Parametric Examination including Brief Survey of Composite and Homogenous Closed Ended Cylindrical Pressure Vessels

Transcription

1 Jacb Nagle Paametc Examnatn ncludng Bef Suvey f Cmste and Hmgenus Clsed Ended Cylndcal Pessue Vessels JACOB NAGLER Faculty f Aesace Engneeng Technn Hafa 3000 ISRAEL syank@tx.technn.ac.l syanktx@gmal.cm Abstact:- Ths ae esents aametc desgn examnatn tgethe wth bef suvey f cmste and hmgenus cylndcal essue vessels wth hemshecal ends. Tw man knds f cmste and hmgenus cylndes gemety wee mdeled and examned: thn and thck cylndes. Stess-stan elatnshs f the cylndes have been deved as a esult f the mdel equatn slutn. Addtnally enfced wang methd effect n thn cylnde stength was esented and examned. Als cmund cylnde they was dscussed and man esults wee esented. Meve thck walled cmste cylndes n la cdnates they has been develed and dslayed tgethe wth the flament wndng methds thugh analytc mdelng and numecal examnatn. Fnally falue ctea f cmsed and hmgenus cylndes wee esented. Keywds:- Cmste Hmgenus Cylnde Pessue vessel Clsed ended Flament wund. I. Intductn Pessue vessels ae n use f a lng tme. Usually the ductn f essue vessels was dne fm nly ne secfc mateal (ceamc wd n ancent ed whle metal ally n mden tmes). Dung the yeas me and me technlgy alcatns have made t ssble t duce vessels fm me than ne mateal. Mltay and aesace ndusty secal ducts equements f hgh stength and lght-weght have lead (tgethe wth the easns) t the use f cmste mateals. F examle n tday's aesace maket 30-50% f the mden alane stuctual cmnents ae made f cmste mateals. In atcula cylndcal the vessels cmnents whch ae usually have been manufactued by flament wndng. Althugh they aea t be smle stuctues cmste and hmgenus (nn-cmste mateal) cylndes ( the knds f essue vessels) ae amng the mst dffcult t desgn. Flament-wund cmste essue vessels have fund wdesead use nt nly f mltay use but als f cvlan alcatns. Pessue vessel s a clsed cntane desgned t hld gases and/ lquds at a essue substantally dffeent fm the ambent essue. The essue dffeental s dangeus and fatal accdents have ccued n the hsty f essue vessel develment and eatn. Cnsequently essue vessel desgn manufactue and eatn ae egulated by engneeng authtes backed by legslatn. F these easns the defntn f a essue vessel vaes fm cunty t cunty but nvlves aametes such as maxmum safe eatng essue and temeatue. Pessue vessels ae used n a vaety f alcatns lke aesace engneeng (alane-cabn fuel-chambe and engnechambe) mechancal engneeng (neumatc and hydaulc esevs) chemcal engneeng (l efney and dstllatn twes) etc. Pessue vessels can theetcally have almst any shae the evalent sectn 1 E-ISSN: Vlume 9 014

2 Jacb Nagle shaes ae f shees cylndes and cnes. A cmmn desgn s a cylnde wth end cas called 'heads'. Head shaes ae fequently ethe hemshecal dshed (t-shecal). Me cmlcated shaes have hstcally been much hade t analyze f safe eatn and ae usually fa me dffcult t cnstuct. Theetcally a shecal essue vessel has axmately twce the stength f a cylndcal essue vessel wth the same wall thckness [1]. Althugh shecal shaes ae dffcult t duce and theefe me exensve s mst essue vessels ae cylndcal wth :1 semelltcal heads end cas n each end. Theetcally almst any mateal wth gd tensle etes that s chemcally stable n the chsen alcatn culd be emlyed. Hweve essue vessel desgn cdes and alcatn standads (see A. - Table A.1.) cntan lng lsts f aved mateals wth asscated lmtatns n temeatue ange. Deste the gwng use f cmste mateals essue vessels whch ae made f steel ae stll elevant. T manufactue a cylndcal shecal essue vessel lled and ssbly fged - ats wuld have t be welded tgethe. Sme mechancal etes f steel acheved by llng fgng culd be advesely affected by weldng unless secal ecautns ae taken. In addtn cuent standads dctate the use f steel wth a hgh mact esstance esecally f vessels used n lw temeatues. In alcatns whee cabn steel wuld suffe csn secal csn esstant mateal shuld als be used. Sme essue vessels whch ae made f cmste mateals such as flament wund cmste usng cabn fbe ae held n lace wth a lyme. Due t the vey hgh tensle stength f cabn fbe these vessels can be vey lght but ae much me dffcult t manufactue. The cmste mateal may be wund aund a metal lne fmng a cmste vewaed essue vessel. In de t event leakng and tect the stuctue f the vessel fm the cntaned medum essue vessels may be lned wth vaus mateals lke metals ceamcs lymes. Ths lne may als cay a sgnfcant tn f the essue lad [ -3]. Pessue vessels may als be cnstucted fm cncete (PCV) the mateals whch ae weak n tensn. Cablng waed aund the vessel wthn the wall the vessel tself vdes the necessay tensn t esst the ntenal essue. A "leak-f steel thn membane" lnes the ntenal wall f the vessel. Such vessels can be assembled fm mdula eces and s have "n nheent sze lmtatns" [4]. Thee s als a hgh de f edundancy thanks t the lage numbe f ndvdual cables esstng the ntenal essue. Analytcal slutn f stess feld develment n thn thck walled nncmste (hmgenus) essue vessels wee made by R. Budyans et al. [5]. In cases whee cmste vessels calculatns ae me cmlcated Gbsn [6] Kaw [7] Klla and Snge [8] Bake et al. [9] and thes have lad the fundamentals f such calculatns. On the ne hand cmste vessels ae lghtweght and wth hghe stength n the elevant dectns than steel alumnum cunte-ats. On the the hand manufactung methds can be me cmlex and exensve than metallc cunteats. Ths ae esents stess-stan elatnshs mdels f cmste and hmgenus cylndcal vessels ncludng the hemshecal ends. II. Analytcal Mdel f Hmgenus Cylndcal Vessel wth Hemshecal Ends The ccula cylnde can be classfed nt tw tyes: thck walled cylnde and thn walled cylnde. A thn walled E-ISSN: Vlume 9 014

