YWMIADH BASILEIOU fifianalush PROSARMOGHS ELASTOPLASTIKWN METALLIKWN KATASKEUWN UPO TO TRISDIASTATO KRITHRIO DIARROHS TRESCA ME TEQNIKES TOU HMIJETIKO

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1 ARISTOTELEIO PANEPISTHMIO JESSALONIKHS TMHMA POLITIKWN MHQANIKWN TOMEAS EPISTHMHS KAI TEQNOLOGIAS TWN KATASKEUWN YWMIADH BASILEIOU PtuqioÔqou PolitikoÔ MhqanikoÔ fifianalush PROSARMOGHS ELASTOPLASTIKWN METALLIKWN KATASKEUWN UPO TO TRISDIASTATO KRITHRIO DIARROHS TRESCA ME TEQNIKES TOU HMIJETIKOU MAJHMATIKOU PROGRAMMATISMOUflfl DIDAKTORIKH DIATRIBH JESSALONIKH 2005

2 YWMIADH BASILEIOU fifianalush PROSARMOGHS ELASTOPLASTIKWN METALLIKWN KATASKEUWN UPO TO TRISDIASTATO KRITHRIO DIARROHS TRESCA ME TEQNIKES TOU HMIJETIKOU MAJHMATIKOU PROGRAMMATISMOUflfl DIDAKTORIKH DIATRIBH Upobl jhke sto Tm ma Politik n Mhqanik n, Tomèac Epist mhc kai TeqnologÐac twn Kataskeu n HmeromhnÐa Proforik c Exètashc: 9 MartÐou, 2005 Exetastik Epitrop An. Kajhght c Q. MpÐsmpoc, Epiblèpwn Kajhght c Q. Mpaniwtìpouloc, Mèloc TrimeloÔc Sumbouleutik c Epitrop c Kajhght c K. Jwmìpouloc, Mèloc TrimeloÔc Sumbouleutik c Epitrop c Kajhght c N. Qaralamp khc, Exetast c An. Kajhght c A. Abdel c, Exetast c Ep. Kajhght c E. Kolts khc, Exetast c Ep. Kajhght c L. PitsoÔlhc, Exetast c

3 Stouc goneðc mou kai sthn mn mh tou kajhght Jos F. Sturm

4 Prìlogoc To kalokaðri tou 1999, o twrinìc epiblèpwn thc diatrib c, Anaplhrwt c Kajhght c tou ErgasthrÐou Metallik n Kataskeu n k. Qr stoc MpÐsmpoc, tìte epiblèpwn thc diplwmatik c mou ergasðac, se suz thsh gia mellontikèc metaptuqiakèc spoudèc, mou prìteine thn ekpìnhsh didaktorikoô sto polô endiafèron jèma tou anelastikoô upologismoô metallik n kataskeu n sta plaðsia tou fainomènou prosarmog c (shakedown analysis). Mèsa apì mia dhmiourgik poreða kai episthmonik anaz thsh aut h arqik prìtash katèlhxe sth shmerin telik morf thc paroôsac diatrib c. Ja jela na euqarist sw jerm ton epiblèponta Anaplhrwt Kajhght k. Qr sto MpÐsmpo gia th suneq kai ousiastik bo jeia pou mou pareðqe kajìlh th di rkeia thc èreunac, melèthc kai suggraf c. EpÐshc euqarist touc Kajhghtèc kai mèlh thc trimeloôc sumbouleutik c mou epitrop c, k. Qar lampo Mpaniwtìpoulo kai k. KÐmwna Jwmìpoulo gia thn upost rix touc. Idiaitèrwc euqarist touc goneðc mou gia thn amèristh hjik sumpar - stas touc sto di sthma twn metaptuqiak n mou spoud n. Tèloc, euqarist touc sunadèlfouc mhqanikoôc upoy fiouc did ktorec kai to sônolo tou didaktikoô proswpikoô tou ErgasthrÐou Metallik n Kataskeu n tou Tm matoc Politik n Mhqanik n tou A.P.J. gia to filikì klðma kai thn kal di jesh pou èdeiqnan p nta ìla aut ta qrìnia sunergasðac kai sumbðwshc ston Ðdio q ro pneumatik c ergasðac. S' autì to shmeðo o suggrafèac ja jele na ekfr sei thn lôph tou gia ton prìwro j nato tou Kajhght J.F. Sturm ton Dekèmbrio tou 200, se hlikða mìlic 2 et n. O J.F. Sturm, EpÐkouroc Kajhght c sthn Epiqeirhsiak 'Ereuna sto panepist mio Tilburg thc OllandÐac, tan o dhmiourgìc tou logismikoô majhmatikoô programmatismoô SeDuMi, to opoðo qrhsimopoi jhke sto upologistikì mèroc thc diatrib c. Prìkeitai gia mia meg lh ap leia ston episthmonikì q ro tou jewrhtikoô all kai upologistikoô majhmatikoô programmatismoô. Eutuq c gia thn episthmonik koinìthta to fifiadvanced Optimization Labflfl sto Panepist mio McMaster tou Kanad, anèlabe thn beltðwsh kai pairetèrw an ptuxh tou k dika SeDuMi upì thn aigðda tou kajhght T. Terlaky kai twn didaktorik n foitht n O. Romanko kai I. Polik.

5 Perieqìmena 1 Eisagwg To fainìmeno prosarmog c sthn epist mh tou politikoô mhqanikoô Istorik anadrom tou fainomènou prosarmog c Stìqoi thc diatrib c Dom thc diatrib c Elastoplastikìthta Basikèc ènnoiec kai sqèseic Oi tanustèc t sewn - paramìrfwsewn Statik kai kinhmatik apodekt pedða Arq twn dunat n èrgwn Paramènousec t seic kai metakin seic Qarakthristik kai montelopoðhsh thc monoaxonik c sumperifor c Elastikì tèleia plastikì montèlo JewrÐa plastik c ro c Krit ria fìrtishc AÐthma mègisthc èklushc enèrgeiac lìgw plastik c paramìrfwshc i

6 ii Perieqìmena 2. Krit ria diarro c JewrÐa plastikìthtac met llwn Krit rio Tresca H mèjodoc peperasmènwn stoiqeðwn (FEM) 21.1 Eisagwg Mìrfwsh me b sh th mèjodo twn metakin sewn Sqhmatismìc kai upologismìc twn mhtr wn isoparametrik n peperasmènwn stoiqeðwn To okt kombo trisdi statoisoparametrikì stoiqeðo (8-node brick element) To fainìmeno prosarmog c (shakedown) Genik perigraf tou fainomènou prosarmog c (shakedown) To statikì je rhma To kinhmatikì je rhma O suntelest c prosarmog c Apaloif tou qrìnou apì to prìblhma prosarmog c H met bash sta peperasmèna stoiqeða Statik an lush prosarmog c Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô Eisagwg Summetrik mhtr a

7 Perieqìmena iii 5. To krit rio Tresca wc sunj kh periorismoô tou eôrouc thc diafor c twn akrot twn idiotim n σ max σ min tou tanust twn t sewn Anagwg tou krithrðou Tresca se sôsthma sunjhk n hmijetikìthtac Apaloif tou isotropèa apì to krit rio Tresca Telik morf twn diakritopoihmènwn problhm twn an lushc prosarmog c Ta probl mata prosarmog c wc probl mata majhmatikoô hmijetikoô programmatismoô SqetikoÐ me ton hmijetikì programmatismì algìrijmoi kai logismikì 64 6 Upologistik ulopoðhsh, arijmhtik paradeðgmata kai e- farmogèc Eisagwg Epilog logismikoô Upologistik ulopoðhsh ProetoimasÐa kai eisagwg dedomènwn sto SeDuMi o Par deigma: Dexamen apoj keushc ugr n Udrostatik pðesh Diafor jermokrasðac o Par deigma: Tom kulðndrou me kôlindro me paqô toðqwma o Par deigma: Antiseismik sqediasmènh sugkollht sôndesh dokoô - upostul matoc Parathr seic kai sqìlia Prot seic antimet pishc adunami n tou algìrijmou epðlushc.. 95

8 iv Perieqìmena 7 Sumper smata EpiteÔgmata thc diatrib c Dunatèc epekt seic ENGLISH SUMMARY 10 BIBLIOGRAFIA 106

9 Kef laio 1 Eisagwg 1.1 To fainìmeno prosarmog c sthn epist mh tou politikoô mhqanikoô To er thma an mia kataskeu eðnai ikan na paral bei ta efarmozìmena fortða kai me poiì perij rio asf leiac, kaj c kai pìte ja katasteð mh leitourgik lìgw uperbolik n anelastik n paramorf sewn kat rreushc, apoteloôse p ntote meðzon jèma thc epist mhc tou politikoô mhqanikoô. Sthn klassik an lush kai ston sqediasmì twn kataskeu n me b sh thn filosofða twn oriak n katast sewn, o trìpoc fìrtishc jewreðtai genik aplìc kai sugkekrimèna wc mða monìtonh, analogik auxanìmenh fìrtish. 'Omwc sthn pr xh, oi kataskeuèc suqn upìkeintai se metaballìmenec mhqanikèc kai jermikèc fortðseic. Ta fortða aut mporeð na epanalamb nontai (kuklikèc fortðseic) na metab llontai me gnwsto istorikì all paramènoun sun jwc entìc sugkekrimènwn orðwn. To gegonìc autì suqn dhmiourgeð dôo eid n duskolðec: a) periorismènh gn sh kai aporrèousa as feia sqetik me to akribèc istorikì thc fìrtishc kai b) shmantikì upologistikì kìstoc se mh-grammik bhmatik an lush. S' autèc tic peript seic thc metaballìmenhc fìrtishc, fortða qamhlìtera tou orðou plastik c kat rreushc dônatai na prokalèsoun astoqða thc kataskeu c eðte lìgw uperbolik c suss reushc plastik n paramìrfwsewn 1

10 To fainìmeno prosarmog c sthn epist mh tou politikoô mhqanikoô (incremental plasticity ratcheting) eðte exaitðac topik c kìpwshc lìgw ennalag c tou pros mou thc metabol c thc plastik c paramìrfwshc (alternating plasticity low-cycle fatigue). Wstìso, ìtan ta metaballìmena fortða an koun se èna sugkekrimèno pedðo fìrtishc, met thn p rodo peperasmènou qrìnou arijmoô kôklwn fìrtishc eðnai dunatìn na stamat sei na anaptôssetai h u- p rqousa plastik paramìrfwsh kai h susswreumènh enèrgeia lìgw plastik n paramorf sewn na parameðnei telik peperasmènh me apotèlesma thn elastik sumperifor thc kataskeu c apì to shmeðo autì kai pèra. To fainìmeno autì anafèretai wc elastik prosarmog (diejn c wc elastic shakedown adaptation). H an lush prosarmog c (shakedown analysis) èqei wc skopì na ektim sei e n mia kataskeu upì dedomèno pedðo fìrtishc (ìqi ìmwc istorikì), katorj nei na petôqei mia tètoia morf elastik c sumperifor c kai na prosdiorðsei ta krðsima ìria thc metabol c tou fortðou. ApoteleÐ t sh twn sôgqronwn kanonism n na enswmat noun mesa touc trìpouc domik c astoqðac, me thn epidðwxh thc ekmet lleushc thc anelastik c paramìrfwshc twn ìlkimwn ulik n me stìqo thn aôxhsh thc fèrousac ikanìthtac fortðou kai kat sunèpeia ton sqediasmì oikonomikìterwn kataskeu n, fèrnontac thn anelastik an lush sto prosk nio akìma kai sthn efarmosmènh mhqanik. Kai parìlo pou h an lush prosarmog c, basismènh sta akrib jewr mata thc klassik c plastikìthtac, jewreðtai fifi aplopoihmènhflfl mèjodoc, h aploôsteush aut epitugq netai periorðzontac thn an lush stic katast - seic astoqðac thc kataskeu c kai ìqi prosjètontac epiplèon proseggðseic kai paradoqèc. San apotèlesma h mejìdoc prosfèrei apodektèc ektim seic thc asf leiac miac kataskeu c par to gegonìc twn fifi qamhl c poiìthtac flfl dedomènwn eisagwg c. S mera h an lush prosarmog c èqei brei efarmog se èna eurô f sma, ekteinìmeno apì thn edafomhqanik mèqri to prìblhma thc olðsjhshc epaf c. Kataskeuèc ìpou to fainìmeno prosarmog c qrhsimopoieðtai wc prìsforh mèjodoc an lushc, anafèrontai endeiktik katwtèrw: - Swlhn seic kai sôndesmoi/gwniak swl nwn - Dexamenèc apoj keushc kai silì

11 Kef laio 1. Eisagwg -DoqeÐa pðeshc - PurhnikoÐ antidrast rec -Katastr mata od n kai sidhrodromikèc r gec 1.2 Istorik anadrom tou fainomènou prosarmog c Apì to èrgo twn prwtopìrwn Bleich [1], Melan [2, ] kaikoiter [4, 5] proèkuyan ta klassik jewr mata prosarmog c, ta opoða antiproswpeôoun mia apì tic shmantikìterec epiteôxeic thc jewrðac thc plastikìthtac sthn Mhqanik s mera. Sthn statik prosèggish me b sh to je rhma tou Melan gnwstoi eðnai o suntelest c fìrtishc kai to pedðo twn paramenous n t sewn kai epidi ketai h megistopoðhsh tou suntelest fortðou. To je rhma tou Melan ousiastik kajorðzei ta plaðsia gia na pragmatopoi sei h kataskeu prosarmog (shakedown). AntÐstoiqa sthn kinhmatik prosèggish me b sh to je rhma tou Koiter, du kì tou jewr matoc tou Melan, gnwstoi eðnai o suntelest c fìrtishc kai ta kinhmatik pedða kai epidi ketai h elaqistopoðhsh tou suntelest. To je rhma tou Koiter orðzei ta plaðsia ste h kataskeu na astoq sei, na mhn pragmatopoi sei dhlad prosarmog (shakedown). Shmantikèc eisforèc sthn efarmog kuklik n mhqanik n kai jermik n fortðsewn se leptìtoiqa kelôfh kai plaisiakèc kataskeuèc, apoteloôn oi ergasðec tou Leckie [8, 9] gia th qr sh tou majhmatikoô programmatismoô, tou Bree [10, 11] gia thn kainotomða qr shc diagramm twn kai twn Maier [12] kai Belytschko [71] gia thn sôndesh me thn mèjodo twn peperasmènwn stoiqeðwn. H klassik jewrða prosarmog c anaptôqjhke ekten c ta epìmena qrìnia ste na sumperil bei kai lla fainìmena, genikìterouc kai poluplokìterouc katastatikoôc nìmouc ulikoô apì ta elastik apìluta plastik ulik thn grammik, aperiìristh kinhmatik kr tunsh, gia na kalôyei tic sôgqronec apait seic (bl. [68, 29, 0, 1, 2, ]). Sugkekrimèna, melet jhke h epirro twn diafìrwn morf n kr tunshc tou ulikoô (hardening) kai h plastikìthta me mh susqetizìmenh ro (non associative flow rule) apì touc Maier [12, 17, 40],

12 Istorik anadrom tou fainomènou prosarmog c Pycko k.l. [41], Stein k.l. [9, 15, 16, 5], Mandel [18], Nguyen [19], Polizzotto k.l. [20], Corigliano k.l. [21], Fuschi [22], Weichert k.l. [42], Heitzer k.l. [4], Corradi k.l. [6] kai Feng k.l. [25]. Probl mata me gewmetrik mhgrammikìthta, melet jhkan apì touc Maier [17, 40], Weichert k.l. [62, 44, 45], Groß-Weege [6], Saczuk k.l. [64], Polizzotto k.l. [46], Stumpf [47] kai Tritsch k.l. [65]. Mia koma shmantik perðptwsh eðnai ekeðnh thc epðdrashc twn jermik n fortðsewn kai thc jermokrasðac sthn epif neia diarro c, me dhmosieôseic apì touc König [66, 4], Prager [67], Karadeniz k.l. [5], Bree [11], Gokhfeld k.l. [29], Groß-Weege k.l. [6], Weichert k.l. [42], Kleiber k.l. [48], kai Xue k.l. [49]. Akìma èqoume ergasðec sto fainìmeno thc por douc plastikìthtac (poroplasticity) apì touc Cocchetti kai Maier [1, 14], stic kataskeuèc me bl bec kai rwgmèc (damaged and cracked structures) apì touc Hachemi kai Weichert [24, 2], Feng k.l. [25], Polizzotto k.l. [26], Siemaszko [27], Huang k.l. [28], Weichert k.l. [45] kai Druyanov k.l. [78]. Epekt seic sta sônjeta ulik (composites) prot jhkan apì touc Tarn k.l. [76], Tirosh [7], Weichert k.l. [8], Carvelli k.l. [75], Hachemi k.l. [8] kai Dvorak k.l. [79, 77]. E- farmogèc thc prosarmog c se pl kec kai kelôfh aforoôn dhmosieôseic apì touc Sawczuk [69, 70], Dang [7], Franco k.l. [80] kai Yan k.l. [81], kaj c kai se plaisiwtèc kataskeuèc apì touc Giambanco k.l. [50] kai Spiliopoulos [51]. EpÐshc epiqeir jhke mia genðkeush twn jewrhm twn prosarmog c kai sthn perioq thc dunamik c p.q. sthn qronik ex rthsh me dun meic adr neiac kai apìsbeshc, apì touc Ceradini [82, 8] kai Corradi kai Maier [6, 7]. H upologistik ulopoðhsh thc an lushc prosarmog c sunteleðtai me th sônjesh k poiac diadikasðac diakritopoðhshc thc kataskeu c sta plaðsia thc mejìdou twn peperasmènwn stoiqeðwn kai qr shc teqnik n beltistopoðhshc apì ton q ro tou majhmatikoô programmatismoô. O Maier sto [40] grammikopoi- ntac thn epif neia diarro c metètreye to prìblhma se prìblhma grammikoô programmatismoô (linear programming) pou epilôetai me ton gnwstì algìrijmo Simplex. E n sto diakritopoihmèno prìblhma, sth statik prosèggish, oi sunart seic diarro c krathjoôn sthn mh grammik morf touc, h majhmatik mìrfwsh tou probl matoc prosarmog c odhgeð se prìblhma kurtoô mhgrammikoô programmatismoô (convex nonlinear programming) me meg lo arijmì metablht n kai mh grammikèc sunj kec me shmantikèc duskolðec ston upologismì. Gi' autìn akrib c to lìgo up rqoun anaptôxeic apotelesmatik n algo-

13 Kef laio 1. Eisagwg 5 rðjmwn kai ulopoi sewn prosarmosmènwn sta eidik qarakthristik tou probl matoc kai enswmatwmènwn pollèc forèc ston Ðdio ton k dika thc mejìdou twn peperasmènwn stoiqeðwn. Oi Stein kai Zhang [9, 5], Groß-Weege [86] kai Heitzer kai Staat [88, 87] anèptuxan exeidikeumènec mejìdouc diadoqikoô tetragwnikoô programmatismoô (sequential quadratic programming) ekmetalleuìmenoi ta eggen qarakthristik tou probl matoc prosarmog c. EpÐshc o Makrodhmìpouloc sto [58] kai [147], prìteine mia mìrfwsh twn problhm twn prosarmog c upì epðpedh kai axonosummetrik èntash, se kajarì prìblhma kwnikoô programmatismoô deôterhc t xhc (second order conic programming) kai gia to statikì kai to kinhmatikì je rhma, gia elastoplastikì ulikì kaj c kai gia di forec morfèc kr tunshc. Mia genðkeush thc prohgoômenhc mìrfwshc kai sthn trisdi stath entatik kat stash ègine apì touc Bisbos, Makrodimopoulos kai Pardalos sto [145]. Genik wstìso, sthn perðptwsh tou kinhmatikoô jewr matoc oi algìrijmoi epðlushc eðnai sun jwc epanalhptikèc mèjodoi perðplokwn problhm twn. ErgasÐec p nw sto jèma autì èqoun dhmosieujeð apì touc Ponter k.l. [56, 57], Zhang [89], Carveli k.l. [90] kai Liu [91]. 1. Stìqoi thc diatrib c Me b sh ta prohgoômena oi stìqoi thc diatrib c orðsthkan wc ex c: Na melethjeð h an lush prosarmog c (shakedown analysis) diakritopoihmènwn metallik n kataskeu n upì trisdi stath entatik kat stash k nontac qr sh tou klassikoô tèleia elastoplastikoô montèlou kai tou krithrðou diarro c Tresca, to opoðo qrhsimopoieðtai suqn stic kataskeuèc apì q luba. To krit rio Tresca parousi zei endiafèron diìti se antðjesh me to krit rio diarro c von Mises emfanðzetai wc mh leðo stic akmèc tou kanonikoô exagwnikoô prðsmatoc pou sqhmatðzei h apeikìnish thc epif neiac diarro c ston q ro twn kôriwn t sewn (lìgoc gia ton opoðo oi perissìteroi ereunhtèc protimoôn to krit rio von Mises). Na qrhsimopoihjeð - sto bajmì tou efiktoô - dokimasmèno logismikì majhmatikoô programmatismoô anafor c.

