FINITE ELEMENT METHODE

Transcription

1 IIE EEME MEHODE

2 A pojt tébn lésült tananago: Anagtchnológá Matals tchnolog Anagtdomán Áamlástchna gép CAD tanönv CAD boo CAD/CAM/CAE ltons példatá CAM tanönv Mééstchna Ménö optmalácó Engnng optmaton Végslm-analís nt Elmnt Mthod

3 Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng Óbda Unvst Donát Bán aclt of Mchancal and Saft Engnng Snt István Unvst aclt of Mchancal Engnng IIE EEME MEHODE Edto: ÁDÁM KOVÁCS Athos: ISVÁ MOHAROS ISVÁ ODA ADRÁS SZEKRÉYES

4 COPYRIGH: -7 Ádám Kovács Andás Séns Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng; István Mohaos Óbda Unvst Donát Bán aclt of Mchancal and Saft Engnng; István Oldal Snt István Unvst aclt of Mchancal Engnng READERS: Ágns Hováthné Vaga István Kppl ajos Pomá Jósf Uj Catv Commons oncommcal-odvs CC BY-C-D hs wo can b podcd cclatd pblshd and pfomd fo non-commcal pposs wthot stcton b ndcatng th atho's nam bt t cannot b modfd ISB PREPARED UDER HE EDIORSHIP O pot Pblshng Hos RESPOSIBE MAAGER: Zssa Vots GRA: Mad wthn th famwo of th pojct ÁMOP--8//A/KMR-9-9 nttld KMR Gépésménö Kao nfomata háttű anaga és tatalm dolgoása KMR nfomaton scnc matals and contnt laboatons of aclts of Mchancal Engnng KEYWORDS: nt Elmnt Mthod ASYS COSMOS/M ods bam plan stss plan stan bam stct asmmtc plat shll SUMMARY: h hsto of th fnt lmnt modl ts mathmatcal fondaton and ts ol n mchancal ngnng dsgn a psntd h ncssa basc contnmmchancal notons and qatons fo ndstandng of th mthod a also dscssd Elmnt tps th most commonl sd n dsgn od bam plan asmmtc thn plat shll a dscbd Statng fom th pncpl of total pncpl ng th dvaton of mat qlbm qaton latd to lna lastc bods dsctd n spac and th stct of coffcnt matcs fo dffnt modls a shown In cas of bam stcts th stct of qaton of moton and th mthod of gnfqnc calclaton a also psntd Undstandng of thotcal chapts a hghl facltatd b th lag nmb of solvd ampls and th dtald dscsson of solton of al poblms obtand fom ndst

5 COES HISORY O IIE EEME MEHOD ICUDIG IS EVOUIO EXESIO AD ROE O APPICAIO I MECHAICA EGIEERIG DESIG Ancnt applcaton Hsto of vaaton of calcls basc dfntons Bachstochon poblm nctonals vaatons Dct mthod 6 Rt-mthod 8 Evolton of modn fnt lmnt mthod oc mthod Moton mthod 5 nt lmnt mthod n ngnng pactc 6 Appnd 6 Pncpals of calcls of vaaton 6 El-agang dffntal qaton OUDAMEA DEIIIOS I COIUUM MECHAICS DIEREIA EQUAIO SYSEM O EASICIY AD IS BOUDARY EEMES PROBEM 5 ndamntal dfntons n contnm mchancs 5 Dffntal qaton sstm and bonda lmnt poblm of Elastct Eqlbm qatons Gomtc qatons 6 Consttton qatons matal qatons Bonda condtons 5 Bonda lmnt mthod EERGY HEOREM O EASICIY CACUUS O VARIAIO IIE EEME MEHOD DEERMIAIO O SIESS EQUAIO I CASE O CO-PAAR ESED EEME Appomat fnctons Knmatcall admssbl dsplacmnt fld Pncpl of vtal ng Pncpl of mnmm potntal ng 5 Pncpl of agang vaaton 7 5 nt lmnt modl basd on dsplacmnt mthod 8 5 Intodcton of vcto flds 8 5 Elastct poblm and th mthod of solton 5 5 nt lmnt appomat dsplacmnt fld 5 6 Dfnton and solton of stffnss mat n cas of co-plana tnsd tss lmnt55 6 Stffnss mat of D tnsd tss lmnt 55 6 Eampl 6 AAYSIS O WO-DIMESIOA RUSSES USIG IIE EEME MEHOD BASED PROGRAM SYSEM 6 wo-dmnsonal ba stcts 6 István Mohaos István Oldal Andás Séns wwwtanonvtah

