7 Finite element methods for the Timoshenko beam problem

Transcription

1 7 Fiit lmt mtods for t Timosko bam problm

2 Rak-54.3 Numrical Mtods i Structural Egirig Cotts. Modllig pricipls ad boudary valu problms i girig scics. Ergy mtods ad basic D fiit lmt mtods - bars/rods bams at diffusio spag lctrostatics 3. Basic D ad 3D fiit lmt mtods - at diffusio spag 4. Numrical implmtatio tciqus of fiit lmt mtods 5. Abstract formulatio ad accuracy of fiit lmt mtods 6. Fiit lmt mtods for Eulr Broulli bams 7. Fiit lmt mtods for Timosko bams 8. Fiit lmt mtods for ircoff ov plats 9. Fiit lmt mtods for Rissr Midli plats. Fiit lmt mtods for D ad 3D lasticity. Etra lctur: otr fiit lmt applicatios i structural girig Rak-54.3 / 6 / JN 339

3 7 Fiit lmt mtods for t Timosko bam problm Cotts. Strog ad ak forms for Timosko bams. Fiit lmt mtods for Timosko bams arig outcom A. Udrstadig of t basic proprtis of t Timosko bam problm ad ability to driv t basic formulatios rlatd to t problm B. Basic koldg ad tools for solvig Timosko bam problms by fiit lmt mtods it lockig fr lmts i particular Rfrcs ctur ots: captr 9. Tt book: captrs Rak-54.3 / 4 / JN 34

4 7. Motivatio for t Timosko bam lmt aalysis T rlvac of bam structurs from rails to ao bams as sigificatly gro du to fuctioal or smart matrials spradig bams from civil girig to filds as spac tcology ad biomcaics! Rak-54.3 / 4 / JN 34

5 7. Strog ad ak forms of Timosko bam lmts t us cosidr a ti straigt bam structur subjct to suc a loadig tat t dformatio stat of t bam ca b modld by t bdig problm i a pla. T basic kimatical assumptios for dimsio rductio of a ti or modratly ti bam calld Timosko bam 9 i.. ormal fibrs of t bam ais rmai straigt durig t dformatio ormal fibrs of t bam ais do ot strc durig t dformatio 3 matrial poits of t bam ais mov i t vrtical dirctio oly 4 ormal fibrs of t bam ais rmai as ormals durig t dformatio u ' yv ' d d Rak-54.3 / 4 / JN 34

6 7. Strog ad ak forms of Timosko bam lmts t us cosidr a ti straigt bam structur subjct to suc a loadig tat t dformatio stat of t bam ca b modld by t bdig problm i a pla. T basic kimatical assumptios for dimsio rductio of a ti or modratly ti bam calld Timosko bam 9 i.. ormal fibrs of t bam ais rmai straigt durig t dformatio ormal fibrs of t bam ais do ot strc durig t dformatio 3 matrial poits of t bam ais mov i t vrtical dirctio oly 4 ormal fibrs of t bam ais rmai as ormals durig t dformatio com tru if t displacmts ar prstd as u y y si y u v y v y cos v : it dotig t dflctio of t bam ctral or utral ais ad dotig t rotatio of t ormal fibrs of t ais to variabls bot dpdig o t coordiat oly. yv ' ' d d Rak-54.3 / 6 / JN 343

7 7. Strog ad ak forms of Timosko bams For liar dformatios t displacmt fild abov implis t aial strai ad trasvrs sar strai as v u y y ' y y y y y '. y Rak-54.3 / 6 / JN 344

8 7. Strog ad ak forms of Timosko bams For liar dformatios t displacmt fild abov implis t aial strai ad trasvrs sar strai as v u y y ' y y y y y '. y W dfiig t bdig momt ad sar forc troug strsss M : M : A yda Q : Qy : da t rgy balac of t pricipl of virtual ork ca b ritt i t form W it W A t V y σ : ε y dv dad V S t y A y bu dv t u ds St t vds; b t [ t y z] T Rak-54.3 / 6 / JN 345

9 7. Strog ad ak forms of Timosko bams For liar dformatios t displacmt fild abov implis t aial strai ad trasvrs sar strai as v u y y ' y y y y y '. y W dfiig t bdig momt ad sar forc troug strsss M : M : A yda Q : Qy : da t rgy balac of t pricipl of virtual ork ca b ritt i t form W it A W A yda ' d t V y σ : ε y dv dad A V y S t y A da ' d y bu dv t u ds St t vds; b t [ t y fd z] T Rak-54.3 / 6 / JN 346