14 Dom thc diatrib c Na suntaqjoôn progr mmata hlektronikoô upologist ikan na analôsoun kataskeuèc upì trisdi stath èntash me th mèjodo twn peperasmènwn stoiqeðwn kai na par goun ta aparaðthta dedomèna eisagwg c ste na trofodot soun to logismikì tou majhmatikoô programmatismoô. Na pragmatopoihjoôn analôseic prosarmog c se montèla kataskeu n ikanoô megèjouc all kai praktik c axðac gia ton mhqanikì ste na diapistwjeð h ikanìthta epðlushc kai na melethjoôn oi upologistikèc epidìseic thc proteinìmenhc mìrfwshc. 1.4 Dom thc diatrib c H diatrib diarjr netai se ept sunolik kef laia, sumperilambanomènou kai tou trèqontoc eisagwgikoô kefalaðou, ìpou gðnetai mia sôntomh nôxh sto fainìmeno prosarmog c kai mia perilhptik istorik anadrom stic shmantikìterec ergasðec gôrw apì autì kaj c kai paratðjentai to kðnhtro kai h kôria epidðwxh ekpìnhshc thc paroôsac. To deôtero kef laio parajètei basikèc ènnoiec kai sqèseic thc elastoplastikìthtac pou qrhsimopoioôntai argìtera kai perigr fetai to elastikì tèleia plastikì montèlo kai to krit rio diarro c Tresca. Sto trðto kef laio parousi zontai stoiqeða apì th mèjodo peperasmènwn stoiqeðwn (Finite Element Method) kai sugkekrimèna h mìrfwsh b sei thc mejìdou twn metakin sewn ste na gðnei eukolìtera katanoht sto epìmeno kef laio h diatôpwsh thc an lushc prosarmog c sta plaðsia thc mejìdou twn peperasmènwn stoiqeðwn. GÐnetai idiaðterh mneða sto okt kombo trisdi stato isoparametrikì stoiqeðo (brick element) mia kai autì ja qrhsimopoihjeð stic arijmhtikèc efarmogèc tou proteleutaðou kefalaðou. To tètarto kef laioaqoleðtai me to fainìmeno prosarmog c kai thn ènnoia thc an lushc prosarmog c. Perigr fontai ta dôo jemeli dh jewr mata twn Melan (statikì je rhma) kai Koiter (kinhmatikì je rhma) kai gðnetai h klassik apaloif tou qrìnou apì to prìblhma prosarmog c kaj c kai h sôndes tou me ta peperasmèna stoiqeða.

15 Kef laio 1. Eisagwg 7 Sto pèmpto kef laio, to opoðo apoteleð kai to tm ma thc diatrib c me th megalôterh shmasða, to krit rio diarro c Tresca morf netai wc sunj kh periorismoô tou eôrouc thc diafor c twn akrot twn idiotim n σ max σ min tou tanust twn t sewn kai epitugq netai mia anagwg tou krithrðou se sôsthma sunjhk n hmijetikìthtac. Sth sunèqeia to prìblhma thc an lushc prosarmog c upì to trisdi stato krit rio Tresca an getai se prìblhma majhmatikoô programmatismoô me grammikèc kai hmijetikèc sunj kec me thn bo jeia kat llhlwn metasqhmatism n. Tèloc anaptôssontai k poiec basikèc ènnoiec sqetik me ton hmijetikì programmatismì kai touc sqetikoôc algìrijmouc kai logismikì. Sto èkto kef laio katarq n gðnetai h epilog tou logismikoô beltistopoðhshc SeDuMi tou Jos F. Sturm sthn platfìrma tou majhmatikoô pakètou MATLAB wc to kat llhlo upologistikì ergaleðo. Sthn sunèqeia analôetai h diadikasða thc upologistik c ulopoðhshc thc protajeðsac mìrfwshc tou probl matoc prosarmog c pou ègine sto prohgoômeno kef laio kai epilôontai treic shmantikoô megèjouc arijmhtikèc efarmogèc me thn qr sh twn proanaferjèntwn pakètwn logismikoô kaj c kai programm twn H/U tou suggrafèa. To dôo pr ta paradeðgmata apoteloôn klassikèc e- farmogèc thc an lushc prosarmog c kai aforoôn mia dexamen apoj keushc ugr n kai thn tom enìc kulðndrou me kôlindro me paqô toðqwma (sôndesh swl na - dexamen c) en to trðto aqoleðtai me mia antiseismik sqediasmènh sugkollht sôndesh dokoô - upostul matoc (MRF connection) apì q luba. To kef laio kleðnei me sqoliasmì twn apotelesm twn twn tri n paradeigm twn ìson afor thn mhqanik mìrfwsh tou probl matoc prosarmog c kai thn sumperifor twn qrhsimopoioômenwn algorðjmwn kaj c kai me prot seic gia antimet pish twn diapistwjèntwn adunami n tou algìrijmou epðlushc. Sto èbdomo kai teleutaðo kef laio, sunoyðzontai ta epiteôgmata thc diatrib c kai ta prwtìtupa shmeða me ta opoða pro getai h epist mh. Par llhla gðnontai k poiec prot seic epèktashc kai peraitèrw diereônhshc twn parousiasjèntwn. Tèloc akoloujeð mia sôntomh perðlhyh thc diatrib c sta Agglik.

16 Kef laio 2 Elastoplastikìthta 2.1 Basikèc ènnoiec kai sqèseic Oi tanustèc t sewn - paramìrfwsewn Sq ma 2.1: Elastoplastikì s ma JewroÔme èna elastoplastikì s ma Ω ston trisdi stato q ro R me sônoro S, sto opoðo askoôntai F b dun meic an mon da ìgkou, F s epifaneiakèc dun meic sto sônoro S s S kai epib llontai metakin seic U sto sônoro S u S me S u S s = S kai S s S u = (Sq. 2.1). 'Estw σ to di nusma twn t sewn sthn trisdi stath entatik kat stash: σ =[σ 11,σ 22,σ,σ 12,σ 2,σ 1 ] t 8

17 Kef laio 2. Elastoplastikìthta 9 'H se tanustik graf : σ = σ 11 σ 12 σ 1 σ 12 σ 22 σ 2 σ 1 σ 2 σ EÐnai gnwstì ìti mporoôme na qwrðsoume ton tanust twn t sewn se dôo mèrh, ston sfairikì udrostatikì kai ston apoklðnonta tanust twn t sewn. O udrostatikìc tanust c twn t sewn σ H ( alli c isotropèac) apoteleðtai apì ta stoiqeða σij H = pδ ij, ìpou δ ij to dèlta tou Kronecker kai p h sfairik udrostatik pðesh, pou dðnetai apì th sqèsh: p = 1 (σ 11 + σ 22 + σ )= 1 I 1 ìpou I 1 = tr(σ) =σ 11 + σ 22 + σ h pr th analoðwth tou tanust twn t sewn. Onom zoume σ D ton apoklðnonta tanust twn t sewn ( diaforetik ektropèa): σ D = s 11 s 12 s 1 s 12 s 22 s 2 s 1 s 2 s MporoÔme na gr youme: σ = σ H + σ D = 1 I 1 I + σ D (2.1) ìpou I to monadiaðo mhtr o ston R. OmoÐwc to di nusma twn paramorf sewn trop n sthn trisdi stath entatik kat stash, ja eðnai: ε =[ε 11,ε 22,ε,ε 12,ε 2,ε 1 ] t 'H se tanustik graf :

18 Basikèc ènnoiec kai sqèseic ε = ε 11 ε 12 ε 1 ε 12 ε 22 ε 2 ε 1 ε 2 ε QwrÐzontac kai p li twn tanust twn trop n se dôo mèrh, ìpwc kai me ton tanust twn t sewn, gr foume: ε = ε H + ε D = 1 I 1 I + εd (2.2) ìpou I 1 = tr(ε) =ε 11 + ε 22 + ε h pr th analoðwth tou tanust twn trop n kai ε D o tanust c: ε D = e 11 e 12 e 1 e 12 e 22 e 2 e 1 e 2 e Statik kai kinhmatik apodekt pedða Oi paramorf seic me tic metakin seic sundèontai me tic sqèseic: ε ij = 1 2 (u i,j + u j,i + u r,i u r,j ) ìpou u i,j = u i x j kai i, j, r =1, 2, Sta plaðsia thc gewmetrik c grammikìthtac oi paramorf seic eðnai apeirost pr thc t xhc kai o ìroc u r, i u r, j mhdenðzetai. Tìte h prohgoômenh sqèsh gr fetai: ε ij = 1 2 (u i,j + u j,i ) (2.) kai o rujmìc paramìrfwshc: ε i,j = 1 2 ( u i,j + u j,i ) ìpou ta sômbola ε kai u dhl noun parag gish wc proc ton qrìno.

19 Kef laio 2. Elastoplastikìthta 11 Oi kinhmatikèc sunoriakèc sunj kec sthn epif neia S u gr fontai: u i = U x S u (2.4) kai u i = v i ìpou v i = V to di nusma twn taqut twn Oi exis seic isorropðac kai oi statikèc sunoriakèc sunj kec sthn epif neia S s, eðnai: σ ij x j = F b x Ω (2.5) σ ij n j = F s x S s (2.6) ìpou n j = n to monadiaðo di nusma k jeto sto sônoro S s. To pedðo twn metakin sewn pou ikanopoieð tic exis seic (2.) kai (2.4) kaleðtai kinhmatik apodektì. En to tasikì pedðo, oi t seic tou opoðou ikanopoioôn tic sunj kec isorropðac se olìklhro ton ìgko tou s matoc kai par llhla tic statikèc sunoriakèc sunj kec, dhl. plhroôn tic sqèseic (2.5) kai (2.6), onom zetai statik apodektì Arq twn dunat n èrgwn Sto trisdi stato s ma ìgkou V, oi posìthtec F s i kai F b i eðnai oi exwterikèc dun meic askoômenec sthn epif neia S s kai oi dun meic an mon da ìgkou, antðstoiqa. To tasikì pedðo σ ij eðnai opoiod pote sônolo, pragmatik n dunat n, t sewn se isorropða me tic dun meic an mon da ìgkou Fi b kai tic epifaneiakèc dun meic Fi s. ParomoÐwc to pedðo paramorf sewn ε ij antiproswpeôei opoiod pote sônolo paramorf sewn sumbat n me tic pragmatikèc dunatèc metakin seic u i twn shmeðwn efarmog c twn exwterik n dun mewn F s i arq twn dunat n èrgwn, ekfr zetai wc: kai F b i. Tìte h

20 Basikèc ènnoiec kai sqèseic S s F s i u i ds s + W ext = W int V F b i u i dv = V σ ij ε ij dv Paramènousec t seic kai metakin seic 'Ena tasikì pedðo ρ pou brðsketai se isorropða me mhdenikèc dun meic an mon - da ìgkou enìc s matoc Ω kai mhdenikèc epifaneiakèc dun meic sto sônoro S s, kaleðtai auto sorropoômeno. EÐnai profanèc ìti èna tètoio pedðo prèpei na ikanopoieð tic sqèseic: ρ ij x j =0 x Ω kai ρ ij n j =0 x S s ìpou n j = n eðnai to monadiaðo di nusma k jeto sto sônoro S s. Se èna elastoplastikì s ma upì dosmènh fìrtish, to pedðo twn pragmatik n t sewn mporeð na grafeð wc: σ = σ e + ρ (2.7) ìpou σ e eðnai to pedðo twn elastik n t sewn pou antistoiqeð sthn dosmènh fìrtish kai ρ to auto sorropoômeno pedðo twn paramenous n t sewn. Onom zoume S r to sônolo twn auto sorropoômenwn paramenous n t sewn. An oi elastikèc t seic sundèontai me tic paramorf seic me th sqèsh: σ e ij = C ijkl ε kl (2.8) ìpou C ijkl o elastikìc tanust c, tìte antðstoiqa me th Ex. (2.7) gia to pedðo twn paramorf sewn mporoôme na gr youme: ε ij = C 1 ijkl σe kl + C 1 ijkl ρ kl + ε p ij (2.9)

21 Kef laio 2. Elastoplastikìthta 1 O pr toc ìroc antiproswpeôei to pedðo twn elastik n paramorf sewn pou antistoiqeð sthn dosmènh fìrtish kai to opoðo eðnai sumbatì me èna pedðo elastik n metakin sewn, èstw u e C 1 ijkl σe kl = 1 2 (ue i,j + ue j,i ) Efìson to olikì pedðo paramìrfwsewn ε eðnai epðshc kinhmatik sumbatì, oi upìloipoi ìroi thc Ex. (2.9) ja proèrqontai apì èna pedðo metakin sewn u r, to opoðo kaleðtai pedðo twn paramenous n metakin sewn. C 1 ijkl ρ kl + ε p ij = 1 2 (ur i,j + u r j,i) (2.10) Kai to sunolikì pedðo metakin sewn ja eðnai: u = u e + u r. Sto [85] apodeiknôetai pwc to pedðo twn plastik n paramorf sewn ε p prosdiorðzei me monadikì trìpo to pedðo twn paramenous n t sewn ρ kai me tic ikanèc sunj kec gia thn apotrop metakin sewn stereoô s matoc kajorðzei epðshc to pedðo twn paramenous n metakin sewn u r. 2.2 Qarakthristik kai montelopoðhsh thc monoaxonik c sumperifor c Elastikì tèleia plastikì montèlo Gia na broôme thn lôsh enìc probl matoc plastik c paramìrfwshc eðnai a- paraðthto na kajorðsoume epakrib c thn sumperifor tou ulikoô, dhlad th sqèsh t sewn-paramorf sewn pou to qarakthrðzei. Gia to lìgo autì dhmiourg jhkan exidanikeumèna montèla sumperifor c, ta opoða ìmwc diathroôn ta shmantik qarakthristik tou ulikoô. To montèlo tou elastikoô tèleia plastikoô ulikoô apoteleð to pio sônhjec gia ton domikì q luba. 'Opwc faðnetai kai sto di gramma t sewn - paramorf sewn tou Sq. 2.2 gia to elastikì tèleia plastikì ulikì, gia mikrèc t seic h apìkrish eðnai kajar elastik. 'Otan ìmwc ft soume sthn t sh diarro c σ y h paramìrfwsh mporeð

22 Qarakthristik kai montelopoðhsh thc monoaxonik c sumperifor c Sq ma 2.2: Elastikì tèleia plastikì montèlo na aux nei aperiìrista qwrðc epiprìsjeth fìrtish. Apì to shmeðo autì tuqìn apofìrtish ja odhg sei se paramènousa plastik paramìrfwsh tou ulikoô. H sqèsh t shc - paramìrfwshc upì monoaxonik fìrtish, ekfr zetai wc: ε = σ E ìtan σ<σ y ε = σ y E + λ ìtan σ = σ y ìpou E eðnai to mètro elastikìthtac kai λ mia jetik posìthta JewrÐa plastik c ro c H sunolik metabol paramìrfwshc dε ij jewreðtai wc to jroisma thc metabol c thc elastik c paramìrfwshc dε e ij kai thc plastik c paramìrfwshc dε p ij (paradoq ajroistik c di spashc - additive decomposition) : dε ij = dε e ij + dε p ij tou Hooke: H elastik paramìrfwsh sundèetai me tic t seic sômfwna me ton nìmo ε e ij = 1+ν E σe ij ν E σe kk δ ij σ e ij = E νe 1+ν εe ij + (1 + ν)(1 2ν) εe kkδ ij

23 Kef laio 2. Elastoplastikìthta 15 ìpou E to mètro elastikìthtac, ν o lìgoc tou Poisson kai δ ij to dèlta tou Kronecker. H plastik paramìrfwsh morf netai me b sh thn upìjesh thc Ôparxhc thc epif neiac diarro c, thc kr tunshc (sthn perðptwsh kratunìmenou ulikoô) kai tou kanìna ro c, o opoðoc kajorðzei th genik morf thc sqèshc t shc - plastik c paramìrfwshc, wc ex c: dε p ij = dλ g σ ij ìpou g h sun rthsh plastikoô dunamikoô kai dλ mia jetik posìthta exart menh apì thn tasik kat stash kai to istorikì fìrtishc. Sthn perðptwsh pou h sun rthsh diarro c tautðzetai me thn sun rthsh plastikoô dunamikoô (f = g), tìte h plastik ro eðnai susqetizìmenh (associative flow rule) kai h parap nw sqèsh gr fetai: dε p ij = dλ f σ ij H dieôjunsh tou dianôsmatoc tou rujmoô plastik c paramìrfwshc dε p ij, eðnai k jeth sthn epif neia diarro c sto shmeðo tou q rou twn t sewn pou antistoiqeð h σ ij (sunj kh kajetìthtac - normality rule) Krit ria fìrtishc H epif neia diarro c se ènan tasikì q ro orðzei to ìrio thc elastik c perioq c. An k poia t sh brðsketai mèsa sto q ro pou perikleðetai apì thn epif neia diarro c, tìte topik èqoume elastik kat stash kai anamènetai apokleistik elastik sumperifor. An ìmwc brðsketai p nw sthn epif neia diarro c anafèretai wc plastik kat stash kai eðnai dunatìn na èqoume eðte elastik eðte plastik sumperifor. An onom soume f thn sun rthsh diarro c, h opoða orðzei thn epif neia diarro c ston tasikì q ro, tìte: f<0: f =0: orðzei thn elastik kat stash orðzei thn plastik kat stash

