Chapter 11 HYDROFORMING

Transcription

1 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Chae HYDROFORMING.. INTRODUCTION. I hydfmig, fluid essue fmig, shee is fmed agais a die by fluid essue. I may cases, a flexible diahagm is laced he shee ad i is he fmed i a female die caviy as shw i Figue.. The advaage f he cess is ha die csuci is simle ad he cess may be ecmical f makig smalle umbes f as. A disadvaage is ha vey high essues may be equied ad he cycle ime is geae ha f samig i a mechaical ess. blak diahagm essue die ( a ) ( b ) Figue.. (a) A yical shee meal a. (b) The aageme f essue, hydfmig i a female die. Hydfmig is als used fm ubula as such as backes f bicycle fames ie fiigs, as shw i Figue.. Axial fce may be alied he ube as well as ieal essue; his ceaes cmessive sess i e dieci s ha elemes f he ube defm wihu hiig ad eaig is delayed. Wih secially desiged fmig machies, a lage umbe f as ca be fmed by his cess a lw cs.

2 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: is ube die high-essue fluid Figue.. Hydfmig a ube cme wih essue ad axial fce. Ahe alicai is fmig ubula as such as vehicle fame cmes. A ud ube is be ad he laced i a die as shw i Figue.. I is he essuized ieally ad fmed a squae seci. ud ube fial squae seci die ube high-essue fluid Figue.. Mehd f hydfmig a be squae-seci cme fm a ud ube. I his chae we d aem he aalysis f a cmlee cess, bu beak he cess dw i elemes f simle gemey ad mdel each f hese seaaely. This will illusae he limis f he cess ad shw he ifluece f fici, gemey ad

3 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: maeial eies. The fee exasi f a ube is fis sudied. Fmig a squae seci fm a ud ube i he s-called high-essue cess is aalysed ad he i is shw hw sme f he cess limis i his cess ca be vecme i a sequeial fmig cess... FREE EXPANSION OF A CYLINDER BY INTERNAL PRESSURE. Exasi f a ud ube wihu chage i legh is aalysed. The ube will defm i lae sai, ie., he sai i he axial dieci will be ze. Iiially he ube will emai cicula ad he adius will icease. The exasi f a cylidical eleme i his mde is illusaed i Figue.4. The sai ad sess saes, f a isic maeial, ae: ε ; βε 0 ; ε φ ; ε ( + β ) ε ε ; φ α ; 0. (.) ie., β 0, α. T, Tφ, φ Figue.4. Eleme f a cicula ube wih ieal essue. If he maeial eies bey he sess sai law, K (ε ), Whe he maeial is defmig we bai fm Equais.8 ad.9, f ; ad ε ε (.)

4 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Fm Seci 7., he icial adii a ay egi f he ube ae, ρ ; ad ρ (.) Fm Equai 7., he h esi is, T.. (.4) ad as his is lae sai, T. φ α. T T F he ube yield, he essue, fm Equais. ad.4, is, T f. (.5) If he ube is iiially f hickess, 0, ad adius, 0, he cue h sai ad hickess ae give by, 0. 0 ε ad 0. exε 0.ex( ε ) (.6) 0 Fm his, we bai he essue chaaceisic f exasi as, 0. K. 0. (.7) 0 Fm he fm f Equai.7, we see ha f a sai hadeig maeial, >0, he essue will ed icease as he maeial defms. O he he had, if he ubula eleme is allwed exad feely, he ube wall will hi ad he adius icease; bh effecs will ed decease he essue. A sme i, he essue will each a maximum as he sig effecs balace. We eed als cside he ssibiliy f he ube wall eckig ad sliig. As his is lae sai defmai, he ladig ah will be alg he veical axis i he sai sace, Figue 5.6. Sliig wuld be execed aximaely whe he h sai has aximaely he value,. Thus he limiig case is whe, 0 ad. K. ex( ) 0 (.8) whee,, is he adius a which he ube becmes usable ad will eck. 4