10 7. Strog ad ak forms of Timosko bams For liar dformatios t displacmt fild abov implis t aial strai ad trasvrs sar strai as v u y y ' y y y y y '. y W dfiig t bdig momt ad sar forc troug strsss M : M : A yda Q : Qy : da t rgy balac of t pricipl of virtual ork ca b ritt i t form W it W M ' d σ : ε dv dad yda ' d A A t y V y A V y Q ' d S t y A da ' d y bu dv t u ds St t vds; b t [ fd t y fd z] T Rak-54.3 / 6 / JN 347

11 7. Strog ad ak forms of Timosko bams r t bam is assumd to b subjct to a vrtical distributd surfac loadig t y t y y z actig o t uppr ad lor surfacs of t bam dfiig a rsultat loadig f : Z t t y t / z dz t / z dz r t itgrals ar tak alog lis ad i t z dirctio for ac o uppr ad lor surfacs S t ad S t rspctivly for collctig t pysical load from surfacs oto t bam ais. Rmarks. Otr loadig typs could b cosidrd as ll. Z t t y Z t Z t For istac distributd surfac loadigs i t dirctio of -ais s Hom rcis 6. as ll as body loads i t bot dirctios ca b also tak ito accout. Surfac loadigs o t d poit facs of t bam at = ca b cosidrd as ll cf. t giv boudary rsultats blo. Rak-54.3 / 6 / JN 348

12 7. Strog ad ak forms of Timosko bams Itgratio by parts i t trm for t itral virtual ork givs t form M M' dq Q' Q d fd Rak-54.3 / 5 / JN 349

13 35 Itgratio by parts i t trm for t itral virtual ork givs t form implyig t forc balac ad boudary coditios i.. t strog form as Rmark. It as assumd tat t bam is subjct to a vrtical distributd surfac loadig i t dirctio of y-ais alog t itrval ; cf. EB-M i Captr 6... β β β β Q Q Q Q M M M M Q M f Q T - M ' T - Q ' Rak-54.3 / 6 / JN ' ' d f d Q Q Q d M M 7. Strog ad ak forms of Timosko bams

14 7. Strog ad ak forms of Timosko bams W takig ito accout t liarly lastic costitutiv rlatios i t form y E y y E ' y G G ' y y Rak-54.3 / 6 / JN 35

15 7. Strog ad ak forms of Timosko bams W takig ito accout t liarly lastic costitutiv rlatios i t form y E y y E ' y G G ' t momt ad sar forc ar giv i trms of dflctio ad rotatio as M EI ' Q GA ' T - QM y y Rak-54.3 / 6 / JN 35

16 7. Strog ad ak forms of Timosko bams W takig ito accout t liarly lastic costitutiv rlatios i t form y E y y E ' y G G ' t momt ad sar forc ar giv i trms of dflctio ad rotatio as M EI ' Q GA ' T - QM ad t strog form ca b ritt as a displacmt formulatio as follos: For a giv loadig fid t dflctio ad rotatio suc tat f : R : R GA ' ' f T - EI '' GA ' T - EI ' M EI ' M GA ' Q GA ' Q y. y Rak-54.3 / 6 / JN 353

17 7. Strog ad ak forms of Timosko bams Modl comparisos. T Timosko bam problm ca b compard to t Eulr Broulli bam problm by ritig i t form T - T - EI ''' f T -β EI '' ' GA T - T -β r t rotatio ca b solvd first from ad t t dflctio follos from T - ic also rvals t diffrc bt t modls sic t Eulr Broulli modl satisfis t coditio ' du to t kimatical assumptio 4 of Captr 6.. Rak-54.3 / 6 / JN 354

18 7. Strog ad ak forms of Timosko bams Modl comparisos. T Timosko bam problm ca b compard to t Eulr Broulli bam problm by ritig i t form EI ''' ' f EI '' GA T -β T -β T - r t rotatio ca b solvd first from ad t t dflctio follos from T - ic also rvals t diffrc bt t modls sic t Eulr Broulli modl satisfis t coditio ' du to t kimatical assumptio 4 of Captr 6.. Q M ' T - T - T - QM EB- O t otr ad sic quatios fially giv a quatio rsmblig t Eulr Broulli quatio cf. : d EI' ''' f d EI d d T - T - Q GA. Rak-54.3 / 6 / JN 355

19 7. Strog ad ak forms of Timosko bams Furtrmor for costat valus of EI ad GA it a sufficitly rgular loadig f t quatio abov ca b ritt i t form EI EI' ''' f f ''. GA Rak-54.3 / 4 / JN 356