24 Qarakthristik kai montelopoðhsh thc monoaxonik c sumperifor c An se èna shmeðo twn t sewn σ ij briskìmeno sthn epif neia diarro c, prostejeð mia metabol t shc h opoða elatt nei thn tim thc f(σ ij ) èqoume apofìrtish kai den shmei netai plastik paramìrfwsh. An antðjeta prostejeð mia metabol, h opoða aux nei thn tim thc f(σ ij ), tìte èqoume fìrtish kai parathreðtai plastik paramìrfwsh. H perðptwsh pou h metabol t shc den prokaleð metabol sthn tim thc f(σ ij ) kai exakoloujoôme na briskìmaste sthn epif neia diarro c, kaleðtai oudèterh fìrtish. Sq ma 2.: Epif neia diarro c gia elastikì tèleia plastikì ulikì Gia èna elastikì tèleia plastikì ulikì (Sq. 2.), ìtan h t sh kineðtai sthn epif neia diarro c, eðnai dunatìn na parathrhjeð elastoplastik paramìrfwsh (fìrtish) all eðnai epðshc pijanìn na mhn proklhjeð (oudèterh fìrtish). IsqÔoun oi sqèseic: f<f c kai dλ =0 : elastik kat stash f = f c kai dλ 0, df = f σ ij dσ ij =0 : oudèterh fìrtish f = f c kai dλ =0, df = f σ ij dσ ij < 0 : apofìrtish ìpou f(σ ij )=f c h epif neia diarro c AÐthma mègisthc èklushc enèrgeiac lìgw plastik c paramìrfwshc An èna elastoplastikì s ma upì monoaxonik fìrtish arqik brðsketai sthn tasik kat stash σ me plastik paramìrfwsh ε p kai exwterikì aðtio

25 Kef laio 2. Elastoplastikìthta 17 prokalèsei fìrtish mèqri thn t sh σ sthn epif neia diarro c, prokal ntac metabol thc t shc kat dσ kai thc plastik c paramìrfwshc kat dε p kai telik apofìrtish pðsw sthn σ (Sq. 2.4 (a) ), to èrgo an mon da ìgkou ja eðnai: (σ ij σij )dεp ij. SÔmfwna me to aðthma Drucker to anwtèrw èrgo eðnai mh arnhtikì: (σ ij σ ij ) εp ij 0 (2.11) Sq ma 2.4: AÐthma mègisthc èklushc enèrgeiac lìgw plastik c paramìrfwshc (a) ston q ro twn t sewn (b) sto epðpedo monoaxonik c t shc - paramìrfwshc 'Opwc faðnetai kai sto Sq. 2.4 (b) h parap nw anisìthta ekfr zei thn idiìthta ìti o rujmìc metabol c thc plastik c paramìrfwshc eðnai jetikìc (arnhtikìc) mìno e n h trèqousa t sh σ den eðnai mikrìterh (megalôterh) apì opoiad pote t sh σ sthn elastik perioq. H èklush enèrgeiac lìgw plastik n paramorf sewn an mon da ìgkou (dissipation) D p orðzetai me thn parak tw sqèsh: D p ( ε p )=σ ij ε p ij Tìte h anisìthta (2.11) mporeð na grafeð wc ex c: D p ( ε p ) σ ij ε p ij (2.12)

26 Krit ria diarro c 2. Krit ria diarro c 2..1 JewrÐa plastikìthtac met llwn Genik h arqik sunj kh diarro c mporeð na grafeð se sqèsh me thn kat stash thc t shc σ ij wc: f(σ ij,k)=0 ìpou f kaleðtai h sun rthsh diarro c kai k mia stajer exart menh apì to ulikì. H jewrða thc plastikìthtac met llwn, h opoða asqoleðtai me thn an lush twn mìnimwn (plastik n) paramorf sewn, qarakthrðzetai apì tic epìmenec treic basikèc paradoqèc: IsotropÐa: Oi arqikèc mhqanikèc idiìthtec se èna shmeðo tou ulikoô eðnai oi Ðdiec se ìlec tic dieujônseic. Plastik asumpiestìthta: H metabol tou ìgkou kat thn plastik paramìrfwsh eðnai amelhtèa kai mporeð na agnohjeð. Mh euaisjhsða sthn udrostatik pðesh: H epðdrash thc udrostatik c pðeshc sthn plastik paramìrfwsh tou ulikoô eðnai as manth. Gia isìtropa ulik h sun rthsh diarro c eðnai anex rthth apì thn kateôjunsh, eðnai dhlad sun rthsh mìno twn kôriwn t sewn σ 1,σ 2 kai σ : f(σ 1,σ 2,σ )=k (2.1) EpÐshc, gia isìtropo ulikì, h antimet jesh opoiwnd pote dôo ek twn kôriwn t sewn den epifèrei tropopoðhsh sthn morf thc sun rthshc diarro c. Tìte h Ex. (2.1) xanagr fetai wc sun rthsh twn analloðwtwn tou tanust twn t sewn: f(i 1,I 2,I )=k (2.14)

27 Kef laio 2. Elastoplastikìthta 19 ìpou I 1,I 2 kai I eðnai h pr th, deôterh kai trðth analloðwth tou tanust twn t sewn σ ij antðstoiqa. Wstìso me b sh thn trðth paradoq gia mh euaðsjhto sthn udrostatik pðesh ulikì, h udrostatik pðesh mporeð na afairejeð apì ton tanust twn t sewn σ ij kai na prokôyei o ektropèac twn t sewn s ij. H ExÐswsh (2.14) paðrnei th morf : f(j 2,J )=k ìpou J 2 kai J eðnai h deôterh kai trðth analloðwth tou tanust tou ektropèa twn t sewn s ij antðstoiqa Krit rio Tresca To pr to krit rio diarro c gia mètalla diatup jhke to 1864 kai eðnai to krit rio Tresca krit rio mègisthc diatmhtik c t shc. To krit rio Tresca dhl nei pwc h diarro epèrqetai ìtan h mègisth diatmhtik t sh ft sei mia krðsimh tim k. Ekfr zontac to krit rio se sqèsh me tic kôriec t seic σ 1,σ 2,σ, h mègisth kat apìluth tim hmidiafor twn tri n kôriwn t sewn ja prèpei na eðnai Ðsh me k sthn diarro. H majhmatik èkfrash tou krithrðou èqei wc ex c: ( 1 max 2 σ 1 σ 2, 1 2 σ 2 σ, 1 ) 2 σ σ 1 = k (2.15) E n h exart menh apì to ulikì stajer k kajorðzetai apì th monoaxonik dokim, isoôtai me to misì thc t shc diarro c σ y se monoaxonik fìrtish. k = σ y 2 Lìgw thc isotropðac tou ulikoô h t sh diarro c se monoaxonik jlðyh kai efelkusmì jewreðtai h Ðdia kat apìluth tim. Lamb nontac upìyin thn sqèsh metaxô twn kôriwn t sewn kai twn analloðwtwn tou tanust twn t sewn, h (2.15) mporeð na grafeð kai wc:

28 Krit ria diarro c f(j 2,θ)=2 J 2 sin(θ + 1 π) 2k =0 (0o θ 60 o ) (2.16) ìpou θ h gwnða omoiìthtac gwnða Lode. 'Opwc eðnai fanerì apì thn parap nw exðswsh, to krit rio Tresca den exart tai apì thn analloðwth I 1, pou shmaðnei anexarthsða apì thn udrostastik pðesh. Profan c h (2.15) gr fetai kai wc sun rthsh twn idiotim n (kôriwn t sewn) s 1,s 2,s tou ektropèa wc ex c: ( 1 max 2 s 1 s 2, 1 2 s 2 s, 1 ) 2 s s 1 = k (2.17) Sto ektropikì epðpedo h (2.17) orðzei èna kanonikì ex gwno, to ex gwno ABGDEZ tou Sq. 2.5 (a). Htrisdi stath apeikìnish tou krithrðou Tresca ston q ro twn kôriwn t sewn eðnai èna kanonikì exagwnikì prðsma ìpwc faðnetai sto Sq. 2.5 (b). Sq ma 2.5: Krit rio Tresca

29 Kef laio H mèjodoc peperasmènwn stoiqeðwn (FEM).1 Eisagwg Tic teleutaðec dekaetðec h mèjodoc peperasmènwn stoiqeðwn (Finite Element Method suntomografik FEM) èqei epikrat sei wc h kat' exoq n mèjodoc an lushc polôplokwn problhm twn thc mhqanik c, me eureða efarmog pou ekteðnetai apì thn an lush kataskeu n sta probl mata tou politikoô mhqanikoô mèqri thn aeronauphgik kai th jermodunamik. KaneÐc mporeð na antilhfjeð th mèjodo wc mia epèktash twn kajierwmènwn mejìdwn an lushc, sthn opoða h kataskeu anaparðstatai wc èna jroisma peperasmènwn diakrit n mer n, apì ìpou prokôptei kai h onomasða thc. Sta arqik st dia an ptuxhc thc mejìdou, ìpwc tan logikì lìgw thc èlleiyhc isqur n upologistik n mèswn, dìjhke èmfash sthn eôresh apotelesmatik n peperasmènwn stoiqeðwn gia thn epðlush sugkekrimènou tôpou problhm twn. Wstìso me thn ragdaða an ptuxh twn h- lektronik n upologist n gr gora ègine antilhpt h genikìthta efarmog c kai apotelesmatikìthta thc mejìdou me sunèpeia thn an ptuxh oloèna megalôterwn kai poluplokìterwn susthm twn peperasmènwn stoiqeðwn. Autì me th seir tou purodìthse thn an ptuxh apotelesmatikìterwn diadikasi n epexergasðac dedomènwn kai teqnik n epðlushc twn exis sewn isorropðac twn peperasmènwn stoiqeðwn. 21

30 22.2. Mìrfwsh me b sh th mèjodo twn metakin sewn Qrhsimopoi ntac thn mèjodo peperasmènwn stoiqeðwn eðnai efiktì na diatup soume kai na epilôsoume tic exis seic isorropðac polôplokwn susthm twn me susthmatikì kai apotelesmatikì trìpo, eôkola ulopoi simo programmatistik se ènan hlektronikì upologist. EÐnai aut akrib c h dunatìthta an lushc genik c morf c kataskeu n me thn eukolða diatôpwshc twn exis sewn kai tic fifikalèc flfl arijmhtikèc idiìthtec twn sqetik n mhtr wn tou sust matoc, pou od ghsan sthn eureða apodoq thc mejìdou. S mera meg la pakèta logismikoô an lushc FEM eðnai ikan na qeiristoôn me lelogismèno upologistikì kìstoc ter stia kai polôploka sust mata peperasmènwn stoiqeðwn q rh sthn an ptuxh apotelesmatik n algorðjmwn kai fusik lìgw thc diajèsimhc shmantik c upologistik c isqôc. H diadikasða an lushc perilamb nei thn mìrfwsh thn mhtr wn twn peperasmènwn stoiqeðwn, thn arijmhtik olokl rwsh gia thn ektðmhsh twn mhtr wn, thn emfôteush twn mhtr wn twn stoiqeðwn se mhtr a pou antistoiqoôn sto sunolikì sôsthma peperasmènwn stoiqeðwn kai thn arijmhtik epðlush tou sust matoc twn exis sewn isorropðac gia thn exagwg twn telik n apotelesm twn..2 Mìrfwsh me b sh th mèjodo twn metakin sewn H shmantikìterh kai me eureða qr sh sta probl mata thc mhqanik c twn kataskeu n, mìrfwsh thc FEM eðnai me b sh thn mejìdo twn metakin sewn. H mèjodoc jewreð thn sunolik kataskeu wc jroisma memonwmènwn peperasmènwn stoiqeðwn. UpologÐzontai ta mhtr a duskamyðac twn stoiqeðwn pou antistoiqoôn stouc kajolikoôc bajmoôc eleujerðac kai to sunolikì mhtr o duskamyðac thc kataskeu c dhmiourgeðtai me thn suneisfor twn mhtr wn duskamyðac twn stoiqeðwn. H epðlush twn exis sewn isorropðac tou sunìlou twn peperasmènwn stoiqeðwn dðnei tic metakin seic stoiqeðou, oi opoðec sthn sunèqeia qrhsimopoioôntai gia ton upologismì twn t sewn. 'Estw èna trisdi stato s ma Ω me sônoro S = S s S u, S s S u = sto opoðo askoôntai F b dun meic an mon da ìgkou, F s epifaneiakèc dun meic sto sônoro S s, F i sugkentrwmèna monaqik fortða kai arqikèc t seic σ 0 kai

31 Kef laio. H mèjodoc peperasmènwn stoiqeðwn (FEM) 2 epib llontai metakin seic U sto sônoro S u. SÔmfwna me thn arq twn dunat n metakin sewn, h isorropða tou s matoc apaiteð gia k je sumbat, dunat mikr metakðnhsh Ū epiballìmenh sto s ma, to sunolikì eswterikì dunatì èrgo na isoôtai me to sunolikì exwterikì dunatì èrgo: V ε T σ dv = V Ū T F b dv + Ū st F s ds + S i Ū it F i (.1) ìpou σ oi pragmatikèc t seic pou anaptôssontai lìgw twn dunat n paramorf sewn ε pou antistoiqoôn stic epiballìmenec dunatèc metakin seic Ū. H isìthta twn dunat n èrgwn sthn Ex. (.1) dhl nei thn isorropða tou s matoc. 'Opwc anafèrame parap nw sthn FEM proseggðzoume to s ma me èna pl joc diakrit n peperasmènwn stoiqeðwn ìpou ta stoiqeða diasundèontai metaxô touc se kombik shmeða apl kìmbouc (nodes) staìriatouc. Oi metakin seic kai epìmenwc kai oi paramorf seic, oi opoðec upologðzontai sto topikì sôsthma suntetagmènwn x, y, z k je stoiqeðou, ekfr zontai wc sunart seic twn metakin sewn twn n peperasmènwn kìmbwn. Opìte gia to stoiqeðo m èqoume: U (m) (x, y, z) =G (m) (x, y, z)u (.2) ε (m) (x, y, z) =B (m) (x, y, z)u (.) ìpou o deðkthc m upodhl nei fifisto stoiqeðo mflfl kai U eðnai to di nusma twn kajolik n metakin sewn (kat thn ènnoia twn bajm n eleujerðac sto kajolikì sôsthma suntetagmènwn) ìlwn twn kìmbwn. H sqèsh t sewn kai paramorf sewn se epðpedo stoiqeðou sumperilambanìmenwn kai twn arqik n t sewn tou stoiqeðou, dðnetai apì thn exðswsh: σ (m) = C (m) ε (m) + σ 0(m) (.4) ìpou C (m) to elastikì mhtr o ulikoô tou stoiqeðou m. Xanagr foume thn Ex. (.1) san to jroisma oloklhrwm twn wc proc ton ìgko V (m) kai thn epif neia S (m) k je peperasmènou stoiqeðou:

32 24.2. Mìrfwsh me b sh th mèjodo twn metakin sewn ε (m) T m V (m) σ (m) dv (m) = m + m V (m) S (m) Ū (m) T F b(m) dv (m) Ū s(m)t F s(m) ds (m) + i Ū it F i ìpou m =1, 2,,N E kai N E eðnai o sunolikìc arijmìc peperasmènwn stoiqeðwn. H parap nw sqèsh me thn bo jeia twn Ex. (.2), (.) kai (.4) metasqhmatðzetai sthn ex c: Ū T { m B (m) T V (m) } { } C (m) B (m) dv (m) U = Ū T G (m) T F b(m) dv (m) m V (m) { } + Ū T G s(m)t F s(m) ds (m) m S (m) { } ŪT B (m) T σ 0(m) dv (m) m V (m) + Ū T F (.5) ìpou F eðnai to di nusma twn exwterik n sugkentrwmènwn dun mewn stouc kìmbouc twn peperasmènwn stoiqeðwn, me to i stoiqeðo tou F na antistoiqeð sto i stoiqeðo tou dianôsmatoc U twn metakin sewn twn kìmbwn. Epib llontac monadiaðec dunatèc metakin seic sthn Ex. (.5), oi exis seic isorropðac tou sust matoc peperasmènwn stoiqeðwn pou antistoiqoôn stic metakin seic twn kìmbwn gr fontai: ìpou KU = R (.6) R = R b + R s R 0 + R c To mhtr o K eðnai to mhtr o duskamyðac tou sust matoc peperasmènwn stoiqeðwn me: K = m B (m) T V (m) C (m) B (m) dv (m) (.7)

33 Kef laio. H mèjodoc peperasmènwn stoiqeðwn (FEM) 25 To di nusma fìrtishc R perilamb nei thn sumbol twn dun mewn an mon da ìgkou: R b = m V (m) G (m) T F b(m) dv (m) tic epikìmbiec dun meic isodônamec proc tic epifaneiakèc dun meic: R s = m S (m) G s(m)t F s(m) ds (m) tic epikìmbiec dun meic isodônamec proc tic arqikèc t seic twn stoiqeðwn: R 0 = m V (m) B (m) T σ 0(m) dv (m) kai ta sugkentrwmèna fortða: R c = F To mhtr o duskamyðac K thc kataskeu c eðnai summetrikì kai jetik orismèno opìte me th qr sh thc klassik c di spashc Gauss-Cholesky, mporeð na grafeð wc: K = LDL T ìpou L èna k tw trigwnikì mhtr o, onomazìmeno suntelest c Cholesky kai D èna diag nio mhtr o kataskeuasmèno me th bo jeia thc mejìdou a- paloif c Gauss. H parap nw di spash dieukolônei se shmantikì bajmì thn epðlush tou sust matoc twn exis sewn isorropðac (.6), h opoða mac dðnei ìlec tic metakin seic twn kìmbwn twn peperasmènwn stoiqeðwn. Sth sunèqeia me thn bo jeia twn Ex. (.) kai (.4) lamb noume kai tic t seic twn peperasmènwn stoiqeðwn. H exðswsh (.6) dhl nei th statik isorropða tou sust matoc twn peperasmènwn stoiqeðwn. Sthn perðptwsh twn qronik metaballìmenwn fortðwn ìpou adraneiakèc dun meic kai pijanèc dun meic apìsbeshc prèpei na lhfjoôn upìyh (dunamikì prìblhma) oi exðswseic isorropðac gr fontai:

34 26.. Sqhmatismìc kai upologismìc twn mhtr wn isoparametrik n peperasmènwn stoiqeðwn MÜ + D U + KU = R ìpou M eðnai to mhtr o m zac thc kataskeu c M = m ρ (m) G (m) T G (m) dv (m) V (m) me ρ (m) thn puknìthta m zac tou stoiqeðou m Ü eðnai to di nusma twn epitaqônsewn twn kìmbwn (dhlad h deôterh par gwgoc wc proc ton qrìno tou U). D eðnai to mhtr o apìsbeshc thc kataskeu c D = m κ (m) G (m) T G (m) dv (m) V (m) me κ (m) thn stajer apìsbeshc tou stoiqeðou m U eðnai to di nusma twn taqut twn twn kìmbwn (dhlad h pr th par - gwgoc wc proc ton qrìno tou U). H sumbol twn dun mewn an mon da ìgkou sto di nusma fìrtishc R sthn perðptwsh aut, dðnetai apì th sqèsh: R b = m V (m) G (m) T [ F b(m) ρ (m) G (m) Ü κ (m) G (m) U ] dv (m) dhlad pia den sumperilamb nontai oi dun meic adr neiac kai apìsbeshc.. Sqhmatismìc kai upologismìc twn mhtr wn isoparametrik n peperasmènwn stoiqeðwn Basik idèa thc mìrfwshc twn isoparametrik n peperasmènwn stoiqeðwn apoteleð h qr sh twn Ðdiwn sunart sewn parembol c (interpolation functions)

35 Kef laio. H mèjodoc peperasmènwn stoiqeðwn (FEM) 27 tìso gia thn gewmetrða ìso kai gia to pedðo twn metakin sewn. Oi sunart seic parembol c eðnai gnwstèc epðshc kai wc sunart seic morf c (shape functions). An jewr soume èna genikì trisdi stato peperasmèno stoiqeðo, oi suntetagmènec opoioud pote shmeðou tou ekfr zontai parametrik sunart sei twn suntetagmènwn twn kìmbwn tou wc ex c: x = q g i x i i=1 y = q g i y i (.8) i=1 z = q g i z i i=1 ìpou x, y, z eðnai oi suntetagmènec se opoid pote shmeðo tou stoiqeðou kai x i,y i,z i me i =1, 2,...,qeÐnai oi suntetagmènec twn q kìmbwn tou stoiqeðou. Oi sunart seic parembol c g i eðnai orismènec sto sôsthma fusik n suntetagmènwn (natural coordinates) tou stoiqeðou me metablhtèc r, s, t h kajemða ek twn opoðwn kumaðnetai apì 1 wc +1. Qarakthristik gia thn sun rthsh parembol c g i eðnai h idiìthta pwc apokt monadiaða tim sto sôsthma fusik n suntetagmènwn ston kìmbo i en mhdenðzetai se ìlouc touc upìloipouc kìmbouc. Q rh stic parap nw sunj kec oi g i dônatai na kataskeuastoôn me susthmatikì trìpo qrhsimopoi ntac sun jwc polu numa deutèrou bajmoô gia stoiqeða ìlwn twn diast sewn kai me euelixða ston arijmì kìmbwn an stoiqeðo. 'Opwc anafèrame sta isoparametrik stoiqeða oi metakin seic opoioud pote shmeðou ekfr zontai parametrik sunart sei twn metakin sewn twn kìmbwn me qr sh twn Ðdiwn sunart sewn parembol c: u = q g i u i i=1 v = q g i v i i=1 w = q g i w i i=1 (.9) ìpou u, v, w eðnai oi topikèc metakin seic se opoid pote shmeðo tou stoiqeðou kai u i,v i,w i me i =1, 2,...,q eðnai oi antðstoiqec metakin seic twn q kìmbwn tou stoiqeðou. Gia na mporèsoume na upologðsoume to mhtr o duskamyðac tou stoiqeðou,

36 28.. Sqhmatismìc kai upologismìc twn mhtr wn isoparametrik n peperasmènwn stoiqeðwn ja prèpei na upologðsoume to mhtr o metasqhmatismoô paramorf sewn - metakin sewn. Oi paramorf seic tou stoiqeðou antloôntai apì thn parag gish twn metakin sewn se sqèsh me tic topikèc suntetagmènec. Efìson oi metakin seic tou stoiqeðou orðzontai sto sôsthma fusik n suntetagmènwn apì thn Ex. (.9) prèpei na susqetðsoume tic parag gouc twn suntetagmènwn twn kìmbwn tou stoiqeðou me tic fusikèc suntetagmènec tou stoiqeðou: r s t = x r x s x t y r y s y t z r z s z t x y z Kai se mhtrwik graf : r = J x x = J 1 r (.10) ìpou J eðnai h Iakwbian, to mhtr o pou sundèei tic parag gouc twn fusik n suntetagmènwn me tic parag gouc twn topik n suntetagmènwn. Me thn bo jeia twn Ex. (.9) kai (.10) mporoôme plèon na kataskeu - soume to mhtr o metasqhmatismoô paramìrfwsewn - metakin sewn B: ε = Bu (.11) ìpou u to di nusma twn kombik n metakin sewn. To mhtr o duskamyðac tou stoiqeðou pou antistoiqeð stouc topikoôc bajmoôc eleujerðac tou stoiqeðou ja eðnai: K = V B T CB dv (.12) Efìson to mhtr o B ekfr zetai sunart sei twn fusik n suntetagmènwn r, s, t ja prèpei kai h olokl rwsh na gðnei stic Ðdiec suntetagmènec. To diaforikì dv sundèetai me tic fusikèc suntetagmènec me th sqèsh: dv =detj dr ds dt

37 Kef laio. H mèjodoc peperasmènwn stoiqeðwn (FEM) 29 Gia ton upologismì tou oloklhr matoc thc Ex. (.12) qrhsimopoieðtai k poia mèjodoc arijmhtik c olokl rwshc (sun jwc h mèjodoc Gauss - Legendre). 'Etsi to mhtr o duskamyðac upologðzetai wc: K = i,j,k β ijk F ijk (.1) ìpou F ijk to mhtr o F = B T CB det J upologismèno sta shmeða deigmatolhyðac (sampling points) r i,s j kai t k tou fusikoô sust matoc suntetagmènwn ètsi ste na epitugq netai h mègisth epijumht akrðbeia sthn arijmhtik olokl rwsh kai β ijk stajeroð suntelestèc b rouc olokl rwshc (weighting factors) exart menoi apì tic timèc twn r i,s j kai t k. Sthn perðptwsh thc arijmhtik c olokl rwshc kat Gauss - Legendre ta shmeða deigmatolhyðac onom - zontai shmeða Gauss (Gauss points). Gr fontac tic metakin seic tou stoiqeðou me th morf u(r, s, t) =Gu, ìpou G to mhtr o twn sunart sewn parembol c, to mhtr o m zac M kai ta dianôsmata fìrtishc ja dðnontai apì tic sqèseic: M = R b = R s = R 0 = V V S V ρ G T G dv G T F b dv G st F s ds B T σ 0 dv.4 To okt kombo trisdi stato isoparametrikì stoiqeðo (8-node brick element) To okt kombo trisdi stato isoparametrikì peperasmèno stoiqeðo (8-node brick element) (Sq..1) orðzetai wc stereì sto q ro me 8 kìmbouc kai treic bajmoôc eleujerðac an kìmbo tic metakin seic u,v,w.

38 0.4. To okt kombo trisdi stato isoparametrikì stoiqeðo (8-node brick element) Sq ma.1: To okt kombo trisdi stato isoparametrikì stoiqeðo To mhtr o ulikoô C gia thn trisdi stath elastikìthta eðnai: C = E(1 ν) (1 + ν)(1 2ν) ν 1 1 ν ν ν 1 1 ν ν 1 ν ν ν 1 ν ν 1 ν ν (1 ν) ν 2(1 ν) ν 2(1 ν) Oi sunart seic parembol c dðnontai apì tic sqèseic: g 1 = 1 (1 + r)(1 + s)(1 + t) 8 g 2 = 1 (1 r)(1 + s)(1 + t) 8 g = 1 (1 r)(1 s)(1 + t) 8 g 4 = 1 (1 + r)(1 s)(1 + t) 8 g 5 = 1 (1 + r)(1 + s)(1 t) 8 g 6 = 1 (1 r)(1 + s)(1 t) 8 g 7 = 1 (1 r)(1 s)(1 t) 8 g 8 = 1 (1 + r)(1 s)(1 t) 8

39 Kef laio. H mèjodoc peperasmènwn stoiqeðwn (FEM) 1 H t xh arijmhtik c olokl rwshc kat Gauss gia ton upologismì twn mhtr wn tou stoiqeðou eðnai sun jwc gðnetai dhlad qr sh 8 shmeðwn deigmatolhyðac shmeðwn Gauss (Sq..2). Sq ma.2: ShmeÐa Gauss sto okt kombo trisdi stato isoparametrikì stoiqeðo

40 Kef laio 4 To fainìmeno prosarmog c (shakedown) 4.1 Genik perigraf tou fainomènou prosarmog c (shakedown) EÐnai gnwstì ìti sto eswterikì enìc s matoc, to opoðo èqei uposteð plastik paramìrfwsh, up rqoun kai met thn apofìrtis tou, paramènousec t - seic (residual stresses). Autì to pedðo paramenous n t sewn eðnai dunatìn na exoudeter sei t seic pou anaptôssontai lìgw epakìloujhc fìrtishc kai kat sunèpeia na periorðsei thn plastik ro. Sugkekrimèna, mporeð to plastikì èrgo na parameðnei peperasmèno gia ìlec tic dunatèc diadromèc fìrtishc mèsa se k poio dosmèno sônolo fortðwn. Autìc o tôpoc apìkrishc tou s matoc anafèretai wc prosarmog (diejn c shakedown adaptation) sto dosmèno sônolo fortðwn. Suqn qrei zetai na apanthjeð to er thma (kai idiaðtera ìtan exet zetai h apìkrish kataskeu n upì metaballìmenh kuklik fìrtish) tou an èna s ma ja pragmatopoi sei prosarmog ìqi. Ac jewr soume mia kataskeu elastoplastikoô ulikoô h opoða fortðzetai me qronik metaballìmena fortða. Apodeqìmenoi arg epiballìmenh fìrtish, dhlad yeudostatikèc sunj kec (quasi-static conditions) kai monadik mh-grammikìthta aut n tou ulikoô, h sumperifor thc kataskeu c eðnai dunatìn na 2

41 Kef laio 4. To fainìmeno prosarmog c (shakedown) kathgoriopoihjeð stic parak tw peript seic: 1. E n to mègejoc twn fortðwn parameðnei se qamhl epðpeda, h kataskeu èqei kajar elastik apìkrish (Sq ma 4.1 (a) ). 2. AntÐjeta an h efarmozìmenh fìrtish eðnai ikan isqur, exantleðtai h dunatìthta paralab c fortðou kai exaitðac mh peperasmènhc dap nhc enèrgeiac lìgw plastik c paramìrfwshc kai astoqi n, h kataskeu ft nei sthn kat rreush (Sq ma 4.1 (b) ).. An oi metabolèc thc plastik c paramìrfwshc se k je kôklo fìrtishc èqoun to Ðdio prìshmo, èpeita apì k poio arijmì kôklwn, oi sunolikèc paramorf seic kai kat sunèpeia oi metakin seic, lìgw suneqoôc suss reus c touc, apoktoôn ikanì mègejoc ste h kataskeu na q sei thn arqik thc morf kai na katasteð mh leitourgik. To fainìmeno autì kaleðtai diejn c incremental plasticity ratcheting (Sq ma 4.1 (g) ). 4. An oi metabolèc thc plastik c paramìrfwshc all zoun prìshmo se k je kôklo fìrtishc, teðnoun na anairèsoun h mða thn llh kai parìlo pou oi sunolikèc paramorf seic thc kataskeu c paramènoun mikrèc, èpeita apì k poio arijmì kôklwn sta shmeða èntonhc katapìnhshc epèrqetai topik kìpwsh tou ulikoô kai astoqða. To fainìmeno autì kaleðtai diejn c alternating plasticity low-cycle fatigue (Sq ma 4.1 (d) ). 5. Met thn p rodo k poiou qrìnou oi plastikèc paramorf seic paôoun na anaptôssontai kai h susswreumènh enèrgeia lìgw twn plastik n paramorf sewn paramènei peperasmènh me apotèlesma h kataskeu apì ed kai pèra na sumperifèretai kajar elastik. 'Opwc èqoume anafèrei to fainìmeno autì kaleðtai prosarmog kai sthn perðptwsh aut lème pwc h kataskeu pragmatopoðhse prosarmog (Sq ma 4.1 (e) ). H sumperifor thc kataskeu c sthn pr th perðptwsh den ephre zei th leitourgikìthta thc kataskeu c efìson den èqoume plastikèc paramorf seic kai topikèc astoqðec (kajar elastik sumperifor ). Wstìso eðnai fanerì pwc den axiopoioôme thn pl rh dunatìthta paralab c fortðou thc kataskeu c.

42 Genik perigraf tou fainomènou prosarmog c (shakedown) Sq ma 4.1: Pijanèc apokrðseic elastoplastik c kataskeu c se metaballìmenh fìrtish

43 Kef laio 4. To fainìmeno prosarmog c (shakedown) 5 Oi epìmenec treic peript seic qarakthrðzontai apì to gegonìc ìti h plastik ro kai oi topikèc diarroèc suneqðzoun na anaptôssontai me apotèlesma oi plastikèc paramorf seic na mhn parameðnoun st simec kai oi susswreumènec astoqðec na epifèroun sunolik astoqða thc kataskeu c. Dhlad up rqoun mèrh thc kataskeu c gia ta opoða h plastik paramìrfwsh suneqðzei na metab lletai kai epomènwc isqôei h epìmenh sunj kh: lim t εp (x, t) 0 (4.1) Sthn teleutaða perðptwsh, ìpou h kataskeu pragmatopoieð prosarmog gia to dedomèno istorikì fìrtishc, èqoume: lim t εp (x, t) =0 (4.2) Gia thn ap nthsh tou erwt matoc an mia kataskeu ja pragmatopoi sei prosarmog ìqi, èqoun anaptuqjeð dôo mèjodoi - jewr mata. To statikì je rhma me metablhtèc to pedðo twn t sewn kai to kinhmatikì je rhma pou ekfr zetai b sei tou pedðou twn taqut twn. Kai ta dôo orðzoun k poio fr gma tou fortðou prosarmog c (shakedown load): to statikì je rhma mac dðnei to k tw fr gma en to kinhmatikì to nw fr gma. 4.2 To statikì je rhma 'Estw ìti mia kataskeu Ω, me mhtr o elastikìthtac ulikoô C kai sun rthsh diarro c f, fortðzetai me qronik metaballìmena fortða P (t) ta opoða metab llontai tuqaða mèsa se èna pedðo fìrtishc L. O Melan sto [2]èdeixe pwc: Statikì Je rhma An up rqei ènac suntelest c α>1 kai èna pedðo paramenous n t sewn ρ (x) anex rthto tou qrìnou, ètsi ste gia ìlec tic fortðseic P(t) L ikanopoioôntai oi sqèseic:

44 To statikì je rhma Ω ρ T C 1 ρ dω < (4.) f[α(σ e (x, t)+ρ (x))] 0 x Ω (4.4) tìte h kataskeu ja pragmatopoi sei elastik prosarmog sto dosmèno pedðo fìrtishc L. Apìdeixh SÔmfwna me ton Lubliner [85] h apìdeixh èqei wc ex c: 'Estw ρ to pragmatikì pedðo paramenous n t sewn thc kataskeu c kai C to mhtr o ulikoô (jetik orismèno). OrÐzoume thn parak tw mh arnhtik posìthta: Y = 1 C 1 (ρ ρ ) 2 dω (4.5) 2 Ω Upojètoume ìti h kataskeu akìmh den èqei pragmatopoi sei prosarmog, opìte to pragmatikì pedðo paramenous n t sewn ρ kai wc ek toôtou kai h Y, mporeð na exart ntai apì to qrìno. ParagwgÐzontac thn (4.5) wc proc ton qrìno kai dedomènou ìti to mhtr o ulikoô C kai to pedðo ρ ex upojèsewc, eðnai anex rthta tou qrìnou, èqoume: Ẏ = Ω C 1 ijkl (ρ ij ρ ij) ρ kl dω (4.6) Efìson ta ρ kai ρ eðnai auto sorropoômena pedða kai to aristerì mèloc thc Ex. (2.10) orðzei èna sumbatì pedðo paramorf sewn, apì thn arq twn dunat n èrgwn prokôptei: Ω (ρ ij ρ ij)(c 1 ijkl ρ kl + ε p ij) dω = 0 (ρ ij ρ ij) C 1 ijkl ρ kl dω = (ρ ij ρ ij) ε p ij dω Ω Ω

45 Kef laio 4. To fainìmeno prosarmog c (shakedown) 7 'Opìte h Ex. (4.6) gr fetai: Ẏ = (ρ ij ρ ij ) εp ij dω = (σ ij σij ) εp ij dω (4.7) Ω Ω ìpou σ ij = σ e ij + ρ ij kai σ ij = σ e ij + ρ ij. Ex upojèsewc, to pedðo σ den parabi zei to krit rio diarro c kai san apotèlesma tou ait matoc mègisthc èklushc enèrgeiac lìgw plastik c paramìrfwshc (bl. Ex. (2.11) ), sumperaðnoume apì thn Ex. (4.7) pwc Ẏ 0, ìpou h isìthta isqôei mìno ìtan apousi zei h plastik ro ( ε p ij =0). All Y 0 opìte ja prèpei na isqôei h isìthta Ẏ =0pou shmaðnei ìti h Y, ra kai to pedðo paramenous n t sewn ρ, den exart ntai apì ton qrìno kai kat sunèpeia h upìjesh mac pwc h kataskeu akìmh den èqei pragmatopoi sei prosarmog eðnai toph. 4. To kinhmatikì je rhma JewroÔme kataskeu Ω me sônoro S, sto opoðo askoôntai F b dun meic an mon da ìgkou, F s epifaneiakèc dun meic sto sônoro S s S kai epib llontai metakin seic U sto sônoro S u S. O Koiter stic ergasðec tou [4] kai [5] èdeixe pwc: Kinhmatikì Je rhma Ikan kai anagkaða sunj kh ste h kataskeu Ω na mhn pragmatopoi sei prosarmog, eðnai h Ôparxh enìc kinhmatik apodektoô pedðou taqut twn v, pou na ikanopoieð th sqèsh v =0sto sônoro S u, ste: F b v dω+ F s v ds u > D p ( ε ) dω (4.8) Ω S u Ω Apìdeixh O Lubliner sto [85] parousi zei thn parak tw apìdeixh:

46 O suntelest c prosarmog c An efarmosteð h arq dunat n èrgwn sto aristerì mèloc thc anisìthtac (4.8), aut metasqhmatðzetai se: σij e ε ij dω > D p ( ε ) dω (4.9) Ω Ω Upojètoume pwc h kataskeu èqei pragmatopoi sei prosarmog kai up rqei èna pedðo paramenous n t sewn ρ anex rthto tou qrìnou. Apì to aðthma thc mègisthc èklushc enèrgeiac lìgw plastik c paramìrfwshc, èqoume: D p ( ε p ) σ ij ε p ij =(σ e ij + ρ ij ) ε p ij kai oloklhr nontac, D p ( ε p ) dω σij e εp ij dω+ ρ ij ε p ij dω (4.10) Ω Ω Ω Me thn efarmog thc arq c twn dunat n èrgwn, lìgw tou ìti to ρ eðnai auto sorropoômeno pedðo kai v =0sto sônoro S u, to teleutaðo olokl rwma tou dexioô mèlouc thc anisìthtac (4.10) mhdenðzetai. Opìte oi anisìthtec (4.9) kai (4.10) èrqontai se antðjesh, pou shmaðnei pwc h arqik mac upìjesh, ìti h kataskeu èqei pragmatopoi sei prosarmog, den eðnai dunatìn naisqôei. 4.4 O suntelest c prosarmog c Pèra apì to er thma tou an mia elastoplastik kataskeu, h opoða fortðzetai me qronik metaballìmena fortða me tuqaðo istorikì sthn perioq fìrtishc L, ja pragmatopoi sei prosarmog ìqi, tðjetai to z thma tou pìso mporoôme na dieurônoume pìso prèpei na periorðsoume thn L ste na eðnai bèbaio pwc h upì exètash kataskeu ja pragmatopoi sei prosarmog sthn nèa perioq fìrtishc α SD L. H diadikasða aneôreshc autoô tou suntelest prosaôxhshc antðstoiqa el ttwshc thc perioq c fìrtishc apoteleð thn an lush prosarmog c (shakedown analysis) kai o Ðdioc o suntelest c α SD kaleðtai suntelest c prosarmog c (shakedown factor).

