8 Finite element methods for the Kirchhoff Love plate problem
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1 8 Fne elemen meods for e Krcoff Love plae problem
2 Rak Numercal Meods n Srucural Engneerng Conens 1. Modellng prncples and boundar value problems n engneerng scences. Energ meods and basc 1D fne elemen meods - bars/rods beams ea dffuson seepage elecrosacs 3. Basc D and 3D fne elemen meods - ea dffuson seepage. Numercal mplemenaon ecnques of fne elemen meods 5. Absrac formulaon and accurac of fne elemen meods 6. Fne elemen meods for Euler Bernoull beams 7. Fne elemen meods for Tmosenko beams 8. Fne elemen meods for Krcoff Love plaes 9. Fne elemen meods for Ressner Mndln plaes 10. Fne elemen meods for D and 3D elasc 11. Era lecure: oer fne elemen applcaons n srucural engneerng Rak / 016 / JN 10
3 8 Fne elemen meods for e Krcoff Love plae problem Conens 1. Srong and eak forms for Krcoff Love plaes. Fne elemen meods for Krcoff Love plaes Learnng oucome A. Undersandng e basc feaures of e Krcoff Love plae problem and abl o derve e basc formulaons relaed o e problem B. Basc knoledge and ools for solvng Krcoff Love plae problems b fne 1 elemen meods and nonconformng elemens n parcular C References Lecure noes: capers Te book: capers Rak / 01 / JN 11
4 8.0 Movaon for Krcoff Love plae elemen analss Plae srucures besde beam srucures are e mos pcal srucural pars n modern srucural engneerng. Krcoff Love plae elemens for nsance are classcal bu sll common n FEM sofare. Rak / 01 / JN 1
5 8.1 Srong and eak forms for Krcoff Love plaes Le us consder a n planar plae srucure subec o suc a loadng and boundar condons a e deformaon sae of e plae can be modeled b e bendng problem. Te basc knemacal assumpons of dmenson reducon for a n plae called Krcoff Love plae 1850 or sorl Krcoff plae.e. K1 normal fbres of e mdsurface reman srag durng e deformaon K normal fbres of e mdsurface do no srec durng e deformaon K3 maeral pons of e mdsurface move n e vercal drecon onl K normal fbres of e mdsurface reman as normals durng e deformaon v u Rak / 01 / JN 13
6 8.1 Srong and eak forms for Krcoff Love plaes Le us consder a n planar plae srucure subec o suc a loadng and boundar condons a e deformaon sae of e plae can be modeled b e bendng problem. Te basc knemacal assumpons of dmenson reducon for a n plae called Krcoff Love plae 1850 or sorl Krcoff plae.e. K1 normal fbres of e mdsurface reman srag durng e deformaon K normal fbres of e mdsurface do no srec durng e deformaon K3 maeral pons of e mdsurface move n e vercal drecon onl K normal fbres of e mdsurface reman as normals durng e deformaon come rue f e dsplacemens are ren as v 0 : u u v ere s e deflecon of e plae mdsurface e onl varable of e problem and furermore dependng on and coordnaes onl no on. Rak / 01 / JN 1
7 8.1 Srong and eak forms for Krcoff Love plaes v u v u Consderng lnear deformaons e dsplacemen feld above mples e srans Rak / 01 / JN
8 8.1 Srong and eak forms for Krcoff Love plaes v u v u Consderng lnear deformaons e dsplacemen feld above mples e srans Defnng e bendng momens ou specfng e sresses a e momen as / / / / / / : : : d M d M d M Rak / 01 / JN
9 8.1 Srong and eak forms for Krcoff Love plaes v u v u Consderng lnear deformaons e dsplacemen feld above mples e srans Defnng e bendng momens ou specfng e sresses a e momen as e energ balance of e prncple of vrual ork can be ren n e form ere e plae s assumed o be subec o a vercal dsrbued surface loadng onl / / / / / / : : : d M d M d M Rak / 01 / JN S S V V e ds dad ds dv dv W W e 0 b u u b ε σ ; : 0 / / n
10 18 Rak / 01 / JN S S ds ds da f da f da d d d : ; / / / / / / 8.1 Srong and eak forms for Krcoff Love plaes
11 19 Rak / 01 / JN. : ; / / / / / / da f da M M M ds ds da f da f da d d d S S 8.1 Srong and eak forms for Krcoff Love plaes