3 Jacb Nagle cylnde s a cylnde whch hs tangental stess (whch s als called 'h stess' 'ccumfeental stess') s nvesely tnal t the thckness and s dectly tnal t the nne adus. Als the tangental stess shuld be develed n secfed lmts. The tanstn fm "thn" t "thck" defntn ccus asymttcally when the nne adus s n lnge 10 tmes ('thumb ule') geate than the thckness f the cylnde. As a esult tangental and adal stesses ae vaed nn-lneaty wth nne adus. In the fllwng sectns these dffeences wll be demnstated begnnng wth thn walled cylnde subject. l t c c l (3) c tc whle the tangental stess n the hemshecal ends s exessed by: t s s (4) s l t s s whee t s s the shecal suface thckness. The lngtudnal stess n the hemshecal ends has the same A. Thn Walled Cylnde wth l( c) c t calculatn by Hemshecal Ends l c c c t t Stess feld develment n thn walled cylnde as aea n Fg. 1 s sncet c s small enugh. descbed badly by Önde [10]. The man assumtns f usng ths methd ae: Usng Eq. (4) calculatn the maxmum 1. Plane sectns eman lane. tangental stess n the shecal ends s:. / tc 10 ('Thumb-ule') ( ts / ). (5) s max wth t c beng unfm cnstant cylnde ts thckness and s the ntenal adus. 3. The aled essue s the ntenal essue nly (snce extenal essue may cause bucklng f the wall). 4. Mateal s lnea-elastc stc and hmgeneus. 5. N vaatn f stess dstbutns thughut the wall thckness. 6. Flud densty s neglgble. 7. N tatn. 8. Hemshecal ends. Tangental stess n the cylnde can be exessed by: L tcl c. (1) (1) c tc whle L s the cylnde length. Hence the maxmum tangental stess s: ( tc / ). () c max tc The lngtudnal stess n the cylnde s calculated as fllws: calculatn as s. By the way Eqs. (3-4) assumest c 1n shecal suface l s 3 E-ISSN: Vlume 9 014

4 Jacb Nagle (a) (b) Fgue 1. Thn cylndcal shell llustatns: a. Thn cylndcal shell wth hemshecal ends half sectn vew. b. Css sectn f hemshecal end ('Head'). It seems that f t s t c then tangental stess n the shecal ends s half f the tangental stess n the cylnde bdy. It's als tue f the lngtudnal stess. Meve lngtudnal stess has n deendency n hemshecal thckness. The elatn between t s andt c s defned by h stan elatns as fllws: (6) S t seems that thckness at deends nly n Pssn's at. Me abut the jnt egn between cylnde and hemshecal ends can be studed fm Bhadu [14]. The stength f the cylnde cmnent s deendent n jnt effcency (lke weldng veted jnts/nals etc). Ths effcency s exessed n the thn laye mdel by usng effcency cnstant wth Eqs. (1 3-4) as fllws: l c c s t t (7) t c c l c s whle c l s dente effcences f tangental lngtudnal and shecal jnts esectvely. On the ne hand t can be bseved fm Eqs. (3-4) that thckness nceasng causes tangental stess deceasng whch s excellent f mateal stength. On the the hand t esults n an ncease n densty whch means that me mass f the acaft cket aylad s added. Realzatn f ths blem may lead t gemetcal tade-ff change n gemetcal ends n the geneal shae whch wuld have a stve effect n deceasng mass and stength. F examle t's well knwn [511] that shecal shae yelds equalty between lngtudnal and tangental stesses. The 'ay' f ths knd f shae takes lace n manufactung methds whch ae vey exensve. Ths late sal f changng ttal gemety wn't be checked n ths essay. Altenatvely usng the mateals and dffeent thckness ats (wang methds) wll be examned. In the next sectn the thckness asect wll be dscussed n the cntext f thck wall cylnde. B. Thck Walled Cylnde wth Hemshecal Ends In case whee the fme assumtn ('thumb ule') f at between thckness and nne adus at desn't exst whee / tc 10 (see Fg..a) we wll use the fllwng cedue. The elevant assumtns f ths case ae: 1. Plane sectns eman lane. s E-ISSN: Vlume 9 014

5 Jacb Nagle. / tc 10 wth t c beng unfm cnstant cylnde thckness and s the ntenal adus. 3. Unfm ntenal ( ) and/ extenal ( ) essue. 4. Defmatn s symmetcal abut z axs. 5. Mateal s lnea-elastc stc and hmgeneus. 6. Flud densty s neglgble. 7. N tatn. 8. Hemshecal ends. Fstly geneal la equatns f element equlbum usng Fg..b wll be wtten: 1 ˆ : F 0. (8) ˆ 1 : F 0 whle and dente the tangental and adal stesses actng nmal t the sdes f the element. F F eesent the adal and tangental bdy fces esectvely. Snce flud densty s neglected and n tatn s exst (assumtns 6 + 7) t can be cncluded that F 0. Meve F shea stess fulflls that 0 due t symmetcal defmatn aund z axs (assumtn ). All these assumtns lead t ne equlbum equatn n ˆ dectn nly: 0 (9) whle stans behave accdng t Hk's law by: u 1 E u 1. (10) E 1 u 1 0 G whee u s the adal dectn dslacement. E G eesent mdulus Yung and shea mdulus esectvely. s Pssn's at. Slvng Eq. (10) f stess feld yelds the fllwng elatns: E E u u 1 1 (11) E E u u 1 1 (a) (b) Fgue. a. Thck cylndcal subjected t bth ntenal and extenal essue. b. Stess element n la cdnates system. Substtutng the abve elatns (11) nt equlbum Eq. (9) esults n the fllwng adal dslacement equatn: u 1u u 0 (1) wth B.C.: The slutns f f stess feld and adal dslacement by slvng Eq. (1) usng (11) ae: 5 E-ISSN: Vlume 9 014

