7 Finite element methods for the Euler Bernoulli beam problem

Transcription

1 7 Fnt lmnt mtods for t Eulr Brnoull bam problm

2 CIV-E6 Engnrng Computaton and Smulaton Contnts. Modllng prncpls and boundary alu problms n ngnrng scncs. Bascs of numrcal ntgraton and dffrntaton 3. Basc D fnt dffrnc and collocaton mtods - bars/rods at dffuson spag lctrostatcs 4. Enrgy mtods and basc D fnt lmnt mtods - bars/rods bams at dffuson spag lctrostatcs 5. Basc D and 3D fnt lmnt mtods - at dffuson spag 6. Numrcal mplmntaton tcnqus for fnt lmnt mtods 7. Fnt lmnt mtods for Eulr Brnoull bams 8. Fnt lmnt mtods for Tmosnko bams 9. Fnt lmnt mtods for D and 3D lastcty. Etra lctur CIV-E6 / 6 / Jarkko Nrann 6

3 7 Fnt lmnt mtods for t Eulr Brnoull bam problm Contnts. Strong and ak forms for Eulr Brnoull bams. Fnt lmnt mtods for Eulr Brnoull bams arnng outcom A. Rcognzng t caractrstcs of fnt lmnt mtods for gr-ordr PDEs B. Undrstandng of t basc proprts of t Eulr Brnoull bam problm and ablty to dr t basc formulatons rlatd to t problm C. Basc knoldg and tools for solng Eulr Brnoull bam problms by fnt lmnt mtods t lmnts n partcular C Rfrncs ctur nots: captr 9. Tt book: captrs.6 3. A.I CIV-E6 / 6 / Jarkko Nrann 7

4 7. Motaton for t Eulr-Brnoull bam lmnt analyss Bam structurs frams trusss bams arcs ar t most typcal structural parts n modrn structural ngnrng and t Eulr Brnoull bam lmnt s t most typcal on n commrcal FEM softar. CIV-E6 / 6 / Jarkko Nrann 8

5 7. Strong and ak forms for Eulr Brnoull bams t us consdr a tn stragt bam structur subct to suc a loadng tat t dformaton stat of t bam can b modld by t bndng problm n a plan. T basc knmatcal dmnson rducton assumptons of a tn bam modl calld Eulr Brnoull bam 75 or ngnrng bam modl.. normal fbrs of t bam as rman stragt durng t dformaton normal fbrs of t bam as do not strc durng t dformaton 3 matral ponts of t bam as mo n t rtcal drcton only 4 normal fbrs of t bam as rman as normals durng t dformaton u y d d CIV-E6 / 6 / Jarkko Nrann 9

6 7. Strong and ak forms for Eulr Brnoull bams t us consdr a tn stragt bam structur subct to suc a loadng tat t dformaton stat of t bam can b modld by t bndng problm n a plan. T basc knmatcal dmnson rducton assumptons of a tn bam modl calld Eulr Brnoull bam 75 or ngnrng bam modl.. normal fbrs of t bam as rman stragt durng t dformaton normal fbrs of t bam as do not strc durng t dformaton 3 matral ponts of t bam as mo n t rtcal drcton only 4 normal fbrs of t bam as rman as normals durng t dformaton com tru f t dsplacmnts ar prsntd as u y y cos : u y ysn y t dnotng t dflcton of t bam cntral or nutral as apparng as t only arabl of t problm t t currnt assumptons and furtrmor dpndng on t coordnat only. y d d CIV-E6 / 6 / Jarkko Nrann

7 7. Strong and ak forms for Eulr Brnoull bams nar dformatons for t dsplacmnt fld abo mpls t aal stran alon u y y y. CIV-E6 / 6 / Jarkko Nrann

8 7. Strong and ak forms for Eulr Brnoull bams nar dformatons for t dsplacmnt fld abo mpls t aal stran alon u y y y. Dfnng t bndng momnt as tout spcfyng t strsss at t momnt M : M z : A y z y da CIV-E6 / 6 / Jarkko Nrann

9 7. Strong and ak forms for Eulr Brnoull bams nar dformatons for t dsplacmnt fld abo mpls t aal stran alon u y y y. Dfnng t bndng momnt as tout spcfyng t strsss at t momnt M : M t nrgy balanc of t prncpl of rtual ork can b rttn n t form W V nt A W z : t σ : ε dv V A dad y z y da b u dv S t y t ds S t t u ds 3D lastcty tory! D + D CIV-E6 / 6 / Jarkko Nrann 3