20 7. Strog ad ak forms of Timosko bams Furtrmor for costat valus of EI ad GA it a sufficitly rgular loadig f t quatio abov ca b ritt i t form EI EI' ''' f f ''. GA Rmark. T sar corrctio factor dpdig o t cross sctioal sap is oft icludd i t Timosko bam formulatios abov:. GA GA Rak-54.3 / 4 / JN 357

21 7. Strog ad ak forms of Timosko bams Furtrmor for costat valus of EI ad GA it a sufficitly rgular loadig f t quatio abov ca b ritt i t form EI EI' ''' f f ''. GA Rmark. T sar corrctio factor dpdig o t cross sctioal sap is oft icludd i t Timosko bam formulatios abov:. GA GA Rmark. If t scod drivativ of t loadig vaiss as it dos for a costat load for istac t quatio abov coicids it t Eulr Broulli bam quatio ic dos ot cssary imply ovr tat t solutios ould coicid sic tr is still t rotatio to b dtrmid from T - for istac ad t boudary coditios of ts to problms ar ot idtical. Rak-54.3 / 4 / JN 358

22 7. Strog ad ak forms of Timosko bams Furtrmor for costat valus of EI ad GA it a sufficitly rgular loadig f t quatio abov ca b ritt i t form EI EI' ''' f f ''. GA Rmark. T sar corrctio factor dpdig o t cross sctioal sap is oft icludd i t Timosko bam formulatios abov:. Rmark. If t scod drivativ of t loadig vaiss as it dos for a costat load for istac t quatio abov coicids it t Eulr Broulli bam quatio ic dos ot cssary imply ovr tat t solutios ould coicid sic tr is still t rotatio to b dtrmid from T - for istac ad t boudary coditios of ts to problms ar ot idtical. Ayay i ligt of t quatio abov t Eulr Broulli bam modl sms to b a good approimatio to t Timosko bam modl as far as it olds tat EI GA f ''. GA GA Rak-54.3 / 4 / JN 359

23 T - From istad o ca dduc tat t Eulr Broulli bam problm is a good approimatio to t Timosko bam problm as far as it olds tat EI '' GA 7. Strog ad ak forms of Timosko bams. Rak-54.3 / 4 / JN 36

24 T - From istad o ca dduc tat t Eulr Broulli bam problm is a good approimatio to t Timosko bam problm as far as it olds tat EI '' GA. From tis coditio a corrspodig iquality formulatd i trms of t a priori problm paramtrs ca b giv: EI. GA 7. Strog ad ak forms of Timosko bams Rak-54.3 / 4 / JN 36

25 From istad o ca dduc tat t Eulr Broulli bam problm is a good approimatio to t Timosko bam problm as far as it olds tat EI '' GA 7. Strog ad ak forms of Timosko bams. From tis coditio a corrspodig iquality formulatd i trms of t a priori problm paramtrs ca b giv: O t otr ad o ca still propos aotr coditio i t form d d T - EI. GA EI d d Q GA. Altogtr t proimity of ts to bam modls is rlatd to t matrial paramtrs of t bam as ll as t rgularity of t loadig or rotatio ad sar forc. Rak-54.3 / 4 / JN 36

26 7. Strog ad ak forms of Timosko bams T ak form of t problm is obtaid from t virtual ork prssios abov or as usual by multiplyig t strog form by tst fuctios variatioal fuctios itgratig ovr t domai ad fially itgratig by parts cf. Captr 6.: fd ˆ [ EI '' GA ' ] ˆ d EI ' ˆ GA ' ' d ˆ GA ' ˆ ˆ ˆ W V ic ar valid for all. [ EI ' ˆ' GA ' ˆ] d GA ' d ˆ' Rak-54.3 / 4 / JN 363

27 7. Strog ad ak forms of Timosko bams T ak form of t problm is obtaid from t virtual ork prssios abov or as usual by multiplyig t strog form by tst fuctios variatioal fuctios itgratig ovr t domai ad fially itgratig by parts cf. Captr 6.: fd ˆ [ EI '' GA ' ] ˆ d EI ' ˆ GA ' ' d ˆ GA ' ˆ [ EI ' ˆ' GA ' ˆ] d GA ' d ˆ' ic ar valid for all. Addig all pics togtr givs bot t rgy balac it rspct to t variatioal spac or spacs EI ' ˆ ' d ˆ ˆ W V GA ' ˆ' ˆ d ad t sstial boudary coditios for a catilvr bam for istac as fd ˆ ; ˆ ˆ. ˆ ˆ W V Rak-54.3 / 4 / JN 364