6 Jacb Nagle 1 u E 1 E In case whee each f these essues ( ) acts alne qualtatve stess dstbutns ae shwn n Fg. 3. Als quanttatve stess dstbutns ae shwn n Fg. 4.a-b f / 8. Snce we deal wth hemshecal ends whch takes eactns fm ntenal and extenal essues ( and ) the lngtudnal dectn needs t be cnsdeed. By smle calculatn we get the fllwng exessn f cylnde lngtudnal stess: lc lc (13). (14) Addtnally lngtudnal stess n the thck cylnde ends s (usng hemshecal suface aea - ): ls ls s s s s s. (15) whle eesents the tangental stess n s the hemshecal end and s s shecal ute adus ( t s ). the Table 1. Maxmum adal stess value accdng t Eq. (13). Intenal and Extenal essues ats max It seems fm Table.1 that the maxmum adal stess value accdng t Eq. (13) s btaned f whle. Othewse ( ) the maxmum adal stess s btaned f and equals. P (a) P Fgue 3. a. Qualtatve stess and dslacement dstbutn sectn ma f thck cylnde subjected t ntenal essue. b. Qualtatve stess and dslacement dstbutn sectn ma f thck cylnde subjected t extenal essue. Seaate dscussn n the maxmum tangental stess whch ccus at ethe the nne utsde edge accdng t essue at wll be elabated hee. Accdng t Ugual and Fenste [13] tangental stess lmtatns f thck cylndes wee studed fstly by Ranv and Pak [15]. The examnatn wll be dvded nt thee ats: (1) Intenal essue nly () Extenal essue nly and bth (3) Intenal & Extenal essues. In cases (1) + () the maxmum tangental stess s btaned at as shwn n Table. but thee ae seveal the cases when bth essues ae actng whle the maxmum tangental stess s btaned at. In de t nvestgate thse cases we wll use the fllwng ntatn accdng t [13]: 1 PR 1 (b) P R 1 R 1 (16) P R Hence 1 PR R S. (17) PR P The maxmum value magntude f tangental stess s btaned at the ed ( S 1) and the ule ( S 0 ) dashed lnes 6 E-ISSN: Vlume 9 014

7 Jacb Nagle n the ute suface egn ( ) as shwn n Fg.4.c. The abslute dslacement dstbutns gahs f bth ntenal and extenal essues ae exhbted n Fg.4.d. These gahs behav fts wth tangental and adal stess behav. Excellent esult cmatblty and sustanng was fund between Ugual & Fenste [13] esults and Fg. 4. Cmasn between thck cylnde t thn cylnde n case whee nly ntenal essue eates (snce bucklng can ccu n thn vessel wth extenal essue) leads t cnclusn that dmnshes wth thckness n thck cylnde because f denmnat and numeat deceasng. By sayng that snce t and s t s btaned. As a esult by usng sme algebac nequalty s btaned. manulatns Thck cylnde Thn cylnde Table. Maxmum tangental stess value accdng t Eq. (13). Case max Intenal essue nly Extenal essue nly Intenal and Extenal essues n the bald lnes Fg.4.c See Fg.4 Intenal and Extenal essues n the dashed lnes Fg.4.c See Fg.4 7 E-ISSN: Vlume 9 014

8 Jacb Nagle (a) (b) (c) (d) Fgue 4. Illustatns f thck walled cylnde subjected t: a. Intenal essue adal and tangental stesses and dslacement dstbutns. b. Extenal essue adal and tangental stesses and dslacement dstbutns. c. Intenal and extenal essues tangental dstbutn. d. Intenal and extenal essues dslacement dstbutn f 0.3. C. Hmgenus Cylndcal Vessel Falue Ctea In ths sectn we wll dscuss the falue ctea f hmgenus (nn-cmste mateal) cylndcal vessels. Accdng t [13] thee ae fve man ctea: 1. Maxmum ncal stess cten assumes that mateal falue ccus when the maxmum ncal stess ( 1 ) n a mateal element exceeds the unaxal tensle stength f the mateal. Pactcally seakng ne shuld check the maxmum stesses ( max max lmax ) accdng t elevant case (Thn-wall Thck wall) and cmae t t the mateal tensle elastc stength (yeld stess - ):. (18) y max max lmax y Altenatvely ths cten can be used n de t calculate the desable essue value ( ).. Maxmum sheang stess cten als knwn as Tesca yeld cten s ften used t edct the yeldng f ductle mateals. Yeld n ductle mateals s usually caused by the slage f cystal lanes alng the maxmum shea stess suface. Theefe a gven nt n the bdy s cnsdeed safe as lng as the maxmum shea stess at that nt s unde the yeld shea stess y (whch s usually half f the metal tensle yeld stess - y / ) btaned fm a unaxal tensle test. Ths cten shuld fulfll: max y. (19) 8 E-ISSN: Vlume 9 014

9 Jacb Nagle Altenatvely t can be used t evaluate essue values ( ). Ths they s used usually wth bttle mateals. 3. Enegy f dsttn they ses that the ttal stan enegy can be seaated nt tw cmnents: the vlumetc (hydstatc) stan enegy and the shae (dsttn shea) stan enegy. It s sed that yeld ccus when the dsttn cmnent exceeds the stan at the tensle yeld nt f a smle tensle test. A fmulated mdel f ths they s n the fllwng fm: (0) l l l y whle all cmnents f shea stess ae ze (assumtn). Altenatvely ths cten can be used n de t calculate the desable essue value ( ). 4. Maxmum ncal stan they was sed by St.Venant. Accdng ths they yeld ccus when the maxmum ncal stan eaches the stan cesndng t the yeld nt dung a smle tensle test. In tems f the ncal stesses t s detemned by the fllwng equatn: l. (1) Altenatvely ths cten can be used n de t calculate the desable essue value ( ). 5. Octahedal sheang stess they wll be defned wth ntes by Wlf et al. [16]. Octahedal lane cuts acss ne f the cnes f a ncal element s that the eght lanes tgethe fm an ctahedn. Fgue 5 llustates the entatn f ne f the eght ctahedal lanes whch ae asscated wth a gven stess state. All 8 lanes have dentcal nmal stesses. Als nmal stesses dn't defm the ctahedn. Meve shea stesses ae als wth dentcal values and they may defm the ctahedn wthut changng ts ttal vlume. The cten mtance s deved fm the fact that ctahedal shea stess s smalle than the hghest y ncal shea stess but t cnsttutes a sngle value that s nfluenced by all thee ncal shea stesses. The cten s gven by the fllwng cndtn: l l y y ct () 3 3 whle all shea stess cmnents equals t ze (assumtn). Ths cten can be aled wth all mateals snce n mateal etes ae nvlved. Fgue 5. Octahedal Planes asscated wth a gven stess state. In cnclusn n de t undestand falue mechansm all falue ctea shuld be checked and cmaed ne aganst each the. Usually ctahedal sheang stess cten and enegy f dsttn cten ae cncded wth each the (see als Ref. [13]). III. Cmund Cylndes Cmund cylndes ae cmbned f tw cylndes whch sused t be me effectve by educng tangental and adal stesses and t hld lage essues. Thee ae seveal dffeent methds f vessels 9 E-ISSN: Vlume 9 014