10 7. Strong and ak forms for Eulr Brnoull bams yda d A fd D + D M d fd D.. alon CIV-E6 / 6 / Jarkko Nrann 4

11 7. Strong and ak forms for Eulr Brnoull bams yda d A M d fd fd r t bam s assumd to b subct to a rtcal dstrbutd surfac loadng t y t y y z actng on t uppr and lor surfacs of t bam dfnng a rsultant loadng f : Z t t y t / z dz t / z dz r t ntgrals ar takn along lns and n t z drcton for ac on uppr and lor surfacs S t and S t rspctly for collctng t pyscal load from surfacs onto t bam as. Rmark. Otr loadng typs could b consdrd as ll. Z t t y Z t Z t CIV-E6 / 6 / Jarkko Nrann 5

12 7. Strong and ak forms for Eulr Brnoull bams Intgraton by parts n t trm for t ntrnal rtual ork gs t form M M M d M M fd f d CIV-E6 / 6 / Jarkko Nrann 6

13 7. Strong and ak forms for Eulr Brnoull bams Intgraton by parts n t trm for t ntrnal rtual ork gs t form M M M d M M fd f d mplyng t forc balanc and boundary condtons.. t strong form as M f EB - M M M or M M M M M M M Q M Q CIV-E6 / 6 / Jarkko Nrann 7

14 7. Strong and ak forms for Eulr Brnoull bams Intgraton by parts n t trm for t ntrnal rtual ork gs t form M M M d M M mplyng t forc balanc and boundary condtons.. t strong form as M f EB - M M M or M M fd f d M M M M M Q M Q T sar forc s dtrmnd by t momnt qulbrum: Q M. CIV-E6 / 6 / Jarkko Nrann 8

15 7. Strong and ak forms for Eulr Brnoull bams Takng nto account t lnarly lastc consttut rlatons n t form y E y y E CIV-E6 / 6 / Jarkko Nrann 9

16 7. Strong and ak forms for Eulr Brnoull bams Takng nto account t lnarly lastc consttut rlatons n t form y E y y E t strong form can b rttn as a dsplacmnt formulaton as follos: For a gn loadng fnd t dflcton : suc tat f : R R EI f EB - EI EI EI EI M M Q Q. CIV-E6 / 6 / Jarkko Nrann 3

17 7. Strong and ak forms for Eulr Brnoull bams Takng nto account t lnarly lastc consttut rlatons n t form y E y y E t strong form can b rttn as a dsplacmnt formulaton as follos: For a gn loadng fnd t dflcton : suc tat f : R R EI f EB - EI EI EI EI M M Q Q T momnt and sar forc ar gn n trms of t dflcton as M EI Q EI I : I z. : A y da. CIV-E6 / 6 / Jarkko Nrann 3

18 7. Strong and ak forms for Eulr Brnoull bams T corrspondng ak form s obtand from t rtual ork prssons abo or as usual by multplyng t strong form by a tst functon aratonal functon ntgratng or t doman and fnally ntgratng by parts: fd ˆ EI d ˆ EI ˆ EI ˆ EI ˆ EI d ˆ EI ˆ d ˆ W. CIV-E6 / 6 / Jarkko Nrann 3

19 7. Strong and ak forms for Eulr Brnoull bams T corrspondng ak form s obtand from t rtual ork prssons abo or as usual by multplyng t strong form by a tst functon aratonal functon ntgratng or t doman and fnally ntgratng by parts: fd ˆ EI d ˆ EI ˆ EI ˆ EI ˆ EI d ˆ EI ˆ d ˆ W. Ts quaton gs t nrgy balanc t rspct to t aratonal spac as EI ˆ d fd ˆ ˆ W and t ssntal boundary condtons for a cantlr bam for nstanc as ; ŵ ŵ. CIV-E6 / 6 / Jarkko Nrann 33

20 7. Strong and ak forms for Eulr Brnoull bams T corrspondng ak form s obtand from t rtual ork prssons abo or as usual by multplyng t strong form by a tst functon aratonal functon ntgratng or t doman and fnally ntgratng by parts: fd ˆ EI d ˆ EI ˆ EI ˆ EI ˆ EI d ˆ EI ˆ d ˆ W. Ts quaton gs t nrgy balanc t rspct to t aratonal spac as EI ˆ d fd ˆ ˆ W and t ssntal boundary condtons for a cantlr bam for nstanc as ; ŵ Rmark. In addton t tral and tst functon spacs ar dtrmnd by t ak form as usual altoug n ts cas t spac H s not an approprat coc anymor du to t scond ordr drats prsnt n t blnar form. ŵ. CIV-E6 / 6 / Jarkko Nrann 34