28 7. Strog ad ak forms of Timosko bams T ak form of t problm is obtaid from t virtual ork prssios abov or as usual by multiplyig t strog form by tst fuctios variatioal fuctios itgratig ovr t domai ad fially itgratig by parts cf. Captr 6.: fd ˆ [ EI '' GA ' ] ˆ d EI ' ˆ GA ' ' d ˆ GA ' ˆ ˆ ˆ W V [ EI ' ˆ' GA ' ˆ] d GA ' d ˆ' ic ar valid for all. Addig all pics togtr givs bot t rgy balac it rspct to t variatioal spac or spacs EI ' ˆ ' d GA ' ˆ' ˆ d ad t sstial boudary coditios for a catilvr bam for istac as ; ˆ ˆ. I additio t trial ad tst fuctio spacs ar dtrmid by t ak form as usual altoug i tis cas t spac cosidrd is a combiatio of to spacs du to to diffrt variabls dflctio ad rotatio prst i t formulatio. fd ˆ ˆ ˆ W V Rak-54.3 / 4 / JN 365

29 7. Strog ad ak forms of Timosko bams Wak form of t Timosko bam problm: t us cosidr a catilvr bam subjct to a distributd load. Fid suc tat a ; ˆ ˆ l ˆ ˆ W ˆ V f W V it t biliar form load fuctioal ad fuctio spacs a ; ˆ ˆ l ˆ W V { { v H H fˆ d EI ' ˆ 'd GA ' ˆ ' ˆd v } }. f E I G A Rak-54.3 / 4 / JN 366

30 367 Wak form of t Timosko bam problm: t us cosidr a catilvr bam subjct to a distributd load. Fid suc tat it t biliar form load fuctioal ad fuctio spacs Rmark. T rgy orm of t problm o a orm icludig to variabls is dfid as Rak-54.3 / 4 / JN ˆ ˆ ˆ ˆ ˆ ; V W l a f V W. d ' d ' ; : v GA EI v v a v a 7. Strog ad ak forms of Timosko bams A G I E f }. { } { d ˆ ˆ ˆd ˆ ' ' 'd ˆ ' ˆ ˆ ; H V v H v W f l GA EI a

31 7. Strog ad ak forms of Timosko bams Brak rcis 7 So tat t biliar form of t Timosko bam problm is lliptic ad cotiuous it rspct to t orm: i a v ; v ii a v ; vˆ ˆ EI' ' d v EI' ˆ'd C v GA v' v' d... v W V GA v' ˆ' v ˆd... vˆ H ˆ v vˆ ˆ W For ic typ of valus of t cross sctioal quatitis EI ad GA t quotit C / apparig i t corrspodig rror stimats ill b larg/small? V. Rak-54.3 / 4 / JN 368

32 7. Fiit lmt mtods for Timosko bams Stadard form fiit lmt mtod for t Timosko bam problm: t t distributd load of a catilvr bam b f. Fid t dflctio ad rotatio approimatios suc tat a ; ˆ ˆ l ˆ ˆ W ˆ V W W V V it t biliar form load fuctioal ad fuctio spacs a v ; ˆ ˆ l ˆ W V fˆ d { v C { C EI ' ˆ 'd v v GA v' ˆ ' ˆd P } k P }. k f E I G A Rak-54.3 / 4 / JN 369

33 7. Fiit lmt mtods for Timosko bams Stadard form fiit lmt mtod for t Timosko bam problm: t t distributd load of a catilvr bam b f. Fid t dflctio ad rotatio approimatios suc tat a ; ˆ ˆ l ˆ ˆ W ˆ V W W V V it t biliar form load fuctioal ad fuctio spacs a v ; ˆ ˆ l ˆ W V { fˆ d v C { C EI ' ˆ 'd v v GA v' ˆ ' ˆd P P Rmark. Tis stadard form fiit lmt mtod is far from optimal: for ti bams t approimatio covrgs trmly sloly to t act solutio du to so calld sar lockig poma ic ill b clarifid i dtail blo. k k } }. f E I G A Rak-54.3 / 4 / JN 37

34 7. Fiit lmt mtods for Timosko bams Eampl for a sar lockig Timosko bam lmt approimatio: Fiit lmt approimatio of t dflctio compard to t act solutio of t dflctio for a ti H/ = / clampd bam. T stadard lockig mtod prstd abov is compard blo to a sar rductio mtod prstd latr o ic as b prov to b lockig fr. Stadard mtod Stadard mtod Rducd itgratio mtod lmts lmts lmts Rak-54.3 / 5 / JN 37

35 37 Rak-54.3 / 4 / JN Sar lockig ca b rvald by drivig a stadard rror stimat i t form ic sos oc agai tat corsivity ad llipticity costats ar prst i t rror stimat affctig t accuracy of t approimatio providd by t mtod. 7. Fiit lmt mtods for Timosko bams k k k c C V W v v C

