Shakedown Analysis of a Composite Cylinder with a Cross-hole

Transcription

1 hakedwn nalyss f a Cmpste Cylnde wth a Css-hle Hafeng Chen *, Wehang Chen, Tanba L, James Ue Depatment f Mechancal Engneeng, Unvesty f tathclyde, Glasgw, G XJ, UK bstact: In ths study, bth the lwe and uppe bund shakedwn lmts f a clsed-end cmpste cylnde wth wthut a css-hle subject t cnstant ntenal pessue and a cyclc themal gadent ae calculated by the Lnea Matchng Methd (LMM). Cnvegence f uppe and lwe bund shakedwn lmts f the cmpste cylndes s sught and shakedwn lmt nteactn dagams f the numecal applcatns dentfyng the egns f evese plastcty lmt and atchet lmt ae pesented. The effects f tempeatue-dependent yeld stess, mateal dscntnutes, cmpste cylnde thckness and the exstence f the css-hle n the shakedwn lmts ae dscussed f dffeent gemety paametes. Fnally, a safety shakedwn envelpe s ceated by fmulatng the shakedwn lmt esults f dffeent cmpste mateals and cylnde thckness ats wth dffeent css-hle szes. Keywds: lwe and uppe bund, shakedwn, lnea matchng methd, cmpste cylnde Intductn Mateals have lagely been kept espnsble f pefmance mpvements n many aeas f stuctues technlgy. The cntnuus develpment f cmputatnal stuctues technlgy and the advanced cmpste mateals have mpved stuctual pefmance, educed peatnal sk, and shtened pductn tme []. On the the hand, ne f the mst mptant easns f usng cmpste mateals s the eductn f weght []. Wth the achevements n aespace ndusty, the stength-t-weght at f engneeng cmpnents has becme a vey mptant desgn cten snce a hgh stength-t-weght at esults n a bette pefmance and geate shea stength. The lwe weght esults n lwe fuel cnsumptn and emssns. * Cespndng auth. Emal: hafeng.chen@stath.ac.uk Tel Fax

2 tength-t-weght at can ncease n case the elastc lmt f mateals s supassed and the allwable accumulated plastc stan cnstants ae assgned. In ths way the desgn f cmpste pessue cylnde subjected t cyclc mechancal and themal lads can be acheved. The nvestgatns f the elastc and elastc-plastc behavu f a unfm cylnde unde cnstant ntenal pessue and cyclc themal lads wth a css-hle ae pesented by the well-knwn Beelke dagam n [3] and [4]. The lcal stess cncentatn s edstbuted aund the mateal bundaes f cmpste cylndes unde cyclc themal lads. It 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 espnse n the fst few cycles f lad. When the elastc shakedwn lmt s exceeded, the cylnde may expeence ethe plastc shakedwn atchetng. In many applcatns, t s allwable f a stuctue t be wthn the elastc shakedwn lmt, but plastc shakedwn altenatng plastcty, unde whch a lcal lw cycle fatgue falue mde ccus, and atchetng that ultmately leads t ncemental plastc cllapse, ae nt pemtted. Cnsequently the shakedwn lmt s a patculaly mptant desgn cndtn t the pessue cylnde. The elastc-plastc behavu f the stuctue needs t be well cmpehended whle usng ths desgn cndtn snce the elastc-plastc eactn s lad path dependent and mst cmmnly smulated by an ncemental Fnte Element nalyss (FE). Ths allws nvestgatn f any type f lad cycle but als eques detaled lad hsty and nvlves sgnfcant cmpute efft. T avd such dffcultes, dect methds ae ncpated nt fnte element analyss n de t evaluate the shakedwn lmt. The mdel s mateal s cnsdeed t be elastc pefectly plastc, and the lad dman ncludng all the pssble lad paths elmnates the necessty t knw the lad hsty patculates n detal. uch methds nclude mathematcal pgammng methds [5-7], the Genealzed Lcal tess tan (GLO) -nde methd [8], the Elastc Cmpensatn Methd (ECM) [9,], and the Lnea Matchng Methd (LMM) [-4]. mng these dect methds, the LMM s cnsdeed t be the mst adaptable methd t pactcal engneeng applcatns that nvlve cmplex cyclc them-mechancal lad cndtns. Othe dect methds eque specfc pgams that ae nt avalable suppted cmmecally, have dffcultes t effectvely analyze cmplex engneeng stuctues. The stable and accuate esults f the LMM n shakedwn analyss have been cnfmed n many ndustal applcatns, ncludng the pblem f the defectve ppelne [] and a supe heate utlet penetatn tube plate [5]. In ths pape, the lnea matchng methd s appled f the shakedwn analyss f a cmpste cylnde wth a css-hle subjected t a cnstant ntenal pessue and cyclc themal lads. The Bee-lke shakedwn lmt dagams f the cmpste cylnde ae pltted f dffeent cmpste