47 Kef laio 4. To fainìmeno prosarmog c (shakedown) 9 Sthn perðptwsh pou den èqoume metabol fortðwn kai sunep c h perioq L apeikonðzetai me èna shmeðo, o suntelest c prosarmog c α SD tautðzetai me ton suntelest asf leiac se mesh analogik auxanìmenh fìrtish (limit analysis). SÔmfwna me ta parap nw o suntelest c prosarmog c mporeð na oristeð wc o mègistoc suntelest c α SD, o opoðoc ja ikanopoieð to statikì je rhma tou Melan (Ex. (4.) kai (4.4) ), dhlad prokôptei apì to prìblhma: max α SD (4.11) s.t. f[α SD (σ e (x, t)+ρ (x))] 0 x Ω, t ρ ij =0 x j x Ω ρ ij n j =0 x S 4.5 Apaloif tou qrìnou apì to prìblhma prosarmog c 'Estw L h perioq fìrtishc sthn opoða an koun ìla ta pijan qronik metaballìmena fortða P (t) pou efarmìzontai sthn kataskeu Ω. EÐnai fanerì pwc gia metablht kuklik fìrtish h perioq fìrtishc L emperièqei peirec dunatèc fortðseic en sthn perðptwsh thc monìtonhc fìrtishc (limit analysis) apl èna kai monadikì fortðo. Sunep c kat thn an lush prosarmog c ja prèpei na epalhjeutoôn oi ikanèc kai anagkaðec sunj kec (statikì je rhma) gia k je mða apì tic peirec dunatèc fortðseic P (t). 'Estwìti h perioq L afor se m anex rthtec monoparametrikèc fortðseic P (k), k =1,...,m. K je anex rthth fìrtish kajorðzetai apì mða par metro pou mporeð na metab lletai metaxô dosmènwn orðwn P (k), P (k) +. KaloÔme N V =2 m topl joctwn koruf n P (1),...,P(N V ) tou pedðou L pou orðzontai me ton trìpo autì. Tìte opoiad pote fìrtish P (t) Lekfr zetai me ènan monadikì kurtì sunduasmì twn P (k): N V P (t) = λ k P (k) k=1 ìpou N V k=1 λ k =1 kai λ k 0, k=1,...,n V (4.12)

48 Apaleif tou qrìnou apì to prìblhma prosarmog c Oi parap nw sqèseic kajorðzoun pwc h perioq fìrtishc L eðnai èna kurtì uperpolôedro tou opoðou oi korufèc apoteloôn tic peript seic fìrtishc P (k) kai gi' autìn to lìgo onom zontai korufèc fìrtishc. Lìgw thc arq c thc epallhlðac oi elastikèc t seic, ja dðnontai apì thn sqèsh: σ e (x, t) = N V k=1 λ k σ e (x, k) (4.1) An h sunj kh tou krithrðou diarro c den parabi zetai stic korufèc fìrtishc tìte den ja èqoume diarro kai gia k je llh dunat fìrtish mèsa sthn L. Pr gmati, qrhsimopoi ntac tic Ex. (4.12) kai (4.1) kai thn kurtìthta thc f èqoume: f[α (σ e (x, t)+ρ (x))] = N V f[ α ( λ k σ e (x, k)+ρ (x))] k=1 = N V f[ λ k (α(σ e (x, k)+ρ (x)))] k=1 N V λ k f[α(σ e (x, k)+ρ (x))] k=1 'Omwc efìson den epèrqetai diarro stic korufèc fìrtishc ja isqôei h sqèsh: f[α(σ e (x, k)+ρ (x))] 0 kai lìgw tou ìti oi suntelestèc λ k eðnai mh arnhtikoð: N V k=1 λ k f[α(σ e (x, k)+ρ (x))] 0 (4.14) Dhlad to prìblhma thc an lushc prosarmog c metasqhmatðzetai sthn parak tw morf :

49 Kef laio 4. To fainìmeno prosarmog c (shakedown) 41 max α SD (4.15) s.t. f[α SD (σ e (x, k)+ρ (x))] 0 x Ω,k=1,...,N V ρ ij =0 x j x Ω ρ ij n j =0 x S Sthn perðptwsh pou - ektìc twn metaballìmenwn epiballìmenwn fortðwn - up rqei kai k poio nekrì fortðo P 0 (p.q. to Ðdio b roc thc kataskeu c), to opoðo prokaleð arqikèc elastikèc t seic σ 0,to prohgoômeno prìblhma gr fetai wc ex c: max α SD (4.16) s.t. f[α SD (σ e (x, k)+ρ (x)) + σ 0 (x)] 0 x Ω,k=1,...,N V ρ ij =0 x j x Ω ρ ij n j =0 x S 4.6 H met bash sta peperasmèna stoiqeða JewroÔme mia kataskeu Ω diakritopoihmènh se N E peperasmèna stoiqeða, ta opoða sunolik diajètoun N G shmeða olokl rwshc Gauss kai N U gnwstec metakin seic (dhlad mh desmeumènouc bajmoôc eleujerðac). H kataskeu upìkeitai se fortðseic P (k), ìpou k =1,...,N V kai me th bo jeia logismikoô an lushc peperasmènwn stoiqeðwn upologðzoume ta dianôsmata twn elastik n t sewn σ e j RNσ, me j =1,...,N G pou anaptôssontai se k je shmeðo Gauss. Profan c h t xh N S tou diakritopoihmènou pedðou twn elastik n t sewn ja eðnai Ðsh me N S = N σ N G. Dedomènou ìti sta plaðsia thc mejìdou peperasmènwn stoiqeðwn h isorropða tou diakritopoihmènou forèa ekfr zetai sta shmeða olokl rwshc (shmeða Gauss), ta shmeða aut apoteloôn fusik epilog gia thn efarmog tou statikoô jewr matoc tou Melan. 'Etsi, sumperilamb nontac kai thn perðptwsh

50 H met bash sta peperasmèna stoiqeða Ôparxhc arqik n t sewn σ 0 sthn kataskeu lìgw k poiou nekroô fortðou, h sqèsh (4.16) gr fetai: max α SD (4.17) s.t. f[α SD (σ e (x, k)+ρ (x)) + σ 0 (x)] 0 x I G,k=1,...,N V ρ (x) S r x I G ìpou I G =1, 2,...,N G to sônolo twn shmeðwn olokl rwshc Gauss twn peperasmènwn stoiqeðwn thc diakritopoihmènhc kataskeu c kai S r to sônolo twn auto sorropoômenwn paramenous n t sewn thc kataskeu c. Qrhsimopoi ntac graf me deðkth, h prohgoômenh sqèsh gr fetai: max α SD (4.18) s.t. f[α SD (σ e j (i)+ρ j )+σ0 j ] 0 j =1,...,N G,i=1,...,N V ρ j S r j =1,...,N G To mhtr o isorropðac H R N U N S thc kataskeu c, dðnetai apì th sqèsh: H =[H 1 :...: H NG ] ìpou H j = β j B T j, j =1,...,N G (4.19) ìpou β j h stajer b rouc olokl rwshc kai B j to mhtr o sôndeshc paramorf sewn - metakin sewn sto j shmeðo Gauss. Opoiod pote tasikì pedðo σ R N S eswterik c fìrtishc: ja par gei to parak tw di nusma f int (σ) =Hσ (4.20) To opoðo sômfwna me thn arq twn dunat n èrgwn, ja prèpei na isorropeð th dosmènh exwterik fìrtish f int = f ext. Kat sunèpeia èna auto sorropoômeno tasikì pedðo, ìpwc to pedðo twn paramenous n t sewn ρ, ja prèpei na ikanopoieð thn sqèsh (null space condition):

51 Kef laio 4. To fainìmeno prosarmog c (shakedown) 4 Hρ =0 (4.21) Epomènwc h (4.18) mporeð na grafeð wc: max α SD (4.22) s.t. f[α SD (σ e j (i)+ρ j )+σ0 j ] 0 j =1,...,N G,i=1,...,N V Hρ =0 ìpou ρ to di nusma twn ρ j me j =1,...,N G. 4.7 Statik an lush prosarmog c Up rqoun treic kathgorðec problhm twn oi opoðec sqetðzontai me thn an lush prosarmog c. Pio sugkekrimèna, èqoume: To prìblhma thc elastik c prosarmog c (elastic shakedown - ESD) To prìblhma thc plastik c prosarmog c (plastic shakedown - PSD) To prìblhma tou elastikoô orðou (elastic limit - ELM) 'Olec oi parap nw kathgorðec problhm twn perilamb noun touc Ðdiouc agn stouc, ènan suntelest pollaplasiasmoô tou pedðou fìrtishc α R + kai èna tasikì pedðo ρ R N S. EpÐshc kai stic treic peript seic autoð oi koinoð gnwstoi eisèrqontai stic aniswtikèc sunj kec pou epib llei to krit rio diarro c kai o suntelest c α apoteleð thn proc beltistopoðhsh grammik sun rthsh stìqo tou probl matoc kai sugkekrimèna proc megistopoðhsh. To tasikì pedðo ρ prèpei na eðnai oi auto sorropoômenec paramènousec t seic sthn perðptwsh thc elastik c prosarmog c en mhdenðzetai sthn perðptwsh tou probl matoc tou elastikoô orðou. Gewmetrik to pedðo ρ kajorðzei mia met jesh twn t sewn tou pedðou fìrtishc L en o suntelest c pollaplasiasmoô α mia omojetik aôxhsh meðwsh tou pedðou fìrtishc.

52 Statik an lush prosarmog c 'Eqoume dh perigr yei stic prohgoômenec paragr fouc to prìblhma thc elastik c prosarmog c (ESD), to opoðo ekfr zetai wc ex c: ESD : max α ESD (4.2) s.t. f[α ESD (σ e j (i)+ρ j )+σ0 j ] 0 j =1,...,N G,i=1,...,N V Hρ =0 H plastik prosarmog (PSD), pou sqetðzetai me thn astoqða exaitðac thc emf nishc topik c kìpwshc lìgw enallag c tou pros mou thc metabol c thc plastik c paramìrfwshc (alternating plasticity low-cycle fatigue), apoteleð aploôsterh perðptwsh apì thn elastik prosarmog efìson paraleðpetai h sunj kh Hρ j =0. H majhmatik tou diatôpwsh eðnai: PSD : max α PSD (4.24) s.t. f[α PSD (σ e j (i)+ρ j )+σ0 j ] 0 j =1,...,N G,i=1,...,N V Sto prìblhma tou elastikoô orðou (ELM) to pedðo t sewn ρ mhdenðzetai, dhlad isqôei h sunj kh: ρ =0. Sunep c to prìblhma beltistopoðhshc metatrèpetai sto monodi stato: ELM : max α ELM (4.25) s.t. f[α ELM σ e j(i)+σ 0 j] 0 j =1,...,N G,i=1,...,N V Antikajist ntac to tasikì pedðo ρ me to pedðo ρ ètsi ste ρ = αρ oi parap nw diatup seic thc elastik c kai plastik c prosarmog c gr fontai: ESD : max α ESD (4.26) s.t. f[α ESD σ e j(i)+ρ j + σ 0 j] 0 j =1,...,N G,i=1,...,N V Hρ =0

53 Kef laio 4. To fainìmeno prosarmog c (shakedown) 45 PSD : max α PSD (4.27) s.t. f[α PSD σ e j (i)+ρ j + σ 0 j ] 0 j =1,...,N G,i=1,...,N V Kai efìson α 0, mporoôme eôkola na metatrèyoume ta trða prohgoômena probl mata seprobl mata elaqistopoðhshc: ESD : min α ESD (4.28) s.t. f[α ESD σ e j (i)+ρ j + σ 0 j ] 0 j =1,...,N G,i=1,...,N V Hρ =0 PSD : min α PSD (4.29) s.t. f[α PSD σ e j (i)+ρ j + σ 0 j ] 0 j =1,...,N G,i=1,...,N V ELM : min α ELM (4.0) s.t. f[α ELM σ e j (i)+σ0 j ] 0 j =1,...,N G,i=1,...,N V E n h perioq fìrtishc L gðnei apl èna shmeðo, dhlad N V = 1, to prìblhma thc elastik c prosarmog c an getai sto prìblhma thc elastoplastik c oriak c an lushc (limit analysis - LMT). Se aut n thn perðptwsh h (4.26) metasqhmatðzetai sthn: LMT : max α LMT (4.1) s.t. f[α LMT σ e j + ρ j + σ 0 j ] 0 j =1,...,N G Hρ =0 Epeid a) to prìblhma thc plastik c prosarmog c den perièqei sunj kec isìthtac kai b) h sunj kh mhdenismoô twn paramenous n t swn sto probl ma tou elastikoô orðou apoteleð ènan polô isqurìtero periorismì apì thn sunj kh isorropðac (null space condition), ja isqôei h parak tw anisotik sqèsh gia

54 Statik an lush prosarmog c touc suntelestèc pollaplasiasmoô fìrtishc twn tri n problhm twn prosarmog c: α ELM α ESD α PSD (4.2)

55 Kef laio 5 Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 5.1 Eisagwg Sto parìn kef laio, pou apoteleð kai ton pur na thc diatrib c, paratðjentai katarq n sthn Par grafo 5.2 orismènec gnwstèc idiìthtec twn summetrik n mhtr wn. Sthn epìmenh par grafo ( 5.) to krit rio Tresca morf netai wc sunj kh periorismoô tou eôrouc thc diafor c twn akrot twn idiotim n σ max σ min tou tanust twn t sewn. Sthn sunèqeia sthn Par grafo 5.4 epitugq netai mia anagwg tou krithrðou se sôsthma sunjhk n hmijetikìthtac (semipositivness constraints), pou apoteleð kai thn b sh twn epiteugm twn thc diatrib c. Sthn Par grafo 5.5 eis getai ènac kat llhloc metasqhmatismìc twn t sewn kai o omìlogoc metasqhmatismìc twn trop n me stìqo thn apaloif tou isotropèa apì to krit rio Tresca. Sthn epìmenh Par grafo ( 5.6) diatup netai me ta anwtèrw ergaleða h telik morf twn diakritopoihmènwn problhm twn thc an lushc prosarmog c. Sthn Par grafo 5.7 ta diakritopoihmèna probl mata an lushc prosarmog c anagnwrðzontai wc probl mata majhmatikoô hmijetikoô programmatismoô (semidefinite programming). H Par grafoc 5.8 afier netai stouc sqetikoôc me ton hmijetikì programma- 47

56 Summetrik mhtr a tismì algìrijmouc kai logismikì. 5.2 Summetrik mhtr a 'Estw S n to sônolo twn pragmatik n summetrik n n n mhtr wn. K je stoiqeðo tou A S n epidèqetai thn gnwst di spash: A = CΛC T (5.1) ìpou to Λ eðnai pragmatikì diag nio mhtr o kai to C eðnai pragmatikì kai orjokanonikì, ikanopoieð dhl. thn sqèsh CC T = C T C = I. Oi diag nioi ìroi tou Λ eðnai oi idiotimèc tou A pou sumbolðzontai wc λ k (A) en oi st lec tou C eðnai ta idiodianôsmata tou A. IsqÔei profan c h sqèsh: Λ = C T AC (5.2) sunèqeia. Orismènec qr simec idiìthtec twn summetrik n mhtr wn paratðjentai sthn 1h Idiìthta A = CΛC T kai B = p I + A B = CDC T (5.) ìpou D = p I + Λ, A, B S n kai p R. ProkÔptei apì to ìti C T BC = p I + Λ = D kai tic (5.1) kai (5.2). 2h Idiìthta λ k ( A) = λ k (A), k =1, 2,...,n (5.4) ProkÔptei mesa apì thn (5.1) kai (5.2). h Idiìthta max{ λ k (A)} = λ min (A), min{ λ k (A)} = λ max (A) (5.5)

57 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 49 ApodeiknÔetai diat ssontac tic idiotimèc tou A kat fjðnousa seir kai pollaplasi zontac epð 1, dhlad : λ max (A)... λ k (A)... λ min (A) λ max (A)... λ k (A)... λ min (A) (5.6) 4h Idiìthta λ max ( A) = λ min (A), λ min ( A) = λ max (A) (5.7) Ta anwtèrw ìria twn idiotim n tou ( A) prokôptoun an qrhsimopoi soume thn (5.4) sthn anisìthta sto dexiì mèloc thc (5.6). Hmijetik orismèna summetrik mhtr a 'Ena stoiqeðo A S n kaleðtai hmijetik orismèno (semipositive definite) an ìlec oi idiotimèc tou eðnai mh-arnhtikèc kai h sunj kh aut sumbolðzetai wc: A 0 λ k (A) 0, k =1, 2,...,n (5.8) To sônolo twn hmijetik orismènwn mhtr wn t xewc n apoteleð èna k no kai sumbolðzetai wc S n +. 5h Idiìthta p λ max (A) p I A 0 (5.9) Apìdeixh: 'Estw B = p I A. Tìte apì thn (5.) prokôptei λ k (B) =p λ k (A). An p λ max (A) tìte λ k (B) 0, k=1, 2,...,n. An B 0, tìte p λ k (A) 0, k =1, 2,...,n kai sunep c p λ max (A). 6h Idiìthta 'Estw ta mhtr a A i S n i, i =1,...,m. To summetrikì uperdiag nio

