educes the evese lastcty lmt. The ncease n lmt lad s an bvus esult, as effectvely the thckness f the e s nceased f the same nne adus. The evese lastcty lmt eductn s caused by the nceased thckness f steel. Ths ncease n thckness (whch causes geate cnductn temeatues n the steel) esults n hghe themal stesses at the mateal bunday. 6

7 θ θ... Wthut hle θ θ... Wthut hle θ θ... Wthut hle (a) 4 (b) (c) Fg. 7 hakedwn lmt nteactn cuves f the cmste cylnde (. 5 ) wth dffeent hle adus ats and dffeent cmste mateal s s s ats: a) b) c) θ θ θ θ (a) (c) θ θ θ θ (b) (d) s Fg. 8 hakedwn lmt nteactn cuves f the cmste cylnde ( ) wth dffeent thckness adus ats and dffeent hle adus ats: a) wthut hle b). c). d). 7

8 5.5 Fmulated hakedwn Lmt Desgn egn. n elastc shakedwn lmt fmulatn f the cmste cylnde s made f the safety f engneeng desgn. The elastc shakedwn desgn egns f cmste cylndes ae shwn n Fg. 9 Whee θ L s the desgn temeatue ange cesndng t the evese lastcty lmt, P L s the desgn ntenal essue eesentng the lmt lad and L s the desgn sle f the atchet lmt cuve. T smlfy, θ L and P L ae assumed t be the duct f thee ndeendent functn f, f, f and g, g, g, esectvely. The aled essue n the X dectn s nmalzed wth esect t P L and the aled temeatue dffeence n Y dectn s nmalzed wth esect t θ L. Theefe, the desgn shakedwn lmts ae fmulated as, θ L θ L f f f (8) P Lg g PL 75 L g (9) () Whee f, f, f, ae the nfluence functns f the desgn temeatue ange cesndng t the evese lastcty lmt,and g, g, g ae the nfluence functns f the desgn ntenal essue eesentng the lmt lad., and stands f the css-hle at, steel t alumnum at and thckness at, esectvely. θ L and P L ae cnstants standng f the calculated evese lastcty lmt and the lmt ntenal essue n case f.5, wthut css hle, whee, θ L 5 C (a) P L. 8MPa (b) nce the shakedwn sle lmt L btaned f the cmste cylnde has smla values f dffeent cases. Thus, f cnsevatve easns, the lwest sle lmt ( L 75 ) s selected as a efeence sle lmt. In de t fnd these nfluence functns, the btaned evese lastc lmts and ntenal essue lmt ae nmalsed and eltted n gahs f functns f and g aganst, and esectvely as shwn n Fg. and Fg.. Tend lnes ae ft t the data btaned fm the shakedwn lmt esults f dffeent cmste mateal at and cylnde thckness at wth dffeent css-hle szes t shw the nfluence functn. Equatns (a-c) and Equatns (a-c) ae the btaned nfluence functns f the desgn temeatue ange cesndng t the evese lastcty lmt and f the desgn ntenal essue eesentng the lmt lad, esectvely. Once θ L the desgn temeatue ange cesndng t the evese lastcty lmt and P L the desgn ntenal essue eesentng the lmt lad ae defned a safety shakedwn envele s ceated as shwn n Fg. 9. ( ) f (. <.).8 (. <.7) (a).659 ( ue alumnum ) f ( ) (b).495 ( ue steel ).659 f (.5.5) (c) ( ) g (. <.) (a).87 + <.56 (..7).4 ( ue alumnum ) g ( ) (b).4 ( ue steel ).7 g.84 (.5.5) (c) θ θ L Fg. 9 Elastc shakedwn desgn egns f cmste cylnde L P P L 8

9 . f. f. f (a). 4 (b)..5.5 (c) Fg. Influence functns f evese lastcty lmts aganst: a) css-hle at b) steel t alumnum at c) thckness at g.4.5 g g (a) (b) (c) Fg. Influence functns f lmt essues aganst: a) css-hle at b) steel t alumnum at c) thckness at 6 CONCLUION The esult btaned f the cmste cylnde wthut css-hle shws that the lmt lad deceases wth the eductn f the steel mateal, wheeas the evese lastcty lmt nceases wth the deceasng vlume f steel. Wth the css-hle addtn, the geneal tend f the shakedwn cuves s smla t the ne wthut css hle - a deceasng lmt lad and nceasng evese lastcty lmt f deceasng vlume f s steel. F steel t alumnum at, the exstence f a hle has lttle effect n the value f evese lastcty lmt, but t causes a eductn n the lmt lad. F mateal at f, the exstence f a hle has a szable effect n the evese lastcty lmt, but macts the lmt lad less sgnfcantly than s f a mateal at f. F a mateal at, the hle s shwn t have neglgble effect n the lmt lad. Ths mles that the sze f the css-hle ased the lcal stess cncentatn whch wll nfluence the fatgue lfe but wll nt geatly affect the glbal esnse when the lmt lad s detemned by the lw yeld stess f the dmnant alumnum mateal. Inceasng the adus at hghly nceases the lmt lad and educes the evese lastcty lmt. safety shakedwn envele s ceated by fmulatng the shakedwn lmt esults f dffeent cmste mateal and cylnde thckness ats wth dffeent css-hle szes. In de t bette undestand the elastc-lastc esnse f the cmste cylnde wth css-hle, futhe nvestgatn and a aametc study ae equed f wde ange f css hle dametes and me vaety f mateal ats. CKNOWLEDGEMENT The auths gatefully acknwledge the sut f the Engneeng and Physcal cences eseach Cuncl f the Unted Kngdm, and the Unvesty f tathclyde dung the cuse f ths wk. 9