3 mateals and thckness ats wth and wthut css-hles. Thee css-hle szes ae cnsdeed, all elatvely small n cmpasn wth the the cylnde dmensns. The bjectve f the nvestgatn s t fmulate a safety shakedwn lmt egn f ndustal pupses usng the calculated shakedwn lmt esults f dffeent cmpste mateal at and cylnde thckness at wth dffeent css-hle szes. Numecal Pcedues The basc assumptn and yeld cndtn f the analyss f shakedwn s pvded n [6]. detaled mathematcal devatn f shakedwn analyss s gven n [7]. F slvng pblems f hgh tempeatue effects, the yeld stess f the mateal s cnsdeed t be tempeatue-dependent. Ths dependence s mplemented at Gauss pnts and elated t evey ladng vetex f the ladng dman. Let a bdy subjected t a cyclc hsty f vayng tempeatue λθ (, t) wthn the vlume x f the stuctue and suface lads λp( x, t) actng ve pat f the stuctue s suface T be cnsdeed. The vaatn s cnsdeed t be ve a typcal cycle t Δt. Hee λ dentes a lad paamete, allwng a whle class f ladng hstes t be taken nt accunt. On the emande f the suface, dented as u, the dsplacement s u. Cespndng t these ladng hstes, a lnea elastc slutn hsty s btaned: θ λˆ λˆ + λ ˆ j j P j () θ whee ˆ j and ˆ ae the elastc slutns cespndng t θ (, t) and P ( x, t), espectvely. P j x F shakedwn cyclc pblems, the cyclc stess hsty dung a typcal cycle espectve f mateal ppetes s gven by t Δt, whee x, t) λˆ ( x, t) + ρ ( x ) j( j j () ρ j dentes a cnstant esdual stess feld n equlbum wth ze suface tactns n T, and cespnds t the esdual state f stess at the begnnng and the end f the cycle. Based upn the knematc theem f Kte [8] and Melan's lwe bund shakedwn theem [9], the LMM pcedue has pved t pduce vey accuate uppe and lwe bund shakedwn lmts [] [6]. 3

4 . Uppe Bund Pcedue Kte's theem [8] states: F all Knematcally dmssble (K) stan ate hstes λ whee T UB ˆ c T j ( x, t) εj ddt D( εj ) ddt jεj c T c c & & & ddt (3) c j dentes a state asscated wth at yeld, then λ λ, whee s UB t as the uppe bund shakedwn theem. s c ε& j (all stan ate hstes that accumulate ve a cycle) λ s the exact shakedwn lmt. Kte's theem s als efeed f They [] shws the fm ( λ λ ) f the uppe bund theem that allws the LMM t be UB UB dsplayed as a pgammng methd. [7] demnstates that the yeld cndtn and the lnea mateal pvde the same stess f stan ate hsty at an ntal K cndtn s: ε& j. s a esult the matchng L j p j (4) p whee j s the asscated stess at yeld. F the vn Mses yeld cndtn, matchng cndtn (4) becmes: y μ 3 & ε (5) whee ε& dentes the vn Mses effectve stan ate and μ dentes shea mdulus. The uppe bund multple 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 pevus teatn. Each step n teatn pvdes bth a knematcally admssble stan ate hsty and an equlbum dstbutn f esdual stess, whle uppe bunds ae geneated such that they cnvege t the mnmum uppe bund.. Lwe Bund Pcedue Melan's theem [9] states: If a tme cnstant esdual stess feld ρ exsts such that supepstn wth nduced elastc stesses λ ˆ ( x, t) fms a safe state f stess eveywhee n the stuctue,.e. LB j 4