21 REATIONS TO THE PREVIOUS WEES

22 . Boundary and ntal alu problms n ngnrng scncs Aally loadd lnarly lastc bar problm n a dsplacmnt form: EAu b EAu b a u u N dffrntal quaton ssntal BC natural BC D gnralzaton. Aally torsonally and transrsally loadd bam uncoupld: tnson : torson : 3 4 bndng : 5 u EAu GJ EI u 6 b EAu r GJ q N T EI Ts on s of a dffrnt form! M EI Q CIV-E6 / 6 / Jarkko Nrann 36

23 4. Wak form D and D gnralzatons D and D at dffuson: EAu d EAb u u D bam: EAu d tnson-comprsson b d EAu d GIt d torson CIV-E6 / 6 / Jarkko Nrann kt d kt f d EI z d bndng Ts on s of a dffrnt form! d f d b d aal T T k f b d r d q d torsonal transrsal 37

24 4. Wak form D modl problm Abstract ak form formalsm. Fnd a soluton a u uˆ l uˆ t t blnar form uˆ V a u uˆ : u AEuˆ d t functonal l uˆ : l uˆ N a : V R buˆ d : S V u S suc tat and t functon spacs for t tral and tst functons rspctly: R S : { d u } V : { d }. CIV-E6 / 6 / Jarkko Nrann 38

25 BAC TO THIS WEE

26 7. Strong and ak forms for Eulr Brnoull bams T ak form of t Eulr Brnoull bam problm: t us consdr a cantlr bam subct to a dstrbutd load. Fnd W s.t. a ˆ l ˆ ˆ W f t t blnar form load functonal and t aratonal spac a ˆ l ˆ fˆ d W { H EI ˆ d } H. f E I CIV-E6 / 6 / Jarkko Nrann 4

27 7. Strong and ak forms for Eulr Brnoull bams T ak form of t Eulr Brnoull bam problm: t us consdr a cantlr bam subct to a dstrbutd load. Fnd W s.t. a ˆ l ˆ ˆ W t t blnar form load functonal and t aratonal spac f a ˆ l ˆ fˆ d W { H EI ˆ d } H. f E I Rmark. For t frst tm t aratonal spac s a subspac of Sobol spac H s scton 7.X c ll ssntally nflunc t fnt lmnt spac.. t form of t fnt lmnt bass functons n practc. CIV-E6 / 6 / Jarkko Nrann 4

28 7. Fnt lmnt formulaton for Eulr Brnoull bams Conformty. It s no clar tat a pcs lnar contnuous fnt lmnt appromaton s not an approprat coc for t currnt bam problm Wy at t lctur. CIV-E6 / 6 / Jarkko Nrann 4

29 7. Fnt lmnt formulaton for Eulr Brnoull bams Conformty. It s no clar tat a pcs lnar contnuous fnt lmnt appromaton s not an approprat coc for t currnt bam problm. Wy at t lctur. Instad a to fnd out c knd of condtons for t polynomal ordr and contnuty accross t lmnts ll satsfy t conformty condton W W H. Ho about a scond ordr k = pcs lnar contnuous appromaton? CIV-E6 / 6 / Jarkko Nrann 43

30 7. Fnt lmnt formulaton for Eulr Brnoull bams Conformty. It s no clar tat a pcs lnar contnuous fnt lmnt appromaton s not an approprat coc for t currnt bam problm. Wy at t lctur. Instad a to fnd out c knd of condtons for t polynomal ordr and contnuty accross t lmnts ll satsfy t conformty condton W W H. Ho about a scond ordr k = pcs lnar contnuous appromaton? Prously t conformty subspac condton as of t form V V H and t as satsfd by smply dfnng t dscrt spac as V { H P k }. CIV-E6 / 6 / Jarkko Nrann 44

31 7. Fnt lmnt formulaton for Eulr Brnoull bams Conformty. It s no clar tat a pcs lnar contnuous fnt lmnt appromaton s not an approprat coc for t currnt bam problm. Wy at t lctur. Instad a to fnd out c knd of condtons for t polynomal ordr and contnuty accross t lmnts ll satsfy t conformty condton W W H. Ho about a scond ordr k = pcs lnar contnuous appromaton? Prously t conformty subspac condton as of t form V V H and t as satsfd by smply dfnng t dscrt spac as V { H P k }. n In practc a prously usd a pcs lnar appromaton c s globally contnuous from lmnt to lmnt. In gnral s ts a suffcnt proprty for satsfyng t condton H? T ansr gn at t lctur. u CIV-E6 / 6 / Jarkko Nrann 45