6 6 nt Elmnt Mthod nt lmnts fo modlng bams 6 h RUSS lmnt popts 65 Bam Elmnt s popts 66 Std solton 66 Rmas 75 5 WO-DIMESIOA BE BARS VARIAIO PROBEM SIESS EQUAIOS AD SOVIG HEM BY IIE EEME MEHOD 76 5 wo-dmnsonal bnt bam lmnt vaaton std 76 5 Solvng th poblm sng fnt lmnt mthod 77 5 h lmnt stffnss mat 78 5 h nt stct stffnss mat 8 5 h complt qatons sstm and th solton 85 5 Rmas 86 6 AAYSIS O WO-DIMESIOA BE BARS USIG IIE EEME MEHOD BASED PROGRAM SYSEM 87 6 Plana bam stcts 87 6 h sd fnt lmnts n modlng 87 6 Popts of th BEAM lmnt 87 6 h sha dfomaton 88 6 h std solton 9 6 Rmas 7 APPICAIO HE PRICIPE O HE MIIMUM POEIA EERGY I IED O HREE-DIMESIOA BE BAR EEMES RIZ MEHOD AD IIE EEME MEHOD 5 7 h-dmnsonal bnt bas vaatonal poblm 5 7 Solvng th poblm sng fnt lmnt mthod 7 Rmas 8 AAYSIS O HREE-DIMESIOA BE BARS USIG IIE EEME MEHOD BASED PROGRAM SYSEM 8 h-dmnsonal bam stcts 8 h sd fnt lmnts n modlng 8 h popts of th BEAMD lmnts 8 h spcal popts of BEAMD lmnts 7 8 h std solton 9 8 Rmas 9 DYAMICS O BEAM SRUCURES MASS MARIX AURA REQUECY AAYSIS 5 9 Etndng of th fnt lmnt mthod 5 9 nt lmnt fomlaton of th lastc bods' natal oscllaton 5 9 atal fqnc calclaton of two-dmnsonal ba stcts sng fnt lmnt mthod 7 9 Dtmnaton of th lmnt mass mat 8 9 Elmnt stffnss mat 9 h sstm total mass and stffnss mat 9 Rmas DYAMIC AAYSIS O HREE-DIMESIOA BARS DEERMIAIO O AURA REQUECY USIG PROGRAM SYSEM BASED O IIE EEME MEHOD wwwtanonvtah István Mohaos István Oldal Andás Séns

7 Contnts 7 Intodcton Popts of th sd fnt lmnts h std dscpton h fnt lmnt solton of th tas 6 5 Rmas 5 IRODUCIO O PAE PROBEMS SUBJEC APPICAIO O PAE SRESS PAE SRAI AD REVOUIO SYMMERIC AXISYMMERIC MODES 55 Basc tps of plan poblms 55 Eqlbm qaton dsplacmnt and dfomaton 55 Constttv qatons 58 Plan stss stat 58 Plan stan stat 6 Basc qatons of plan lastct 6 Compatblt qaton 6 A s stss fncton 6 av s qaton 6 Bonda val poblms 6 5 Eampls fo plan stss 65 5 Dtmnaton of th tacton on th bondas of a sqa shap plat 65 5 Analss of a tangntall loadd plat 67 6 h govnng qaton of plan poblms sng pola coodnats 68 7 Asmmtc plan poblms 7 7 Sold ccla clnd and thc-walld tb 7 7 Rotatng dss 75 8 Bblogaph 79 MODEIG O PAE SRESS SAE USIG EM SOWARE SYSEMS MODEIG AAYSIS O PROBEM EVAUAIO 8 nt lmnt solton of plan poblms 8 na th nod tangl lmnt 8 Intpolaton of th dsplacmnt fld 8 Calclaton of th stffnss mat 85 Dfnton of th loads 86 Eampl fo th lna tangl lmnt plan stss stat 88 Qadatc s nod tangl lmnt 97 5 Isopaamtc fo nod qadlatal 97 5 Intpolaton of th gomt 97 5 Intpolaton of th dsplacmnt fld 5 Calclaton of stan componnts Jacob mat and Jacob dtmnant 5 h mpotanc of th Jacob dtmnant ampl 55 Calclaton of th stss fld 5 56 Calclaton of th stffnss mat 6 57 Calclaton of th foc vcto 7 6 mcal ntgaton th Gass l 9 6 On dmnsonal Gass l 6 wo dmnsonal Gass l 7 Eampl fo th sopaamtc qadlatal 8 Qadatc sopaamtc qadlatal István Mohaos István Oldal Andás Séns wwwtanonvtah