5 f ( λ ˆ ( x, t) ρ ( x )) (6a) LB j + j then λlb λ s (6b) Melan's theem can als be efeed t as 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 usng the same pcedue by maxmzng the lwe bund lad paamete λ LB unde the cndtn whee f any ptentally actve lad/tempeatue path, the stesses esultng fm the supepstn f ths cnstant esdual stess feld ρ j wth the themalmechancal elastc stess λ LB j ˆ nwhee wll vlate the tempeatue-dependent yeld cndtn. Hence, as the abve uppe bund teatve pcess pvdes a sequence f esdual stess felds, t s pssble t evaluate a lwe bund at each step f the teatn by scalng the elastc slutn s that λ ˆ + ρ satsfes the yeld cndtn eveywhee. The lwe bund f shakedwn lmt LB j j multple can be wtten as: λ s LB max λ LB.3 Iteatn teps f LMM hakedwn nalyss vey sgnfcant advantage f the methd cmes fm the 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 wthn the cde BQU wth use subutne UMT. Essentally, BQU caes ut a cnventnal step-by-step analyss and, thugh the use f the use subutne, each ncement s entepeted n tems f an teatn f the methd. t each ncement, the use subutne UMT allws a dynamc pescptn f the Jacban whch defnes the elatnshp between ncements f stess and stan. Fg. pesents a flw chat shwng the + teatn steps n BQU f estmatng the shakedwn lmt usng the uppe and lwe bund theem. detaled teatn f lwe bund and uppe bund shakedwn lmt s gven n [6]. 3 Cmpste Cylnde Gemety The gemetcal shape and the mateal ppetes f the cmpste cylnde wth a css-hle ae as shwn n Fg. and Table, espectvely. The cmpste thck cylnde has an nne laye f steel and an ute laye f alumnum., adus f the cmpste cylnde, espectvely. m, ae the nne adus, mddle adus, and ute 5

6 The aea suundng the hle, whch can be an nstumentatn tappng a pt f the flud enty ext, s expected t be the mst ctcal egn snce ths s a stuctue dscntnuty causng the se f the lcal stess cncentatn. T mpve the mechancal pefmance f ths ctcal egn, the mateal suundng the hle s selected t be the same hgh pefmance steel as the nne ptn f the cmpste cylnde. The thckness f the cylndcal shape steel nset s equal t the half thckness f the cmpste cylnde. The shakedwn esults ae btaned f thee dffeent adus ats:.5,.75,.. Thee css-hle adus ats ae als mdelled:.,.,. 3. The maxmum adus ats defned n ths pape meet the equement f ME B&P Cde ectn III Dvsn, n whch the lmtatn f shuld be less equal t / f pefated cylndcal shells []. The analyss s pefmed f thee cmpste mateal ats: s,, 3, whee s and stand f 3 the vlume f steel and alumnum, espectvely. F bette cmpasn f esults, n all the cases the nne adus s chsen t be 3mm whle length s L 9mm. 4 Fnte Element Mdellng The cmpste cylndes ae analyzed usng BQU type C3D nde quadatc bck elements wth educed ntegatn scheme. The cmpste cylndes wth css-hles have thee planes f symmety. Hence, t mnmze the sze f the mdel, these symmety bunday cndtns ae appled t a quate sectn f the mdel. clse 3D vew f a cmpste cylnde wth csshle s shwn n Fg. 3. The man cylnde be and the hle be ae unde cnstant ntenal pessue. The cut end f the cylnde s cnstaned n de t keep the plane sectn plane dung ladng. The clsed-end bunday cndtn s acheved by applyng unfm axal thust t the end f the cylnde. The hles ae assumed t have pen-ended bunday cndtn. The appled cyclc themal ladng s pduced by assumng that the utsde suface f the cylnde s at ambent tempeatue whle the ntenal suface tempeatue s fluctuatng fm ambent t hghe values. Thee themal stess extemes ae adpted f ths cyclc lad hsty: - Fstly, accdng t dffeent themal cnductvtes f the steel and alumnum, a themal stess s pduced by the mst sgnfcant nnlnea themal gadent alng the thckness. Ths mst sgnfcant themal lad s calculated by a steady-state themal analyss; 6

