Bab III Metode Elemen Hingga Pada Shell

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1 III Metode eme Hgg Pd Se III. eor tt eor ett merpk g g petg dr k mtemt g megkj g tr g perpd tegg d regg dm ed et. Hmpr em memk t et (ett dm p g r megk per etk (deormto tdk mee t tertet mk per etk k g ed g dep. t ed de oe g r ed teret k er etk t erdeorm egg tm tegg d regg dm. Per etk tergtg pd kogr geometr dr ed teret d pd t mek. Dm teor ett g d g et er t ked dm g tr tegg d regg ert er d per etk k g g r dgk. Se t teor ett kk megggp ert omoge d otropk t t g er m d t et jg m dm eg r. III. ompoe egg egg er mm ddek eg ro tr g g ekerj pd t eeme deg eeme t edr. Dm tet g g ekerj tegk r terdp r det tegg orm (Norm Stre er mtemt ddek eg erkt : ΔF σ m (III- Δ A ΔA ompoe dr tegg g ekerj ejjr deg dg det : egg geer (Serg Stre g ddek : τ m ΔA ΔV ΔA (III- ed tegg pd ed et ervr dr ttk t kettk. Pd ked tg dme dpt dtk σ( d τ(. Utk III-

2 meggmrk ked tegg dpt dm t eeme g ke tk ergg dm etk kotk (d d d Zw σ σ τ τ d Yv τ τ τ σ Gmr III. ompoe tegg pd t ttk eeme d d d Z τ τ d d X τ τ τ X Gmr III. ompoe tegg pd dg Dm tk tegg orm terdr dr tg ot k σ σ σ d egg geer terdr dr em ot k τ τ τ τ τ τ. Deg mempertk keetmg g F d jg mempertk dme eeme ke tk ergg g kt tj d (d(d(d egm tegg orm tegg geer pd dg g ejjr er merk d m. F (III- (τ kr (d(d (τ k (d(d τ τ (III- Utk tegg geer g g tegk r dpt dr dm etk dg (d dme deg mer keetmg mome terdp ttk O mk dperoe : III-

3 Mo τ (d.d(d τ (d.d(d (III-5 Dm er ertrt perm teret dpt det deg tegg d eg mome. Deg meederk dperoe : τ τ deg r g m dpt dktk τ τ (III- (III- τ τ (III- etg perm teret det km tm k tegg. eer kompoe tegg tg dme demrg ttk ed et dtetk oe em kompoe teor tegg k tg tegg orm d em tegg geer k dtk deg mtrk: σ τ τ σ τ σ τ (III-7 τ τ σ Pd pet te ked tegg ert tg dme tetp pd pet g tp g memk ketegr etr memp ked tegg tg dme g tdk empr k kompoe tegg pd permk g ejjr dg XY m deg o (. Dm pet et ked tegg dme erper petg. Pd ked σ τ τ deg demk mtrk teor tegg mejd : σ τ σ τ τ τ σ dm τ (III-7 III. ompoe Regg d Perpd Regg er mm ddek eg ro tr per pjg d pjg em g dtk dm etk mtemt : ΔL ε (III-8 L re ed et teret er etk kt g r etp ttk pd megm perpd et g ke. Deg metk perpd t tro dm r eg v w kt dpt tk III-

4 ( v ( ( (III-9 g mejk w kompoe perpd jg merpk g etk. Utk meggk perpd d per etk kt dpt mej kem Gmr III. re keer ed et er etk mk eeme ke d d d jg k er etk k pjg d dt - dt tr permk g em k - k jg k er (Gmr III. Z w Z w π/ - γ π/ - γ Dd O X O X Y v d d Dd d Dd Y v π/ - γ Gmr III. Deorm t eme Deg memt pd per etk g ke m dm r X d Δd ε (III- d dm pertm Δd dtk dm k ked deret tor Δ d d mk dpt dtk v w ε ε ε (III- Akt pegr tegg geer permk eeme teret k erptr. Proek eeme teret pd m XY epert gmr ddek eg Regg geer (kt dtor dt III-