8 8 nt Elmnt Mthod 9 Bblogaph MODEIG O AXISYMMERIC SAE BY EM SOWARE SYSEMS MODEIG AAYSIS O PROBEM EVAUAIO nt lmnt solton of asmmtc poblms Asmmtc lna tangl lmnt 6 Eampl fo th applcaton of asmmtc tangl lmnt 8 Asmmtc sopaamtc qadlatal lmnt 5 Eampl fo th applcaton of asmmtc sopaamtc qadlatal lmnt5 6 Bblogaph MODEIG O HI-WAED SHES AD PAES IRODUCIO O HE HEORY O SHE IIE EEME MODES Plat and shll thos h basc qatons of Kchhoff plat tho Dsplacmnt fld Stan componnts 5 Stss fld focs and momnts n th mdplan 5 h qlbm and govnng qaton of thn plats 7 nt lmnt qatons of thn plats 5 Basc qatons of th tchncal tho of thn shlls 5 Gomtcal qatons 5 Stss sltants and copls qlbm qatons 55 Dsplacmnt fld stan componnts 57 Appomatons wthn th tchncal tho of thn shlls 58 5 Majo stps n th fnt lmnt modlng of shlls 59 6 Bblogaph 6 5 MODEIG O I-PAE HI-WAED SHES UDER I-PAE AD RASVERSE OAD BY IIE EEEM MEHOD BASED SOWARE SYSEMS 6 5 Plat lmnts sbjctd to bndng 6 5 angla plat bndng lmnt o och tangl lmnt 6 5 Eampl fo th applcaton of th och tangl plat lmnt 67 5 Incompatbl ctangla shap plat lmnt 7 55 Eampl fo th applcaton of th ncompatbl ctangl shap lmnt 7 56 Compatbl ctangla shap plat lmnt Plats nd n-plan and tansvs load 8 58 Bblogaph 8 6 MODEIG O SPAIA HI-WAED SHES BY IIE EEME MEHOD-BASED SOWARE SYSEMS 8 6 Smpl flat shll lmnts 8 6 Spposton of th lna tangl and och bndng plat lmnts 8 6 Eampl fo th combnaton of th lna tangl and och tangl lmnts 87 6 Bblogaph 9 7 MODEIG O CURVED AD DOUBY-CURVED SHES BY IIE EEME MEHOD BASED SOWARE SYSEMS 95 7 Cvd shll lmnts 95 7 hn-walld clndcal shll lmnt 95 7 Asmmtc shll poblms concal shll lmnt 7 hc-walld shll lmnts 8 wwwtanonvtah István Mohaos István Oldal Andás Séns

9 Contnts 9 75 A shll-sold tanston lmnt 76 Bblogaph 5 8 AAYSIS O D PROBEMS WIH IIE EEME BASED PROGRAM SYSEMS IRODUCIO O D EEMES 7 8 Hahdon lmnts 7 8 Hahdon lmnt wth 8 nods 8 8 Hahdon lmnt wth nods 8 Hahdon lmnt wth nods 8 Pntagon lmnts 8 tahdon lmnts 8 tahdon lmnt wth nods 6 8 tahdon lmnt wth nods 7 8 tahdon lmnt wth nods 9 8 Hachc bass fnctons 8 Dfnton of stffnss mat and nodal loads 8 mcal Gass ntgaton mthod 8 Dfnton of stffnss mat n cas of D lmnts 8 Dvaton of nodal loads fom dstbtd foc sstm on volm 8 Dvaton of nodal loads fom dstbtd foc sstm on sfacl 9 AAYSIS O D PROBEMS WIH IIE EEME BASED PROGRAM SYSEMS APPICAIO O D EEMES 5 9 Caton of gomtc modl 5 9 Edtng th ognal gomt 5 9 Modlng sds and cons 6 9 Modlng nloadd pats 6 9 Modlng smmtc pats 7 9 Dfnng th msh 8 9 Inflnc of lmnt s 8 9 Inflnc of lmnt tp 9 Bonda condtons 9 oads 9 Constants 5 PRICIPES ABOU MODEIG ACCURACY AD APPICABIIY COMPARISO O DIERE IIE EEME MODES AAYSIS O RESUS 8 Modlng bams 8 Analss of a bam wth ccla coss scton 8 Modlng of thn-walld bams 5 Modlng of thn-walld opn coss scton bams 5 Modlng of thc-walld clnds tbs 58 EVAUAIO AD APPICAIO O COMPUAIOA RESU I DESIG AD QUAIICAIO REAED MECHAICA EGIEERIG ASKS REAIOSHIP BEWEE IIE EEME MEHOD AD SADARDIZED SREGH BASED DESIG 6 Pcson of nt Elmnt Mthod 6 Estmaton of o n cas of h-tp appomaton 65 Calclaton of o n cas of h-tp appomaton 7 p-tp appomaton 75 István Mohaos István Oldal Andás Séns wwwtanonvtah