7 - ecndly, a themal stess ccung at the hghest unfm tempeatue s appled due t the mateal msmatch. Ths themal stess s adpted knwng that themal expansns between the steel and alumnum ae sgnfcantly dffeent; - Fnally, a ze themal stess feld s selected t smulate a unfm ambent tempeatue f the whle cylnde. When the ambent tempeatue 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 tempeatue dffeence Δ θ between the nne suface and ute suface f the cmpste cylnde. Hence these themal and mechancal lad path extemes can be chaactezed by the maxmum tempeatue dffeence Δ θ and the ntenal pessue p. The efeence cnstant elastc alumnum mechancal stess can be calculated by the ntenal pessue p p y MPa whle the efeence tempeatue dffeence Δθ Δθ C detemnes the efeence cyclc themal elastc stesses. When the tempeatue-dependent yeld stess (T ) s adpted, the actual lad fact s updated n an teatve way dung the calculatn. The adpted tempeatue-dependent yeld stess s gven n equatn (7) f steel and pesented n Table f alumnum: ( T ).4 ( MPa / C ) T (7) Y Y Y 5 esults and Dscussns 5. Uppe and Lwe Bund esults wth Tempeatue Dependent and Independent Yeld tesses Based upn the knematc theem f Kte [8], the LMM pcedue has pved t pduce hghly accuate uppe bund [] and lwe bund shakedwn lmts [6]. The cnveged values f bth uppe and lwe bunds shakedwn lmts f the cmpste cylnde ae shwn n Fg. 4 s whee mateal ats, cylnde and css-hle adus ats ae,. 75,., espectvely. n nteactn dagam cnsstng f shakedwn lmt f dffeent ats f vayng themal lad and cnstant mechancal lad s als pesented. Ths lmt s dvded nt tw egns; evese plastcty lmt B *B*, and atchet lmt BC B*C. Elastc shakedwn wll nt ccu f the lad appled supasses the evese plastcty lmt B/ *B*. In ths case the pemanent stans settle nt a clsed cycle, a cndtn als knwn as cyclc altenatng plastcty. These pemanent plastc stans wll ncease ndefntely f the appled cyclc lad level s beynd 7

8 the atchet lmt BC/ B*C. Ths s knwn as atchetng ncemental plastc cllapse. The pnt C cespnds t the lmt lad f the appled mechancal lad. Thee ae sgnfcant dffeences between the evese plastcty lmt *B* adptng tempeatue-dependent yeld stess and the evese plastcty lmt B cnsdeng tempeatue-ndependent yeld stess. Hence t s mptant t adpt tempeatue-dependent yeld stess f a stuctue assessment unde hgh tempeatue vaatns. Hweve, n de t smplfy the calculatns, the tempeatue-ndependent yeld stess can be adpted when the vaatn f peatng tempeatue appaches t ze the tempeatue vaes wthn a lmted ange. The tempeatue effects n the yeld stess may be gned n such cndtns. Fg. 4b shws typcal uppe and lwe bund sequences cnvegng afte 7 teatns f lad pnt (Fg. 4a) cnsdeng tempeatue-ndependent yeld stess, and f lad pnt *(Fg. 4a) cnsdeng tempeatue-dependent yeld stess. It can be bseved that bth the uppe bund and lwe bund cnvege t the exact shakedwn lmt pvng that LMM pduces hghly accuate uppe bund and lwe bund shakedwn lmt esults. F the smplfcatn f dscussn, the esults n the next sectn nly shw the uppe bund shakedwn lmt f the tempeatuendependent yeld stess. In de t vefy the accuacy f the LMM, fu lad cases (labelled D, E, F and G n Fg.4a) Δθ Δθ wth cyclc themal lads f. 5 Δ, θ. 35, Δ. Δθ Δθ 7 θ and θ. 7 espectvely, have been pefmed usng BQU step-by-step analyses. The plastc stan hstes epesentng the maxmum plastc stan ange f the cyclc ladng cases D, E, F and G ae shwn n Fg.5. Lad cases D (Fg.5a) and F (Fg.5b) exhbt shakedwn mechansm as the calculated equvalent plastc stan stp changng afte lad cycles. The calculated equvalent plastc stan f the lad case E (Fg.5a) cnveges t a clsed cycle afte abut 9 lad cycles shwng a evese plastcty mechansm, and the lad case G (Fg.5b) shws a stng atchetng mechansm, wth the equvalent plastc stan nceasng at evey cycle. Thus, the esults n Fg.5 btaned usng BQU step-bystep analyss cnfm the accuacy f the pedcted shakedwn lmts by the LMM. Futhe benefts f the LMM can be fund cnsdeng the cmputng tme necessay t geneate the shakedwn cuves. The tme that the LMM needed t geneate the pnts n the atchetng bunday was less than % f that needed f the abve fu lad cases t cmplete usng the BQU step-by-step analyse. Δ Δθ 5. Effect f the cmpste mateal at 8