10 Jacb Nagle ductn such as clng & heatng wang methds essue cmessn methds shnkng methds extusn etc. The mdel calculatn hee assumes that bth cylndes ae made f tw dffeent mateals wth aate mdulus Yung and Pssn's at n the nsde E and utsde E cylndes esectvely. It s assumed that hemshecal ends wll nt be ncluded n ths sectn. Suse that extenal adus f the nne cylnde s lage than ntenal adus f the ute cylnde by dffeence f as shwn n Fg.6.a. (a) (b) Fgue 6. Cmund cylndes sectns: a. Befe nsetn. b. Afte nsetn. Ths henmenn causes the ute cylnde t 'feel' an ntenal essue. As a esult f Newtn's thd law the nne cylnde 'feels' an extenal essue. Due t the adal ntefeence between bth ute and ntenal cylndes an ntenal essue cntact s develed whch causes t the fllwng equlbum f adal dslacement: whle R s the nmnal adus. Eq. (3) has been btaned usng Eq. (13) f secal cases (ntenal and extenal adal dslacements) wheeas have been altenatvely swtched t nteface essue ( c ) esectvely. Inteface essue cntact can be slated usng Eq. (3) n the fllwng fm: c (4) R R R R E R E R (3) whle all ntenal stesses ( l ) shuld be calculated accdng t cylnde essue behav (ntenal essue extenal essue). Paamete tmzatn blem f cmund cylndes wee slved analytcally and numecally f tw and thee layes wth vaus cnstants by Majzb & Ghm [17] and by Maje & Patl [18] esectvely. Ref. [18] gves exact cedue f estmatng the tmal de f the layes tgethe wth tangental stess exessns. In addtn when the cmund cylnde s subjected t a wkng essue (ntenal essue) a eactng essue s develed at the cntact suface f the matng cylndes and s gven by [17]: (5) R R E1 R E R E R E The essues accdng t Eqs. (4-5) act as extenal essues f the nne cylnde and as ntenal essues f the ute cylnde. Hence we get the fllwng 10 E-ISSN: Vlume 9 014

11 Jacb Nagle elatns f ntenal and extenal essue vessel: Intenal laye: (ntenal essue) Wkng essue Oute laye: 0 c c (6) whle stesses n bth cylndes wll be calculated usng Eqs. (13 6). The genealzed they f cmund cylnde wth n layes wuld be develed hee accdng t fllwng cedue. Cnsde a geneal cmund cylnde wth cncentc n cylnde layes as llustates belw n Fg.7. Each cylnde laye adus s nted by: whle j ange j n fulflls1 j n. Theefe the nmnal adus j s eesented by j 4... Each adjacent a layes adal ntefeence wll be nted by k 1 k... n 1 n whle k n. essues. Ths sectn esents nly the fst at examnatn (see exlanatn belw). The slutn f ntenal essue at the nne laye s shwn n Table. 3 belw. Ntce that slutn f cmund cylndes wth extenal essue nly (b) s the same as n ths case (a) wth ne dstnctn - calculatn shuld begn fm ute laye t the nne laye. F me nfmatn see Ref. [17-18]. The case f bth extenal and ntenal essue eatng (c) s equed me cmlcated calculatns wth cnvegence cesses. k 5 n Fgue 6. Cmund cylnde wth n cylnde layes. Als each laye has ts wn mateal etes such as ( E )( E )...( E )...( E ). 1 1 j j n n Futhe analytc examnatn f the blem wuld be dvded nt thee man ats: (a) ntenal essue nsde the nne laye ( ) (b) extenal essue at ute laye ( ) (c) bth ntenal and extenal 11 E-ISSN: Vlume 9 014

12 Jacb Nagle Table 3. Geneal slutn f cmund cylnde subjected t ntenal essue. Layes Radal ntefeence ( ) Cntact essue ( c ) 1 1 (gven) u1 u E 3 E3 4 3 k 1 k uk 1 u k 1 k k k 1 k k 1 k Shuld be calculated accdng Eq. (13) E 3 E3 4 3 Layes and due t cntact essue E1 1 E 3 1 and 1 (Substtute n Eq.(13): c ) Laye 1 Laye 1 3 and Laye (Substtute n Eq.(13): Laye 0 3 c ) and (Substtute n Eq.(13): Laye Laye 1 ) k k k 1 k k 1 k c k 1 k k 1 k k 1 k c m m 1 m m 1 k k whle 3 m n and k n IV. Cmste Cylndes wth Hemshecal Ends Ths sectn esents analyss methds f cmste cylnde. Cylndes whch ae fmed usng cmste mateals have been nvestgated f the last fu decades. Tw eseaches fm NASA Catn Nemeth and Mkulas [19] have been nvestgated stffness and the lamnated - cmste cylnde aametes f bucklng henmenn. Futheme Tattng [0] has examned desgn aametes f vaable stffness cmste cylndes. Snce we ae dealng wth cmste mateal t wll be useful t mentn that thee ae fu man tyes f cmste mateals stuctues: fbe sandwch hneycmb and mult hases. F me nfmatn see Ref. [91]. Ths sectn wll be fcused n the fllwng subjects: We wund thn cylndes wth hemshecal ends. Thck walled cmste cylndes n la cdnates. Flament wndng methd estmatn f cmsed cylnde. Falue ctea f cmste mateals. A. We Wund Thn Cylndes wth Hemshecal Ends We wund thn cylndes whch ae als knwn as 'We wund thn cylndes' 'Cmste enfcement'[3] ae used t ncease stength f thn cylndes t wthstand hgh ntenal essues wthut excessve ncease n wall thckness as aea n Fg. 8. By usng ths methd weght and cst can be saved. Sme ways f dng t use wund wth hgh tensle steel tae we unde tensn. Accdng t Hen [] the enfcement causes the cylnde t 'feel' an ntal cmessve h stess whch must be vecme by the stesses wng t ntenal essue befe the mateal s subjected t tensn. It emans at ths stage befe the 1 E-ISSN: Vlume 9 014