10 EFEENCE. N, hmed K.,, tuctues technlgy f futue aesace systems, Cmutes and tuctues, 74, Thus, H.G..J., & Bemans, C., 997, Desgn fabcatn and testng f a cmste baket f aesace alcatns, Cmste tuctues, 8, Makulsawatudm, P., Mackenze, D., and Hamtn,., 4, hakedwn behavu f thck cylndcal vessels wth css hles, Pc. Instn Mech. Engs, 8, Pat E: J. Pcess Mechancal Engneeng 4. Camlle, D., Mackemze, D., Hamltn,., 9, hakedwn f a Thck Cylnde Wth a adal Csshle, Junal f Pessue essel Technlgy (), - 5. Lu YH, Cavell, Mae G., 997, Integty assessment f defectve essuzed elnes by dect smlfed methds. Intenatnal Junal f Pessue essels and Png, 74, u, D.K., Yan,.M., Nguyen-Dang, H., 4, mal dual algthm f shakedwn analyss f stuctues. Cmut. Methds l. Mech. Eng, 9, taat M., Hetze M.,, LI a Euean Pject f FEM-based Lmt and hakedwn nalyss, Nuclea Engneeng and Desgn, 6, eshad,., 995, Inelastc Evaluatn f Mechancal and tuctual cmnents Usng the Genealzed Lcal tess tan Methd f nalyss, Nucl. Eng. Des., 5, Mackenze, D., Byle, J. T., Hamltn, and. & h, J., 996, Elastc cmensatn methd n shell-based desgn by analyss, Pceedngs f the 996 ME Pessue essels and Png Cnfeence, 8,. -8. Mackenze, D., Byle, J.T., Hamltn,.,, The elastc cmensatn methd f lmt and shakedwn analyss: a evew, Tans IMechE, Junal f tan nalyss f Engneeng Desgn, 5, Chen, H.F., Pnte.,, hakedwn and lmt analyses f -D stuctues usng the Lnea Matchng Methd, Intenatnal Junal f Pessue essels and Png, 78, Chen, H.F. and Pnte,...,, Methd f the Evaluatn f a atchet Lmt and the mltude f Plastc tan f Bdes ubjected t Cyclc Ladng, Euean Junal f Mechancs, /lds, (4), Chen, H.F., Pnte,.. and nswth,.., 6, The Lnea Matchng Methd aled t the Hgh Temeatue Lfe Integty f tuctues, Pat : ssessments nvlvng Cnstant esdual tess Felds, Intenatnal Junal f Pessue essels and Png, 8(), Chen, H.F., Pnte,.. and nswth,.., 6, The Lnea Matchng Methd aled t the Hgh Temeatue Lfe Integty f tuctues, Pat : ssessments beynd shakedwn nvlvng Changng esdual tess Felds, Intenatnal Junal f Pessue essels and Png, 8(), Chen, H.F. & Pnte,..., 6, Lnea Matchng Methd n the evaluatn f lastc and cee behavus f bdes subjected t cyclc themal and mechancal ladng, Intenatnal Junal f Numecal Methds n Engneeng, 68,.- 6. Kte W T, 96, Geneal theems f elastc lastc slds, Pgess n sld mechancs J.N.neddn and.hll, eds. Nth Hlland, mstedam,, Chen, H.F.,, Lwe and Ue Bund hakedwn nalyss f stuctues Wth Temeatue-Deendent Yeld tess, Junal f Pessue essel Technlgy, (), - 8. Pnte,... & Chen, H.F.,, mnmum theem f cyclc lad n excess f shakedwn, wth alcatn t the evaluatn f a atchet lmt, Euean Junal f Mechancs - /lds,, ME, 7, Ble and Pessue essel Cde, The mecan cety f Mechancal Engnees, New Yk.. Chen, H.F. & Pnte,..., 9, tuctual ntegty assessment f sue heate utlet enetatn tube late, Intenatnal Junal f Pessue essels and Png, 86, 4 49