9 The shakedwn lmt nteactn cuves f a cmpste cylnde wth vayng mateal at cnfguatns (Fg. 3) ae pesented n Fg. 6. The appled pessue n X-axs s nmalzed wth espect t the efeence ntenal pessue whle the themal lad n Y-axs s nmalzed by usng the efeence tempeatue dffeence Δθ Δθ C. Fg. 6 shws that the lmt lad f the cmpste cylnde educes when the vlume f steel mateal s deceasng, wheeas the evese plastcty lmt s nceased wth smalle s. The eductn n the lmt lad s appxmately n pptn t the lss f steel mateal. The nceasng evese plastcty lmt s due t the dffeence n themal cnductvtes f the steel and alumnum. s the vlume f alumnum nceases, a lage pptn f the cylnde wll have lage themal cnductvty, whch leads t a lwe themal elastc stess ange. Hence, when the vlume f alumnum nceases the evese plastcty lmt nceases. hakedwn lmt nteactn cuves f the cmpste cylnde (. 5 ) wth css-hle f dffeent cmpste mateal ats and dffeent css-hle ats ae pesented n Fg. 7 whch shws that wth the addtn f a csshle, the geneal tend f the shakedwn cuves s smla t Fg. 6. Bth fgues shw a deceasng lmt lad and nceasng evese plastcty lmt f deceasng vlume f steel. It s wth ntng that f the pue mateal cases, the evese plastc lmt s detemned by the maxmum themal stess due t the tempeatue gadent whle the evese plastc lmt f the cmpste mateal s defned by the maxmum themal stess due t the mateal msmatch. 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 the hle damete s small. detaled dscussn f the effects f the hle damete s gven n sectn Effect f the Hle Damete Css-hles n cmpste cylndes ae stuctual dscntnutes whch ncease elastc stess due t lcal stess cncentatn. The nfluence f css-hle sze,.,.,. 3 n the shakedwn lmt nteactn cuve s shwn n Fg. 8 f dffeent mateal at cnfguatns. Fg. 8a shws that f a mateal at f 3, the addtn f a hle has a lage mpact n the evese plastcty lmt, whch demnstates the dmnance f ths stess ase t the mechansm. The addtn f a hle s shwn t have neglgble effect n the lmt lad. When the mateal at s, the lmt lad s detemned by the lage pptn f the alumnum mateal due t ts 3 9

10 lwe yeld stess. The ntductn f a hle has much less effect n the lmt lad than ths small mateal at. Fg. 8b demnstates that f a mateal at f, the addtn f a hle has a szable effect n the evese plastcty lmt, but mpacts the lmt lad less sgnfcantly than Fg. 8c f a mateal at f 3. Ths s because when the mateal at educes t, the stess cncentatn fm the hle becmes cmpaable wth the stess cncentatn due t the mateal msmatch. When the sze f hle nceases, bth the lmt lad and evese plastcty lmt deceases. Fg. 8c shws that f a mateal at f 3, the addtn f a hle has lttle effect n the value f evese plastcty lmt, but causes a eductn n the lmt lad. The eductn n mateal by an nceasng hle damete s the cause f the eductn n lmt lad. Thee s lttle effect f the hle sze n the evese plastcty lmt 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 Cmpste Cylnde Thckness Fg. 9 pesents the effects f the adus at n the shakedwn lmt nteactn cuve. Thee dffeent elatve thcknesses.5,.75,. f the cmpste cylnde wth a fxed mateal at f wee analyzed. Inceasng ths adus at geatly nceases the lmt lad and educes the evese plastcty lmt. The ncease n lmt lad s an bvus esult, as effectvely the thckness f the ppe s nceased f the same nne adus. The eductn n the evese plastcty lmt s caused by the nceased thckness f steel. Ths ncease n thckness (whch causes geate cnductve tempeatues n the steel) esults n hghe themal stesses at the mateal bunday. 5.5 Fmulated hakedwn Lmt Desgn egn n elastc shakedwn lmt fmulatn f the cmpste cylnde s made f the safety f engneeng desgn. The elastc shakedwn desgn egns f cmpste cylndes ae shwn n Fg., whee Δ θ L s the desgn tempeatue ange cespndng t the evese plastcty lmt, P L s the desgn ntenal pessue epesentng the lmt lad and L s the desgn slpe f the atchet lmt cuve. In de t smplfy the fmulatn, Δ thee ndependent functns f, θ L P and L ae assumed t be the pduct f, L f, f ; g 3, g, g,and 3 h, h,