10 nt Elmnt Mthod Convgnc n sngla locatons 76 5 Modlng mstas 76 Evalaton of th calclatd slts 76 Stsss bond ld stngth 76 Sngla locatons 77 Standadd mthods and th EM 77 wwwtanonvtah István Mohaos István Oldal Andás Séns

11 HISORY O IIE EEME MEHOD ICUDIG IS EVOUIO EXESIO AD ROE O APPICAIO I ME- CHAICA EGIEERIG DESIG Ancnt applcaton nt lmnts: compl and mostl n cas of ctan condtons nsolvabl poblms can b smplfd b thm h basc da s to ba p th gomt of th bod nto fnt smpl shapd lmnts ths th poblm bcoms solvabl B ths wa nstad of applng lss bt mo dffclt stps smpl bt mo mathmatcal calclaton wll b cad ot n od to fnd th solton Applcaton of dsctaton on gomtcal poblms Ac lngth and aa of a ccl g Volm of a clnd and sph Oth compl gomts a b g : Appomaton of th aa of th a ccl In od to calclat th aa of a ccl plat th gomt mst b bon p to n dntcal lmnts as t s sn on g a h appomatd val of and ts o fncton latd to th dsctaton s shown on g : 6 6 ncos sn 6 6 ncos sn and n n Eo % n n István Oldal SZIE wwwtanonvtah

12 nt Elmnt Mthod Eo[%] n o p g : Val of π and ts o n th fncton of n dsctaton s Ch ng-chh Chns ngn AD 8 dtmnd b th s of ctangls that th appomatd val of π s btwn 596 and 597 Hsto of vaaton of calcls basc dfntons Bachstochon poblm Bnoll fomd a poblm n 696 whch ntatd th volton of vaaton of calcls n od to fnd th solton h poblm: wo ponts A and B a gvn on a plan hs ponts a locatd at dffnt hghts and not on th sam vtcal ln t s consd co-plana vtcal cvs connctng th two ponts If a patcl s lasd fom pont A wthot ntal vloct and th ffct of fcton on whch cv wold t dscnd to pont B wthn th shotst tm h qston s whth sch a fncton among th cvs sts whch allows th patcl to complt th moton n th shotst possbl tm and f t dos how s t possbl to dtmn t? P P g : Bachstochon poblm wwwtanonvtah István Oldal SZIE

13 Hsto of lmnt mthod h dmandd fncton ntscts ponts P and P ths: and Accodng to th Consvaton of Eng: mv mg h vloct: ds v dt h nfntsmal ac lngth: ds d d smplfng wth th mass and sbstttng and : d dt d dt g 5 Sttng th qaton: d dt d d dt d g 6 d d g 7 d dt B spaatng th vaabls th dmandd tm to n th ac lngth: ' d 8 g t s fnd that fncton whch satsfs th and povds mnmm to 8 h solton s a cclod: 9 t c acsn c c c István Oldal SZIE wwwtanonvtah