5 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: FORMING A CYLINDER TO A SQUARE SECTION. A cmm eai is fmig a ud ube i a squae die as illusaed i Figue.; i he middle f he a, he ube will defm i lae sai. Vaius sages f he cess ae shw i Figue.5. (a) (b) A 0 + d d A + d (c) 0 Figue.5. Exadig (a) a cicula ube i (b) a squae die by fluid essue; (c) shws a iceme i he cess a he ce. I Figue.5(c), he ube has bee aially exaded s ha he wall is uchig he die u he i, A. The cac legh is,, ad assumig ha he hickess is small cmaed wih he adius, he cue cac legh is, 0. Duig a iceme i he cess, he cac legh iceases, + d, ad he ce adius deceases + d, whee, d, is a egaive quaiy, ie., d d. I he cac ze, he ube will be essed agais he die wall by he ieal essue. If he maeial slides alg he die, fici will se he mi ad he esi will chage. A sme i, he esi will be isufficie sech he wall ad hee will be a sickig ze as shw i Figue.6. 5

6 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: slidig sickig s (a) T ube wall µ. die wall T + T T (b) ds s Figue.6. (a) Pa f he ube wall i cac wih he die duig fmig f a cylide a squae seci. (b) Eleme f he ube wall i he slidig ze. The equilibium equai f he eleme i Figue.6(b) is, T + dt T µ.. ds. +, dt µ. (.9) ds The defmai cess is sable as lg as he esi iceases wih sai s we assume hee ha he esi i he usued ce will ciue icease as he adius becmes smalle. Refeig Seci.6, he sle f he esi vesus sai cuve f lae sai is siive f sais f, ε. Thus i Figue.6, he esi yield he ube will icease as he ube wall his. F he ube i cac wih he die, he geaes esi will be a he age i.equai.9 shws ha due fici, he esi deceases liealy wads he cee-lie; he disibui is shw i Figue.7. T he igh f he i whee slidig ceases, he esi i he ube wall is less ha ha equied f yieldig ad hee is fuhe defmai i his sickig egi. The ciical i is whee he hickess is, s. F a maeial beyig he sess sai law, K (ε ) fm Equais., he h sess is give by, 6

7 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: K ε As his is a lae sai cess, ε 0 ε The esi a he ciical i i Figue.7 seaaig slidig ad sickig is, 0 T. s. s K s (.0) s I he sickig ze, hee is slidig ad he sle f he esi cuve i Figue.7 will be less ha, µ.. wall esi T s dt ds φ µ. T slidig s sickig Figue.7. Disibui f esi i ha a f he ube i cac wih he die wall duig fmig f a ud ube a squae seci. The disibui f hickess i he wall ca be deemied by a icemeal aalysis. I his wk, he exeme cases, eihe wih fici a he die wall wih sickig fici alg he eie cac legh will be csideed. 7

8 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Tube Fmig i a Ficiless Die. If cac bewee he ube ad he die is ficiless, a ay isa, he esi ad als he hickess a ay i aud he ube will be uifm. The cue eimee legh f e-quae f he ube i Figue.5 is, π π π + + ( 0 ) 0 (.) As hee is chage i vlume f he ube maeial, π π , 0 (.) 4 4 π 0 π As his is a lae sai cess, he h sai is, ε 0 ε ad f a maeial beyig he sess, sai law, Kε, he h sess is 0 K (.) The essue equied defm he usued ce f adius,, ca be deemied fm Equai Tube Fmig wih Sickig Fici ( Vey High Fici). I he ube sicks he die wall as s as i uches i, he f a cicula ube ha iiially jus fis iside he die as shw i Figue.5(a), he hickess a he fis i f cac will be, 0. As he ube becmes gessively aached he wall, he hickess a he age i his will decease s ha a A, i Figue.5(c), i has he value,. F a ui legh eedicula he lae f he diagam, he vlume f maeial i he ac, AA, is, π.. /. This vlume will emai uchaged duig he iceme, ad as his is lae sai, equaig he vlumes befe ad afe he iceme, we bai, 8