5 Y (εd (j /j d C" (j v/j d d A" A' A C " O' φ ' v (εd O d (j /j d (j v/j d Gmr III. Regg d perpd t eeme X Dr gmr dperoe : γ γ ' γ " (III. γ v d d v γ (III- d d Deg r g m dperoe γ γ (III- v w γ γ (III-d Regg teor dpt ddek eg erkt: ε γ γ γ ε γ ε γ ε γ (III- III. Hkm Hooke Utk trktr g et er Hkm Hooke g meggk tegg d regg dtk eg : σ.ε (III- t dpt dtk d g g er tr kompoe tegg d regg pd t tertet epert Gmr III. erkt : III-5

6 egg Regg Der t Der Pt Gmr III. Hg egg d Regg Dm Mod tt Yog (om Yog ;77-89 Imw Iggr. Jk tegg orm trk ekerj dm r X perpjg ε dkt oe perpedek ter jd regg dm r XYZ : σ σ σ ε ε μ ε μ (III-5 Dm μ d Ro Poo (Smeo De Poo ; 78-8 Imw Per g ddek : Regg Lter μ Regg Ak (III- P D P D Gmr III.5 Per Pjg Ak d Lter Jk tegg orm dm r (σ σ σ ekerj erm pd eeme g ke teret mk Hkm Hooke dper mejd : σ σ σ ε μ μ (III-7 σ σ σ ε μ μ (III-7 σ σ σ ε μ μ (III-7 III-

7 Hkm Hooke jg erk pd tegg geer dm k m pd der et er tegg geer erdg r deg regg geer : τ G.γ (III-8 Dm G d Mod ttt geer (Ser Mod o tt. Jk egg geer ekerj erm pd permk eeme teret perm mejd : γ τ γ τ γ τ (III-9 G G G Dm g tr Mod tt d Mod tt Geer (G dpt dperoe dr g erkt (erdr moeko ;99 : G ( μ (III- III.5 Perm Metode eme Hgg tk gkg. Umm Utk meg etk geometr gkg deg eeme gg kt dpt meggk erg etk pedekt g ered. Pedekt g pg eder deg meggk eeme dtr dm etk egtg t egempt. Pd eeme egempt m epert pd eeme egtg t kom per mm d per od kompoe memr (tegg dg d kompoe etr (etr pet. Cr tk memek eeme mpr deg meggk kom eeme egempt per er (er dpemet retge g dkemgk oe meo tk m tegg dg d eeme egempt MZC (Meo Zekewd Ceg tk etr pet. Deg kom mk pd etp ttk od k terdpt m per od terdp m ok. Gmr III. ompoe Memr pd eeme egempt III-7

8 Gmr III. ompoe etr pd eeme egempt III. Akt G Memr III.. eme Segempt Je eeme egempt per er (er dpemet retge g dkemgk oe Meo. Sejt kt k merk kekk d e ttk od ekve tk eeme. t tj e eeme egempt deg te t epert g dpertk dm Gmr III.(. Dm gmr teret jg dkk koordt etrod tp dme g ddeek eg : (III- Gmr III.7 Segempt per er dm d ertrt trt d eteg er d eteg tgg. Per mm eeme terdr dr tr dm dg -. Jd : III-8

9 III-9 { } v (III- tk od d dgk pd ttk ttk dt dm dr kr w d ejt erw r deg jrm jm. Pd etp ttk od terjd d tr (dm r d mk vektor per ttk od k mejd : { } 8 v... v...q q q q (III- g per m tk eeme d : v mk dpt kt t w g teret er dm d. erdrk kt et eeme td eg egempt per er. Deg g per mtrk g k mejd : g (III- Mtrk dpt dtg tk etp ttk od g medptk : g g g g (III-5 Deg meg mtrk mejd k kt peroe : (III- Opertor pe g k mejd :

10 III- (III-7 d mtrk k mejd : (III-8 Iver dr d : (III-9 emd ver dr dpt dtg eg : (III- Deg memperoe - k kt dptk g etk per dm. Jd: ( ( ( ( ( ( ( ( g ( ( ( ( ( ( ( ( (III- Perm dpt e dgkt :