11 h 3 espectvely. The X dectn s the appled pessue L P and the Y dectn s the appled tempeatue dffeence L θ Δ. Theefe, the desgn shakedwn lmts ae fmulated as (8) (9) () Whee f, f, f 3, ae the nfluence functns f the desgn tempeatue ange cespndng t the evese plastcty lmt, g, g, g 3 ae the nfluence functns f the desgn ntenal pessue epesentng the lmt lad, and h, h, h 3 ae the nfluence functns f the desgn slpe f the atchet lmt cuve., and stand f the css-hle at, steel t alumnum at and thckness at, espectvely. θ L Δ, L P and L ae cnstants standng f the calculated evese plastcty lmt, the lmt ntenal pessue and the slpe f the atchet lmt cuve n case f 5., wthut a css-hle, whee, C L 53 Δθ (a) P L 8MPa 3. (b) MPa C L / ο (c) In de t fnd these nfluence functns, the btaned evese plastcty lmts, lmt ntenal pessue and the slpe f the atchet lmt cuve ae epltted n gaphs f functns f, g and h aganst, and espectvely as shwn n Fg., Fg. and Fg.3. Tend lnes ae ftted t the data btaned fm the shakedwn lmt esults f dffeent cmpste mateal at and cylnde thckness at wth dffeent css-hle szes t shw the nfluence functn. L L g g g P P 3 Δ Δ L L f f f 3 θ θ L L h h h 3

12 Equatns (a-c), (3a-3c) and (4a-4c) ae the btaned nfluence functns f the desgn tempeatue ange cespndng t the evese plastcty lmt, the desgn ntenal pessue epesentng the lmt lad, and the desgn slpe f the atchet lmt cuve, espectvely. Once L θ Δ, L P and L, ae defned, a safety shakedwn envelpe s ceated as shwn n Fg.. < < +.7) ( ) ( ) ( f (a) + ) (.495 3) 3 ( alumnum ) (.659 steel pue pue f (b).5) ( f (c) < + < +.7) ( ) ( ) ( g (3a) + ) (.433 3) 3 ( alumnum ) (.4 steel pue pue g (3b).5) ( g (3c)

13 h h ( ) (. (.3 ( pue ( pue ( 3 <.3) <.7) alumnum ) 3) steel ) (4a) (4b) h +.65 (.5.5) (4c) 6 Cnclusn The Lnea Matchng Methd has been vefed by step-by-step analyses, shwng that t gves vey accuate shakedwn lmts f the cmpste cylnde wth a css hle. The esult btaned usng the LMM f the cmpste cylnde wthut a css-hle shws that the lmt lad deceases wth the eductn f the steel mateal, wheeas the evese plastcty 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 a css-hle - a deceasng lmt lad and nceasng evese plastcty s lmt f deceasng vlume f steel. F steel t alumnum at 3, the exstence f a hle has lttle effect n the value f evese plastcty 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 plastcty lmt, but s mpacts the lmt lad less sgnfcantly than f a mateal at f 3. F a mateal at, 3 the hle s shwn t have neglgble effect n the lmt lad. Ths mples that the sze f the csshle ased the lcal stess cncentatn whch wll nfluence the fatgue lfe but wll nt geatly affect the glbal espnse when the lmt lad s detemned by the lw yeld stess f the dmnant alumnum mateal. Inceasng the cylnde adus at hghly nceases the lmt lad and educes the evese plastcty lmt. safety shakedwn envelpe s ceated by fmulatng the 3

14 shakedwn lmt esults f dffeent cmpste mateal and cylnde thckness ats wth dffeent css-hle szes. cknwledgements The auths gatefully acknwledge the suppt f the Engneeng and Physcal cences eseach Cuncl f the Unted Kngdm, and the Unvesty f tathclyde dung the cuse f ths wk. efeences. N, hmed K.,, tuctues technlgy f futue aespace systems, Cmputes and tuctues, 74, pp Thus, H.G..J., & Bemans, C., 997, Desgn fabcatn and testng f a cmpste backet f aespace applcatns, Cmpste tuctues, 38, pp 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 3(), 3-5. Lu YH, Cavell, Mae G., 997, Integty assessment f defectve pessuzed ppelnes by dect smplfed methds. Intenatnal Junal f Pessue essels and Ppng, 74, pp u, D.K., Yan,.M., Nguyen-Dang, H., 4, pmal dual algthm f shakedwn analyss f stuctues. Cmput. Methds ppl. Mech. Eng, 93, pp taat M., Hetze M.,, LI a Eupean Pject f FEM-based Lmt and hakedwn nalyss, Nuclea Engneeng and Desgn, 6, pp eshad,., 995, Inelastc Evaluatn f Mechancal and tuctual cmpnents Usng the Genealzed Lcal tess tan Methd f nalyss, Nucl. Eng. Des., 53, pp Mackenze, D., Byle, J. T., Hamltn, and. & h, J., 996, Elastc cmpensatn methd n shell-based desgn by analyss, Pceedngs f the 996 ME Pessue essels and Ppng Cnfeence, 338, pp Mackenze, D., Byle, J.T., Hamltn,.,, The elastc cmpensatn methd f lmt and shakedwn analyss: a evew, Tans IMechE, Junal f tan nalyss f Engneeng Desgn, 35, pp