9 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: π Fm abve, π 0, ad, d d, hece, [( + d) + d ]( + d) d 4 π d (.4) Iegaig wih he iiial cdiis f, 0, a, 0, we bai, 4 π 0 (.5) 0 The h sai ad essue ciue he defmai ca he be deemied fllwig he same aach as i Seci... The wall hickess will be uifm vayig fm he iiial hickess a he cee-lie f he die a miimum a he usued ce adius. F a give ce adius, he ce hickess will be less ha f he ficiless case ad failue culd ccu ealie i he cess.... Failue i Fmig a Squae Seci. The abve aalysis eglecs he effec f ubedig ude esi a he age i, A, i Figue.5. This will be simila he effecs descibed i Seci 0.5, ad cause addiial hiig ad wk-hadeig f he ube wall. I lae sai fmig f a ube f uifm seci, he cess is ssible vided ha he esi i he defmig wall ciues icease wih fmig. If i eaches a maximum, eckig ad failue f he ube will ake lace. As idicaed i Seci 5.4, f lae sai, he limi f a we law hadeig maeial is aximaely whe, ε. I fmig he ce adius, he essue equied will icease as he adius deceases, as shw by Equai.5. Eve hugh vey high essue equime may be used, he limiai ce adius is usually sigifica ad his, gehe wih he fmig limi sai, ae he fis higs ha shuld be calculaed i elimiay desig..4. CONSTANT THICKNESS FORMING. I lae sai exasi, as idicaed abve, he maeial will sli aximaely whe he h sai eaches he value f he sai-hadeig idex,. The sai ah is illusaed i Figue.8(a). T bai he equied sai i he a, high saihadeig maeial is used, bu his exacebaes he blem f high essue f fmig. 9

10 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema ε , ISBN: ε 0 ε 0 φ (a) (b) ε φ Figue.8. Sai ahs f (a) lae sai ad (b) csa hickess fmig. Lage sais ae eially ssible i cesses such as hse illusaed i Figue., ad i wuld be advaageus chse a lw sai-hadeig maeial. T avid sliig, a csa hickess sai ah, β, wuld be he bjecive ad his sai ah gehe wih he fmig limi f a lw maeial is illusaed i Figue.8(b). The desig f cesses ha achieve csa hickess defmai is easy, bu we cside sme cases f simle gemey..4.. Csa Thickess Defmai f a Tube Exaded by Ieal Pessue. I a csa hickess cess, he h ad axial esis ad sesses wuld be equal ad sie. The h esi, fm Equai.4, is, T ; he axial esi aci, which is cmessive, is, Tφ. T achieve his, a axial fce mus be alied he ube as illusaed i Figue.9; he axial fce is, F π. Tφ π.. π.. (.6) Fm Equais.8 ad.9, he effecive sai ad sess ae, ε ε ad (.7) 0 whee,, is give by Equai.4. F a give exasi i a maeial beyig he sess sai law, K (ε ), ad ig ha he hickess emais csa, we bai he essue equied ciue he cess as, 0 K (.8) 0 Cmaig his wih Equai.7, we see ha he essue is educed f csa hickess exasi, cmaed wih lae sai, bu f cuse a axial fce is equied ad his will d wk as well as he essue. 0

11 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: F T, Tφ, φ F Figue.9. Cylidical eleme defmig ude csa hickess cdiis wih a ieal essue ad axial cmessi..4.. Effec f Fici Axial Cmessi. T achieve axial cmessi a eleme, a fce is alied he ed f he ube as shw i Figue.. The effec f his fce is lcal because fici bewee he ube ad he die will cause i dimiish wih disace fm he i f alicai f he fce. The case f a simle ube is shw i Figue.0. µ.q q q z dz T φ T φ + dt φ Figue.0. Effec f fici he axial cmessi f a ube.

12 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: I his diagam, a luge he lef alies cmessi a ube. A sme disace, z, he esi e side f a eleme f widh, dz, is, T φ, ad he he side, T φ + dt φ. Aud his elemeal ig hee is a cac essue, q, ad a fici sess, µ.q. The equilibium equai f he eleme is, ( T + dt ) π. T π.. q dz π. µ. φ φ φ +, dt φ µ. q. dz (.9) This shws ha he esi, aci iceases, ie., becmes me esile as, z, iceases. I sas as ms cmessive a he luge ad he cmessi deceases liealy wih disace fm he ed f he ube. F his eas i is ssible bai axial cmessi i he middle f as such as shw i Figue., bu i she as, such as Figue., axial cmessi ca be effecive i eveig hiig ad eckig. I as f cmlicaed shae, such as i Figue., i is fud ha esue ha hiig des ccu i ay ciical egis, sme laces will becme hicke, bu his is usually acceable..4. LOW-PRESSURE OR SEQUENTIAL HYDROFORMING. I fmig as such as i Figue., a echique has evlved ha avids he use f vey high essues. T fm a squae seci, a val ube is cmessed duig clsue f he die ad he ieal fluid ly seves kee he ube agais he die as shw i Figue.. cicula ube die ecagula seci fluid (a) (b) Figue.. Fmig (a) a val ube i (b) a squae ecagula seci i a lwessue hydfmig cess.