14 nt Elmnt Mthod wh c c constants and can b dtmnd fom condton hs poblm whch s pactcall tmng a scala qantt dw th attnton of th vaaton of calcls and statd ts volton nctonals vaatons Smla poblms as th Bachstochon oftn appa n natal- and socal scnc W fqntl fac nds qantts whch a dfnd b fnctons h smplst ampl s th solton of an ndtmnat ntgal whch dpnds on th chosn fncton At th sam tm an ac lngth sfac volm o th potntal ng of a bam bas th sam manng hs qantts a calld as fnctonals An abta st mappd to th st of al nmbs s namd fnctonal o opato It s a spcfc cas of th gnal mathmatc dfnton whn a st of fnctons s mappd to th st of al nmbs and namd as fnctonal t: f R R gvn fncton and R R possbl fncton whch s contnosl dffntabl on ts agmnt C[ ] and ntscts P and P ponts whch a fd at th bonda of th doman: hn lt s assgn all to: I [ ] f ' d al nmb hs w can dfn I as a fnctonal In most cass th poblm s to tm th vals of th fnctonal h tm val can b th absolt o local If th thn I ~ s absolt mnmm I fnctonal s vald on th nt agmntm fncton ~ and I I ~ If th I fnctonal s onl vald on a ctan pat of th agmntm and I I~ thn I ~ s local mnmm h classcal dfnton of th vaaton of calcls s analog wth th calcls agang ntodcd th vaaton dnotd b δ and dfnd th ls smlal as t s n th calcls t s amn th classcal dfnton: wwwtanonvtah István Oldal SZIE

15 Hsto of lmnt mthod 5 h vaaton of ~ fncton s and w now that and dsappas n th and ponts of th doman whl btwn thm s abta povds a sm of pmttd fnctons whch nclds th solton as wll h vaaton of th fnctonal s dfnd as: P P g : Classcal agang dfnton of vaaton I f d h solton of ~ can b obtand f th fnctonal has a mnmm Accodng to th vaaton dfnton I hs s ncssa condton of th tm val In cas of a fnctonal-mnmm basd mthod th solton of th poblm can b dfnd as an act o an appomat whch s analog to th absolt and local mnmm tho Eact solton f th fncton whch povds mnmm to th fnctonal s chosn among all possbl stng fncton o fnd that spcfc fncton s onl possbl n v smpl cass In most cass th act solton cannot b fond snc t s nth possbl to solv th qatons analtcall no amnng nfnt fnctons to s whch on povds mnmm Stll th poblm mst b solvd vn f w a nabl to povd th act solton hn an appomat solton mst b fond István Oldal SZIE wwwtanonvtah

16 6 nt Elmnt Mthod Appomat solton f th fncton whch povds mnmm to th fnctonal s not chosn among all possbl stng fncton Dct mthods w catd to fnd appomat solton h so calld El s bon lns w th lmnts of El s vaaton mthod hs mthod b wldng th accssos of modn mathmatcs gand attnton and bcam th fondaton of th dct mthods of vaaton of calcls Dct mthod h fst poblm of th vaaton of calcls was th dtmnaton of tmng fnctons whch povds tm vals to th fnctonal ndamntals of El s mthod: lt s consd th poblm analog wth tma poblm of fnctons whch dpnd on fnt vaabls hs pmttd fnctons hav fnt n dscbng vaabl and th ntgal whch s dfnd as fnctonal wll b sbstttd wth an appomat val If n convgs to nfnt thn th appomat val convgs to th val of th ntgal El s solton: th s of lna contnos El s bon ln fnctons fo ach pat of th doman t s dvd th ntval to n dntcal pat and gv abta al nmbs n hn th lngth of th lns: t n h ponts of t t nt n a connctd wth bon lns and t cats a contnos fncton whch stat and nd ponts a fd n P and P P P t +t g 5 : El s bon lns wwwtanonvtah István Oldal SZIE

17 Hsto of lmnt mthod 7 hn th fnctonal can b modfd as: I n n f t t t : n If th appomat val of I n fnctonal qals to th tm val of an El s bon ln thn n th t t nt n ponts th dvatv: I n j j n 5 t s ntodc th patal dvatv of a fncton wth spct to th vaabl dnotd b low nd: f f : f ' f : ' hn anoth dnotaton fo th basc fncton s: f j : j j f j t j t hs: I n j f t f ' f ' j j j 6 Sttng th qaton of 5 and 6: f f ' j ' j f 7 j t If n t and th ss of th bon lns convgs to a two tms dffntabl fncton thn fom 7 d f ' f ' ' 8 d fom 7 o n anoth fom István Oldal SZIE wwwtanonvtah