36 7. Fiit lmt mtods for Timosko bams Sar lockig ca b rvald by drivig a stadard rror stimat i t form ic sos oc agai tat corsivity ad llipticity costats ar prst i t rror stimat affctig t accuracy of t approimatio providd by t mtod. t us cosidr t ratio of ts costats i dtail; i t cas of a quadragular cross sctio for simplicity cf. Brak rcis 7: EI EBH 3 GA C v C k c EBH C ~ ma EI GA GA k k ~ mi EI GA v W EI for V H Rak-54.3 / 4 / JN 373

37 7. Fiit lmt mtods for Timosko bams Sar lockig ca b rvald by drivig a stadard rror stimat i t form ic sos oc agai tat corsivity ad llipticity costats ar prst i t rror stimat affctig t accuracy of t approimatio providd by t mtod. t us cosidr t ratio of ts costats i dtail; i t cas of a quadragular cross sctio for simplicity cf. Brak rcis 7: EI EBH C ~ 3 GA ma EI GA mi EI GA C v C k c EBH C ~ ma EI GA GA GA EI k ~ mi EI GA 6 H k v W EI if H for. V H Rak-54.3 / 4 / JN 374

38 7. Fiit lmt mtods for Timosko bams T atur of t problm ad c t rror stimat as ll dpds o t dimsios of t bam t tickss paramtr ic ca b clarly s by dividig t ak form of t problm by t bdig stiffss givig t form ' ˆ' d GA ' ˆ' ˆ d EI f d ˆ : EI gd ˆ ˆ W ˆ V. Rak-54.3 / 4 / JN 375

39 7. Fiit lmt mtods for Timosko bams T atur of t problm ad c t rror stimat as ll dpds o t dimsios of t bam t tickss paramtr ic ca b clarly s by dividig t ak form of t problm by t bdig stiffss givig t form Tis sos tat i t tickss limit i practic for ti bams t cofficit of t sar trm i t formulatio abov blos up GA EI ' ˆ' d GA ' ˆ' ˆ d EI 6 for H H gd ˆ ˆ W ˆ V. ic implis tat t ol sar trm blos up ulss t rst of t itgrad is abl to balac t baviour by covrgig to zro as follos: ' ˆ' ˆ ' ˆ W ˆ V '. ' Rmark. T limit cas coms tru i t Eulr Broulli bam problm ic is actually ko to b t limit of t Timosko bam problm as dscribd by t argumtatio abov. f d ˆ : EI Rak-54.3 / 4 / JN 376

40 7. Fiit lmt mtods for Timosko bams I t tickss limit for all ti bams i practic t fiit lmt approimatio strivs for satisfyig t Eulr Broulli coditio as ll: ' '. Tis lads to lockig ovr ic ca b s i computatios as a trmly lo covrgc toards t act solutio t tir t bam t lor t covrgc rat. I t limit cas for a clampd bam it liar lmts t approimatio locks as fully as possibl:. Rak-54.3 / 4 / JN 377

41 7. Fiit lmt mtods for Timosko bams I t tickss limit for all ti bams i practic t fiit lmt approimatio strivs for satisfyig t Eulr Broulli coditio as ll: ' '. Tis lads to lockig ovr ic ca b s i computatios as a trmly lo covrgc toards t act solutio t tir t bam t lor t covrgc rat. I t limit cas for a clampd bam it liar lmts t approimatio locks as fully as possibl:. Rmark. T lockig poma is ot limitd to t liar lmts oly. Hovr for lo ordr lmts t lockig ffct taks a ladig rol ad tis ffct ca b asily udrstood by cosidrig t approimatios of t dflctio ad rotatio: P ' P Rak-54.3 / 4 / JN 378

42 7. Fiit lmt mtods for Timosko bams I t tickss limit for all ti bams i practic t fiit lmt approimatio strivs for satisfyig t Eulr Broulli coditio as ll: ' '. Tis lads to lockig ovr ic ca b s i computatios as a trmly lo covrgc toards t act solutio t tir t bam t lor t covrgc rat. I t limit cas for a clampd bam it liar lmts t approimatio locks as fully as possibl:. Rmark. T lockig poma is ot limitd to t liar lmts oly. Hovr for lo ordr lmts t lockig ffct taks a ladig rol ad tis ffct ca b asily udrstood by cosidrig t approimatios of t dflctio ad rotatio: P ' P ' ad o t coditio mas tat t dflctio drivativ a picis costat fuctio sould coicid it t picis liar cotiuous rotatio forcig t bot variabls to b global costats zro for clampd boudaris. Rak-54.3 / 4 / JN 379