13 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: I his cess, he eihey f he ube emais aximaely csa ad eaig is likely be a limiig fac. The mechaics f he cess ca be illusaed by csideig he case f a cicula ac f a ube beig fmed i a ce as shw i Figue.. I his case he cess is symmeic abu he diagal; his is quie he same as ha shw i Figue., bu will illusae he icile. T B T M B B A T (a) (b) A M A V Figue.. Defmai f he ce f a ube i a lw-essue hydfmig cess. (a) The fces acig, ad (b), he bedig mme diagam he ac, AB. T I fmig he ube i he ce, he wall is be a shae adius a, B, ad saigheed a, A. The bedig mme diagam is shw i Figue.(b). The sess disibui a, B, is shw i Figue., f a igid, efecly lasic maeial. S T B, ve M B S Figue.. Sess disibui a he ce f a ube beig defmed as i Figue..

14 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: As shw i Seci 0.4, he bedig mme equied defm he ube a, B, is, M B S T T M T T B B 4 y y (.0) whee, S, is he lae sai flw sess, T y, he yield esi, ad, M, he mme bed he wall i he absece f esi cmessi i.e. he fully lasic mme deemied i Seci Thus by alyig cmessi he ube wall, he ube ca be be easily a he ce ad he age i; a lasic hige will fm a, B, bed he ube, ad e a, A, saighe i. I his simle aalysis, bedig will ly ccu a, A, ad, B, ad he egi bewee will emai uchaged. This is vey ealisic as maeial is eve cmleely igid, efecly lasic, bu i is bseved ha i his cess, he adius f cuvaue i he usued ce is csa; i is leas a he i, B. I efmig his eai, he ube is filled wih fluid, usually wae, ad sealed as i is laced i he die as i Figue.. As he ieal vlume dimiishes i fmig he val he squae seci, fluid is exelled fm he ube. The essue is egulaed usig a essue cl valve ad is maiaied a a sufficie level kee he ube agais he die walls ad eve wiklig. Oce he die is clsed, he essue may be iceased imve he shae. I may be see ha he veall eimee f he ube des chage vey much duig fmig ad heefe sliig is avided..5. SUMMARY. I his chae, seveal examle f fmig usig fluid essue have bee examied. Thee ae may he alicais f hese icile ad he fllwig facs shuld be be i mid i csideig fluid fmig. Vey high fluid essues ae equied fm small fille adii. Fmig equime becmes exesive as he essues becme high. Vey high sais ca be achieved if cmessive fces ae alied he a as well as essue ad aximaely csa hickess defmai baied. Fmig essues ca be educed if clled bucklig ude cmessive fces is baied. 4

15 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Execises Ex... A cicula ube wih a adius R ad hickess is defmed i a die wih a squae css seci hugh high essue hydfmig. Fid he elai bewee he ce adius ad he ieal essue i he w cases: a) Thee is fici a he die meal ieface. b) The ube maeial fully sicks he die suface. The maeial beys effecive sess-sai law: K ( ε + ε ). Ex... A mild seel ube f 80 mm diamee ad hickess f 4 mm is be exaded by ieal essue i a squae seci. The maximum essue available is 64MPa. The maeial has a sess sai cuve f ( ε ) Deemie he miimum ce adius ha ca be achieved. Ex... I a ube hydfmig cess, a squae seci was fmed fm a cicula ube. The iiial ube had a adius f R. The cue ce adius is. The maximum hickess alg he ube was. The ce hickess was. The maeial sai hades accdig he elai K ( ε + ε ). Assumig ha he hickess vaies liealy alg he ube wall, calculae he aveage fici cefficie bewee he ube wall ad die suface. Ex..4. Fid he essue equied exad a cicula ube fm iiial adius a fial adius. The maeial hades as Kε, f w cases a) Tube eds ae fee mve axially. b) Tube eds ae esiced fm axial mvemes. Calculae als he essue a isabiliy. Ex..5. A hi walled us wih a adius R, ad css-secial adius, ad a hickess, is subjeced ieal essue. Fid he sess disibui i he us ad he ieal essue a fis yield ad ha cesdig full yieldig f he us. 5