11 dm ( (III- ( ( ( ( ( ( ( ( (III- Utk megtg regg kt per merk g etk per terdp d. gmp g g dtk dm koordt g tdk ert d. Jd kt r meggk r ert dr tr pr epert erkt : Opertor dere er d k mejd : (III- d (III-5 Deg memkk opertor ke dm mtrk k kt peroe mtrk eg erkt : d ( ( ( ( ( ( ( ( ( ( ( etk g e eder dr perm d : ( ( ( ( ( (III- ( (III-7 Not d mejkk tr pr dr terdp d eg erkt : III-

12 (III-8 kt mk mter ortotropk terdp m d g tegg-regg dpt dtk er mm eg : (III-9 egg dpt dr deg meggk : ( (III- ekk eeme deg md dpt dtg me perm erkt : dv t dd (III- Dm perm mtrk d : Ak kt peroe : v d g (III- metr Φ (III- Φ Φ Dm : Φ ( Φ III-

13 III- Φ H tegr perm (III- g dttk dm perm(iii- k megk : C metr C t (III- dm C C Akr ddpt : (III-5

14 III- e III. Mtrk ekk eeme egempt per er metr t.. t... t metr t.. t.. 5. t III.7 Akt Mome Letr eme dpt dgk tk memodek ked regg kot pd pet g megm etr d eeme-eeme jg memk gg g emg d egkp. Oe kre t eeme k memerk g koverge. eme pd gmr.8 det egempt MZC kre dtemk oe MeoZekew d Ceg.

15 q q q w w Gmr III.8 Segempt MZC Sepert eeme g eje eeme memk t per mm t w (tr dm r. Jd: w (III- Dm gmr jg dkk per ttk od: q w w w (III-7 { q q q } Per td dm q w / dkk deg tj tk meeek ptr dt deg r pot perptr ttk od. G ttk od g megk per d : { p p p } { p M M } p (III-8 Not mejkk g dm r edgk M d M d mome dm r d Gmr III.9 Segtg P Fg per g dp tk eeme deg met po egtg P d : III-5

16 w 5 7 Dm etk koordt tr : 8 9 w (III-9 Yg merpk g kk egkp g terdr dr ep k deg d k pgkt empt. D ggp kt dpt merk g etk per mejd: Dm : [ ] (III-5 8 ( ( ( ( ( ( 8 (III-5 D ( ( ( 8 N d ( (III-5 dm tk dt-dt egempt epert pd kdrter Opertor dere er mm d d eg erkt : d (III-5 Yg tdk megdg. Mtrk regg per mm dpt dtk deg : d (III-5 III-

17 III-7 : ( ( ( ( ( ( ( ( ( ( ( (III-55 egg mm g e dpt dtg deg : { } q M M M M φ (III-5 Jd tk mter otropk : λ μ μ μ μ λ mk t ( ( ( ( ( ( 8 λ μ μ μ (III-57 Yg erp mtrk X. Mtrk kekk eeme dpt dtg dr : A d d da (III-58

18 e III. Mtrk kekk egempt MZC t ( ( μ μ λ III-8

19 III.8 Perm Ioprmetrk III.8. eme Ioprmetrk drter (Q eme egempt per er merpk dk dr eeme oprmetrk kdrter (Q g dtjk dm gmr erkt: q 8 / q q 7 q 5 / q q q q ( q q 8 q 5 q 7 v q q q q Gmr III.9 eme Q : (. Segempt dk. ( Pg oprmetrk Per g dtjk dm gmr teret d : { v} (III-59 Pd etp ttk od terdpt tr r d jd vektor per ttk od d : III-9

20 { q q... q } { v... v } q (III- 8 Fg etk per d: Dm etk mtrk: Dm (III- v v v v v v (III- q (III- (III- Per mm merpk tr pd etp ttk kt per q ke ttk od. Jk dederk mk g dpt dt eg erkt : Dm Adp ( ( (III-5 d derk dm te erkt e III. oordt ttk od tk eeme Q I Deg r g m g regg per tk eeme Q dpt dtk eg erkt : Dm ε q III-

21 d Smtrk dpt jg dtk dm etk : DG DG D G DG Utk ktor-ktor epert erkt : J [ ( J ( J ] D G (III- (III-7 J [ ( J ( J ] D G J [ ( J ( J ] D G J [ ( J ( J ] D G J [ ( J ( J ] D G J [ ( J ( J ] D G J [ ( J ( J ] D G D G [ ( J ( J] (III-8 J Mtrk kekk eeme Q (deg te kot t dpt dtk dm koordt Crte: t ( ( dd (III-9 A Dm koordt tr rm kekk k mejd : III-