18 8 nt Elmnt Mthod f d d f 9 ' hs s a dffntal qaton whch blongs to th f basc fncton and namd as El- agang dffntal qaton h solton s whch povds th mnmm of th fnctonal hs th and 8 qatons man dffnt dfnton of an tama poblm n cas of a gvn fnctonal hs qaton can b also dvd b amnng th tma of a on-paamtc sm of fnctons 6 tabl howv El s bon lns tho s th fondaton of th dct mthods that s wh th classcal dfnton was amnd n dtals Accodng to th bon lns lt s consd th ss of th pmttd fnctons dnotd as latd to th absolt tma val n th th stps qals to a whch povds a fncton n ach stp h obtand fncton ss mst b amnd f t s convgnt whth ts tma stnc s satsfd o not hs mthod can b sd as: An appomat mthod fo poblms n vaaton of calcls If th bonda fncton has an tm val and satsfs 8 ths th solton of a dffntal qaton can b tansfom to a vaaton poblm hs s calld as th dct mthod of vaaton of calcls Rt-mthod In th Rt-mthod th dct mthod of vaaton of calcls s appld to fnd an appomat solton In conta wth th fnt lmnt mthod h th complt doman s modld wth on fncton n ss st on a nom and a n ss on al nmbs Dfnton: t complt f to all lmnt th s a a a n n ss whch abta appomats t n ss s Dfnton: t I b a fnctonal and n ss th agmntm of I t s consd that I mnmabl on th lna combnaton of n st f: A lna nom sts whch s pat of th agmntm I and n s complt on t All lna combnaton of n also pat of th agmntm of I In all cass of n sts an I n mnmal lmnt wh n : DI n a wwwtanonvtah István Oldal SZIE

19 Hsto of lmnt mthod 9 Accodng to Rt s thom f a fnctonal can b mnmd on th lna combnaton of a n st thn I mst hav a mnmal lmnt whch s contnos and n that cas n whch s th ss of th mnmal fncton of I s th mnmng ss of I fnctonal Rt mthod n lastct t s choos th potntal ng of a flbl bod as a fnctonal B th s of n fnt paamt an appomat fncton of a nmatcall admssbl dsplacmnt fld s catd dfntons s at and chapts Accodng to th thom of mnmm potntal ng th potntal ng s mnmal n cas of al dsplacmnt h nmatcall admssbl dsplacmnt fld s appomatd b a fncton ss: a an a a an n h dvd potntal ng nclds th sam nmb of n paamt: n a a a h potntal ng s a fnctonal ths ts tma s fond n cas of hs: a a an 5 a a a n Snc ths ntgal dpnds on th paamts th tma s obtand b th dvaton of th potntal ng wth spct th paamts and qalng to o: a a István Oldal SZIE wwwtanonvtah

20 nt Elmnt Mthod a n paamts of ths w obtan a lna algbac n dg sstm of qaton whch povds th n ss atall th fncton ss s abta chosn th solton s onl appomatd h mpotant lmnt of th mthod that th dsplacmnt fld mst b nmatcall admssbl thfo t wold satsf th nmatc bonda condtons In cas of compl poblms th dtmnaton of th fnctons a qt dffclt ths th mthod s lmtd to smpl poblms h fnt lmnt mthod has th advantag to smplf th gomt of th bod to smpl lmnts ths th appomat fnctons can b fond asl Evolton of modn fnt lmnt mthod oc mthod In th al s th jt plans appad and th hgh tmnal and opatng vloct dmandd mo compl stcts sch as th swpt and dlta wngs h al mthods to dsgn ths spcal wngs appad to b slss snc th nlablt of th calclaton cold not b compnsatd b saft coffcnts d to th ncasng pc of th appld matals opatonal costs A sddn nd aos fo a labl and pcs calclaton mthod fo compl gomts v appld fst th foc mthod whch s basd on th classc lastct that th dsplacmnts w calclatd fom th qlbm of th focs H pblshd hs fst pap abot th jt plans wth swpt wngs n 97 In cas of Dlta wngs poblms appad wth th foc mthod ths anoth appoach had to b sd fo th solton Moton mthod Paalll wth th foc mthod oth mthods basd on th dsplacmnts w bng sachd n od to pt t nto pactc In 956 a sach gop of th Bong Compan ld b n pblshd a poblm solvd b a nw mthod h mthod basd on th solton of a stffnss mat dvd fom a nmatcall admssbl dsplacmnt whch nclds th bascs of th cnt modn fnt lmnt mthod In th followng dcads nw soltons w fond fo and D poblms wth lag dsplacmnt and vaos nds of gomtc matal and oth non-lnats Aft th cognng th mpotanc of th analss of convgnc and th paalllsm btwn th mat qatons and lastct pncpls th fnt lmnt mthod was pt on a nw fondaton n th 6s whch was calld as: calcls of vaaton h nw mthod whch basd on th vtal dsplacmnt bcam almost th ltmat solton ath wd h poblms of appld mathmatcs and th soltons a stll bng dvlopd as compt scnc s constantl volvng h fnt lmnt mthod s wdl appld on constctonal thmo- and fld mchancal poblms ncldng lna- non-lna and SI poblms as wll Snc th compts and th pogams apdl dvlopd th bcam s-fndl and hghl sfl tools fo th ngns Howv th lac of thotcal bacgond sltd nappopat choc of bonda condtons and modls whch fald to povd th solton of th al poblm wwwtanonvtah István Oldal SZIE