15 . Chen, H.F., Pnte.,, hakedwn and lmt analyses f 3-D stuctues usng the Lnea Matchng Methd, Intenatnal Junal f Pessue essels and Ppng, 78, pp Chen, H.F. and Pnte,...,, Methd f the Evaluatn f a atchet Lmt and the mpltude f Plastc tan f Bdes ubjected t Cyclc Ladng, Eupean Junal f Mechancs, /lds, (4), pp Chen, H.F., Pnte,.. and nswth,.., 6, The Lnea Matchng Methd appled t the Hgh Tempeatue Lfe Integty f tuctues, Pat : ssessments nvlvng Cnstant esdual tess Felds, Intenatnal Junal f Pessue essels and Ppng, 83(), pp Chen, H.F., Pnte,.. and nswth,.., 6, The Lnea Matchng Methd appled t the Hgh Tempeatue Lfe Integty f tuctues, Pat : ssessments beynd shakedwn nvlvng Changng esdual tess Felds, Intenatnal Junal f Pessue essels and Ppng, 83(), pp Chen, H.F. & Pnte,..., 9, tuctual ntegty assessment f supe heate utlet penetatn tube plate, Intenatnal Junal f Pessue essels and Ppng, 86, Chen, H.F.,, Lwe and Uppe Bund hakedwn nalyss f stuctues Wth Tempeatue-Dependent Yeld tess, Junal f Pessue essel Technlgy, 3(), - 7. Pnte,... & Chen, H.F.,, mnmum theem f cyclc lad n excess f shakedwn, wth applcatn t the evaluatn f a atchet lmt, Eupean Junal f Mechancs - /lds,, pp Kte W T, 96, Geneal theems f elastc plastc slds, Pgess n sld mechancs J.N.neddn and.hll, eds. Nth Hlland, mstedam,, pp Melan, E. 936, Thee statsch unbestmmte systeme aus deal-plastchem baustff, tzubgsbe. kad. Wss. Wen, Math.-Natuwss. K., bt., 45, pp ME, 7, Ble and Pessue essel Cde, The mecan cety f Mechancal Engnees, New Yk. 5

16 Table Captns Table Mateal ppety paametes f the steel and alumnum Table Tempeatue-dependent yeld stess f alumnum 6

17 Table. Mateal ppety paametes f the steel and alumnum Type Yung s mdulus E (GPa) Pssn s at ν Ceffcent f themal expansn α ( C ) Yeld stess (MPa) y Themal Cnductvty k (W/mK) Densty (Kg/mm 3 ) teel lumnum Table Tempeatue-dependent yeld stess f alumnum Tempeatue (ºC) ( T ) (MPa) y 7

18 Fgue Captns Fg. LMM flw dagam f + teatn step Fg. Gemetcal shape f the cmpste cylnde Fg. 3 Quate fnte element mdels f dffeent mateal ats Fg. 4 a) Uppe and lwe bunds shakedwn lmt nteactn cuves f the cmpste cylnde b) the cnvegence cndtn f teatve pcesses f shakedwn analyss (pnt and *, s subjected t cyclc themal lads nly) (,. 75,. ) Fg. 5 BQU vefcatn usng step by step analyss f (a) the evese plastcty lmt (b) the atchet lmt Fg. 6 hakedwn lmt nteactn cuves f the cmpste cylnde f dffeent cmpste mateal at wthut a css-hle Fg. 7 hakedwn lmt nteactn cuves f the cmpste cylnde (. 5 ) f dffeent cmpste mateal at wth dffeent css-hle at: a). 3 b). c). Fg. 8 hakedwn lmt nteactn cuves f the cmpste cylnde (. 5 ) wth dffeent hle adus ats and dffeent cmpste mateal ats: a) s 3 s s b) c) 3 s Fg. 9 hakedwn lmt nteactn cuves f the cmpste cylnde ( ) wth dffeent thckness adus ats and dffeent hle adus ats: a) wthut hle b). c). d). 3 Fg. Elastc shakedwn desgn egns f cmpste cylndes Fg. Influence functns f evese plastcty lmts aganst: a) css-hle at b) steel t alumnum at c) thckness at 8