22 t Dm etk tegr merk ( ( J ( dd (III-7 t k j R j R j ( ( J ( (III-7 j k j k j k Dm RjRk d ktor oot tegr G-Legedre mtrk kemtk mtrk mter J Jo III.9 Perkt Mtrk ekk Cgkg Utk perkt eeme gkg dm k d kom dr eeme pet etr d eeme tegg dg (gmr III.. Utk eeme pet etr terdr dr DOF t perpd trver ert d rot tk tp od. Sedg tk eeme tegg dg terdr dr perpd dm r dg per od. Gmr III. Cgkg gg dr etr d memr Dr gg teret mk gkg memp 5 DOF t tg perpd d d rot. Utk mtrk kekk gkg dpt dtk eg erkt [ ] [ ] [] [ ] m { d} { d } m { F } { Fm } (III-7 III-

23 Utk d d F d mg-mg mtrk kekk perpd t rot od d g t mome pd ttk od g t mome pd ttk od. Skrp d m d mome (edg d memr. Perkt mtrk kekk ejt deg mempertgk rot gkg eg koekwe ertm DOF per od. Mk dr per (III-7 dpt dtk kem [ ] [ ] { d } { F } [] [ ] m { d m} { Fm } (III-7 θ Mtrk dr per (III-7 megekprek tem koordt ok. Utk ejt mk mtrk teret dtrormk mejd tem koordt go. Jk mtrk trorm [ ] dket mk : { } [ ]{ } d (III-7 o d go Utk etp od g tr DOF ok d go dpt dtk : θ θ θ o o o o o o θ θ θ go go go go go go (III-75 Utk j d o r tr m ok d m go j. Mk tk trorm [ ] mtrk tk empt od : [ ] [ d ] [ ] d [ d ] [ ] d Dm mtrk [ d ] deg meggk per (III-75 deg kr. (III-7 Deg meggk trorm mtrk mk mtrk kekk d vektor e g dtrorm derk erkt: [ ] [ ] [ ok ][ ] [ ] [ ] [ F ] go (III-77 F (III-78 go ok III-

24 III-. eor dm e D dm g kt megkk dg eedro g oprmetr H tk mejd empt eeme g degkgk tk dr e mm. Perkt dr eeme gkg SHQ8 d erp deg tekk g dgk d dm memperoe eeme dm perkt pet. gmp kod ked dmodk kre d tr tm d v terjd pd mg-mg ttk od dr eeme gkg. Jd Deg demk mtrk C tk t ttk od dt dr eeme eg empt memp d g koom ddg eem eg erkt: (-79 Deg r g m mtrk C tk t ttk od teg ok dr eg empt mejd:

25 (-8 deg m dpemet pd mg-mg dr dep ttk od eeme SHQ8 mk (8(5 dpemet ok. Sepert eeme pet PQ8 eeme gkg g mm SHQ8 k drmk er gg. jkk g geometr dr eeme SHQ8 g m koordt eg ttk d 8 8 m (-8 g m d perm (-8 Jd deg demk w kete oe er er kwdrt erdrk eeme. Seg tm t m d d r ko dr t vektor V permk merpk te dr gkg pd ttk od vektor g dperoe eg j k V j k m j (-8 k j d k d d permk dr gkg. D t progrm kompter g mp koordt dr j d k t r ko V g r der eg dt. Dpemet pd etp ttk pd eeme gkg dm r go jd { v w} (-8 Sek dpemet ok terdr t tr g m d dm r go epert jg d rot keα d β ektr d k mert gr ggg ok d. q { v w α β } (...8 (-8 III-5

26 Dpemet go dm dr dpemet ok d 8 v v w w D dm rm mo 8 [ α] μ β μ medk mtrk g erkt: (-85 m μ m (-8 oom er - egt dr ko r vektor mert gr ggg g ked V ; d koom memp ko r tk vektor pertm mert gr ggg V epert g dtjkk d dm gmr vektorvektor ert ortogo ke vektor V d tk t m p tk r dr rtrer. Utk megt p kt dpt V e V (-87 L V V V (-88 Jk V d pre oe e. tr mm ok ' d v' d dm r V d V kre rot-rot g od β d α - d ' β v α ' (-89 Dpemet go pd etp ttk der oe tm ked d perm (-85 Per eeme pet etk dpemet erg d perm (-85 dtrver ke dm etk mtrk m m Utk medptk mtr perk mk (...8 (-9 d A (-9 III-

