ࡱ> )`8bjbjUU ???0llll(((<$F$F$F$FGl<{fHRIRIRIRIKKKjllllll$hI(KKKKKllRIRIo5edededKlRI(RIjedKjeded.|6(}RI|H jeo$F\|F$K0{}cX4}(}dKKedKKKKKUdKKK{KKKK<<<'d<<<'<<<llllll `esti susret Hrvatskoga druatva za mehaniku Rijeka, 29-30. svibnja 2014. METODA USVOJENIH DEFORMACIJA ZA 9- VORNE ELEMENTE MINDLINOVIH PLO A Ribari, Dragan Sa~etak: Ovdje je prikaz novog 9- vornog etverostrani nog elementa za analizu umjereno debelih i tankih plo a po Mindlinovoj teoriji. Deformacija smicanja unutar elementa interpolirana je direktno izrazom koji je linearan u glavnom prirodnom smjeru i kvadratan u sporednom, umjesto da je derivirana iz polja pomaka i rotacija kao u standardnoj metodi pomaka. Pri tome stupnjevi slobode ostaju vorni pomaci i vorne rotacije, a ne novi parametri smicanja kao u mjeaovitoj metodi. Matrica krutosti elementa se zbog toga izvodi jednako kao i za elemente opisane standardnim interpolacijama pomaka i rotacija, minimiziranjem ukupne potencijalne energije. Ovdje su prikazani rezultati ponaaanja novog elementa, prednosti i nedostaci u odnosu na druge kona ne elemente razvijene na konceptu samo interpoliranih pomaka i rotacija. Klju ne rije i: Mindlinova teorija plo a, kona ni elementi, vezana interpolacija, metoda usvojenih deformacija smicanja 1. UVOD Za modeliranje umjereno debelih plo a po Mindlinovoj teoriji, metoda usvojenih deformacija primjenjena na kona ne plo aste etverostrani ne elemente s po etiri vora, pokazala je dobre rezultate po brzini i stabilnosti konvergencije to nome rezultatu razli itih modela plo a [1,2,5]. S druge strane 9- vorni etverostrani ni elementi bazirani na vezanoj interpolaciji polja pomaka i polja rotacija pokazali su se uspjeanima u opisivanju slo~eno distordiranih mre~a sa zakrivljenim stranicama [3]. Ovdje se metoda usvojenih deformacija ~eli primjeniti i na 9- vornim etverostrani nim elementima, tako da se funkcija deformacija smicanja unutar domene elementa reducira na minimalni polinomni stupanj, jer je to preduvjet izbjegavanja efekta  lockinga , posebno kod rijetkih mre~a vorova. Takoer, parametri koji opisuju smicanje ostaju vorni pomaci i rotacije, ime su zadovoljeni kriteriji stabilnosti ( count conditions ), a matrica krutosti elementa mo~e biti izvedena minimiziranjem funkcionala ukupne energije kona nog elementa. 2. MINDLINOVA TEORIJA PLO A I USVOJENE ORJENTACIJE Mindlinova teorija opisuje umjereno debele i u krajnjem slu aju tanke plo e, tako da povezuje transverzalni pomak srednje ravnine modela (w) i rotacije presjeka plo e oko dva koordinatna smjera (x,y) s deformacijom smicanja slijedeim poznatim izrazima  EMBED Equation.3 . (1) Istovremeno zakrivljenost deformirane ravnine ovisna je samo o rotacijama presjeka plo e  EMBED Equation.3 , (2) te se konstitutivne jednad~be u slu aju izotropno linearno elasti nih plo a, izvode za definirane materijalne veli ine Youngovog modula (E), Poissonova koeficijenta () i iz njih izvedenog modula smicanja (G), te faktora korekcije smicanja (uobi ajeno k=5/6), u formi  EMBED Equation.3  (3)  EMBED Equation.3 . (4) t je u gornjim izrazima debljina plo e, Sx i Sy rezultante naprezanja smicanja, Mx, i My rezultante momenata od normalnih naprezanja, odnosno Mxy tangencijalnih naprezanja u pripadajuim presjecima plo e. 3. VEZANA INTERPOLACIJA ZA DISTORDIRANI 9- VORNI ELEMENT Da bi 9- vorni plo asti element mogao zadovoljiti stanje istog savijanja na  patch-testu u prizvoljno distordiranoj formi, dokazano je [3] da mora imati interpolacijski izraz za transverzalni pomak w bi-kvarti ne polinomne forme u prirodnim koordinatama, a da pri tome interpolacijski izrazi za rotacije presjeka (x,y) mogu ostati bi-kvadratni polinomi. Proairenje interpolacije pomaka do tra~ene forme izvodi se uvoenjem 16 dodatnih  bubble internih stupnjeva slobode. Kako se deformacije smicanja (1) izvode iz interpolacija pomaka i rotacija, smicanja svakog od dva prirodna smjera (Sl.1.) postaju kubne funkcije po glavnom smjeru i kvarti ne po sporednom. Ravnote~a momenata plo e sa silama smicanja uvjetuje ni~u razinu polinoma smicanja u odnosu na polinome dobivene iz (2), pa je to razlog efekta  lockinga u situaciji kad debljina plo e te~i tankom limitu (Kirchhoffova teorija plo a). Stoga je po~eljno reducirati razinu polinoma deformacije smicanja, a to se mo~e izvesti dijelom izborom  bubble parametara tako da eliminiraju polinomne lanove treeg i drugog reda u glavnom prirodnom smjeru, ostavljajui samo linarnu promjenu. Tako se formulira nova interpolacija za pomake koju nazivmo vezana interpolacija, jer ona u polje pomaka uvidi rotacijske stupnjeve slobode.  Sl. 1. Distortdirani 9- vorni etverostrani ni plo asti element  odnos izmeu globalnih i prirodnih koordinata Na primjer, izraz za deformaciju smicanja uzdu~ stranice 1-2-3 elementa u prirodnim koordinatama ima linearnu formu (5) zavisnu od prirodne varijable  , a pri =-1.  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  (5) Ovakvom redukcijom polinoma deformacije smicanja obuhvaen je samo glavni prirodni smjer, dok u sporednom ostaje zarobljena kvarti na razina polinoma i ona ostavlja prisutnim problem  lockinga kod relativno rijetkih mre~a modela kona nih elemenata. Daljnja redukcija polinoma nije viae mogua adekvatnom zamjenom  bubble parametara, jer je prethodna redukcija uklonila njih 12, a preostala etiri poprimaju nul-vrijedosti po svim vornim linijama. Ideja ove analize je da se daljnja redukcija provede usvojenim poljem smicanja koji e u sporednom smjeru imati minimalnu polinomnu razinu kroz tri reducirana smicanja po vezanoj formi sli noj (5) za svaku od tri vorne linije. Ta minimalna razina je kvadratna:  EMBED Equation.3  i  EMBED Equation.3 , Deformacije smicanja u prirodnim smjerovima treba potom transformirati u globalne koordinate. 4. NUMERI KI PRIMJERI 4.1  Patch testovi Element razvijen metodom usvojenih deformacija smicanja nosi oznaku Q9-AS ( assumed shear strain ) i najprije je testiran na standardnim  patch modelima sa po pet elemenata razli itih klasa distordiranosti unutar pravokutne vanjske konture [3]. 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