43 38 ockig fr rducd itgratio fiit lmt mtod for t Timosko bam problm: t t distributd loadig of a catilvr bam b giv as. Fid suc tat it t biliar form load fuctioal ad variatioal spacs }. { } { d ˆ ˆ ˆd ˆ ' ' ˆ'd ' ˆ ˆ ; P C V P v v C v W f l P v GA P EI v a k k Rak-54.3 / 4 / JN ˆ ˆ ˆ ˆ ˆ ; V W l a f V V W W 7. Fiit lmt mtods for Timosko bams A G I E f

44 38 ockig fr rducd itgratio fiit lmt mtod for t Timosko bam problm: t t distributd loadig of a catilvr bam b giv as. Fid suc tat it t biliar form load fuctioal ad variatioal spacs T oly diffrc compard to t stadard mtod is t rductio oprator t projctio of t sar trm to a lor ordr polyomial spac: }. { } { d ˆ ˆ ˆd ˆ ' ' ˆ'd ' ˆ ˆ ; P C V P v v C v W f l P v GA P EI v a k k Rak-54.3 / 4 / JN ˆ ˆ ˆ ˆ ˆ ; V W l a f V V W W }. { : P r r S r r v r P v S P k 7. Fiit lmt mtods for Timosko bams A G I E f

45 7. Fiit lmt mtods for Timosko bams T trm slctiv rducd itgratio origiats from t fact tat t projctio opratio i t biliar form ca b accomplisd i practic by usig a lor ordr itgratio rul for t umrical itgratio of t sar trm as follos: T igst polyomial ordr of t sap fuctios i t sar trm is k ad c t igst polyomial ordr of t ol itgrad is k P ' P ' ˆ' ˆ P k k k Rak-54.3 / 4 / JN 38

46 7. Fiit lmt mtods for Timosko bams T trm slctiv rducd itgratio origiats from t fact tat t projctio opratio i t biliar form ca b accomplisd i practic by usig a lor ordr itgratio rul for t umrical itgratio of t sar trm as follos: T igst polyomial ordr of t sap fuctios i t sar trm is k ad c t igst polyomial ordr of t ol itgrad is k P ' P ' ˆ' ˆ P k k k ic ould d k+ Gauss itgratio poits for accurat itgratio sic k poits Gauss quadratur itgrats accuratly polyomials of ordr k il k+ poits is oug for accurat itgratio of polyomials of ordr k+. Rak-54.3 / 4 / JN 383

47 7. Fiit lmt mtods for Timosko bams T trm slctiv rducd itgratio origiats from t fact tat t projctio opratio i t biliar form ca b accomplisd i practic by usig a lor ordr itgratio rul for t umrical itgratio of t sar trm as follos: T igst polyomial ordr of t sap fuctios i t sar trm is k ad c t igst polyomial ordr of t ol itgrad is k P ' P ' ˆ' ˆ P k k k ic ould d k+ Gauss itgratio poits for accurat itgratio sic k poits Gauss quadratur itgrats accuratly polyomials of ordr k il k+ poits is oug for accurat itgratio of polyomials of ordr k+. I rducd itgratio scm t sar trm is summd up ovr k poits istad of k+: GA P ' P ˆ' ˆd k i G GA ' ˆ' ˆ Rmark. Tis quadratur of k poits itgrats accuratly also t bdig trm of t biliar form sic t sap fuctio drivativs tri ar of ordr k ad c t ol itgrad is of ordr k < k. i G G G G i i i. Rak-54.3 / 4 / JN 384

48 7. Fiit lmt mtods for Timosko bams I t stadard form Timosko bam lmt mtod t fiit lmt approimatio of t sar forc is computd from t dflctio ad rotatio approimatios by simply usig t dfiitio of t sar forc: Q GA '. I t rducd itgratio mtod istad t projctio oprator is applid: Q GAP ' S. Rak-54.3 / 4 / JN 385

49 7. Fiit lmt mtods for Timosko bams I t stadard form Timosko bam lmt mtod t fiit lmt approimatio of t sar forc is computd from t dflctio ad rotatio approimatios by simply usig t dfiitio of t sar forc: Q GA '. I t rducd itgratio mtod istad t projctio oprator is applid: Q GAP ' S. Rmark. T projctio oprator abov ad also i t biliar form is applid sstially to t sap fuctios rlatd to t rotatio sic t drivativ of t dflctio approimatio is of ordr k ad c blogs to t projctio spac: P ' S k ' P k Rak-54.3 / 4 / JN 386

