6 Finite element methods for the Euler Bernoulli beam problem
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1 6 Fnt lmnt mtods for t Eulr Brnoull bam problm
2 Rak-54.3 Numrcal Mtods n Structural Engnrng Contnts. Modllng prncpls and boundary valu problms n ngnrng scncs. Enrgy mtods and basc D fnt lmnt mtods - bars/rods bams at dffuson spag lctrostatcs 3. Basc D and 3D fnt lmnt mtods - at dffuson spag 4. Numrcal mplmntaton tcnqus of fnt lmnt mtods 5. Abstract formulaton and accuracy of fnt lmnt mtods 6. Fnt lmnt mtods for Eulr Brnoull bams 7. Fnt lmnt mtods for Tmosnko bams 8. Fnt lmnt mtods for rcoff ov plats 9. Fnt lmnt mtods for Rssnr Mndln plats. Fnt lmnt mtods for D and 3D lastcty. Etra lctur: otr fnt lmnt applcatons n structural ngnrng Rak-54.3 / 6 / JN 87
3 6 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. Undrstandng of t basc proprts of t Eulr Brnoull bam problm and ablty to drv t basc formulatons rlatd to t problm B. Basc knoldg and tools for solvng 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 Rak-54.3 / 4 / JN 88
4 6. Motvaton 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. Rak-54.3 / 4 / JN 89
5 6. 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 calld Eulr Brnoull bam 75.. 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 mov n t vrtcal drcton only 4 normal fbrs of t bam as rman as normals durng t dformaton u yv d d Rak-54.3 / 4 / JN 9
6 6. 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 calld Eulr Brnoull bam 75.. 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 mov n t vrtcal drcton only 4 normal fbrs of t bam as rman as normals durng t dformaton com tru f t dsplacmnts ar prsntd as u v y v y cos v : u y ysn y t dnotng t dflcton of t bam cntral or nutral as apparng as t only varabl of t problm t t currnt assumptons and furtrmor dpndng on t coordnat only. yv d d Rak-54.3 / 4 / JN 9
7 6. Strong and ak forms for Eulr Brnoull bams Consdrng lnar dformatons t dsplacmnt fld abov mpls t aal stran u y y y. Rak-54.3 / 4 / JN 9
8 6. Strong and ak forms for Eulr Brnoull bams nar dformatons for t dsplacmnt fld abov 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 Rak-54.3 / 6 / JN 93
9 6. Strong and ak forms for Eulr Brnoull bams nar dformatons for t dsplacmnt fld abov 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 vrtual 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 vds S t t u ds 3D lastcty tory! D + D Rak-54.3 / 6 / JN 94
10 6. Strong and ak forms for Eulr Brnoull bams yda d A fd D + D M d fd D.. alon Rak-54.3 / 6 / JN 95
11 6. 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 vrtcal 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 rspctvly 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 Rak-54.3 / 5 / JN 96
12 6. Strong and ak forms for Eulr Brnoull bams Intgraton by parts n t trm for t ntrnal vrtual ork gvs t form M M M d M M fd f d Rak-54.3 / 4 / JN 97
13 6. Strong and ak forms for Eulr Brnoull bams Intgraton by parts n t trm for t ntrnal vrtual ork gvs 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 Rak-54.3 / 4 / JN 98
14 6. Strong and ak forms for Eulr Brnoull bams Intgraton by parts n t trm for t ntrnal vrtual ork gvs 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. Rak-54.3 / 4 / JN 99
15 6. Strong and ak forms for Eulr Brnoull bams Takng nto account t lnarly lastc consttutv rlatons n t form y E y y E Rak-54.3 / 4 / JN 3
16 6. Strong and ak forms for Eulr Brnoull bams Takng nto account t lnarly lastc consttutv rlatons n t form y E y y E t strong form can b rttn as a dsplacmnt formulaton as follos: For a gvn loadng fnd t dflcton : suc tat f : R R EI f EB - EI EI EI EI M M Q Q. Rak-54.3 / 4 / JN 3