58 To krit rio Tresca (block-diagonal) mhtr o A A A A = A n eðnai hmijetik orismèno e n kai mìnon e nìla ta upomhtr a tou A i eðnai hmijetik orismèna. 5. To krit rio Tresca wc sunj kh periorismoô tou eôrouc thc diafor c twn akrot twn idiotim n σ max σ min tou tanust twn t sewn 'Estw o adi statoc tanust c twn t sewn Σ sthn genik trisdi stath entatik kat stash: Σ = 1 σ y σ = 1 σ y σ 11 σ 12 σ 1 σ 12 σ 22 σ 2 (5.10) σ 1 σ 2 σ Oi kôriec t seic (kai idiotimèc) tou Σ, σ i, i =1, 2, sundèontai me tic kôriec t seic (kai idiotimèc) tou σ, σ i, i =1, 2, me tic sqèseic: σ i = 1 σ y σ i ìpou i =1, 2, (5.11) 'Opwc eðdame sthn (2.15) h majhmatik èkfrash tou krithrðou Tresca eðnai: ( 1 max 2 σ 1 σ 2, 1 2 σ 2 σ, 1 ) 2 σ σ 1 σ y 2 (5.12) Kai me th bo jeia thc (5.11): ( 1 max 2 σ 1 σ 2, 1 2 σ 2 σ, 1 ) 2 σ σ 1 1 σ y σ y 2

59 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 51 max ( σ 1 σ 2, σ 2 σ, σ σ 1 ) 1 (5.1) Ja deðxoume ìti h (5.1) eðnai isodônamh me thn èkfrash: σ max σ min 1 (5.14) Apìdeixh H apìdeixh gðnetai polô apl, jewr ntac xeqwrist tic èxi dunatèc diat xeic wc proc to mègejoc twn σ 1, σ 2, σ. P.q. an isqôei σ 2 σ σ 1 ja èqoume σ max = σ 2 kai σ min = σ 1. Opìte: max ( σ 1 σ 2, σ 2 σ, σ σ 1 ) = σ 1 σ 2 = σ 2 σ 1 = σ max σ min ParomoÐwc katal goume sthn Ðdia isìthta kai gia tic upìloipec pènte dunatèc peript seic. Epomènwc to krit rio Tresca dônatai na ekfrasteð wc proc thn mègisth kai thn el qisth idiotim tou adi statou tanust twn t sewn Σ : f Tresca (σ) = σ max σ min 1 (5.15) 5.4 Anagwg tou krithrðou Tresca se sôsthma sunjhk n hmijetikìthtac Sta epìmena apodeiknôoume k poiec prot seic kai sqèseic pou ja qrhsimopoi soume gia thn mìrfwsh tou probl matoc prosarmog c se prìblhma hmijetikoô

60 Anagwg tou krithrðou Tresca se sôsthma sunjhk n hmijetikìthtac programmatismoô. Basikì L mma An a, b R tìte: a + b 1 w, z R : w a, z b kai w + z =1 (5.16) Apìdeixh E n isqôei to pr to skèloc thc isodunamðac (5.16) tìte efìson a + b 1 up rqei mia mh arnhtik metablht δ R + tètoia ste: a + b + δ =1 Jètoume: w = a δ kai z = b δ Tìte mporoôme na gr youme: w a = 1 2 δ 0 w a z b = 1 2 δ 0 z b kai w + z = (a + 1 ) 2 δ + (b + 1 ) 2 δ = a + b + δ =1 En an isqôei to deôtero skèloc thc isodunamðac (5.16), prosjètontac tic anisìthtec w a, z b kat mèlh, paðrnoume: w + z a + b 1 a + b

61 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 5 Basik Prìtash IsqÔei hisodunamða: λ max (X) λ min (X) 1 w, z R : w I X 0, z I + X 0, (5.17) w + z =1 Apìdeixh Lìgw thc 4hc idiìthtac (5.7) mporoôme na gr youme: λ max (X) λ min (X) =λ max (X)+λ max ( X) Apì to basikì l mma (5.16) èqoume: λ max (X)+λ max ( X) 1 w, z R : w λ max (X), z λ max ( X), w + z =1 All h 5h idiìthta (5.9) mac dðnei: w λ max (X) w I X 0 z λ max ( X) z I ( X) 0 z I + X 0 Apì ta parap nw gðnetai mesa faner h isqô thc (5.17). Me b sh thn isodunamða (5.17), mporoôme na an goume to krit rio Tresca se periorismoôc hmijetikìthtac. Pr gmati, h sqèsh (5.15) mporeð na grafeð wc:

62 Apaloif tou isotropèa apì to krit rio Tresca λ max ( Σ) λ min ( Σ) 1 w, z R : w I Σ 0, z I + Σ 0, (5.18) w + z =1 5.5 Apaloif tou isotropèa apì to krit rio Tresca 'Estw to mhtr o Q R 6 6 me st lec k jetec metaxô touc: Q = 1 σ y (5.19) Q 1 : LÔnontac thn QQ 1 = Q 1 Q = I brðskoume to antðstrofì tou mhtr o Q 1 = σ y (5.20) ìpou σ y h t sh diarro c tou ulikoô sthn monoaxonik dokim. OrÐzoume ton metasqhmatismì: ˆσ = Qσ σ = Q 1 ˆσ (5.21)

63 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 55 me ˆσ =[p, t 1,t 2,t,t 4,t 5 ] t kai σ =[σ 11,σ 22,σ,σ 12,σ 2,σ 1 ] t to di nusma twn t sewn. Ektel ntac tic pr xeic sthn Ex. (5.21) brðskoume: p = 1 ) ( σ y 2 σ 11 2 σ 22 2 σ t 1 = 1 ( 1 σ y 2 σ 11 + σ 22 1 ) 2 σ t 2 = 1 ) ( σ y 2 σ σ (5.22) (5.2) (5.24) t = 1 σ y σ12 (5.25) t 4 = 1 σ y σ2 (5.26) t 5 = 1 σ y σ1 (5.27) 2 σ 11 = σ y ( p 1 t 1 1 ) t 2 2 σ 22 = σ y ( p + 2 ) t 1 2 σ = σ y ( p 1 t ) t 2 (5.28) (5.29) (5.0) σ 12 = σ y 1 t (5.1) σ 2 = σ y 1 t 4 (5.2) σ 1 = σ y 1 t 5 (5.) Apì thn di spash tou tanust twn t sewn (bl. Ex. (2.1) ), o apoklðnwn tanust c twn t sewn upologðzetai ston nèo metasqhmatismì: σ D = σ 1 I s ii = σ ii 1 1 I I 1 s ij = σ ij i j

64 Apaloif tou isotropèa apì to krit rio Tresca ( s 11 = σ y 1 t 1 1 ) t 2 (5.4) 2 s 22 = σ y t 1 (5.5) ( s = σ y 1 t ) t 2 (5.6) s 12 = σ y 1 t (5.7) s 2 = σ y 1 t 4 (5.8) s 1 = σ y 1 t 5 (5.9) O adiastopoihmènoc ektropèac loipìn, mporeð na grafeð me th bo jeia tou prohgoômenou metasqhmatismoô wc ex c: Σ D = 1 σ y σ D = 1 t 1 1 t 2 1 t 1 t t t 1 1 t 4 1 t 5 1 t 4 1 t t 2 (5.40) = F 1 t 1 + F 2 t 2 + F t + F 4 t 4 + F 5 t 5 'Opou F 1, F 2, F, F 4, F 5 R ta mhtr a: F 1 = F 2 =

65 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 57 F = F 4 = F 5 = Efìson to krit rio Tresca eðnai anex rthto thc udrostatik c pðeshc, mporoôme na agno soume thn sumbol tou isotropèa ston tanust twn t sewn kai na krat soume mìno ton ektropèa. Tìte h sqèsh (5.18) gr fetai: f Tresca ( Σ D ) 1 w, z R : w I Σ D 0, z I + Σ D 0, (5.41) w + z =1 Ekfr zontac thn z wc sun rthsh thc w: z =1 w, èqoume dôo hmijetikèc sunj kec gia touc èxi agn stouc t 1,t 2,t,t 4,t 5,z oi opoðec eðnai isodônamec me to krit rio Tresca. AntÐstoiqa gia tic paramorf seic eis goume ton metasqhmatismì: ε = Q t ˆε ˆε = Q t ε (5.42) me ˆε =[θ, n 1,n 2,n,n 4,n 5 ] t, ε =[ε 11,ε 22,ε,ε 12,ε 2,ε 1 ] t to di nusma twn trop n kai to mhtr o Q R 6 6 na èqei oristeð prohgoumènwc me thn sqèsh (5.19). Ektel ntac tic pr xeic sthn Ex. (5.42) brðskoume:

66 Apaloif tou isotropèa apì to krit rio Tresca ) θ = σ y ( ε 11 ε 22 ε n 1 = σ y ( 1 2 ε 11 + ε 22 1 ) ε ( n 2 = σ y 1 ε ) ε (5.4) (5.44) (5.45) n = σ y 1 ε 12 (5.46) n 4 = σ y 1 ε 2 (5.47) n 5 = σ y 1 ε 1 (5.48) ε 11 = 1 2 ( σ y 2 θ 1 2 n 1 ε 22 = 1 ) 2 ( σ y 2 θ + n 1 ε = 1 2 ( σ y 2 θ 1 2 n 1 + ) 2 n 2 ) 2 n 2 (5.49) (5.50) (5.51) ε 12 = 1 σ y n (5.52) ε 2 = 1 σ y n4 (5.5) ε 1 = 1 σ y n5 (5.54) Kai apì thn di spash tou tanust twn trop n (bl. Ex. (2.2) ), o ektropèac twn trop n, upologðzetai ston nèo metasqhmatismì: ε D = ε 1 I 1 I e ii = ε ii 1 I 1 e ij = ε ij i j e 11 = 1 σ y ( 1 2 n 1 ) 2 n 2 (5.55) e 22 = 1 σ y n 1 (5.56)

67 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 59 e = 1 σ y ( 1 2 n 1 + ) 2 n 2 (5.57) e 12 = 1 σ y n (5.58) e 2 = 1 σ y n4 (5.59) e 1 = 1 σ y n5 (5.60) Sthn ousða oi exis seic metasqhmatismoô twn t sewn (5.21) kai twn paramorf sewn (5.42) eis goun ènan kat llhlo metasqhmatismì twn stoiqeðwn twn dianôsmatwn twn t sewn kai twn paramorf sewn antðstoiqa, ste na gðnei o diaqwrismìc touc sto udrostatikì kai apoklðnon mèroc, efìson telik mìno to apoklðnon tm ma summetèqei sto krit rio diarro c. 5.6 Telik morf twn diakritopoihmènwn problhm twn an lushc prosarmog c To di nusma twn paramenous n t sewn ρ mporeð epðshc na analujeð se isotropikì kai ektropikì mèroc me th bo jeia tou metasqhmatismoô: ˆρ = Qρ ρ = Q 1 ˆρ me ˆρ =[ˆρ p : ˆρ t ] t ìpou ˆρ p =[p ρ ] t R N G kai ˆρ t =[t ρ 1,t ρ 2,t ρ,t ρ 4,t ρ 5 ] t R N S N G. Opìte h sunj kh isorropðac twn paramenous n t sewn Hρ =0,dÔnatai na grafeð: Ĥ p ˆρ p + Ĥt ˆρ t =0 ìpou Ĥ =[Ĥp : Ĥt] =HQ 1. Se sunèpeia me ta prohgoômena, mporoôme plèon na morf soume to prìblhma (4.28) thc elastik c prosarmog c (ESD) wc k twji:

68 Telik morf twn diakritopoihmènwn problhm twn an lushc prosarmog c ESD : min α ESD (5.61) s.t. m i j = α ESD vj i + v0 j + ˆρt j Ĥ ˆρ =0 wj i + zi j =1 M i j =mat(m i j) X i j = wi j I Mi j 0 Yj i = zi j I + Mi j 0 ìpou: j =1,...,N G deðkthc pou fifitrèqeiflfl sta sunolik N G shmeða Gauss i =1,...,N V deðkthc pou fifitrèqeiflfl stic sunolik N V korufèc tou pedðou fìrtishc v i j RN S to di nusma twn adi statwn elastik n t sewn (ektropikì mèroc) sto j-shmeðo Gauss lìgw fìrtishc sthn i-koruf tou pedðou fìrtishc v 0 j RN S to di nusma twn adi statwn elastik n t sewn (ektropikì mèroc) sto j-shmeðo Gauss lìgw thc nekr c fìrtishc ˆρ t j RN S N G ˆρ j R N S Ĥ R N U N S wj i,zi j R to di nusma twn paramenous n t sewn (ektropikì mèroc) sto j-shmeðo Gauss to di nusma twn paramenous n t sewn (isotropikì kai ektropikì mèroc) sto j-shmeðo Gauss to mhtr o isorropðac thc kataskeu c pollaplasiasmèno apì dexi me to mhtr o Q 1 bohjhtikoð gnwstoi gia thn ulopoðhsh tou krithrðou diarro c Tresca se k je j-shmeðo Gauss kai i-koruf fìrtishc mat(m) metatrop dianôsmatoc m R 6 se summetrikì mhtr o M R

69 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 61 H metatrop twn dianusm twn twn t sewn se adi stata tasik mhtr a anex rthta thc udrostatik c pðeshc, gðnetai ìpwc eðdame prohgoumènwc me thn bo jeia thc sqèshc (5.40): mat(v) = 1 mat(σ) = σ y 1 t 1 1 t 2 1 t 1 t t t 1 1 t 4 1 t 5 1 t 4 1 t t 2 En ta t 1,t 2,t, t 4,t 5 upologðzontai apì ta dianôsmata twn t sewn σ me qr sh twn exis sewn (5.2) - (5.27). AntÐstoiqa bèbaia mporoôme na morf soume to prìblhma (4.29) thc plastik c prosarmog c (PSD) wc ex c: PSD : min α PSD (5.62) s.t. m i j = α PSDvj i + v0 j + ˆρt j wj i + zj i =1 M i j =mat(mi j ) X i j = wj i I M i j 0 Yj i = zi j I + Mi j 0 Kai to prìblhma (4.0) tou elastikoô orðou (ELM) gr fetai: ELM : min α ELM (5.6) s.t. m i j = α ELM vj i + v0 j ˆρ j =0 wj i + zi j =1 M i j =mat(mi j ) X i j = wj i I M i j 0 Yj i = zi j I + Mi j 0

70 Ta probl mata prosarmog c wc probl mata majhmatikoô hmijetikoô programmatismoô En to prìblhma (4.1) thc oriak c an lushc (LMT) diatup netai wc: LMT : min α LMT (5.64) s.t. m j = α LMT v j + v 0 j + ˆρt j Ĥˆρ =0 w j + z j =1 M j =mat(m j ) X j = w j I M j 0 Y j = z j I + M j Ta probl mata prosarmog c wc probl mata majhmatikoô hmijetikoô programmatismoô 'Opwc parathroôme ta anwtèrw probl mata (5.61) - (5.64) èqoun eniaða dom. Katarq n oi basikoð mac gnwstoi eðnai o suntelest c asf leiac a kai to diakritopoihmèno pedðo paramenous n t sewn ρ. Epiplèon emfanðzontai oi bohjhtikoð gnwstoi w i j,zi j, mi j, Mi j, Xi j, Yi j. Ta eniaða qarakthristik twn anwtèrw problhm twn an lushc prosarmog c eðnai ta ex c trða: - H proc elaqistopoðhsh sun rthsh (sun rthsh stìqoc - objective function) eðnai grammik wc proc touc agn stouc kai m lista eðnai exairetik apl afoô emperièqei mìno ton gnwsto a. - Oi gnwstoi ofeðloun katarq n na ikanopoioôn èna sôsthma grammik n exis sewn. - Tèloc prèpei na ikanopoioôntai kai mia seir sunjhk n hmijetikìthtac mhtr wn pou eðnai grammikèc sunart seic twn agn stwn.