13 Jacb Nagle maxmum allwable stess n the cylnde s exceeded. Calculatng the tensn that s develed n the tae dung the wndng cess n de t ensue that the maxmum h stess n the cylnde wll nt exceed a cetan value when the ntenal essue s aled wll be efmed hee. Ths sub sectn takes ste ahead wth genealzatn f Hen's they by egadng t hemshecal ends and ntal ntenal essue. Oute Tae Cveng cylnde tae Cylnde Fgue 7. Sectn f a thn cylnde wth an extenal laye f tae wund n wth a tensn. whle eesents the cmessve tae stess 1 s the ntal ntenal essue - ntenal cylnde adus. t t eesent the thcknesses cylnde tae f cylnde and tae esectvely. The equlbum afte changng ntenal essue s: tae Lt fnal tae cylnde Lt fnal cylnde L f (8) t f 1 tae (7) whle f eesents ntal stess afte t addng subtactng cylnde essue. Equatng dffeences f cylnde stan and tae stan n the tangental dectn leads t the fllwng elatn: The cylnde bdy ntal state equlbum s: tae tae l tae tae tae l tae fnal fnal ntal ntal tae Etae whle cylnde cylnde cylnde l cylnde cylnde cylnde l cylnde cylnde tae fnal fnal ntal ntal E cylnde. (9) ae the Pssn's at f cylnde and tae esectvely. Next ste s t detemne usng enegy f dsttn cten (0): cylnde fnal l cylnde fnal cylnde fnal l cylnde fnal cylnde y fnal. (30) 13 E-ISSN: Vlume 9 014

14 Jacb Nagle Theefe by slvng Eqs. (7-30) tgethe wth lngtudnal exessn (3) we get the fllwng elatns f cylnde tae egn n Table. 4. Table 4. Geneal slutn f we wund thn cylndes case. Case cylnde tae lcylnde Intal Fnal 1 tae ntal tae t cylnde f 3 f y 4t cylnde 3t cylnde t cylnde fnal tcylnde E tae Etae 1 f ttae E cylnde Ecylnde tcylnde t tae Etae ttae 1 E t cylnde cylnde 1 f tae tae cylnde ttae Ecylnde t cylnde E 1 E tae cylnde t t E tae cylnde t t tae ltae t t 1 cylnde cylnde fnal cylnde f f It's nly left t examne the hemshecal ends stess behav (esecally when vey hgh essues act we want t stength the hemshecal ends). In smla way t the cylndcal bdy the hemshecal ntal state equlbum s: cylnde t t 1 tae f tae (31) whle s eesents the cmessve stess and 1 s the ntal ntenal essue. The equlbum afte changng ntenal essue s: ( t ) t t. (3) s tae stae stae cylnde 1 fnal s cylnde s f f fnal Nw by equatng between stan dffeences f hemshecal ends and hemshecal tae egn we btan the fllwng elatn: tae tae s l s tae s tae tael s tae fnal fnal ntal ntal stae Estae cylnde cylnde s l s cylnde s cylnde cylndel s cylnde fnal Next stage s t detemne fnal s cylnde fnal E scylnde ntal ntal scylnde usng enegy f dsttn cten (0) such as: s cylnde fnal ls cylnde fnal ls cylnde fnal ls cylnde y fnal. (33). (34) 14 E-ISSN: Vlume 9 014

15 Jacb Nagle Theefe by slvng Eqs. (31-34) tgethe wth lngtudnal exessn (4) we get the fllwng exessns f shecal ends cylnde tae egn that ae summazed n Table. 5. Table 5. Geneal slutn f we wund thn cylndes case. Case s cylnde tae Intal Fnal tae ( t ) t 1 s stae stae ntal t scylnde f 3 f y 4t scylnde 3t scylnde s See Eq. (35) f cylnde t s scylnde fnal ( t ) t stae stae l s cylnde l s tae t t 1 scylnde f scylnde t 1 stae t stae t f stae t stae E s tae ntal E stae scylnde t t ( t ) t t Estae ( t ) t 1 E t f cylnde scylnde fnal s stae s cylnde fnal stae stae stae scylnde cylnde 1 f tscylnde scylnde stae stae scylnde stae stae scylnde t Estae ( t ) t 1 E t 1 scylnde tae 1 f In de t smlfy ths they t can be assumed that 1 0. The easn f ths assumtn s because dung wndng cess at the begnnng f gas\lqud nsetn ntal amunt f gas\lqud essue (ntal state) s usually neglected ( nt exsts). Meve the whle develment f Eqs. (7-35) eesents tanstn fm ne state t anthe. Anthe smlfcatn f calculatn can be efmed by assumng ne equlbum state but nly as lng as the cylnde mateal has nt yelded. Smlfed stess-stan elatns f ne equlbum state aea n Kunz ntes [3]. Hweve ths methd can als be used f any the wang cesses nt nly f cmste wa. (35) 15 E-ISSN: Vlume 9 014

16 Jacb Nagle (a) Fgue 9. Thck walled cmste cylnde. (b) Fgue 8. Flament wndng f cylndcal tanks. a Helcal layes cve cylndes lus dmes. b H les cve cylnde sectn nly. These hts ae taken fm Kunz ntes [3]. B. Thck walled cmste cylndes n la cdnates In ths sub-sectn a la cdnates aach t cmste cylnde wll be ntduced and develed. Suse we have a geneal cylnde wth nne adus and ute adus. Addtnally the cylnde s made f cmste mateals as llustated n Fg. 9. Snce aveaged tangental elastcty mdulus E and aveaged adal elastcty mdulus E ae functns f fbes numbe they ae nt necessaly equal. The stess elatns n la cdnates ae: E E E E (36) whle stess and stan feld n me geneal fm ncludng dectns and stffness chaactestcs ae gven by: 1 E 1 E. (37) E E 1 1 E E 1 1 Substtutng stesses accdng t Eq. (37) n equlbum Eq. (9) yelds: E E E. (38) E E E E 0 Substtutng axs-symmetc la cdnates stans exessns accdng t Eq. (10) nt Eq. (38) leads t: E E E u. (39) E E E u u E 0 In case whee E E ae cnstants we get the smle Eule' slutn t Eq. (39) whch s: 16 E-ISSN: Vlume 9 014