17 6. Strong and ak forms for Eulr Brnoull bams Takng nto account t lnarly lastc consttutv rlatons n t form y E y y E t strong form can b rttn as a dsplacmnt formulaton as follos: For a gvn 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 gvn n trms of t dflcton as M EI Q EI I : I z. : A y da. Rak-54.3 / 4 / JN 3
18 6. Strong and ak forms for Eulr Brnoull bams T corrspondng ak form s obtand from t vrtual ork prssons abov or as usual by multplyng t strong form by a tst functon varatonal functon ntgratng ovr 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. Rak-54.3 / 4 / JN 33
19 6. Strong and ak forms for Eulr Brnoull bams T corrspondng ak form s obtand from t vrtual ork prssons abov or as usual by multplyng t strong form by a tst functon varatonal functon ntgratng ovr 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 gvs t nrgy balanc t rspct to t varatonal spac as EI ˆ d fd ˆ ˆ W and t ssntal boundary condtons for a cantlvr bam for nstanc as ; ŵ ŵ. Rak-54.3 / 4 / JN 34
20 6. Strong and ak forms for Eulr Brnoull bams T corrspondng ak form s obtand from t vrtual ork prssons abov or as usual by multplyng t strong form by a tst functon varatonal functon ntgratng ovr 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 gvs t nrgy balanc t rspct to t varatonal spac as EI ˆ d fd ˆ ˆ W and t ssntal boundary condtons for a cantlvr 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 s not an approprat coc anymor du to t scond ordr drvatvs prsnt n t blnar form. ŵ. Rak-54.3 / 6 / JN 35
21 6. Strong and ak forms for Eulr Brnoull bams T ak form of t Eulr Brnoull bam problm: t us consdr a cantlvr bam subct to a dstrbutd load. Fnd W s.t. a ˆ l ˆ ˆ W t t blnar form load functonal and t varatonal spac f a ˆ l ˆ fˆ d W { v EI ˆ d v v }. f E I Rak-54.3 / 4 / JN 36
22 6. Strong and ak forms for Eulr Brnoull bams T ak form of t Eulr Brnoull bam problm: t us consdr a cantlvr bam subct to a dstrbutd load. Fnd W s.t. a ˆ l ˆ ˆ W t t blnar form load functonal and t varatonal spac f a ˆ l ˆ fˆ d W { v EI ˆ d v v }. f E I Rmark. For t frst tm t varatonal spac s a subspac of Sobolv spac c ll ssntally nflunc t fnt lmnt spac. Accordngly contnuty corcvty and rror stmats ll b formulatd t rspct to t norm basc prncpls for t analyss ll rman t sam ovr. Rak-54.3 / 6 / JN 37
23 6. Strong and ak forms for Eulr Brnoull bams Brak rcs 6 So tat t blnar form of t Eulr Brnoull bam problm s llptc and contnous t rspct to t norm: a v v a v uˆ EIv vd EIv uˆd C v v v W uˆ v uˆ W. For c typ of valus of t cross sctonal quantts E and I t quotnt C / apparng n t corrspondng rror stmats ll b larg/small? Rak-54.3 / 6 / JN 38
24 6. 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. Instad av to fnd out c knd of condtons for t polynomal ordr and contnuty accross t lmnts ll satsfy t conformty condton W W. o about a scond ordr k = pcs lnar contnuous appromaton? Rak-54.3 / 4 / JN 39
25 6. 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. Instad av to fnd out c knd of condtons for t polynomal ordr and contnuty accross t lmnts ll satsfy t conformty condton W W. o about a scond ordr k = pcs lnar contnuous appromaton? Prvously t conformty subspac condton as of t form V V and t as satsfd by smply dfnng t dscrt spac as V { v v P k }. Rak-54.3 / 6 / JN 3
26 6. 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. Instad av to fnd out c knd of condtons for t polynomal ordr and contnuty accross t lmnts ll satsfy t conformty condton W W. o about a scond ordr k = pcs lnar contnuous appromaton? Prvously t conformty subspac condton as of t form V V and t as satsfd by smply dfnng t dscrt spac as v V { v v P k }. n In practc av prvously usd a pcs lnar appromaton c s globally contnuous from lmnt to lmnt. In gnral s ts a suffcnt proprty for satsfyng t condton cf. Coff rcs 5.? u Rak-54.3 / 6 / JN 3