19 Fg. Influence functns f lmt pessues aganst: a) css-hle at b) steel t alumnum at c) thckness at Fg. 3 Influence functns f the desgn slpe f the atchet lmt cuve aganst : a) css-hle at b) steel t alumnum at c) thckness at 9

20 ssgn teatn numbe + F k, n (n vetces f the lad hsty) + y Lnea matchng: μ k ε Obtan the Jacban [J] + that elates t the ncements f stess k Calculate cnstant esdual stess Defne stan ate asscated wth n vetces at the lad hsty Calculate the shakedwn lmt multple: λ UB (Kte's uppe bund theem) λ LB (Melan's lwe bund theem) Cnvegence cndtn: λ λ + UB λ UB UB λub λlb e e λ LB N Yes Fg. LMM flw dagam f + teatn step

21 p MPa, θ C teel nset aea m L () t θ + Δθ ( t) p MPa, θ Fg. Gemetcal shape f the cmpste cylnde

22 (a) s 3 s (b) s (c) 3 Fg. 3 Quate fnte element mdels f dffeent mateal ats

23 .6.4 Δθ Δθ E B..8 * D * B F G.6 Uppe bund wth cnstant yeld stess (MPa).4 Lwe bund wth cnstant yeld stess (MPa). Uppe bund wth tempeatue dependent yeld stess P Lwe bund wth tempeatue dependent yeld stess C P (a) 3 hakedwn lmt multple.5 Uppe bund wth cnstant yeld stess (MPa) Uppe bund wth tempeatue dependent yeld stess.5.5 Lwe bund wth cnstant yeld stess (MPa) Lwe bund wth tempeatue dependent yeld stess Iteatns (b) Fg. 4 a) Uppe and lwe bunds shakedwn lmt nteactn cuves f the cmpste cylnde b) the cnvegence cndtn f teatve pcesses f shakedwn analyss (pnt and *, subjected s t cyclc themal lads nly) (,. 75,. ) 3

24 ..5 Equvalent plastc stan Cyclc lad case E shwng evese plastcty Cyclc lad case D shwng shakedwn Equvalent plastc stan Cyclc lad case G shwng atchetng Cyclc lad case F shwng shakedwn Numbe f cycles (a) Numbe f cycles (b) Fg. 5 BQU vefcatn usng step by step analyss f (a) the evese plastcty lmt (b) the atchet lmt 4

25 Δθ Δ θ 3 P p Fg. 6 hakedwn lmt nteactn cuves f the cmpste cylnde f dffeent cmpste mateal at wthut css-hle 5

26 Δ θ Δ θ 3 P (a) 3 Fully steel Fully alumnum p Δ θ Δ θ (b) 3 Fully alumnum Fully steel p P Δ θ Δ θ (c) 3 3 Fully steel Fully alumnum p P Fg. 7 hakedwn lmt nteactn cuves f the cmpste cylnde (. 5 ) f dffeent cmpste mateal at wth dffeent css-hle at: a). 3 b). c). 6

27 Δθ Δ θ.3.. Wthut hle Δθ Δ θ.3.. Wthut hle Δθ Δ θ.3.. Wthut hle P p p P p P (a) (b) (c) Fg. 8 hakedwn lmt nteactn cuves f the cmpste cylnde (. 5 ) wth dffeent hle adus ats and dffeent cmpste mateal ats: a) s 3 s s b) c) 3 7

28 Δθ Δ θ p P Δθ Δ θ p P Δθ Δ θ (a) (c) p P Δθ Δ θ (b) (d) p P s Fg. 9 hakedwn lmt nteactn cuves f the cmpste cylnde ( ) wth dffeent thckness adus ats and dffeent hle adus ats: a) wthut hle b). c). d).3 8

29 Δ θ Δ θ L L P L P Fg. Elastc shakedwn desgn egns f cmpste cylndes 9

30 . f. f f (a) (b) (c) Fg. Influence functns f evese plastcty lmts aganst: a) css-hle at b) steel t alumnum at c) thckness at 3

31 . g.4 g.5 g (a) (b) (c) Fg. Influence functns f lmt pessues aganst: a) css-hle at b) steel t alumnum at c) thckness at 3

32 . h.5 h h (a) (b) (c) Fg. 3 Influence functns f the desgn slpe f the atchet lmt cuve aganst: a) css-hle at b) steel t alumnum at c) thckness at 3