17 Jacb Nagle u A A k1 k 1 whle A 1 A ae cnstants. It s easy t E E E E E 4 (40) bseve that f E 0 we get k1. E E E E k1 In de t fnd enclsed fmula f adal dslacement and stess feld ne shuld use B.C. as assumed n (1) tgethe wth (37) n de t calculate A 1 A cnstants. Ths calculatn can be ease usng the fllwng ntatn: k11 k1 k1 k 1 Ek1 E A 1 Ek E k11 k1 k1 k 1 E A k1 E Ek E. (41) 1 b b A b3 b 4A Slvng system (41) leads t the fllwng cnstants defntn: 1 b b 4 1 b3 b1 A1 A. (4) E b1b 4bb3 E b1b 4bb3 b1 b Althugh t s equed that the detemnant wll fulfll: 0. In case whee b b 3 4 E 0 (when all fbes ae gven n tangental and adal dectn nly) we have: whle k11 k1 E k1 k A 1 k11 k1 (43) 1 k A 1 k 1 1 k1 k1 k11 k11 1 A E k1 k A k11 k1 k1 k11 k11 k1 k1 k11 E. (44) Hence the exessns f adal dslacement n the geneal fm whee E 0 s: k1 k1 k11 k11 1 k 1 k u. (45) E k k k11 k k k k k k k 1 whle stess shuld be deved usng Eqs. (10 37). Ntce that snce we deal wth hllw cylnde the case whee 0 s nnelevant. The lngtudnal stesses calculatns f cylnde and shecal ends wll be efmed usng Eqs. (14-15) esectvely. Geneally accuate calculatn s acheved by cnsdeng shecal ends usng lngtudnal mdulus as exlaned by Gbsn [6]. If the shecal ends ae made f cmste mateal the adal and tangental stesses shuld be calculated seaately wth cmste methd calculatns FEM (fnte element methds) [4-5]. Fm hee we wll ass t u next dscussn abut wndng methd estmatn f cmsed cylnde. C. Flament wndng methd estmatn f cmsed cylnde Ths sub sectn eques me deely vevew f the cmste mateal they. The easn f that s deved fm the cmste mateal cmlex behav. Intally we wll stat wth undestandng the wndng methd whch s used t duce the cmsed cylnde. Accdng t Önde [10] 17 E-ISSN: Vlume 9 014

18 Jacb Nagle thee ae thee knds f wndng attens as shwn n Fg.10 and wll be elabated hee. The fst atten s called 'H Wndng' whch s knwn as tangental ccumfeental wndng. H wndng s chaactezed by hgh helcal angle f 90 [deg].each full tatn f the mandel advances the band delvey by ne full bandwdth as shwn n Fg. 10.a. The secnd atten s called 'Helcal Wndng' whch s dne by mandel tatn at a cnstant seed whle the fbe feed caage tansveses back and fth at a seed egulated t geneate the desed helcal angles as shwn n Fg. 10.b. The thd atten s called 'Pla Wndng' whch s dne by fbe assng tangentally t the la enng at ne end f the chambe eveses dectn and asses tangentally t the ste sde f the la enng at the the end usng the mandel am tatn abut the lngtudnal axs as shwn n Fg. 10.b. It s used t wnd almst axal fbes n dmed end tye f essue vessels. On vessels wth aallel sdes a subsequent ccumfeental wndng wuld be dne. Clea bsevatn n these thee wndng attens leads t helcal wndng cess advantage wth geat vesatlty. Usually cylnde vessel cmste ae duced by means f helcal wndng. In each atten ne can vay wndng tensn wndng angle and/ esn cntent n each laye f enfcement untl desed etes such - thckness and stength f the cmste ae acheved. The etes f the fnshed cmste can be vaed by the tye f wndng atten selected. Tadng ff wnd angle and ccuts t clse the attens can ceate any cmbnatn f damete and length by wund. (a) (b) (c) Fgue 10. Wndng attens fm Önde [10]. a. H wndng. b. Helcal wndng. c. Pla wndng. In de t undestand bette the mtance f the flament wund cmste n ths feld we wll take a bef lk n the fllwng tables fm Ref. [10]. Accdng t Table A. n the Aendx t seems that the mateal etes f flament wund cmste ae smla t thse f stanless steel ttanum ally wth beneft f lw densty. Meve thee ae many alcatns f flament wund cmste methd accdng t Table. A.3. Afte ths bef ntductn f flament wund cmste we have enugh mtvatn t g fwad wth flament wund cmste vessel mechancs. Fstly a bef ntductn f cmste mateal stuctue wll be dslayed. Cmste mateals ae usually hetegeneus and anstc mateal. The unqueness s n E-ISSN: Vlume

19 Jacb Nagle the mechancal etes behav whch s changed fm nt t nt; nt n the matx has dffeent etes when cmaed t the fbe. Als etes n ne dectn ae nt necessaly the same n the dectn. These cmehensns ae fced mcscc and macscc bsevatns. The buldng blck f the cmste mateal stuctue s the lamna. The lamna usually cnssts f fbe/matx cnfguatns. The mcmechancs vew deals wth mechancal behav f cnsttuent mateal nteactns between these knds f mateals and the esultng behav f sngle lamna n a lamnate as shwn n Fg.11. Hweve macscc vew cncentates n the gss mechancal behav f cmste mateals stuctues (lamnate) wthut egad t cnsttuent mateals and the ntenal nteactns as shwn n Fg.11. Dung the fllwng dscussn stess-stan elatnshs between mcscc and macscc mechancs wll be develed. Geneal nfmatn cncenng cmste mateals and atculaly abut lamna stess-stan elatnshs can be fund n Ref. [6-9]. Fgue 11. Mc-mechancs and Mac-mechancs esectves f cmste mateal (Wkeda). Thn Flament Wund Pessue Vessel Suse we have a thn flament wund essue vessel wth nn-ncal cdnates ( x y ) (bdy cdnates) and ncal mateal cdnates (1 ) as shwn n Fg Fgue 1. Lamna entatn n nn-ncal cdnates Fgue 13. Flament wund essue vessel n ncal mateal cdnates In de t fnd stess-stan elatnshs cdnates tansfm fm bdy cdnates ( x y ) nt ncal mateal cdnates (1 ) wll be efmed. The cdnate's entatn dffeence can be bseved n Fg. 13. Als t's assumed that nly ntenal essue acts (extenal essue may cause bucklng). Usng Eqs. (1 3) f thn cylnde leads t: x y xy 0. t c t (46) c whle these equatns ae assumed t be based n gemety and statc equlbum. Snce stan and stess ae alng fbe dectn we wll ass t ncal mateal cdnates (1 ) by usng thtc lamna they [6]. The stess-stan tansfmatn elatnshs fm lamna cdnates t ncal cdnates ae: E-ISSN: Vlume