27 6. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or vn an functon? Rak-54.3 / 4 / JN 3
28 6. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or vn an functon? It can b son tat contnuty accross t lmnt dgs s a suffcnt condton for t stnc of t ak drvatv as long as t funcon as a ak drvatv locally n ac lmnt d = 3: R d v and vc v. Rak-54.3 / 6 / JN 33
29 6. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or vn an functon? It can b son tat contnuty accross t lmnt dgs s a suffcnt condton for t stnc of t ak drvatv as long as t funcon as a ak drvatv locally n ac lmnt d = 3: R d v and vc v. Snc a fnt lmnt appromaton s oftn a functon c s a polynomal n ac lmnt v Pk and nc nfntly smoot n ac lmnt du to t fact tat P k C t mans tat contnuty accross t lmnt dgs s an ssntal condton to b rqurd from t appromaton. Rak-54.3 / 6 / JN 34
30 6. Fnt lmnt formulaton for Eulr Brnoull bams Contnuty. Is a contnuous functon an functon or vn an functon? It can b son tat contnuty accross t lmnt dgs s a suffcnt condton for t stnc of t ak drvatv as long as t funcon as a ak drvatv locally n ac lmnt d = 3: R d v and vc v. Snc a fnt lmnt appromaton s oftn a functon c s a polynomal n ac lmnt v Pk and nc nfntly smoot n ac lmnt du to t fact tat P 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 drvatv of a functon accross t lmnt dgs s a suffcnt condton for t stnc of t scond ak drvatv as long as t functon as a scond ordr ak drvatv locally n ac lmnt: R d v and vc v. Rak-54.3 / 6 / JN 35
31 6. Fnt lmnt formulaton for Eulr Brnoull bams Conformng fnt lmnt mtod for t Eulr Brnoull bam problm: t us consdr a cantlvr bam subct to a loadng f. Fnd suc tat W a a v ˆ l ˆ W W ˆ l ˆ fˆ d W { v { v C EIv ˆd ˆ W v v v f v } v P 3 }. E I Rak-54.3 / 4 / JN 36
32 6. Fnt lmnt formulaton for Eulr Brnoull bams Conformng fnt lmnt mtod for t Eulr Brnoull bam problm: t us consdr a cantlvr bam subct to a loadng f. Fnd suc tat W a a v ˆ l ˆ W W ˆ l ˆ fˆ d W { v { v C EIv ˆd ˆ W v v v f v } v P 3 }. E I C Rmark. contnuty.. contnuty of t functon and ts drvatvs accross t lmnt dgs ll b satsfd by applyng t trd ordr rmt sap functons c ssntally dffr from t prvously usd agrang sap functons. Rak-54.3 / 4 / JN 37
33 38. d u d u n Rak-54.3 / 4 / JN Prvously for t contnuous pcs polynomal fnt lmnt appromaton of ordr k t dgrs of frdom r t nodal valus agrang ntrpolaton: 6. Fnt lmnt formulaton for Eulr Brnoull bams
34 39. d u d u n Rak-54.3 / 4 / JN Prvously for t contnuous pcs polynomal fnt lmnt appromaton of ordr k t dgrs of frdom r t nodal valus agrang ntrpolaton: No a pcs cubc trd ordr.. k = 3 fnt lmnt appromaton ll b usd t nodal valus of bot t functon and ts drvatvs takn as dgrs of frdom rmt ntrpolaton: d d n 6. Fnt lmnt formulaton for Eulr Brnoull bams
35 3. d u d u n Rak-54.3 / 4 / JN Prvously for t contnuous pcs polynomal fnt lmnt appromaton of ordr k t dgrs of frdom r t nodal valus agrang ntrpolaton: No a pcs cubc trd ordr.. k = 3 fnt lmnt appromaton ll b usd t nodal valus of bot t functon and ts drvatvs takn as dgrs of frdom rmt ntrpolaton:.. d d d d n 6. Fnt lmnt formulaton for Eulr Brnoull bams
36 3 3 3 Rak-54.3 / 4 / JN In ts approac four sap functons ll b rlatd to ac lmnt; to to ac nd of ac ntrval : : : : : : : : : 6. Fnt lmnt formulaton for Eulr Brnoull bams
37 3 3 3 Rak-54.3 / 4 / JN In ts approac four sap functons ll b rlatd to ac lmnt; to to ac nd of ac ntrval : : : : : : : : : 6. Fnt lmnt formulaton for Eulr Brnoull bams