12 0 Inegraon b pars ce n e erm for e nernal vrual ork gves e form Rak / 01 / JN da f da ds da f da T M Mn v v v M dv / ; : Srong and eak forms for Krcoff Love plaes. : ; / / / / / / da f da M M M ds ds da f da f da d d d S S
13 1 Inegraon b pars ce n e erm for e nernal vrual ork gves e form Rak / 01 / JN dv / ; : 0 da f ds s ds da ds da f da ds da f da T s Mn n n Mn M n M M Mn v v v M dv dv dv 8.1 Srong and eak forms for Krcoff Love plaes. : ; / / / / / / da f da M M M ds ds da f da f da d d d S S
14 8.1 Srong and eak forms for Krcoff Love plaes mplng e srong form.e. e force balance dv dv M f R KL - M Rak / 01 / JN
15 8.1 Srong and eak forms for Krcoff Love plaes mplng e srong form.e. e force balance dv dv M and boundar condons afer one negraon b pars for clamped smpl suppored and free boundares respecvel Mn s Mn n M 1 c n nnf Mn n M Mn f β n nns Mn s dv M n Q s s c c ere e ndces 1 and refer o e sdes of a boundar angle a a corner pon c on e free boundar. c f s R n KL - M f Rak / 01 / JN 3
16 8.1 Srong and eak forms for Krcoff Love plaes mplng e srong form.e. e force balance dv dv M f R KL - M and boundar condons afer one negraon b pars for clamped smpl suppored and free boundares respecvel Mn s Mn n M 1 c n nnf Mn ere e ndces 1 and refer o e sdes of a boundar angle a a corner pon c on e free boundar. Te momen equlbrum gves e sear force as Q dv M. β Mn n M n nns c f s Mn s dv M n Q s s c c n f Rak / 01 / JN
17 8.1 Srong and eak forms for Krcoff Love plaes Takng no accoun e lnearl elasc consuve relaons follong e assumpons of e plane sress sae.e. 0 Rak / 01 / JN 5
18 6 Takng no accoun e lnearl elasc consuve relaons follong e assumpons of e plane sress sae.e. gves e sress componens n erms of deflecon as Rak / 01 / JN. ; 1 1 E E G G E E E E Srong and eak forms for Krcoff Love plaes
19 7 Takng no accoun e lnearl elasc consuve relaons follong e assumpons of e plane sress sae.e. gves e sress componens n erms of deflecon as Te momen and sear force componens can be no gven n erms of deflecon n e form Rak / 01 / JN. ; 1 1 E E G G E E E E Srong and eak forms for Krcoff Love plaes
20 8 Rak / 01 / JN E D D D Q D D Q D M D M D M c mpl a dsplacemen formulaon for e equlbrum equaons and boundar condons no ren ou ere oever. 8.1 Srong and eak forms for Krcoff Love plaes
21 8.1 Srong and eak forms for Krcoff Love plaes Assumng consan maeral values and consan ckness gves e equlbrum as a compac barmonc dsplacemen formulaon: For a gven loadng f : R fnd e deflecon suc a D : R f ; armonc operaor barmonc operaor e correspondng boundar condons from above ren n erms of deflecon. KL - Rak / 01 / JN 9
22 8.1 Srong and eak forms for Krcoff Love plaes Assumng consan maeral values and consan ckness gves e equlbrum as a compac barmonc dsplacemen formulaon: For a gven loadng f : R fnd e deflecon suc a D : R f ; armonc operaor barmonc operaor e correspondng boundar condons from above ren n erms of deflecon. KL - Te correspondng eak form s obaned from e vrual ork epresson above or as usual b mulplng e srong form KL-M b a es funcon negrang over e doman and fnall negrang b pars and collecng e erms n fron of es funcons and under e correspondng negrals: Rak / 01 / JN 30
23 8.1 Srong and eak forms for Krcoff Love plaes fda ˆ dv dv dv M d ˆ M n ds ˆ dv M d ˆ Rak / 01 / JN 31
24 8.1 Srong and eak forms for Krcoff Love plaes fda ˆ dv dv dv dv M d ˆ M n ds ˆ M n ds ˆ dv M d ˆ M n ds ˆ M : d ˆ Rak / 01 / JN 3
25 8.1 Srong and eak forms for Krcoff Love plaes : D fda ˆ ˆ W b : dv dv dv 1 D 0 M n ds ˆ M n ds ˆ M n ds ˆ 1 0 dv dv / M d ˆ dv M d ˆ M n ds ˆ M n ds ˆ κ : / /. / M : d ˆ κ ˆ T D b κ d In e las equal above e ensor noaon uled for e negraon b pars pase as been canged o a vecor noaon Vog noaon nroducng e curvaure vecor κ. Rak / 016 / JN 33