20 Jacb Nagle 1 cs sn sn x sn cs sn y 1 sn sn xy cs (47) 1 cs sn sn x sn cs sn y 1 / sn sn xy / cs MATLAB gam. Whle all stesses ae nmalzed by. Als stans f the fst tc ncal secnd ncal and shea ae nmalzed by esectvely. Ntce tce1 tce tcg1 that stan numecal calculatn s based n the fllwng data: E1 E and Substtutng elatns (46) nt (47) leads t the ncal mateal stesses and stans as fllws: sn 1 cs sn (48) tc 4 In de t fnd stan elatns we wll use the fllwng tansfmatn nstead f (47): 1 S11 S1 1 S1 S 1 S 66 (49) 1 whle whee the cmlances and the engneeng cnstants ae elated by the fllwng equatns: 1 1 S11 S E1 E. (50) S1 S1 S66 E E G1 Hence sn 1 cs 1 E1 E 1 cs 1 sn (51) 1 E E sn 4G 1 Resultant numecal analyzng f Eqs. (48 51) leads t the fllwng gahs usng Fgue 14. Nmalzed mateal ncal stesses Vs. Fgue 15. Nmalzed mateal ncal stans Vs. The fllwng cnclusns have been deduced usng Fg.14-15: Maxmum fst ncal stess and stan ( 1max 1max ) tgethe wth mnmum secnd ncal stess 0 0 ) ae btaned at ( mn E-ISSN: Vlume

21 Jacb Nagle Mnmum fst ncal stess and stan ( 1mn 1mn ) tgethe wth maxmum secnd ncal stess 0 ) ae btaned at 0. ( max Fst ncal stess ( 1 ) and secnd ncal stess ( ) gahs ae meet 0 0 at (mutual angle). Mnmum ( 1mn 1mn ) and maxmum ( 1max 1max ) shea stesses and stans ae btaned 0 0 at esectvely. Shea stess ( 1 ) gets lwest value elatve t fst ( 1 ) and secnd ( ) ncal stesses. Maxmum secnd ncal stan 0 ) s btaned at 38 whle ( max ts mnmum value ( mn ) s btaned 0 at 50. The abslute stan values ae fulflls: 11 n each angle accdng t ntal assumtns. Whle all stesses and stans ae nmalzed accdng t the emak abve. Shecal ends ae dvded nt tw gus: cmstes and hmgenus. The case f hmgenus shecal ends has been teated aleady n Sec.1. If the shecal ends ae made f cmste mateals ne shuld use 'The nettng they' whch s beynd ths essay dscussn. F me nfmatn abut ths methd see Ref. [6-9].. Thck Flament Wund Pessue Vessel Thck-walled cmstes vessels n theedmensnal cdnates wee studed n detaled by Zu [30]. Thee ae tw tyes f methds f analyzng stess-stan elatnshs n thck-walled cmstes cylnde: (1) dect slutn methd and () stess functn methd. These methds ae based n Lekhntk s they [31-3] as bught by Önde [10] Tsukv and Dach [33]. I suggest the eade t ead abut these methds n Zu study [30]. Fm hee we wll ass t dscuss u fnal subject n falue ctea f cmste mateals. D. Falue ctea f cmste mateals Falue ctea f cmste mateals have been dscussed by [34-36] and thes. Thee ae tw man ctea f falue f cmste lamnates. Maxmum Stess ( stan) ctea ths they emhaszes mde f falue but neglects the effect f stess nteactns. Ths cten s cnsdeed t be qute cnsevatve (ndeendent cten). The fllwng cndtns must be satsfed n tensn and/ shea: F G S (5) 1 t t 1 t whle Ft Gt St ae the lmtng tensle and shea stengths alng ncle mateal dectns esectvely. Meve n cmessn case the fllwng cndtn must be satsfed: F G (53) 1 c whle Fc Gc ae the lmtng cmessve stengths alng ncle mateal dectns esectvely. Tsa-Wu falue they (nteactve cten). Ths they ncludes stess nteactns n the falue mechansm and edcts fst ly falue but t eques sme effts t detemne aametes. The mathematcal fmulatn f ths cndtn usng [34] s: F11 1 F1 1 F (54) F66 1 F1 1 F 1 whee F F F Ft Fc GtGc S F F F F F Ft Fc Gt Gc c (55) E-ISSN: Vlume

22 Jacb Nagle whle thee dmensnal genealze mdel f Tsa-Wu they s esented by Bhavya et al. [36]. T sum t u nteactve thees whch ae based n cuve fttng lke the Tsa-Wu cten ae efeable at edctng falue f a sngle lamna than ndeendent cten. Hweve n de t dentfy and undestand falue mechansm bth methds shuld be used and cmaed aganst each the cmbned wth FEM analyss. V. Cnclusn Cmstes and hmgenus cylndes wth and wthut shecal ends wee nvestgated. Each knd f cylnde was examned f thck and thn gemetes accdng t the fllwng de. Analytcal mdel f hmgenus cylndcal vessel wth hemshecal ends: Thn walled cylnde wth hemshecal ends. Thck walled cylnde wth hemshecal ends. Hmgenus cylndcal vessel falue ctea. Cmund cylndes. Analytcal mdel f cmste cylndcal vessel wth hemshecal ends: We wund thn cylndes wth hemshecal ends. Thck walled cmste cylndes n la cdnates. Flament wndng methd estmatn f cmsed cylnde: Thn flament wund essue vessel. Thck flament wund essue vessel. Falue ctea f cmste mateals. Each f these subjects was esented and dscussed. Addtnally all subjects wee als examned analytcally nt ncludng thck flament wund essue vessel subject. Numecal nvestgatn usng MATLAB gam was dne f thn flament wund essue vessel subject. Meve qualtatve and quanttatve tests wee esented f hmgenus thck cylnde stess feld. Each methd has ts wn advantages and dsadvantages n cst manufactung stength and weght as aea n ths text and the elevant efeences. Acknwledgement All fgues n ths atcle have been duced by the auth f ths study nt ncludng Fgues ( ) whch have the wn aate emssns. Fg. 8 and Fg.10 use emssns have been ganted thanks t Pf. Rchad Kunz and Pf. D. Onu Sayman esectvely. Fg. 11 has been ganted emssn by Wkeda fee-use ule. Refeences [1] E. J. Hean Mechancs f Mateals 1: An Intductn t the Mechancs f Elastc and Plastc Defmatn f Slds and Stuctual Mateals 3 d Ed. Chate 9: Buttewth-Henemann [] NASA Tech Befs Makng a Metal- Lned Cmste Ovewaed Pessue Vessel Ma [3] O. Fetas Mantenance and Rea f Glass-Lned Equment Chemcal Engneeng. J Jul [4] D. Feye and J. Havey Hgh Pessue Vessels [5] R. Budynas. And J. Nsbett Shgley's Mechancal Engneeng Desgn New Yk: McGaw-Hll th Ed. [6] R. F. Gbsn Pncles f Cmste Mateal Mechancs New Yk: McGaw-Hll [7] Kaw A. K. Mechancs f Cmste Mateals New Yk: Tayl and Fancs006 nd Ed. [8] L. P. Klla and G.S. Snge Mechancs f Cmste Stuctues Cambdge Unvesty Pess 003. [9] A. Bake S. Duttn and D. Kelly Cmste Mateals f Acaft Stuctues AIAA Educatn Sees nd Ed E-ISSN: Vlume 9 014