32 7. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or n an functon? H H CIV-E6 / 6 / Jarkko Nrann 46

33 7. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or n an functon? H It can b son tat contnuty accross t lmnt dgs s a suffcnt condton for t stnc of t ak drat as long as t funcon as a ak drat locally n ac lmnt d = 3: H R d H and C H. CIV-E6 / 6 / Jarkko Nrann 47

34 7. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or n an functon? H It can b son tat contnuty accross t lmnt dgs s a suffcnt condton for t stnc of t ak drat as long as t funcon as a ak drat locally n ac lmnt d = 3: H R d H and C H. Snc a fnt lmnt appromaton s oftn a functon c s a polynomal n ac lmnt Pk and nc nfntly smoot n ac lmnt du to t fact tat P H k C t mans tat contnuty accross t lmnt dgs s an ssntal condton to b rqurd from t appromaton. CIV-E6 / 6 / Jarkko Nrann 48

35 7. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or n an functon? H It can b son tat contnuty accross t lmnt dgs s a suffcnt condton for t stnc of t ak drat as long as t funcon as a ak drat locally n ac lmnt d = 3: H R d H and C H. Snc a fnt lmnt appromaton s oftn a functon c s a polynomal n ac lmnt Pk and nc nfntly smoot n ac lmnt du to t fact tat P H k C t mans tat contnuty accross t lmnt dgs s an ssntal condton to b rqurd from t appromaton. Accordngly ts mans tat contnuty of t drat of a functon accross t lmnt dgs s a suffcnt condton for t stnc of t scond ak drat as long as t functon as a scond ordr ak drat locally n ac lmnt: R d H and C H. CIV-E6 / 6 / Jarkko Nrann 49

36 7. Fnt lmnt formulaton for Eulr Brnoull bams Conformng fnt lmnt mtod for t Eulr Brnoull bam problm: t us consdr a cantlr bam subct to a loadng f. Fnd suc tat W a a ˆ l ˆ W W ˆ l ˆ fˆ d W { H { C EI ˆd ˆ W f } P 3 }. E I CIV-E6 / 6 / Jarkko Nrann 5

37 7. Fnt lmnt formulaton for Eulr Brnoull bams Conformng fnt lmnt mtod for t Eulr Brnoull bam problm: t us consdr a cantlr bam subct to a loadng f. Fnd suc tat W a a ˆ l ˆ W W ˆ l ˆ fˆ d W { H { C EI ˆd ˆ W f } P 3 }. E I C Rmark. contnuty.. contnuty of t functon and ts drats btn lmnts ll b satsfd by applyng t trd ordr Hrmt sap functons c ssntally dffr from t prously usd agrang sap functons. CIV-E6 / 6 / Jarkko Nrann 5

38 5. d u d u n Prously for t contnuous pcs polynomal fnt lmnt appromaton of ordr k t dgrs of frdom r t nodal alus agrang ntrpolaton: 7. Fnt lmnt formulaton for Eulr Brnoull bams CIV-E6 / 6 / Jarkko Nrann

39 53. d u d u n Prously for t contnuous pcs polynomal fnt lmnt appromaton of ordr k t dgrs of frdom r t nodal alus agrang ntrpolaton: No a pcs cubc trd ordr.. k = 3 fnt lmnt appromaton ll b usd t nodal alus of bot t functon and ts drats takn as dgrs of frdom Hrmt ntrpolaton: d d n 7. Fnt lmnt formulaton for Eulr Brnoull bams CIV-E6 / 6 / Jarkko Nrann

40 54. d u d u n Prously for t contnuous pcs polynomal fnt lmnt appromaton of ordr k t dgrs of frdom r t nodal alus agrang ntrpolaton: No a pcs cubc trd ordr.. k = 3 fnt lmnt appromaton ll b usd t nodal alus of bot t functon and ts drats takn as dgrs of frdom Hrmt ntrpolaton:.. d d d d n 7. Fnt lmnt formulaton for Eulr Brnoull bams CIV-E6 / 6 / Jarkko Nrann

41 In ts approac four sap functons ll b rlatd to ac lmnt; to to ac nd of ac ntral : : : : : : : : : 7. Fnt lmnt formulaton for Eulr Brnoull bams CIV-E6 / 6 / Jarkko Nrann

42 In ts approac four sap functons ll b rlatd to ac lmnt; to to ac nd of ac ntral : : : : : : : : : 7. Fnt lmnt formulaton for Eulr Brnoull bams CIV-E6 / 6 / Jarkko Nrann