26 8.1 Srong and eak forms for Krcoff Love plaes Weak form for e Krcoff Love plae problem: Le a clamped plae be subec o a dsrbued loadng f L R. Fnd W suc a a ˆ l ˆ ˆ W e blnear form load funconal and varaonal spaces a ˆ l ˆ W { v H κ ˆ fd ˆ T D b v κ d 0 v n 0 }. Rak / 01 / JN 3
27 8.1 Srong and eak forms for Krcoff Love plaes Weak form for e Krcoff Love plae problem: Le a clamped plae be subec o a dsrbued loadng f L R. Fnd W suc a a ˆ l ˆ ˆ W e blnear form load funconal and varaonal spaces a ˆ l ˆ W { v H κ ˆ fd ˆ T D b v κ d 0 v n 0 }. Remark. Te varaonal space s once agan a subspace of e c ll affec essenall e correspondng fne elemen spaces. H space Rak / 016 / JN 35
28 8.1 Srong and eak forms for Krcoff Love plaes Weak form for e Krcoff Love plae problem: Le a clamped plae be subec o a dsrbued loadng f L R. Fnd W suc a a ˆ l ˆ ˆ W e blnear form load funconal and varaonal spaces a ˆ l ˆ W { v H κ ˆ fd ˆ T D b v κ d 0 v n 0 }. H Remark. Te varaonal space s once agan a subspace of e space c ll affec essenall e correspondng fne elemen spaces. Remark. For smplc e plae as been assumed o be full clamped. Oer boundar condons could ave been consdered as ell oever: nonero dsplacemens forces and momens could ave been prescrbed on e boundares. Rak / 016 / JN 36
29 8.1 Srong and eak forms for Krcoff Love plaes Break eercse 8 So a e blnear form of e Krcoff Love plae problem s ellpc and connuous respec o e norm for : a v v a v ˆ κ v T κ ˆ D T b D κ vd b κ vd C H v v v W For c pe of values of quanes E and e quoen C / appearng n e correspondng error esmaes ll be large/small? ˆ 0 1 v ˆ W. Rak / 01 / JN 37
30 8. Fne elemen meods for Krcoff Love plaes Conformng fne elemen meod for e Krcoff Love plae problem: Le a clamped plae be subec o a dsrbued load f L R. Fnd s.. a ˆ l ˆ ˆ W e blnear form load funconal and varaonal spaces a v ˆ l ˆ W κ ˆ fˆ d T W { v H D b κ vd v 0 v n 0 }. W Rak / 01 / JN 38
31 8. Fne elemen meods for Krcoff Love plaes Conformng fne elemen meod for e Krcoff Love plae problem: Le a clamped plae be subec o a dsrbued load f L R. Fnd s.. a W ˆ l ˆ ˆ W 1 e blnear form load funconal and varaonal spaces a v ˆ l ˆ W W fˆ d { κ ˆ T v H D b κ vd v 0 v n { v C v 0 v n 0 v K P5 K }. W In order o oban a conformng compable fne elemen meod e regular 1 requremen of order H as o be sasfed. I can fulflled b a C connuous fne elemen appromaon cf. subsecon 6. c leads rangular elemens o pecese ff order qunc polnomals Argrs rangle 1968: 0 }. Rak / 01 / JN 39
32 8. Fne elemen meods for Krcoff Love plaes Te Argrs rangle as 1 degrees of freedom funcon values plus e frs and second dervaves a corners plus e normal dervaves a edge mdpons c 1 ogeer quaranee connu across e elemens: C L1 L L3 : v a 13 a 3 K ref a 1 a Rak / 015 / JN 0
33 8. Fne elemen meods for Krcoff Love plaes Te Argrs rangle as 1 degrees of freedom funcon values plus e frs and second dervaves a corners plus e normal dervaves a edge mdpons c 1 ogeer quaranee connu across e elemens: L L L 1 L... L 9 3 : : C v a v a 13 v a 13 a 3 K ref a 1 a Rak / 015 / JN 1
34 8. Fne elemen meods for Krcoff Love plaes Te Argrs rangle as 1 degrees of freedom funcon values plus e frs and second dervaves a corners plus e normal dervaves a edge mdpons c 1 ogeer quaranee connu across e elemens: L L L L 1 10 L... L 9... L 3 : 18 : C v a v a v : a 13 v a v 13 a v a 13 a 3 K ref a 1 a Rak / 015 / JN
35 8. Fne elemen meods for Krcoff Love plaes Te Argrs rangle as 1 degrees of freedom funcon values plus e frs and second dervaves a corners plus e normal dervaves a edge mdpons c 1 ogeer quaranee connu across e elemens: L L L L L L... L 9... L... L 3 : 18 1 : : : C v a v a v v n a a 13 v a v 13 a 1 3. v a 13 a 3 a 13 a 3 K ref a 1 a 1 a Rak / 015 / JN 3