50 7. Fiit lmt mtods for Timosko bams I t stadard form Timosko bam lmt mtod t fiit lmt approimatio of t sar forc is computd from t dflctio ad rotatio approimatios by simply usig t dfiitio of t sar forc: Q GA '. I t rducd itgratio mtod istad t projctio oprator is applid: Q GAP ' S. Rmark. T projctio oprator abov ad also i t biliar form is applid sstially to t sap fuctios rlatd to t rotatio sic t drivativ of t dflctio approimatio is of ordr k ad c blogs to t projctio spac: P Q k ' Pk GA P GA P v' P ' GA ˆ ' ˆd ' P ' S GA v' P ˆ ' P ˆd. Rak-54.3 / 4 / JN 387

51 388 Rak-54.3 / 4 / JN A simpl ampl of applyig t projctio: I ordr to calculat t projctio of a liar polyomial say a polyomial dfid locally o lmt as t dfiitio of t projctio abov as to b applid it k = i its local form o lmt i.. P b a v 7. Fiit lmt mtods for Timosko bams. } { : P r r v r P v P P v P r r S r r v r P v S P k

52 389 Rak-54.3 / 4 / JN A simpl ampl of applyig t projctio: I ordr to calculat t projctio of a liar polyomial say a polyomial dfid locally o lmt as t dfiitio of t projctio abov as to b applid it k = i its local form o lmt i.. No for all r... simply mas for all costats. O t otr ad itslf is simply a costat say c. Hc o ca fially rit t itgral coditio i a cocrt form ad obtai t projctd polyomial ic is o just t costat c: P b a v 7. Fiit lmt mtods for Timosko bams d d b a c P v d b a d c r d r b a d r c r d r v d r P v P v. } { : P r r v r P v P P v P r r S r r v r P v S P k

53 7. Fiit lmt mtods for Timosko bams Rducd itgratio is t oly ostadard fatur of t mtod drivd abov. Hc t typical agrag sap fuctios ca b usd as a basis for obtaiig a cotiuous fiit lmt approimatio. Cotiuous approimatio for t dflctio ad rotatio it liar lmts implis for istac tat t momt approimatio is picis costat ad discotiuous t sar forc approimatio is picis costat ad discotiuous. Rak-54.3 / 4 / JN 39

54 7. Fiit lmt mtods for Timosko bams Rducd itgratio is t oly ostadard fatur of t mtod drivd abov. Hc t typical agrag sap fuctios ca b usd as a basis for obtaiig a cotiuous fiit lmt approimatio. Cotiuous approimatio for t dflctio ad rotatio it liar lmts implis for istac tat t momt approimatio is picis costat ad discotiuous t sar forc approimatio is picis costat ad discotiuous. Rmark. Rgardig t systm quatios o sould ot tat for ods for istac to variabls rsults ukos: i d i i d i; d i i d i i i i. Rak-54.3 / 4 / JN 39

55 7. Fiit lmt mtods for Timosko bams Rducd itgratio is t oly ostadard fatur of t mtod drivd abov. Hc t typical agrag sap fuctios ca b usd as a basis for obtaiig a cotiuous fiit lmt approimatio. Cotiuous approimatio for t dflctio ad rotatio it liar lmts implis for istac tat t momt approimatio is picis costat ad discotiuous t sar forc approimatio is picis costat ad discotiuous. Rmark. Rgardig t systm quatios o sould ot tat for ods for istac to variabls rsults ukos: T sam is tru for ac lmt: usig quadratic lmts implis 3 ods pr lmt ad c 6 ukos pr lmt 3 for t dflctio 3 for t rotatio. i d i i d i; d i i d i i i i i i N N 3 N i i i i 3 3. Rak-54.3 / 4 / JN 39

56 393 Rak-54.3 / 4 / JN I trms of local sap fuctios t local fiit lmt approimatios ca b ritt i t form : : T i i i T i i i N N N N N N d N d N 7. Fiit lmt mtods for Timosko bams

57 394 Rak-54.3 / 4 / JN I t sar part of t biliar form t dflctio ad rotatio ar coupld ic implis t cross trms ito t local stiffss matri: I trms of local sap fuctios t stiffss matri trms ca b ritt as 7. Fiit lmt mtods for Timosko bams. d d d ' ' d ' d ' ' i i T T T T T T f J P GA P J EI J P GA GA J rf rf rf rf rf N F N N N N N N N N. d d d F F F

58 395 Rak-54.3 / 4 / JN Wit local dgrss of frdoms giv as gt. F F F F F T d 7. Fiit lmt mtods for Timosko bams