43 57 Eac lmnt ll g ts contrbuton to t stffnss matr and forc ctor as 7. Fnt lmnt formulaton for Eulr Brnoull bams n d f l F n q p d EI a p p p q p q p pq CIV-E6 / 6 / Jarkko Nrann

44 58 Eac lmnt ll g ts contrbuton to t stffnss matr and forc ctor as In practc only four sap functons ar nonzro n ac lmnt and nc 7. Fnt lmnt formulaton for Eulr Brnoull bams. T d d d d F F F F d F n d f l F n q p d EI a p p p q p q p pq CIV-E6 / 6 / Jarkko Nrann

45 7. Fnt lmnt formulaton for Eulr Brnoull bams T global stffnss matr and forc ctor can no b assmbld n a usual ay. C Rmark. Hrmt typ trd ordr contnuous dflcton appromaton mpls by takng lmnts drats tat rotaton appromaton s pcs quadratc and contnuous momnt appromaton s pcs lnar and dscontnuous sar forc appromaton s pcs constant and dscontnuous. CIV-E6 / 6 / Jarkko Nrann 59

46 7. Fnt lmnt formulaton for Eulr Brnoull bams T global stffnss matr and forc ctor can no b assmbld n a usual ay. C Rmark. Hrmt typ trd ordr contnuous dflcton appromaton mpls by takng lmnts drats tat rotaton appromaton s pcs quadratc and contnuous momnt appromaton s pcs lnar and dscontnuous sar forc appromaton s pcs constant and dscontnuous. Rmark. In a smlar mannr on can dduc tat a typcal contnuous agrang typ quadratc fnt lmnt appromaton ould lad to pcs lnar dscontnuous rotaton. Takng tn t global drat of t rotaton ould lad on lmnt bordrs to Drac dlta functons c ar not squar-ntgrabl. Hnc t blnar form of t problm ould not b dfnd for ts typ of tral functons and t problm ould not b solabl. Ts argumntaton gs a farly ntut ustfcaton for t contnuty rqurmnt mpld by t conformty condton W W C H. CIV-E6 / 6 / Jarkko Nrann 6

47 u : : 7.X Innr products norms and Sobol spacs Sobol norms and Sobol spacs: For and. nnr product norm and spac ar dfnd rspctly as u : u d : { R s dfnd n u : R } : H CIV-E6 / 6 / Jarkko Nrann 6

48 u : H H u : u u d u u and nnr product norm and smnorm rspctly as : : 7.X Innr products norms and Sobol spacs Sobol norms and Sobol spacs: For and. nnr product norm and spac ar dfnd rspctly as u : u d : { s dfnd R : n H u : R } : H CIV-E6 / 6 / Jarkko Nrann 6

49 63 and nnr product norm and smnorm rspctly as H H : : d : u u u u u k H k H. :... :... d... : k k k k k k k k k k k k u u u u u u u H k H and nnr product norm and smnorm rspctly as : } n s dfnd { : : d : : H u u u Sobol norms and Sobol spacs: For and. nnr product norm and spac ar dfnd rspctly as R R u : 7.X Innr products norms and Sobol spacs CIV-E6 / 6 / Jarkko Nrann

50 7.X Innr products norms and Sobol spacs Sobol spacs ar no dfnd as follos: H H k : { : { s dfnd n s dfnd n k { u u }. d} / } CIV-E6 / 6 / Jarkko Nrann 64

51 7.X Innr products norms and Sobol spacs Sobol spacs ar no dfnd as follos: H H k Sobol nnr products and norms can b gnralzd to gr dmnsons: For and t u u : { : { : : R n n s dfnd n s dfnd n u u d u u u u : R u u y y u y k { y u u u d} }. yy yy d. / } CIV-E6 / 6 / Jarkko Nrann 65

52 7.X Innr products norms and Sobol spacs Sobol spacs ar no dfnd as follos: Sobol nnr products and norms can b gnralzd to gr dmnsons: For and t For H H k u u : { : { : : R n u u d u : R m m u : u u : d. n s dfnd s dfnd n u u u n u : R u u y y t u y k { y u u u d} }. yy yy d. / } u u y u y y y CIV-E6 / 6 / Jarkko Nrann 66

53 7.X Innr products norms and Sobol spacs Coff rcs 7 Fnd t drat of t functon a a sn a 3/ t. So tat nr. CIV-E6 / 6 / Jarkko Nrann 67

54 QUESTIONS? ANSWERS ECTURE BREA!