36 8. Fne elemen meods for Krcoff Love plaes Te sape funcons of e reference elemen correspondng o ese degrees of freedom are deermned b a se of unqueness condons for : N 1 P5 Kref L N N 1...1; 0 L N Rak / 01 / JN
37 5 Rak / 01 / JN Te sape funcons of e reference elemen correspondng o ese degrees of freedom are deermned b a se of unqueness condons for : 8. Fne elemen meods for Krcoff Love plaes ; ref ref c c c c c c c c c c c c c c c c c c c c c N K P N N N L K P N N L
38 6 Rak / 013 / JN Te fne elemen appromaon s locall of e form a 1 a a 3 a 1 a 3 a 13 K ref 8. Fne elemen meods for Krcoff Love plaes a n N a n N a n N a N a N a N a N a N a N d N e n K e e
39 8. Fne elemen meods for Krcoff Love plaes Nonconformng ncompable lo order fne elemen meod for e Krcoff Love plae problem: Te Morle rangle as 6 degrees of freedom funcon values a corners and normal dervaves a edge mdpons c mpl a e fne elemen deflecon appromaon provded b e elemen s dsconnuous from elemen o elemen: L L L 1 L 5 L 3 L 6 : : v a v n a a 3 a 13 a 1 K ref a 3 a 1 a Rak / 015 / JN 7
40 8. Fne elemen meods for Krcoff Love plaes Nonconformng ncompable lo order fne elemen meod for e Krcoff Love plae problem: Te Morle rangle as 6 degrees of freedom funcon values a corners and normal dervaves a edge mdpons c mpl a e fne elemen deflecon appromaon provded b e elemen s dsconnuous from elemen o elemen: L L L 1 L 5 L 3 L 6 : : v a v n a Te sape funcons of e reference elemen correspondng o ese degrees of freedom are deermned b a se of unqueness condons for : N P L N K 1 0 N c c 1 c c c c a 3 a 13 a 1 5 L N K ref c a 3 a 1 6 a Rak / 015 / JN 8
41 8. Fne elemen meods for Krcoff Love plaes Nonconformng Morle fne elemen meod for e Krcoff Love plae problem: Le a clamped plae be subec o a dsrbued load f L R. Fnd suc a a a l ˆ W W ˆ l ˆ ˆ { v L fˆ d n elem e 1 e ˆ W K v κ ˆ K κ dk connuous a edge mdpons m; T ; D b P K; v s connuous a corners c v n s v c 0 v n m } W. Rak / 015 / JN 9
42 8. Fne elemen meods for Krcoff Love plaes Nonconformng Morle fne elemen meod for e Krcoff Love plae problem: Le a clamped plae be subec o a dsrbued load f L R. Fnd suc a a a W W ˆ l ˆ ˆ l ˆ { v L fˆ d n elem e 1 e ˆ W K v κ ˆ K κ dk connuous a edge mdpons m; T ; D b P K; v s connuous a corners c v n v c } W. Remark. Te deflecon appromaon of s elemen Morle rangle 1968 s no connuous; e connu of e deflecon s sasfed onl a e corner nodes le e connu of e normal dervave s sasfed a e edge mdpons. Hence e essenal boundar condons ll no be full sasfed: e deflecon ll be fed a e corner nodes e normal dervave a e edge mdpons. 0 v n m s Rak / 015 / JN 50
43 8. Fne elemen meods for Krcoff Love plaes Argrs rangle: connu and ellpc of e blnear form ogeer e Galerkn orogonal mpl a sandard error esmae follong Cea s lemma: C v v W c mples a more quanave esmae assumng a soluon smoo enoug H k 1 c 5 for 5. k c k k 1 6 Rak / 01 / JN 51
44 8. Fne elemen meods for Krcoff Love plaes Argrs rangle: connu and ellpc of e blnear form ogeer e Galerkn orogonal mpl a sandard error esmae follong Cea s lemma: C v v W c mples a more quanave esmae assumng a soluon smoo enoug H k 1 c 5 for 5. k c k k 1 6 Remark. In order o full ule e g polnomal order of s elemen n e acual convergence rae e eac soluon of e problem sould be eremel s regular. In pracce oever ver ofen olds a. H s 6 Rak / 01 / JN 5
45 8. Fne elemen meods for Krcoff Love plaes Argrs rangle: connu and ellpc of e blnear form ogeer e Galerkn orogonal mpl a sandard error esmae follong Cea s lemma: C v v W c mples a more quanave esmae assumng a soluon smoo enoug H k 1 c 5 for 5. k c k k 1 6 Remark. In order o full ule e g polnomal order of s elemen n e acual convergence rae e eac soluon of e problem sould be eremel s regular. In pracce oever ver ofen olds a. H s Morle rangle: Error analss follos a nonsandard roue of nonconformng meods Srang s II lemma c gves b assumng requred regular as usual an error esmae n e form 1975 : 6 n e 1 elem : e c f. K 3 0 Rak / 01 / JN 53
