A Mixed Formulation Triangular Mindlin Plate Finite Element with Cubic Displacements and Quadratic Moments
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1 Rcn Adancs in Enginring Mchanics Srcrs and Urban lanning A Mid Forlaion rianglar Mindlin la Fini Eln ih Cbic isplacns and Qadraic Mons AAM ÓSA VALENIN-VASILE UNGUREANU IOAN LUCIAN CÎRSOLOVEAN MIRCEA HORNEł Ciil Enginring parn ransilania Unirsi of Braşo 5 h rnli Sr Braşo ROMANIA adadosa@ahooco ngran@nibro hp:nibroconsrciin Bildings Srics parn ransilania Unirsi of Braşo 5 h rnli Sr Braşo ROMANIA lco@ahooco ircahorn@nibro hp:nibroconsrciin Absrac: - his papr prsns a si-nod rianglar Mindlin pla fini ln basd on a id forlaion h ransrsal displacns ihin h ln ar inrpolad b a copl cbic polnoial in ric coordinas h roaion filds ar dscribd b copl qadraic polnoials h indpndn srss fild is gin b a8-β polnoial rlaion ih qadraic ons In h absnc of disribd on loads on h ln srfac h shar forcs ar in qilibri ih h ons h rsld ln is locking fr and of corrc rank h prsnd nrical apls sho a high accrac of h rsls hich ar coparabl ih knon lns fro aailabl corcial srcral analsis sofar K-Words: - fini lns Mindlin pla riangl id forlaion Inrodcion Bn fini ln applicaions bnding pla lns ar h os frqnl sd in srcral analsis Hor h FEM horis hich ar a h basis of h os par of hs lns canno b fll cord b h nginring dcaional procss h dlopn of h si-nod Mindlin pla rianglar fini ln prsnd in his papr has an dcaional prpos h qaions inold in h forlaion of h ln ar sipl In h forlaion of h ln adancd isss lik pararic fncions rdcd and slci ingraion bbbl fncions or linkd inrpolaion ar aoidd b incrasd copaional ffors ar accpd in ordr o siplif h forlaion h ln can b ndrsood ih h basic pla bnding hor and iho adancd fini ln knoldg h ln is fr of shar locking passs h consan bnding and shar pach ss and has corrc rank h prsnd nrical apls sho good prforancs Mindlin la Eqaions For h pla of figr h displacns of a poin ( ar: ( ( ( ( h plan of h rfrnc ss is h dian plan of h pla and ar h roaions of h noral in h and rspcil plan ( ; h srains ar pr bnding srains: ISBN:
2 ( (3 and ransrsal shar srains: (4 Figr : isplacns and roaions For an isoropic linar lasic arial h nsions ar: E ( ( ( (5 h srss rslans sd in bnding pla applicaions ar: d d d d d ( a b Figr : a nsions b Srss rslans Hr is h hicknss of h pla and ar bnding ons is h orsional on and ar h shar forcs ina ina ina ± ± ± (7 h ransrsal shar nsions and sall ar sall hir ariaion is qadraic on h hicknss of h pla 5 5 a a (8 Fro h qaions (3-( rsls: ( kg kg (9 or: Hr ( 3 E and k inrodcs h ffc of nonnifor shar dforaions on h hicknss of h pla For isoropic arial k5 3 h 3 Eln h ln has si nods ih a oal nbr of 8 dgrs of frdo Nods 4 5 ar in h iddl of h sraigh dgs of h ln h coordinas of h nods ar: ( ( h displacns of h nods ar: ( a Figr 3: h 3 ln Rcn Adancs in Enginring Mchanics Srcrs and Urban lanning ISBN:
3 Rcn Adancs in Enginring Mchanics Srcrs and Urban lanning h inrnal displacn fild is: α 3 α 3 α 3 α 8 ( hr 3 3 ( ( 3 ( ( 3 3 and conains h fncions of a copl polnoial of dgr hr in and 3 and 3 ar is driais hich ar incopl dgr o polnoials h roaions and ar gin b h driais of h ransrsal displacns copld b linar ransrsal shar dforaions dscribd b si indpndn parars α α B adding h rs α α and 8 α 7 α h roaion filds bco copl dgr o polnoials h nknon parars α(α α α 8 can b drind fro h nodal displacns of h ln h rlaions bn h nodal roaions and hich ar corial qaniis and h inrnal roaions and hich ar slops ar: and h rlaion bn h nodal displacns and h inrnal displacn fild parars α is: a Cα ( Whr α α a α α 8 3 ( C 3 ( 3 ( ( ( 7 8 Rsls α C a Using h abo rlaion h displacn fild bcos: C a Na ( Hr N conains h displacn inrpolaion fncions h ransrsal displacns dpnd boh on h nodal displacns and roaions I can b shon ha h displacns on h coon dg of o nighboring lns ar copaibl h displacn drid srains ar: or: Ba C (3 hr 3 ( 3 ( ( 4 4 and 3 h ln 3 prsnd in [3] is basd an a displacn forlaion and ss rlaions ( (3 o dfin h displacn filds As i can b obsrd in papr [3] 3 ln has fair n high prforancs spciall in h cas of hick and odral hick plas dspi h fac ha i is no locking-fr h displacn drid srains gin b rlaion (3 conain parasiic shar rs corrsponding o h qadraic polnoial rs inrodcd b α 7 and α 8 In ordr o liina h ffc of hs rs h displacn drid srain fild is rplacd b a srss drid on h indpndn nsion fild is gin b a 8-β qadraic polnoial rlaion: Hr ( and ( ( β β S β β8 (4 In h absnc of disribd on loads on h ln srfac h shar forcs ar in saic qilibri ih h bnding ons: and a ISBN:
4 Rcn Adancs in Enginring Mchanics Srcrs and Urban lanning h ln siffnss ari is gin b: k Q H Q (5 Hr Q S BdV H S SdV ( V V h nodal forcs rsld fro h loads q disribd on h A srfac of h ln ar: f N qda (7 A h ingrals ar copd b a si poin qadrar rl h nsions in h ln ar copd in h si nods b: ( SH Qa (8 4 Nrical Eapls 4 ach s h 3 ln passs h consan bnding and shar pach ss gin b MacNal and Hardr [] abl : Mai ip displacns ( Cas A ip Mon Cas B ip Forc Cas C Unifor ransrsal load Ba hor Uniforl Loadd Sqar la A sqar pla of sid L is considrd A qadran of h pla is odlld b shs In h abl cnr ransrsal displacns ar prsnd for hard sipl sppor on h sids ( n and o L raios abl 3 conains cnr ransrsal displacns for h fll clapd cas ( n h ln is copll fr of shar locking In h figrs 7 and 8 bnding ons orsional ons and shar forcs ar prsnd for a clapd hick pla (L h rfrnc als gin in h caprs of hs figrs r copd sing 88 shs of 3 lns 4 Canilr Ba A canilr ba odlld b o 3 lns fll clapd a on nd is sbjcd o o load cass as shon in figr 4 and a nifor ransrsal load cas on h hol srfac o aoid h aniclasic crar ffc oisson raio of h arial is akn o b ro As i can b sn in abl h ip displacns ar acl copd b h fini ln odl Figr 5: h odl of a qadran of a sqar pla ( sh; E9 3 L; q Figr 4: Canilr ba - E 7 (kn ; abl : Cnr displacns of h niforl loadd sipl sppord pla Msh L - L -7 3 [] 3 [] Sris ISBN:
![Rcn Adancs in Enginring Mchanics Srcrs and Urban lanning abl 3: Cnr displacns of h niforl loadd clapd pla Msh L - L -7 3 [] 3 [] 4 4 8 8 3 3 Sris 544 544 54 54 54 4 4858 4997 534 543 499 8 5](/docs-images/77/76048003/images/5-0.jpg "53 53 53 49 3 583 37 4 53 Figr 8: Shar forcs for h niforl loadd clapd sqar pla Rfrnc als: 37 493 Figr : Bnding ons for h niforl loadd clapd sqar pla Rfrnc als: 38 5 5 Conclsion In his papr a")
![prforancs ih h bs knon lns of is p Figr 7: orsional ons for h niforl loadd clapd sqar pla Rfrnc al: 773 Rfrncs: [] Zinkiic OC alor RL: h Fini Eln Mhod ol ( h d 5 [] MacNal R: Fini Elns: hir](/docs-images/77/76048003/images/5-2.jpg "dsign and prforanc Marcl ckr Inc994 [3] osa A Alqaain HA A sd of Mindlin pla fini lns h 4 h Inrnaional Confrnc on Copaional Mchanics and Viral Enginring COMEC Braşo L Libris ISBN:")
5 Rcn Adancs in Enginring Mchanics Srcrs and Urban lanning abl 3: Cnr displacns of h niforl loadd clapd pla Msh L - L -7 3 [] 3 [] Sris Figr 8: Shar forcs for h niforl loadd clapd sqar pla Rfrnc als: Figr : Bnding ons for h niforl loadd clapd sqar pla Rfrnc als: Conclsion In his papr a si-nod Mindlin pla rianglar fini ln is prsnd h ln is basd on a id forlaion h inrnal displacns of h ln ar dscribd b a copl cbic polnoial for h ransrsal displacns and qadraic roaions in ric coordinas h inrnal bnding on filds ar qadraic and ar in qilibri ih h shar forcs h ln passs h consan bnding and shar pach ss and is fr of locking Fro h nrical apls rsls ha h ln has coparabl prforancs ih h bs knon lns of is p Figr 7: orsional ons for h niforl loadd clapd sqar pla Rfrnc al: 773 Rfrncs: [] Zinkiic OC alor RL: h Fini Eln Mhod ol ( h d 5 [] MacNal R: Fini Elns: hir dsign and prforanc Marcl ckr Inc994 [3] osa A Alqaain HA A sd of Mindlin pla fini lns h 4 h Inrnaional Confrnc on Copaional Mchanics and Viral Enginring COMEC Braşo L Libris ISBN:
A STUDY OF MINDLIN PLATE FINITE ELEMENTS
Th 4h Inrnaional Confrnc Copaional Mchanics and Viral Enginring COMEC - OCTOBER Brasov Roania A STUY OF MINLIN LATE FINITE ELEMENTS Ada osa Hadan Ahd Af Alqaain Univrsi TRANSILVANIA Brasov ROMANIA adadosa@ahooco
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