71 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 6 Ta anwtèrw trða qarakthristik epitrèpoun plèon thn èntaxh twn diakritopoihmènwn problhm twn prosarmog c sthn kathgorða twn problhm twn majhmatikoô hmijetikoô programmatismoô. To tupikì prìblhma hmijetikoô programmatismoô (semidefinite programming - SDP) me gnwstouc to di nusma x R n èqei thn ex c morf : SDP : min c T x (5.65) s.t. Ax = b F(x) 0 ìpou F(x) =G 0 + x 1 G 1 + x 2 G x n G n me dedomèna ta c R n, b R m, A R m n kai ta k k mhtr a G 0,G 1,G 2,..., G n S k. To k k summetrikì mhtr o F(x) eðnai grammik sun rthsh twn agn stwn tou probl matoc. Suqnìtata to F(x) èqei uperdiag nia (block-diagonal) morf me diag nia upomhtr a ta F 1,F 2,..., F s katall lwn diast sewn pou me th seir touc eðnai to kajèna grammik sun rthsh enìc uposunìlou twn agn stwn. Tìte sômfwna me thn 6h idiìthta thc anwtèrw Paragr fou 5.1 h (5.65) paðrnei thn morf : SDP : min c T x (5.66) s.t. Ax = b F ρ (x) 0, ρ =1, 2,...,s H shmasða twn sunjhk n pou orðzoun pìte k poio sugkekrimèno mhtr o eðnai jetik hmijetik orismèno anagnwrðsthke apì nwrðc sthn jewrða elègqou. O qarakthrismìc apì ton Lyapunov to 1890, thc stajerìthtac thc lôshc miac grammik c diaforik c exðswshc, proôpìjetai mia tètoia sunj kh, h opoða kaleðtai kai grammik anisìthta mhtr ou (linear matrix inequality). Oi Luré, Postnikov kai Yakubovic me tic ergasðec touc stic dekaetðec tou fl40, fl50 kai fl60 (bl. [101]) pagðwsan thn shmasða twn grammik n anisot twn mhtr wn sthn jewrða elègqou. Oi Bellman kai Fan faðnontai wc oi pr toi pou mìrfwsan

72 SqetikoÐ me ton hmijetikì programmatismì algìrijmoi kai logismikì èna prìblhma hmijetikoô programmatismoô sto [102] to 196. Stic arqèc tou fl70 oi Donath kai Hoffman [104] kai katìpin oi Cullum, Donath kai Wolfe [10] kai o Lovász [105] qrhsimopoðhsan probl mata programmatismoô idiotim n mesa susqetizìmena me ton hmijetikì programmatismì. EkeÐnh thn qronik perðodo, h pio apotelesmatik gnwst mèjodoc gia probl mata hmijetikoô programmatismoô tan h elleiyoeid c mèjodoc kai oi Grötschel, Lovász kai Schrijver [106] diereônhsan se b joc thn efarmog thc se probl mata sunduastikoô programmatismoô qrhsimopoi ntac thn gia na proseggðsoun thn lôsh problhm twn grammikoô kai hmijetikoô programmatismoô sugqrìnwc. O Fletcher [107, 108] anazwpôrwse to endiafèron gia ton hmijetikì programmatismì sta plaðsia tou mh grammikoô programmatismoô sthn dekaetða tou fl80, gegonìc pou od ghse se mia seir dhmosieôsewn apì touc Overton kai Womersley [109]. Oi kaðriec ergasðec twn Nesterov kai Nemirovski [110, 111] kai Alizadeh [124] èdeixan pwc h nèa geni twn algìrijmwn basismènwn stic mejìdouc eswterikoô shmeðou pou efarmìsthkan apì ton Karmarkar [11] ston grammikì programmatismì, mporeð na epektajeð kai sto prìblhma tou hmijetikoô programmatismoô. EpÐshc oi Goemans kai Williamson [115, 116] apèdeixan ìti to prìblhma max-cut ston sunduastikì programmatismì mporeð na proseggisteð polô ikanopoihtik me th bo jeia tou hmijetikoô programmatismoô. Apotèlesma aut n tan h an ptuxh enìc ter stiou endiafèrontoc gia ton hmijetikì programmatismì kai h paragwg enìc shmantikoô arijmoô rjrwn, ìpwc epibebai netai apì ton Wolkowicz sthn lðsta apì bibliografikèc anaforèc me jèma ton hmijetikì programmatismì, pou diathreð ston diadiktuakì tou tìpo [114], me pl joc megalôtero twn 1000 eggraf n kai me tic perissìterec apì autèc metagenèsterec tou SqetikoÐ me ton hmijetikì programmatismì algìrijmoi kai logismikì Tic pio sun jeic kathgorðec algorðjmwn epðlushc tou probl matoc tou hmijetikoô programmatismoô apoteloôn oi mèjodoi pou akoloujoôn mða diadrom (path-following methods), oi mèjodoi el ttwshc dunamikoô (potential-reduction

73 Kef laio 5. Anagwg thc an lushc prosarmog c se prìblhma hmijetikoô programmatismoô 65 methods) kai mèjodoi basismènec se leðec mh leðec morf seic mh grammikoô programmatismoô. Oi pr tec dôo kathgorðec k noun qr sh twn mejìdwn eswterikoô shmeðou en oi teleutaðec emperièqoun kai llouc trìpouc je rhshc kai prosèggishc tou probl matoc. Oi pr toi pou qrhsimopoðhsan tic mejìdouc eswterikoô shmeðou gia probl mata hmijetikoô programmatismoô, tan oi Nesterov kai Nemirovski [111], Jarre [12] kai Alizadeh [124]. 'Opwc oi perissìterec mèjodoi beltistopoðhshc kai oi mèjodoi eswterikoô shmeðou eðnai epanalhptik c fôshc. Dhlad, dosmènhc mðac eswterik c, mh bèltisthc lôshc prospajoôn na par goun mia seir beltiwmènwn lôsewn mèsw miac epanalhptik c diadikasðac. Arqik oi mèjodoi eswterikoô shmeðou epikentr nontan se epitreptèc mejìdouc stic opoðec ìlec oi epanal yeic eðnai epitreptèc all h anaz thsh p nw sto jèma tou shmeðou ekkðnhshc od ghse sthn an ptuxh twn mh epitrept n mejìdwn eswterikoô shmeðou (bl. Mizuno [125]). Autèc oi mèjodoi xekinoôn me mia pijan mh epitrept eswterik lôsh kai kinoôntai epanalhptik proc thn bèltisth lôsh tou probl matoc, e n up rqei. 'Otan ìmwc to arqikì prìblhma eðnai mh epitreptì mh fragmèno h mèjodoc parousi zei duskolða sto na to antilhfjeð. Mia enallaktik teqnik, gnwst wc auto-du k emfôteush (self-dual embedding), pou prwtoparousi sthke apì touc Ye, Todd kai Mizuno [118], enswmat nei to prwteôon kai to du kì tou prìblhma se èna megalôtero auto-du kì sôsthma. LÔnontac to enswmatwmèno sôsthma, epilôetai kai to arqikì prìblhma en pia eðnai polô eukolìtero na anagnwrðsei kaneðc tuqìn duskolðec pou emfanðzontai. H idèa thc auto-du k c emfôteushc epekt jhke sthn epðlush perissìtero genik n, kurt n kai upì sunj kec, problhm twn beltistopoðhshc me dôo diaforetikoôc trìpouc. Gia thn perðptwsh thc beltistopoðhshc upì kurtèc kwnikèc sunj kec, sumperilambanìmenou kai tou hmijetikoô programmatismoô, oi Luo, Sturm kai Zhang [120] prìteinan èna auto-du kì emfuteuìmeno montèlo. En gia sumbatikì kurtì programmatismì me aniswtikèc sunj kec, oi Andersen kai Ye [128, 129] anèptuxan èna diaforetikoô tôpou auto-du kì emfuteuìmeno montèlo basismèno sto aplopoihmèno montèlo twn Xu, Hung kai Ye [126] gia grammikì programmatismì. O Sturm [10] ston k dika SeDuMi, qrhsimopoieð to auto-du kì emfuteuìmeno montèlo gia epðlush problhm twn summetrikoô kwnikoô programmatismoô en kai o Andersen [11] anèptuxe to pakèto logismikoô MOSEK ep nw sto montèlo pou pragmateôthke mazð me ton Ye. Kai ta dôo

74 SqetikoÐ me ton hmijetikì programmatismì algìrijmoi kai logismikì pakèta jewroôntai ristec ulopoi seic twn mejìdwn eswterikoô shmeðou ston kurtì programmatismì. To pakèto SP twn Vanderberghe kai Boyd [12] eðnai èna apì ta pr ta ergaleða logismikoô pou anaptôqjhke gia hmijetikì programmatismì. Sthn jewrða elègqou to pakèto SP mporeð na qrhsimopoihjeð èmmesa mèsw tou LMI- TOOL twn El Ghaoui, Nikoukhah kai Delebecque [1] to MRCT twn Dussy kai El Ghaoui [14]. Pio sôgqronoi kai gr goroi epilôtec probl matwn hmijetikoô programmatismoô eðnai ta SDPA apì touc Fujisawa, Kojima kai Nakata [15], CSDP apì ton Borchers [16], SDPHA apì touc Brixius kai Potra [17], SDPT apì touc Toh, Todd kai Tütüncü [18] kai DSDP5 apì touc Benson kai Ye [146]. To logismikì SeDuMi (pou ofeðlei thn onomasða tou ston ìroself-dual Minimization) ulopoieð thn teqnik thc auto-du k c emfôteushc se probl mata beltistopoðhshc se auto-du koôc omogeneðc k nouc perilhptik summetrikoôc k nouc kai eðnai prìgramma epèktash (toolbox addon) tou gnwstoô majhmatikoô pakètou MATLAB. To prìblhma tou hmijetikoô programmatismoô eðnai eidik perðptwsh thc beltistopoðhshc se summetrikoôc k nouc. H teqnik thc auto-du k c emfôteushc, ousiastik k nei efikt th lôsh sugkekrimènwn problhm twn beltistopoðhshc se mða f sh, odhg ntac eðte sthn bèltisth lôsh eðte se pistopoðhsh thc mh epitreptìthtac. To SeDuMi epitrèpei thn epðlush problhm twn beltistopoðhshc me grammikèc, tetragwnikèc kai hmijetikèc sunj kec. EpÐshc ekmetalleôetai thn tuqìn Ôparxh arai n mhtr wn (sparsity) pou sun jwc sundèontai me tic sunj kec poll n problhm twn beltistopoðhshc, gia thn apodotikìterh epðlush problhm twn meg lhc klðmakac.

75 Kef laio 6 Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 6.1 Eisagwg Sto kef laio autì parousi zetai h upologistik ulopoðhsh ìswn anaptôqjhkan sto prohgoômeno kef laio gia thn anagwg tou probl matoc prosarmog c se prìblhma hmijetikoô programmatismoô. Sthn Par grafo 6.2 gðnetai h epilog tou logismikoô majhmatikoô programmatismoô pou ja qrhsimopoihjeð. Sthn epìmenh Par grafo ( 6.) akoloujeð h perigraf thc sunolik c diadikasðac me th morf enìc diagr mmatoc ro c. Sthn sunèqeia (Par grafoc 6.4) parousi zetai analutik h kat llhlh morfopoðhsh kai proetoimasða twn dedomènwn gia thn eisagwg touc sto sqetikì logismikì majhmatikoô programmatismoô. Stic epìmenec treic Paragr fouc ( ) epilôontai is rijmec arijmhtikèc efarmogèc pou aforoôn mia dexamen apoj keushc ugr n, mia tom kulðndrou me kôlindro me paqô toðqwma kai mia antiseismik sqediasmènh sugkollht sôndesh dokoô - upostul matoc antðstoiqa. Gia k je mia efarmog parajètontai ta qarakthristik tou fusikoô probl matoc, ta qrhsimopoioômena montèla peperasmènwn stoiqeðwn kai leptomer apotelèsmata twn analôsewn prosarmog c pou diex qjhsan. 67

76 Upologistik ulopoðhsh Stic dôo teleutaðec Paragr fouc 6.8 kai 6.9 sqoli zontai ta apotelèsmata twn tri n paradeigm twn wc proc tic upologistikèc epidìseic tou algorðjmou epðlushc kaj c kai thn genik sumperifor thc proteinìmenhc mìrfwshc kat thn upologistik ulopoðhs thc. Par llhla gðnontai k poiec prot seic antimet pishc adunami n tou algìrijmou epðlushc. 6.2 Epilog logismikoô 'Opwc eðdame sto prohgoômeno kef laio to prìblhma thc an lushc prosarmog c kai stic treic kathgorðec tou kaj c kai to prìblhma thc oriak c an lushc me thn mìrfwsh pou parousi same katal gei se prìblhma hmijetikoô programmatismoô. To logismikì SeDuMi tou Jos F. Sturm, ìpwc ja doôme sta epìmena, èqei thn dunatìthta epðlushc problhm twn akrib c tètoiac morf c. Kai epeid to logismikì autì apoteleð prìgramma epèktash (toolbox addon) tou gnwstoô majhmatikoô pakètou MATLAB, prosfèrei meg lh euelixða kai euqrhstða wc proc ton trìpo eisagwg c dedomènwn kaj c kai dunatìthta sunergasðac me exwterikèc routðnec elègqou grammènec se gl ssa programmatismoô FORTRAN C. To gegonìc pwc kat thn paroôsa perðodo ta form eisagwg c problhm twn (problem input formats) twn SeDuMi kai SDPA eðnai ta pio diadedomèna kai uposthrizìmena apì trðta logismik ston q ro tou hmijetikoô programmatismoô eðnai qarakthristikì thc apodoq c thc opoðac tugq nei o k dikac tou Kajhght Jos F. Sturm sthn episthmonik koinìthta. Gia touc lìgouc autoôc to SeDuMi kai h platfìrma tou MATLAB epilèqjhkan wc ta plèon prìsfora ergaleða gia thn upologistik ulopoðhsh tou probl matoc. 6. Upologistik ulopoðhsh H sunolik diadikasða paragwg c kai proetoimasðac twn dedomènwn kaj c kai exagwg c twn telik n apotelesm twn sunoyðzetai sto di gramma ro c pou akoloujeð:

77 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 69 Shmei netai pwc tìso gia to prìblhma thc plastik c prosarmog c ìso kai gia to prìblhma tou elastikoô orðou mporeð na qrhsimopoihjeð opoiod pote up rqon axiìpisto logismikì FEM kai den apaiteðtai idiaðteroc k dikac, afoô ta anagkaða dedomèna periorðzontai stic elastikèc t seic sta shmeða Gauss pou

78 ProetoimasÐa kai eisagwg dedomènwn sto SeDuMi antistoiqoôn stic korufèc thc perioq c fìrtishc. Oi t seic autèc an koun sta sun jh exagìmena thc an lushc FEM. Sthn perðptwsh aut apaiteðtai mìnon to logismikì pou ja diab zei ta exagìmena thc an lushc FEM kai ja ulopoieð thn an ptuxh twn dedomènwn eisagwg c sto SeDuMi. Sthn perðptwsh ìmwc thc elastik c prosarmog c kai thc oriak c an lushc apaiteðtai h trofodìthsh tou SeDuMi me tic omogeneðc sunj kec isorropðac gia tic paramènousec t seic (null space condition). Epeid oi suntelestèc tou anwtèrou sust matoc exis sewn den an koun sta sun jh exagìmena miac an lushc FEM (ta mhtr a isorropðac den sqhmatðzontai par èmmesa sthn klassik an lush FEM), apaiteðtai ènac idiaðteroc ereunhtikìc k dikac FEM pou ja sqhmatðzei kai ja parèqei to mhtr o isorropðac H. Sta plaðsia thc paroôsac diatrib c sunt qjhkan: 'Ena prìgramma anagn sthc apotelesm twn elastik c an lushc FEM kai metasqhmatismoô/an ptuxhc twn dedomènwn eisagwg c sto SeDuMi. 'Enac ereunhtikìc k dikac FEM pou parèqei tìso tic elastikèc t seic ìso kai ta mhtr a H j gia ton sqhmatismì twn omogen n exis sewn isorropðac (null space condition). Wc tôpoi stoiqeðwn perilamb nontai to okt kombo trisdi stato isoparametrikì stoiqeðo (brick element) kai to tetr edro. 6.4 ProetoimasÐa kai eisagwg dedomènwn sto SeDuMi To SeDuMi lônei - metaxô llwn - probl mata beltistopoðhshc thc morf c: min c t x (6.1) s.t. Ax = b x i R x j R + ìpou i =1,...,K.f ìpou j =K.f+1,...,K.f+K.l x n S n cone ìpou n =1,...,length(K.s) S n cone := {x n R K.sk K.s k :mat(x n ) 0}

79 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 71 'Opou: K.f K.l K.s length(v) mat(v) O arijmìc twn eleôjerwn metablht n O arijmìc twn mh arnhtik n metablht n Di nusma me stoiqeða tic diast seic twn hmijetik orismènwn mhtr wn Oarijmìc twn stoiqeðwn tou dianôsmatoc v Kataskeu tou mhtr ou M R n n apì to dianôsma v R n2 me pl rwsh tou M kat st lec apì ta stoiqeða tou dianôsmatoc v 'H gr fontac pio periektik tic sunj kec sthn (6.1): min c t x (6.2) s.t. Ax = b x K K = R K.f R K.l + (S cone... S cone ) ìpou oi diast seic twn hmijetik n sunjhk n (S cone... S cone ) perièqontai sto di nusma K.s. Oi grammikèc sunj kec tou probl matoc thc elastik c prosarmog c eðnai dh sthn morf Ax = b, epìmenwc ja prèpei na ekfr soume kai tic hmijetikèc sunj kec me an logo trìpo. Dhlad : X i j = wi j I Mi j X i j = wj i I mat(m i j) X i j = wi j I mat(α ESD vj i + v0 j + ρ j) X i j = wj i I α ESD mat(vj) i mat(vj) 0 mat(ρ j ) DiaqwrÐzontac ta gnwst megèjh apì ta zhtoômena paðrnoume: X i j + α ESDmat(v i j ) wi j I +mat(ρ j)= mat(v 0 j ) (6.)

80 ProetoimasÐa kai eisagwg dedomènwn sto SeDuMi AnaptÔssontac se mhtrwik graf paðrnoume: x 11 x 12 x 1 x 21 x 22 x 2 x 1 x 2 x w w w (i) (j) (i) (j) + α ESD + = 1 tv 1 1 t v 2 1 t v 1 t v 5 1 tρ 1 1 t ρ 2 1 t ρ 1 t ρ 5 1 tv0 1 1 t v0 2 1 t v0 1 t v0 5 1 t v 2 tv 1 1 t v 5 1 t v 4 1 t v 4 1 tv t v 2 1 t ρ 1 t ρ 5 2 tρ 1 1 t ρ 4 = 1 t ρ 4 1 tρ t ρ 2 (j) 1 t v0 1 t v tv0 1 t v0 4 1 t v0 4 1 tv t v0 2 (i) (0) (j) (j) OmoÐwc kai gia thn deôterh sunj kh mporoôme na gr youme: Y i j α ESD mat(v i j) z i j I mat(ρ j )=mat(v 0 j) (6.4) Kai se mhtrwik graf : y 11 y 12 y 1 y 21 y 22 y 2 y 1 y 2 y z z z (i) (j) (i) (j) = α ESD 1 tv 1 1 t v 2 1 t v 1 t v 5 1 tρ 1 1 t ρ 2 1 t ρ 1 t ρ 5 1 tv0 1 1 t v0 2 1 t v0 1 t v0 5 1 t ρ 2 tρ 1 1 t v 2 tv 1 1 t v 5 1 t v 4 1 t v 4 1 tv t v 2 1 t ρ 5 1 t ρ 4 = 1 t ρ 4 1 tρ t ρ 2 1 t v0 2 tv0 1 1 t v0 5 1 t v0 4 (j) 1 t v0 4 1 tv t v0 2 (i) (0) (j) (j) To di nusma twn agn stwn x sthn perðptwsh tou elastikoô shakedown

81 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 7 ja eðnai thc morf c: x = p ρ. t ρ. a ESD w. z. x 11 x 21 x 1 x 12 x 22 x 2 x 1 x 2 x. y 11 y 21 y 1 y 12 y 22 y 2 y 1 y 2 y. N G 5N G 1 N V N G N V N G 9N V N G 9N V N G

82 ProetoimasÐa kai eisagwg dedomènwn sto SeDuMi To mhtr o A thc sunj khc eisagwg c dedomènwn tou SeDuMi, Ax = b ja emfanðzei thn akìloujh morf :

83 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 75 Sto anwtèrw skarðfhma, h diagrammismènh perioq dhl nei mhdenik stoiqeða, ta tm mata me gkrðza diag nio upodeiknôoun diag nia upomhtr a en ìla ta upìloipa tm mata antistoiqoôn se upomhtr a me meg lo posostì mhdenik n stoiqeðwn, dhlad arai (sparse matrices). Oi pr tec N U grammèc tou mhtr ou A ekfr zoun thn sunj kh isorropðac twn paramenous n t sewn Hρ =0. Oi epìmenec N V N G grammèc antiproswpeôoun tic grammikèc sunj kec twn bohjhtik n metablht n wj i + zi j = 1. AkoloujoÔn 9N V N G grammèc pou antistoiqoôn sthn pr th om da hmijetik n sunjhk n (6.) en oi teleutaðec 9N V N G grammèc sthn deôterh om da hmijetik n sunjhk n (6.4). Sunep c to di nusma b ja eðnai thc morf c: b = v 0 j. v 0 j. N U N V N G 9N V N G 9N V N G Efìson h sun rthsh stìqoc eðnai Ðsh me c t x = α ESD,eÐnai fanerì pwc o mìnoc mh mhdenikìc ìroc tou dianôsmatoc c, eðnai ekeðnoc pou antistoiqeð ston suntelest prosarmog c kai ja isoôtai me 1. c t =[ ] }{{}}{{} N G +5N G N V N G +N V N G +9N V N G +9N V N G Me parìmoio trìpo eis gontai kai ta dedomèna stic peript seic tou probl matoc thc plastik c prosarmog c kaj c kai tou probl matoc tou e- lastikoô orðou.