46 8. Fne elemen meods for Krcoff Love plaes Ts resul can be ren n e form 3 c. and furer eended o an esmae n a eaker norm: c. 0 3 Rak / 01 / JN 5
47 8. Fne elemen meods for Krcoff Love plaes Ts resul can be ren n e form 3 c. and furer eended o an esmae n a eaker norm: c. 0 3 For regular anoer resul ave been proved 01: For all ere ess an suc a for all sasfng 0 H f. 0 f L Rak / 01 / JN 55
48 8. Fne elemen meods for Krcoff Love plaes Ts resul can be ren n e form and furer eended o an esmae n a eaker norm: For regular anoer resul ave been proved 01: For all ere ess an suc a for all sasfng For e follong resuls ave been proved 01: 3 c. c H f. 0 c s f L H 0 f H l ˆ : f I ˆ ; a con. ln.nerpolaor I s.. H b nerpreng e loadng funconal n e form f and f. 1 s 1 I 0 0 ˆ a 1 1 H ˆ a Rak / 01 / JN 56
49 8. Fne elemen meods for Krcoff Love plaes Eample for e convergence rae of e Morle rangle: Convergence order of e Morle elemen appled o a clamped Krcoff Love square plae subec o a unform loadng: log log c f log. 3 0 log e log Rak / 01 / JN 57
50 8. Fne elemen meods for Krcoff Love plaes Eample for e convergence rae of e Morle rangle: Convergence order of e Morle elemen appled o a clamped Krcoff Love square plae subec o a unform loadng: log log c f log. 3 0 log e log Remark. For plong sould be noed a N denong e number of elemens n e mes olds a log 1 1/ N N 1 log N Rak / 01 / JN 58
51 8. Fne elemen meods for Krcoff Love plaes Eample for adapve mes refnemens an error ndcaor developed for e Morle elemen 008: Smpl suppored unforml loaded Krcoff Love plae; resdual based error ndcaors; auomaed Delauna rangulaons. a Energ error and mes for e eraon sep 3. Rak / 01 / JN 59
52 8. Fne elemen meods for Krcoff Love plaes Eample for adapve mes refnemens an error ndcaor developed for e Morle elemen 008: Smpl suppored unforml loaded Krcoff Love plae; resdual based error ndcaors; auomaed Delauna rangulaons. a Energ error and mes for e eraon sep 3. b Energ error and mes for e eraon sep 11. Rak / 01 / JN 60
53 8.X Curved boundares revsed In general soluon domans curved boundares are no andled n er eac form n fne elemen formulaons.e. e geomer of e doman s appromaed suc as e problem varables. In mos cases e geomer of e doman s mapped beeen e acual and reference elemen b polnomals: F e l : eref el ; Fe p q p q Pm el. If e polnomal order m of e geomer mappng s equal o e polnomal order of e elemen m = k dencal or smlar sape funcons of order k e mappng or elemen s called soparamerc le for m < k e erm subparamerc s used and for m > k e erm used s superparamerc. Isoparamerc mappng elemen sape funcons s e mos ofen used one; n 1D case e formula s N 1 l l 1 N. l Remark. Inegrals n fne elemen forms are compued over e mapped geomer. Hence a possbl non-consan Jacoban mar of e ger order geomer mappng mples an era erm a small oever n error esmaes. e ref eref F e1 F e F el e 1 e l e Rak / 01 / JN 61
54 8.X Curved boundares revsed Coffee eercse 8 Solve e plae bendng problem of a clamped unforml loaded crcular plae b usng Comsol Plae Elemens four dfferen meses. Invesgae e convergence beavour b abulang: e deflecon bendng momens and sear force a e mdpon r = 0 bendng momens and sear force on e boundar r = a. Compare e fne elemen appromaon o e eac values provded b e Krcoff-Love plae model e radus s denoed b a: M r r r f0a 16 f0a 6D 1 3 r / a 1 r / a Q r r M f0r. r f0a r / a Rak / 01 / JN 6
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. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.
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