59 7. Fiit lmt mtods for Timosko bams T lmt stiffss matri ad forc vctor ca b ritt i a compact form as usual by usig t biliar form ad load fuctioal; for istac as i j [ i j] a i ; j or altrativly it local sap fuctios as abov. GA ' 'd i j Rak-54.3 / 4 / JN 396

60 7. Fiit lmt mtods for Timosko bams T lmt stiffss matri ad forc vctor ca b ritt i a compact form as usual by usig t biliar form ad load fuctioal; for istac as i j [ i j] a i ; j or altrativly it local sap fuctios as abov. GA ' 'd Rmark. T stiffss matri is sstially composd of products of sap fuctios ad tir first drivativs ic ca b asily computd by stadard mas spcially for lo ordr lmts as dpictd blo for a to lmts cas. i j 3 / ' 3 ' / 3 3 ' Rak-54.3 / 4 / JN 397

61 7. Fiit lmt mtods for Timosko bams Error aalysis for t rducd itgratio Timosko bam lmt dos ot follo t stadard procdur: t problm is aalyzd i t mid from cf. Captr 7.X ic rsults a rror stimat for t sar forc as ll as ritt do blo. Ellipticity coditio or stability coditio ill b provd i t mid form by utilizig so calld Babuska Brzzi coditio ic is oft calld t if sup coditio. Rak-54.3 / 4 / JN 398

62 7. Fiit lmt mtods for Timosko bams Error aalysis for t rducd itgratio Timosko bam lmt dos ot follo t stadard procdur: t problm is aalyzd i t mid from cf. Captr 7.X ic rsults a rror stimat for t sar forc as ll as ritt do blo. Ellipticity coditio or stability coditio ill b provd i t mid form by utilizig so calld Babuska Brzzi coditio ic is oft calld t if sup coditio. Aftr all t rror stimat of t mtod is optimal it rspct to t polyomial ordr ad t rgularity of t act solutio: k Q Q c k Q k k k Wit crtai additioal assumptios a. rror stimat ca b drivd: k c Q k k k. Rak-54.3 / 4 / JN 399

63 7. Fiit lmt mtods for Timosko bams Error aalysis for t rducd itgratio Timosko bam lmt dos ot follo t stadard procdur: t problm is aalyzd i t mid from cf. Captr 7.X ic rsults a rror stimat for t sar forc as ll as ritt do blo. Ellipticity coditio or stability coditio ill b provd i t mid form by utilizig so calld Babuska Brzzi coditio ic is oft calld t if sup coditio. Aftr all t rror stimat of t mtod is optimal it rspct to t polyomial ordr ad t rgularity of t act solutio: k Q Q c k Q k k k Wit crtai additioal assumptios a rror stimat ca b drivd: k c Q k k k Rmark. T most accurat poit valus ar obtaid it tis mtod i t d poits ad itgratios poits of ac lmt i.. at Gauss obatto poits tis olds tru for k > ic is dscribd i t folloig ampl... Rak-54.3 / 4 / JN 4

64 7. Fiit lmt mtods for Timosko bams Eampl for t accuracy of t rducd itgratio Timosko bam fiit lmt mtod : For a ti H/ = / clampd bam it t loadig. f 3 Solid lis for t covrgc of t fiit lmt dflctio approimatio; i t H orm for t lmt dgrs k = 3. Circls for t suprcovrgt poits; it to lmts for t lmt dgrs k = 345. Rak-54.3 / 4 / JN 4

65 7.X Mid formulatio Rducd itgratio fiit lmt mtod for t Timosko bam problm is aalyzd i so calld mid form i ic t sar forc Q GA ' is tak as a idpdt variabl. T ak form of t problm is t of t folloig form: For a giv loadig f fid suc tat W V QS EI ' ˆ' d QQd ˆ Q ˆ' ˆ d QGA ˆ ' d fd ˆ Qˆ S. ˆ W ˆ V Rmark. I t corrspodig fiit lmt formulatio mid mtod t fiit lmt subspacs for t dflctio ad rotatio rmai t sam il t sar forc approimatio is a subspac of : S { r r Pk }. T projctio oprator dos ot plicitly appar i t mid mtod. Rak-54.3 / 4 / JN 4

66 7.X Mid formulatio Coff rcis i Driv t ak form of t mid formulatio 7.X by startig from t strog form T - T - of t Timosko bam problm togtr it t dfiitio of t sar forc i. T - QM ii Formulat t corrspodig mid fiit lmt mtod. Rak-54.3 / 4 / JN 43

67 QUESTIONS? ANSWERS ECTURE BREA!