A STUDY OF MINDLIN PLATE FINITE ELEMENTS

Transcription

1 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 Alqaain Af projc offic Tafila JORAN alqaain@ahooco Absrac: Bn fini ln applicaions bnding pla lns ar h os frqnl sd in srcral analsis Hovr h advancd opics hich ar a h basis of h os par of hs lns canno b fll covrd b h nginring dcaional procss This papr conains a sd of Mindlin pla fini lns in ordr o find or vn rforla lns sch ha hir prsnaion bcos as sipl as possibl Isss lik pararic fncions rdcd and slciv ingraion bbbl fncions or linkd inrpolaion ar avoidd b incrasd copaional ffors ar accpd if h siplif h forlaion A displacn forlaion riangl in ric coordinas as fond hih corrc rank rdcd lvl of locking and a rlaiv sipl forlaion Th prsnd nrical apls sho an accrac of h rsls hich is coparabl ih knon lns fro availabl corcial srcral analsis sofar Kords: fini lns Mindlin pla riangl displacn forlaion INTROUCTION Th dvlopn of h si nod Mindlin pla rianglar fini ln prsnd in his papr has an dcaional prpos Th caions involvd in h forlaion of h ln ar sipl Th ln can b ndrsood ih h basic pla bnding hor and iho advancd fini ln knoldg Th lns pass h consan bnding and shar pach ss and hav corrc rank Th prsnd nrical apls sho good prforancs MINLIN LATE ECUATIONS For h pla of figr h displacns of a poin ar: v Th plan of h rfrnc ss is h dian plan of h pla and ar h roaions of h noral in h and rspcivl plan ; v Figr : isplacns and roaions 4

2 5 Th srains ar pr bnding srains: v v and ransvrsal shar srains: v 4 For an iohrop linar lasic arial h nsions ar: E 5 Th srss rslans sd in bnding pla applicaions ar: d d d d d a b Figr : a Tnsions b Srss rslans Hr and ar bnding ons = is h orsional on and ar h shar forcs ina ina ina 7 Th ransvrsal shar nsions and sall ar sall Thir variaion is qadraic on h hicknss of h pla 5 5 a a 8 Fro h qaions - rsls: kg kg 9 or: ε σ Hr E and k inrodcs h ffc of nonnifor shar dforaions on h hicknss of h pla For isohropic arial k=5

3 THE T ELEMENT Th ln has si nods Nods 4 5 ar in h iddl of h sraigh dgs of h ln Th coordinas of h nods ar: T and T Th displacns of h nods ar: T a Figr : Th T ln Th inrnal displacn fild is: 8 hr and conains h fncions of a copl polnoial of dgr hr in and and ar is drivaivs hich ar incopl dgr o polnoials Th roaions and ar givn b h drivaivs of h ransvrsal displacns copld b linar ransvrsal shar dforaions dscribd b si indpndn parars B adding h rs 8 7 and 8 7 h roaion filds bco copl dgr o polnoials Th nkno parars = 8 T can b drind fro h nodal displacns of h ln Th rlaions bn h nodal roaions hich ar vcorial qaniis and h inrnal roaions hich ar slops ar: and Th rlaion bn h nodal displacns and h inrnal displacn fild parars is: 8 Or: a C Rsls a C Using h abov rlaion h displacn fild bcos: Na a C Hr N conains h displacn inrpolaion fncions Th ransvrsal 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 Th displacn drivd srains ar: Ba a C ε

4 hr 4 4 and Th qilibri of an ln can b prssd as: ka f 4 Whr k B Bdv V T is h siffnss ari of h ln and f N qda A T ar h nodal forcs rsld fro h loads q disribd on h A srfac of h ln Ingral 5 on h vol V of h ln is copd b h idpoin qadrar rl hich is ac for qadraic ingrands and rdcs h shar locking ffc inrodcd b h and 7 roaion rs For nifor ransvrsal load q bcos: f = qa * T 8 4 NUMERICAL EXAMLES 4 ach s Th T ln passs h consan bnding and shar pach ss givn b MacNal and Hardr [] 4 Canilvr ba A canilvr ba odld b o lns fll clapd a on nd is sbjcd o hr load cass as shon in figor 4 To avoid h aniclasic crvar ffc oisson raio of h arial is akn o b ro Figr 4: Canilvr ba - E = 7 kn ; = Ba hor T ln Tabl : Mai ip displacns Cas A Concnrad Mon Cas B Concnrad Forc Cas C Unifor ni ransvrsal load Uniforl loadd sqar pla A sqar pla of sid L is considrd A qadran of h pla is odlld b 44 shs In h abl cnr ransvrsal displacns ar prsnd for hard sipl sppor on h sids = n = and o L raios Tabl conains cnr ransvrsal displacns for h clapd cas = = Alhogh h ln is no copll fr of shar locking h rsls ar good spciall for hick plas In h figrs 7 and 8 bnding ons orsional ons and shar forcs ar prsnd for a fll calpd hick pla L= = = n = 7

5 Figr 5: Th odl of a qadran of a sqar pla N=; E=9 ν= L=; q= Tabl : Cnr displacns of h niforl loadd sipl sppord pla Msh N L= L= -7 T [] T [] 4 8 Sris Tabl : Cnr displacns of h niforl loadd clapd pla Msh N L= L= -7 T [] T [] 4 8 Sris Figr : Bnding ons for h niforl loadd clapd sqar pla Rfrnc vals: 8 5 8

6 Figr 7: Torsional ons for h niforl loadd clapd sqar pla Rfrnc val: 77 Figr 8: Shar forcs for h niforl loadd clapd sqar pla Rfrnc vals: CONCLUSION In his papr a si nod Mindlin pla rianglar fini ln is prsnd Th inrnal displacns of h ln ar dscribd b a copl cbic polnoial for h ransvrsal displacns and qadraic roaions in ric coordinas Th ln passs h consan bnding and shar pach ss Alhogh h ln is no copll fr of shar locking fro nrical apls rsls ha i has coparabl prforancs ih h bs knon lns of is p REFERENCES [] Zinkiic OC Talor RL: Th Fini Eln Mhod vol h d 5 [] MacNal R: Fini Elns:Thir dsign and prforanc Marcl ckr Inc994 9