23 Jacb Nagle [10] A. Önde Fst Falue Pessue f Cmste Pessue Vessels Maste Thess Dkuz Eylul Unvesty 007. [11] E. J. Heen Mechancs f Mateals Geat Btan: B.H d Ed.. [1] R. G. Budynas Advanced Stength and Aled Stess Analyss nd Ed. New Yk: McGaw-Hll [13] Ugual A. C. and Fenste S.K. Advanced Stength and Aled Elastcty New Jesey: Pentce Hall 4 th Ed. Jan.003. [14] S. Bhadu Pessue Cylndes New Yk: McGaw-Hll Ch [15] T. Ranv and Pak R. On the numecal value f the Tangental Stess n Thck Walled Cylndes J.Al. Mech. Mach [16] L. Wlf M. S. Kazm N. E. Tdeas Intductn t Stuctual Mechancs M.I.T. Deatment f Nuclea Engneeng. Mach 003. [17] G. H. Majzb and A. Ghm Otmzatn f Cmund Pessue Cylndes J. A. M.M.E. Vl [18] A. A. Maje and S. A. Patl Mnmzatn f mateal vlume f thee laye cmund cylnde havng same mateals subjected t ntenal essue J. I.J.E.S.T. Vl.3 N [19] M. P. Nemeth and M. M. Mkulas Smle Fmulas and Results f Bucklng-Resstance and Stffness Desgn f Cmessn-Laded Lamnated-Cmste Cylndes NASA TP Aug [0] B. F. Tattng Analyss and Desgn f Vaable Stffness Cmste Cylndes P.H.D. Vgna Plytechnc Insttute [1] D.O.D. Ceamc Matx Cmstes MIL-HDBK Vl.5 Jun [] E. J. Hean Mechancs f Mateals 1: An Intductn t the Mechancs f Elanstc and Plastc Defmatn Slds and Stuctual Mateals B.H.Ch.9. 3 d Ed [3] R. K. Kunz Cmste Renfcement f Cylndcal Pessue Vessels MAE 661 Lamnated Cmste Mateals Cuse' ntes Mece Unvesty 008. [4] Nagesh Fnte-element Analyss f Cmste Pessue Vessels Defense Scence Junal Vl. 53 N [5] J. M. Stanszewsk T. A. Bgett M. Keefe and B. Pwes An Imved Desgn Methdlgy f Mdelng Thck-Sectn Cmste Stuctues Usng a Multscale Aach A.R.L [6] S. Gha A. Glshan and A. Ayb Theetcal Analyss and Fnte Element Smulatn f Behav f Lamnated Hemshecal GRP Dme Subjected t Intenal Pessue Intenatnal Cnfeence n Cmute and Sftwae Mdelng vl [7] Z. M. Kab Fnte element analyss f cmste essue vessels wth a lad shang metallc lne Junal f Cmste Stuctues Vl [8] J. S. Pak and C. S. Hng Analyss Of Flament Wund Cmste Stuctues Cnsdeng The Change f Wndng Angles Thugh The Thckness Dectn Junal f Cmste Stuctues Vl [9] K. A. Ry Stength Analyss and Desgn f Multlayeed Thck Cmste Shecal Pessue Vessels W.R.D.C [30] Zu L. Desgn and Otmzatn f Flament Wund Cmste Pessue Vessels MSc.Thess Nethelands BOX-Pess Ch [31] Lekhntsk S. G. Anstc lates New Yk: Gdn and Beach Tanslated by S. W. Tsa and T. Chen. [3] S. G. Lekhntsk They f elastcty f an anstc elastc bdy San Fancsc: Hlden-Day Inc.; Tanslated by P. Fen. E-ISSN: Vlume

24 Jacb Nagle [33] I. Tsukv and B. Dach Elastc defmatn f cmste cylndes wth cylndcally thtc layes Junal f Cmste Stuctues Vl [34] S. M. Shakb Analyss f cmste lamnated stuctues Emam Hssen Unvesty Pess [35] S. S. Kuma and A. S. Kuma. Desgn and Falue analyss f Gedesc Dme f a Cmste Pessue Vessel I.J.E.R.T. ISSN: Vl. 1 Iss. 7 Se. 01. [36] S. Bhavya P. Rav Kuma and Sd. Abdul Kalam. Falue Analyss f a Cmste Cylnde I.O.S.R-J.M.C.E ISSN: Vl. 3 Iss Aendx Table A.1. Amecan & Euean Pessue Vessels Standads. N.\Standad Amecan Ble and Pessue Vessel Cde Euean Standads 1 ASME BPVC Sec. I - Pwe Bles ASME BPVC Sec. VIII Dv. 1 - Pessue Vessels EN Unfed Pessue Vessels PD Unfed fusn welded essue vessels ASME BPVC Sec. VIII Dv. - Pessue Vessels (altenatve ules) ASME BPVC Sec. VIII Dv. 3 - Pessue Vessels (hgh essue vessels) ASME BPVC Sec. X - Fbe Renfced Pessue Vessels Table A.. Pety cmasn fm Ref. [10] E-ISSN: Vlume

25 Jacb Nagle Table A.3. Flament wund ducts: Alcatns Vs. Resn systems used fm Ref. [10] 5 E-ISSN: Vlume 9 014