27 III-7 m m (-9 jd A (-9 D A (-9 kr rm k dgk tk memperoe mrk m g kote d e-e tk eeme ok SHQ8. Mtrk Jo g dperk d dm orm d J (-95 t dptk tr d dm mtrk J dr perm (-8: Iver mtrk J d J J * (-9 t memerk tr tertet dr dpemet go dpt dt pd perm (-85 erke deg koordt ok. I g dtrk erdrk t vektor koom dr em r eg erkt: w v m m m m m m w w w v v v β α 8 (-97 rorm tr pd koordt go g dperk dm Mtrk Jo t dterpk. Oe kre t

28 III-8... * * *... w J J J w (-98 Megk r d dm perm kt memperoe w v g g e e d d gm gm em em m d m d g g e e d d w w w v v v β α 8 (-99 * * J J * ( J d * * J J * ( J e (- * * J J * ( J g Utk eeme SHQ8 kt megggp em tpe dr tegg g tdk o eg erkt: w w v v wv γ γγ (- Mett ver dr vektor regg g ked kt rkt d g dr mtrk dr perm (-99 g d g d e m g e m g m d e m d e g g m e m e d d (- Serp deg eeme pet kt dpt megok r d mtr g megk tk meemk A (- Smtre A d d dr perm (- d (- jd A (- ketk meetk mrk kekk tk eeme gkg. mk erkt dtekk d dm r me g k dpertmgk:

29 { σ σ τ τ τ } σ ' (-5 ' ' ' ' D tegg g e d ' ' ' ' { γ γ γ } ' (- ' ' ' ' ' ' D regg te dgk. L mrk regg tegg tk t otropk mejd m w tk eeme PQ8. Utk meggk tegg ok d dm vektor ' t kepd tegg go d dm vektor kt dpt meggk mrk trorm regg eg erkt: ' gmp deret g ketg dr mrk r dp kre ' ' ' tdk tk mejd terkp d vektor ' tk tj megev pd t ttk pegtegr kt memerk ko r tk vektor-vektor V V d V d ttk. I kk t eg kejt rt perm: ( J orm e ( JJ orm ee e D dm ekpre vektor e (-7 ( J orm deret g pertm dr mrk jo t g dormr kepd pjg t d eter. etk megtg d dm r ok jg erg g ' (-8 Mrk Ak er m r kre pegp deret g ketg Sekrg kt d p tk mermk mrk kekk tk eeme SHQ8 g meggk mrk eg erkt: ( ' ' d d d ( A ' ' ( A ' ' J d d d (-9 D mrk A ' d d mp dr kr 5 tetp g er r g dkk deg [ ( A ' A ' / ( ' ] J d d ' (- III-9

30 Itegr pd perm (- r dev e omor meggk d po pegtegr d etp Ar d. D dm proe ktor d / dkk deg / dr deret g ketg J d jg er kot g m. Jd Deg demk kt er eekt memperoe ktor d / d dm ked g-g dr mrk Mrk m g kote tk eeme SHQ8 d M ρ J d d d ρ ( A ( A J d d d (- Igt w mrk F A d F d mp dr kr tetp g ked memp r tk dkk deg megtegrk perm (- me kete g megk M ρ [ [ ' / ( ] '] J d d (- A A A re ktor d / er eekt d / ket d ked gg dr M mrk er trto d er-er pemtr ertrt-trt. e-e od etr kre g t d eeme SHQ8 dtemk deg mrk F A eg erkt: p( t ( ( t J d d d A ( t J d d (- A D dm ekpre vektor e (t dmk tk er kompoe g (per vome t g ert ergm me kete dr gkg. Jd; Deg demk ( t { } (- Per eeme pet g t tdk meek etp t ekve od. Sete dpemet od d dm vektor q(t te dperoe mk eeme dm r ok σ '( t q( t ' q( t (-5 III-

31 epert r dtetk ttk oto tk pegtegr merk. III-