16 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: ψ R 6

17 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Chae SOLUTIONS Ex.. a) F he ficiless case: The cue adius f he ube a he ce is L. The ube wall hickess will be uifm. Fm vlume csacy R ( π π ( R ) + ) R e-aagig 4 4 π. ( ) π π R F he lae sai cdii ε ε l( ) ad l 4 4 π K [ ε + l( ( ) )] π π R Ieal essue is give by: K )] 4 4 π R [ ( ) ] π π π π R 4 4 π [ ε + l( ( ). b) F he full sickig case: The elai bewee he ce adius ad he ce hickess ca be baied fm he icemeal elai d π π d + ( d)( d) -d ) R 4 ( π d d -d R 4 ( ) ( ) π Cicumfeeial sess 68

18 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: R K[ ε + ( )l( )] π The ieal essue is give by: ) π 4 ( 4 R ( ) K[ ε + ( )l( )] R π Ex.. The miimum ce adius is limied eihe by eckig by he maximum fluid essue available. Sice 4 4 π, ( ) π π R 0 * ε ε l, ε 0..7, 7mm. The ieal essue equied fm ce adius is give i Ex... Subsiuig he maeial eies ad die gemey i he equai, we bai: (mm) (mm) (MPa) (MPa) Theefe, he miimum ce adius is mm, ad is limied by he maximum essue available. 69

19 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Ex.. I de calculae he iiial ube hickess, we use vlume csacy. ( + ) L + π π R whee L ( R ) [ ( )( R ) + + ] R π The fces ha exised a each ed f he wall ca be calculaed as: F F K[ ε + l( )] K[ ε + l( )] The fial ieal essue a he cue fmig adius ca be fud fm: F hus F T calculae he aveage fici cefficie fm fce balace µ av L F F µ av F F L F q L F 70

20 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Ex..4 a) F he fee eds, hee is sess i he axial dieci, 0. F hi cylide 0. ε l( ) ε ε ε l( ) l( ) Effecive sess ad sais ae: ad ε ε The ieal essue is calculaed as K [l( )] / T calculae he exaded adius a isabiliy, le d 0. / This leads : l( ) e b) F esiced eds, hee is sai i he axial dieci, ε 0. ε l( ) ε ε l( ) l( ) Effecive sess ad sais ae: ad ε ε 7

21 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: K [ l( )] The ieal essue is calculaed as ) + K [l( ( )] T calculae he exaded adius a isabiliy, le d 0. This leads : l( ) e / 7

22 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: Ex..5 Slui: + ρ ρ Rsecψ R ψ Fce balace i he z-dieci (he dieci f he axis f he us): π R ( R csψ ) ] π ( R csψ ) csψ [ eaagig : R csψ ( ) R csψ subsiue ge : ( )( Rsecψ ) R csψ [ ( ) R csψ ( R csψ ) ] csψ z-axis R- cs ψ ψ P R which is csa ve he css seci. The maximum ccus a ψ 0, he ie adius f he us. R max ( ) R f yieldig ccu, usig he v Mises yield ciei: + y subsiuig ad aagig, he essue iiiae yieldig is give by: y [ + + ( ) ] R R y / 7

23 Mechaics f Shee Meal Fmig, Secd Edii, by Zdzislaw Maciiak, Jh Duca ad Jack Hu Publishe: Buewh-Heiema , ISBN: The miimum ccus a ψ π, he ie adius f he us. R + max ( ) R + Similaly, f yieldig ccu a he ue adius, usig he v Mises yield ciei: + y subsiuig ad aagig, he essue f full yieldig is give by: y [ + ( ) ] R + R + y / 74