38 33 Rak-54.3 / 4 / JN Eac lmnt ll gv ts contrbuton to t stffnss matr and forc vctor as 6. 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
39 34 Rak-54.3 / 4 / JN Eac lmnt ll gv ts contrbuton to t stffnss matr and forc vctor as In practc only four sap functons ar nonzro n ac lmnt and nc 6. 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
40 6. Fnt lmnt formulaton for Eulr Brnoull bams T global stffnss matr and forc vctor can no b assmbld n a usual ay. C Rmark. rmt typ trd ordr contnuous dflcton appromaton mpls by takng lmnts drvatvs tat rotaton appromaton s pcs quadratc and contnuous momnt appromaton s pcs lnar and dscontnuous sar forc appromaton s pcs constant and dscontnuous. Rak-54.3 / 4 / JN 35
41 6. Fnt lmnt formulaton for Eulr Brnoull bams T global stffnss matr and forc vctor can no b assmbld n a usual ay. C Rmark. rmt typ trd ordr contnuous dflcton appromaton mpls by takng lmnts drvatvs 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 drvatv of t rotaton ould lad on lmnt bordrs to Drac dlta functons c ar not squar-ntgrabl. nc t blnar form of t problm ould not b dfnd for ts typ of tral functons and t problm ould not b solvabl. Ts argumntaton gvs a farly ntutv ustfcaton for t contnuty rqurmnt mpld by t conformty condton W W C. Du to conformty t rror analyss can b carrd out by standard tcnqus: Rak-54.3 / 4 / JN 36
42 Error stmats. contnuty and corsvty of t blnar form togtr t Galrkn ortogonalty mply an rror stmat follong t Ca s lmma 6. Fnt lmnt formulaton for Eulr Brnoull bams C v v W Rak-54.3 / 6 / JN 37
43 Error stmats. contnuty and corsvty of t blnar form togtr t Galrkn ortogonalty mply an rror stmat follong t Ca s lmma 6. Fnt lmnt formulaton for Eulr Brnoull bams C v v W from c gt a mor quanttatv stmat assumng a smoot soluton k c 3 for k c k k 4 3. Rak-54.3 / 6 / JN 38
44 Error stmats. contnuty and corsvty of t blnar form togtr t Galrkn ortogonalty mply an rror stmat follong t Ca s lmma 6. Fnt lmnt formulaton for Eulr Brnoull bams C v v W from c gt a mor quanttatv stmat assumng a smoot soluton k c 3 for k c k k 4 3. Abov as ll as n arlr rror stmats av usd a rsult from appromaton tory for t ntrpolaton rror of polynomals statng tat a polynomal ~ of ordr ntrpolats a functon t t follong accuracy: ~ m c k k m k k. Rak-54.3 / 6 / JN 39
45 Error stmats. contnuty and corsvty of t blnar form togtr t Galrkn ortogonalty mply an rror stmat follong t Ca s lmma 6. Fnt lmnt formulaton for Eulr Brnoull bams C v v W from c gt a mor quanttatv stmat assumng a smoot soluton k c 3 for k c k k 4 3. Abov as ll as n arlr rror stmats av usd a rsult from appromaton tory for t ntrpolaton rror of polynomals statng tat a polynomal ~ of ordr ntrpolats a functon t t follong accuracy: ~ m c k k m k k. Rmark. It can b son tat t rmt fnt lmnt appromaton gvs accurat nodal valus for t dflcton and ts drvatv rotaton of t Eulr Brnoull bam problm: and for all nods. Rak-54.3 / 6 / JN 33
46 6.X Sobolv mbddng & trac Abov contnuty as son to mply an drvatv undr crtan crcumstancs. Wt crtan assumptons t oppost mplcaton olds as ll Sobolv mbddng torm: k C m m k d / t R For nstanc n D cas an functon s contnuous l n D cas nstad a functon as to b rgular n ordr to b contnuous. If a functon s rgular nsd ts doman o rgular t s on t boundary of t doman? If t doman s boundd and ts boundary s C rgular tn t trac of a functon v on t boundary s an functon Trac torm: Tv d. m C k Tv T : c v v C r s lnar. If t furtr olds tat tn. Tv v Rak-54.3 / 4 / JN 33
47 6.X Sobolv mbddng & trac Coff rcs 9 Fnd t drvatv of t functon v a a sn v a 3/ t. So tat nvr. Rak-54.3 / 4 / JN 33
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