84 o Par deigma: Dexamen apoj keushc ugr n 6.5 1o Par deigma: Dexamen apoj keushc ugr n H melèth doqeðwn pðeshc kai dexamen n lìgw twn idiaðterwn qarakthristik n thc, apoteleð ènan apì touc tomeðc ìpou h an lush prosarmog c èqei brei meg lh efarmog. Ja exet soume èna trisdi stato montèlo miac dexamen c apoj keushc ugr n basismèno se antðstoiqh efarmog pou parousi zetai sto egqeirðdio odhgi n gia thn efarmog tou fifisqediasmoô mèsw An lushc flfl ( Design By Analysis ) [14] tou Eurwpa koô KanonismoÔ gia ton sqediasmì doqeðwn pðeshc (pren 1445-). To kurðwc kèlufoc thc dexamen c apoteleðtai apì trða tm mata: dôo kulðndrouc sta p nw kai k tw kra me ènan kìlouro k no an mes touc, en oi sundèseic twn kulðndrwn me ton k nogðnontai me th bo jeia dôo daktôliwn. H gewmetrða kai oi diast seic thcupì exètash dexamen c faðnontai sto Sq To ulikì tou kelôfouc eðnai q lubac tôpou X6CrNiTi sômfwna me ta eurwpa k prìtupa pren en oi daktôlioi enðsqushc eðnai kataskeuasmènoi apì P25GH sômfwna me ta eurwpa k prìtupa pren gia qalôbdinec kataskeuèc upì sunj kec pðeshc. H t sh diarro c tou kelôfouc dðnetai Ðsh proc 224 MPa, to mètro elastikìthtac 19 GPa, o suntelest c grammik c jermik c diastol c / C kai h puknìthta tou ulikoô 790 kg/m. Oi daktôlioi antðstoiqa diajètoun t sh diarro c 202 MPa, mètro e- lastikìthtac 209 GPa, suntelest grammik c jermik c diastol c / C kai puknìthta 7850 kg/m. H diakritopoðhsh thc kataskeu c ègine me th qr sh okt kombwn trisdi statwn isoparametrik n peperasmènwn stoiqeðwn (brick elements). Kataskeu sthkan dôo dðktua peperasmènwn stoiqeðwn gia na up rqei h dunatìthta sôgkrishc twn epidìsewn twn epilôsewn. To pr to apoteleðtai apì 576 peperasmèna stoiqeða (bl. Sq. 6.2) en to deôtero diajètei puknìtero dðktuo kai apoteleðtai sunolik apì 1060 stoiqeða (bl. Sq. 6.). Me ekmet lleush thc summetrðac tou montèlou kataskeu sthke mìno h mis dexamen kai ègine h kat llhlh dèsmeush twn bajm n eleujerðac twn kìmbwn pou brðskontai sto epðpedo summetrðac. Oi kìmboi sto anoiqtì pèrac tou stenoô

85 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 77 Sq ma 6.1: 1o Par deigma: Dexamen apoj keushc ugr n (diast seic se cm)

86 o Par deigma: Dexamen apoj keushc ugr n Sq ma 6.2: DÐktuo peperasmènwn stoiqeðwn (kanonikì dðktuo)

87 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 79 Sq ma 6.: DÐktuo peperasmènwn stoiqeðwn (puknì dðktuo)

88 o Par deigma: Dexamen apoj keushc ugr n kulðndrou (stok tw mèroc thc dexamen c) èqoun ìlouc touc bajmoôc eleujerðac touc desmeumènouc. To Ðdio b roc thc dexamen c sumperilambanomènhc thc mìnwshc (thc opoðac to i.b. isoôtai me 220 N/m 2 thc exwterik c epif neiac thc dexamen c) jewr jhke nekr fìrtish. To Ðdio b roc tou kalômmatoc thc dexamen c kai thc mìnws c tou efarmìzetai stouc kìmbouc tou nw krou tou meg lou kulðndrou kai epðshc perilamb netai sthn nekr fìrtish. JewroÔme ìti h dexamen dèqetai dôo eid n fortðseic anex rthtec metaxô touc kai h an lush prosarmog c ègine qwrist gia k je mia apì autèc Udrostatik pðesh H pr th fìrtish kai an lush prosarmog c aforoôn sthn udrostatik pðesh p H pou askeð to apojhkeumèno ugrì. H udrostatik pðesh wc gnwstìn dðnetai apì th sqèsh ρgh, ìpou ρ to eidikì b roc tou ugroô (ed lamb netai Ðso me 1000 kg/m, dhlad to eidikì b roc tou neroô), g h epit qunsh thc barôthtac (9.81 m/sec 2 ) kai h to Ôyoc tou ugroô. O mèsoc ìroc thc mègisthc pl rwshc thc dexamen c, antistoiqeð se st jmh ugroô Ôyouc h max =0.975 h tot, en o mèsoc ìroc thc el qisthc pl rwshc antistoiqeð se st jmh ugroô Ôyouc h min = h tot, ìpou h tot to sunolikì Ôyoc thc dexamen c. Dhlad h udrostastik pðesh sto eswterikì thc dexamen c metab lletai metaxô twn orðwn: ρgh min p H ρgh max Gia ton upologismì twn elastik n t sewn se k je koruf fìrtishc lìgw thc udrostatik c pðeshc el fjhsan oi timèc tou PÐnaka A.1 : PINAKAS A.1 Korufèc fìrtishc Koruf Udrostatik pðesh (N/m 2 ) 1 ρgh min 2 ρgh max

89 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 81 Stouc PÐnakec A.2 kai A. paratðjentai ta apotelèsmata twn analôsewn gia to prìblhma tou elastikoô orðou, thc elastik c kai thc plastik c prosarmog c kai gia ta dôo dðktua peperasmènwn stoiqeðwn: PINAKAS A.2 AnalÔseic se kanonikì dðktuo (udrostatik pðesh) Prìblhma Grammikèc Hmijetikèc Diast seic Mh mhdenik sunj kec sunj kec A(m n) stoiqeða tou A ELM 9,216 18,42 175, ,21 77,616 ESD 12,744 18,42 178,62 211,969 1,080,208 PSD 9,216 18,42 175, ,61 580,68 PINAKAS A.2 AnalÔseic se kanonikì dðktuo (udrostatik pðesh) Prìblhma Arijmìc Du kì Olikìc Tim epanal yewn di keno qrìnoc (sec) suntelest α ELM E ESD E PSD E PINAKAS A. AnalÔseic se puknì dðktuo (udrostatik pðesh) Prìblhma Grammikèc Hmijetikèc Diast seic Mh mhdenik sunj kec sunj kec A(m n) stoiqeða tou A ELM 16,960,920 22,240 9, ,544 ESD 2,426,920 28,706 90,081 1,987,024 PSD 16,960,920 22,240 81,601 1,059,664 PINAKAS A. AnalÔseic se puknì dðktuo (udrostatik pðesh) Prìblhma Arijmìc Du kì Olikìc Tim epanal yewn di keno qrìnoc (sec) suntelest α ELM E ESD E PSD E

90 o Par deigma: Dexamen apoj keushc ugr n Diafor jermokrasðac H deôterh fìrtish kai h antðstoiqh an lush prosarmog c aforoôn sthn omoiìmorfh aôxhsh jermokrasðac ΔT. H jermokrasða prin thn pl rwsh thc dexamen c jewreðtai Ðsh proc 20 C en me thn dexamen se qr sh lamb netai Ðsh proc 60 C. Dhlad h jermokrasða sto eswterikì thc dexamen c metab lletai metaxô twn orðwn: 20 C ΔT 60 C Gia ton upologismì twn elastik n t sewn se k je koruf fìrtishc lìgw thc diafor c jermokrasðac el fjhsan oi timèc tou PÐnaka A.4 : PINAKAS A.4 Korufèc fìrtishc Koruf JermokrasÐa (N/m 2 ) Stouc pðnakec A.5 kai A.6 paratðjentai ta apotelèsma twn analôsewn gia to prìblhma tou elastikoô orðou, thc elastik c kai thc plastik c prosarmog c kai gia to dôo dðktua peperasmènwn stoiqeðwn: PINAKAS A.5 AnalÔseic sekanonikì dðktuo (diafor jermokrasðac) Prìblhma Grammikèc Hmijetikèc Diast seic Mh mhdenik sunj kec sunj kec A(m n) stoiqeða tou A ELM 9,216 18,42 175, ,21 8,600 ESD 12,744 18,42 178,62 211,969 1,086,192 PSD 9,216 18,42 175, ,61 586,52

91 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 8 PINAKAS A.5 AnalÔseic se kanonikì dðktuo (diafor jermokrasðac) Prìblhma Arijmìc Du kì Olikìc Tim epanal yewn di keno qrìnoc (sec) suntelest α ELM E ESD 11.59E PSD E PINAKAS A.6 AnalÔseic se puknì dðktuo (diafor jermokrasðac) Prìblhma Grammikèc Hmijetikèc Diast seic Mh mhdenik sunj kec sunj kec A(m n) stoiqeða tou A ELM 16,960,920 22,240 9, ,194 ESD 2,426,920 28,706 90,081 1,977,674 PSD 16,960,920 22,240 81,601 1,050,14 PINAKAS A.6 AnalÔseic se puknì dðktuo (diafor jermokrasðac) Prìblhma Arijmìc Du kì Olikìc Tim epanal yewn di keno qrìnoc (sec) suntelest α ELM 11.15E ESD E PSD E o Par deigma: Tom kulðndrou me kôlindro me paqô toðqwma Sto parìn par deigma ja melethjeð mia tom enìc kulindrikoô akrofôsiou me ènan kôlindro me paqô toðqwma, prosomoi nontac sthn ousða mia diakl dwsh swl nwshc (pipe junction). To montèlo autì parousi zetai kai sto egqeirðdio odhgi n gia thn efarmog tou fifisqediasmoô mèsw An lushc flfl ( Design By Analysis ) [14] tou Eurwpa koô KanonismoÔ gia ton sqediasmì doqeðwn pðeshc (pren 1445-).

92 o Par deigma: Tom kulðndrou me kôlindro me paqô toðqwma H fìrtish afor se rop M x pou askeðtai sto kro tou akrofôsiou kat thn dieôjunsh tou xona X tou kajolikoô sust matoc suntetagmènwn kai pðesh p se olìklhro to eswterikì tou kulðndrou me paqô toðqwma (Sq. 6.4). To gewmetrikì montèlo kataskeu sthke me b sh tic prodiagrafèc tou probl matoc sto [14] (bl. kai Sq ma 6.5). To m koc tou kelôfouc eðnai 500 mm en to m koc tou akrofôsiou 80 mm. To ulikì tou kelôfouc eðnai q lubac P265GH sômfwna me ta eurwpa k prìtupa pren gia qalôbdinec kataskeuèc upì sunj kec pðeshc, en to akrofôsio eðnai kataskeuasmèno apì 11CrMo9-10 sômfwna me to eurwpa kì prìtupo pren Gia thn kataskeu tou kann bou tou diktôou twn peperasmènwn stoiqeðwn (bl. Sq ma 6.6) ègine ekmet lleush thc summetrðac tou montèlou kai qrhsimopoi jhkan sunolik 1188 okt komba, trisdi stata isoparametrik stoiqeða (brick elements). Oi sunoriakèc sunj kec pou efarmìsthkan sto montèlo eðnai h summetrða kat m koc tou YZ epipèdou, opìte kai h askoômenh rop èqei mis tim thc problepìmenhc. Oi kìmboi kai sta dôo kra tou kôriou kulðndrou eðnai desmeumènoi kat thn dieôjunsh thc stef nhc all epitrèpetai h aktinwt metakðnhsh. Oi kìmboi sto èna kro tou kôriou kulðndrou desmeôthkan pl rwc kat thn diam kh dieôjunsh me touc kìmbouc tou llou krou suzeugmènouc epðshc sthn diam kh dieôjunsh, diathr ntac tic epðpedec diatomèc epðpedec. H efarmozìmenh rop M x sto akrofôsio montelopoi jhke me th morf zeôgouc dun mewn askoômeno epð kampthc perioq c sto nw kro tou akrofôsiou, prosomoiwmènhc me th bo jeia stoiqeðwn polô megalôterhc duskamyðac ènanti twn upoloðpwn. Sunep c h askoômenh rop katanèmetai exðsou stouc exart menouc kìmbouc thc koruf c tou akrofôsiou mèsw aut c thc kampthc z nhc. Wc apotèlesma thc efarmog c kajar c rop c sthn koruf tou akrofôsiou qwrðc strèblwsh h diatom thc sôndeshc paramènei epðpedh. Wc korufèc thc perioq c fìrtishc, el fjhsan oi timèc: PINAKAS B.1 Korufèc fìrtishc Koruf Rop M x (Nmm) PÐesh p (N/mm 2 ) 1 17, , ,.90

93 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 85 Sq ma 6.4: Fusikì prìblhma Sq ma 6.5: GewmetrÐa montèlou

94 o Par deigma: Tom kulðndrou me kôlindro me paqô toðqwma Sq ma 6.6: DÐktuo peperasmènwn stoiqeðwn

95 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 87 Prìkeitai dhlad gia mia mh tetragwnik c morf c perioq fìrtishc (non box-shaped load domain). To di gramma thc perioq c fìrtishc faðnetai sto Sq. 6.6 ìpou ston orizìntio xona apeikonðzetai h pðesh p kai ston katakìrufo h rop M x anhgmènh wc proc thn tim thc rop c thc pr thc koruf c M x,1. To Ðdio b roc thc kataskeu c èpaixe kai p li ton rìlo thc arqik c elastik c fìrtishc. Sq ma 6.7: Perioq fìrtishc H an lush gia to prìblhma tou elastikoô orðou, thc elastik c kai thc plastik c prosarmog c, èdwse ta parak tw apotelèsmata: PINAKAS B.2 AnalÔseic prosarmog c Prìblhma Grammikèc Hmijetikèc Diast seic Mh mhdenik prosarmog c sunj kec sunj kec A(m n) stoiqeða tou A ELM 28,512 57, , ,241 1,254,528 ESD,52 57, , ,265 2,905,12 PSD 28,512 57, , ,761 1,881,792 PINAKAS B.2 AnalÔseic prosarmog c (sunèqeia) Prìblhma Arijmìc Du kì Olikìc Tim epanal yewn di keno qrìnoc (sec) suntelest α ELM E ESD E PSD E

96 o Par deigma: Antiseismik sqediasmènh sugkollht sôndesh dokoô - upostul matoc 6.7 o Par deigma: Antiseismik sqediasmènh sugkollht sôndesh dokoô - upostul matoc Exet zetai h sugkollht sôndesh dokoô - upostul matoc apì q luba pou faðnetai sto Sq ma 6.8. H sôndesh apoteleðtai apì mia dokì W6x150 sundeìmenh m' ènan stôlo W14x11. Prìkeitai gia mia tupik sôndesh me sun jeic diatomèc se kamptik plaðsia (Moment-Resisting Frames) gia thn plhj ra twn ne terwn metallik n kataskeu n sthn Kalifìrnia twn H.P.A. Autì to montèlo sôndeshc melet jhke apì ton Ricles sto [142] sta plaðsia ereunhtikoô progr mmatoc tou panepisthmðou Lehigh gia thn èreuna twn aitðwn twn shmantikoô arijmoô peript sewn emf nishc yajur c astoqðac sugkollht n sundèsewn pou parathr jhkan kat ton seismì tou Northridge to 1994 stic H.P.A. Ta pèlmata thc dokoô sundèontai sto pèlma tou stôlou me epifaneiakèc kat m koc sugkoll seic. H k tw lepðda upost rixhc afairèjhke, kìphke kai enisqôjhke me mia raf sugkìllhshc, en h nw lepðda upost rixhc afèjhke sth jèsh thc kai par llhla enisqôjhke me kleist raf sugkìllhshc. O kormìc thc dokoô sundèjhke me to pèlma tou stôlou qrhsimopoi ntac kat m koc sugkìllhsh me sumplhrwmatikèc rafèc gôrw apì ìlec tic akmèc thc lepðdac diatmhtik c enðsqushc. H lepðda aut leitoôrghse kai wc mèso anègershc kat thn kataskeu (dia thc qr shc koqlðwn anègershc) kai wc lepðda upost rixhc gia thn kat m koc sugkìllhsh tou kormoô thc dokoô. Akìma topojet jhkan lepðdec sunèqeiac sto Ôyoc twn pelm twn thc dokoô kai stic dôo pleurèc tou kormoô tou upostul matoc. H elastik an lush ègine jewr ntac wc sunj kec st rixhc, rjrwsh kai kôlish sth b sh tou stôlou kai to dexiì kro thc dokoô antðstoiqa. Hseismik fìrtish prosomoi jhke me thn efarmog orizìntiou monaqikoô fortðou sthn koruf tou upostul matoc. To statikì montèlo me tic epiballìmenec sunoriakèc sunj kec parousi zetai sto Sq ma 6.9. H diakritopoðhsh tou montèlou ègine me 149 okt komba trisdi stata isoparametrik stoiqeða (brick elements). Gia thn kataskeu tou diktôou twn peperasmènwn stoiqeðwn qrhsimopoi jhkan progr mmata tou suggrafèa se sunduasmì me to gnwstì pakèto Finete Ele-

97 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 89 ment Analysis ANSYS. To telikì trisdi stato dðktuo peperasmènwn stoiqeðwn faðnetai sto Sq ma Sq ma 6.8: Sugkollht sôndesh dokoô - upostul matoc

98 o Par deigma: Antiseismik sqediasmènh sugkollht sôndesh dokoô - upostul matoc Sq ma 6.9: Statikì montèlo

99 Kef laio 6. Upologistik ulopoðhsh, arijmhtik paradeðgmata kai efarmogèc 91 Sq ma 6.10: DÐktuo peperasmènwn stoiqeðwn