21 Hsto of lmnt mthod 5 nt lmnt mthod n ngnng pactc h spad of fnt lmnt mthod fndamntall changd th classcal pocss of podcton snc t was mplmntd nto th podcton chan g 6 vés Pototíps lgátása Póbaüm mgfll Gátás nm fll mg g 6: Smplfd modl of classcal podcton h podcton and opaton of th pototp consms consdng cost dng podcton hs phass q matals machnng spcal condtons fo th opaton tst pmntal tools and gs atall assstanc wth spcal slls a also dmandd to ca ot th podcton and th tst of th pototp hs costs a onl balancd f th podcton has th gat volm o th manfactd pcs a smpl pnsv hs cost s lvantl dcasd b fnt lmnt smlaton g 7 vés Végslm smlácó mgfll Pototíps süségs nm Gátás nm fll mg gn Pototíps lgátása nm fll mg Póbaüm mgfll g 7: nt lmnt add modl of podcton h qd nmb of pototps s dcd b th fnt lmnt smlaton and n cas th poblm can b asl modld thn th pototp podcton mght b vn nglctd In sch a cas th mass podcton can bgn and onl th o ss mst b tstd dng opaton h smlaton povds hlp dng th tchnologcal dsgn as wll and not onl n th stngth chc Dffnt softwa s avalabl to modl moldng fogng dp dawng pocsss ths th hgh cost podcton mthods also bcom chap W can stat that th fnt lmnt mthod appld n th dsgn s spad to man flds: Stngth thmodnamc fld magntc amnaton of th pc dng nomal opaton condtons hs hlps to mpov th qalt of th podct and dc th cost dcng wght István Oldal SZIE wwwtanonvtah

22 nt Elmnt Mthod Ral-tm smlaton of th podct dng th manfactng pocss n od to achv an optmal cost fo a pop manfactng tchnq Smlaton of tools whch povds addtonal nfomaton abot tool lf and optmal opaton condtons h fnt lmnt mthod s not onl spad n th podcton bt n oth scntfc flds as wll Analogosl wth th manfactng th qd pototps and pmnts can b gatl dcd ths th dsgn s chap fast and mo pcs 6 Appnd 6 Pncpals of calcls of vaaton fncton ' fnctonal h vaaton of th fncton o ts small scal ptbaton: h fst vaaton of th fnctonal: ' 6 ' otal dvatv: ' d d d' 7 ' In cas of fnctonal th followng qatons a vald: 8 9 n n n d d It s vald to fncton that: d d d d wwwtanonvtah István Oldal SZIE

23 Hsto of lmnt mthod 6 El-agang dffntal qaton t f R R a gvn fncton bas fncton and R R s an admssbl fncton whch s contnosl dvatv along ts agmntm C[ ] ntscts P and P fd ponts on th bonda and vald to th followng statmnts: t s assgn to all fnctons th I [ ] f ' d al nmb t s fnd that ctan fncton to whch th I [] fnctonal staton In cas of ctan condtons t tas on tm vals t R R abta fncton wth fd ponts of 5 and R al nmb hn w can dfn an R R paamtc st of fnctons: 6 = P P b sbstttng 6 nto : g 8 : Solton of a vaaton poblm and th vad cv István Oldal SZIE wwwtanonvtah

24 nt Elmnt Mthod wwwtanonvtah István Oldal SZIE ' ' ' d f d f I 7 In cas of a gvn fncton th I fnctonal onl dpnds on I can b onl staton satsfng th qd bonda condtons f d di 8 and 9 Accodng to th dffntal ls of th paamtc ntgals: ' ' ' ' d f d f d f f d di W appl th patal ntgaton l on ' ' f : ' ' ' ' d f d d f d f h fst mmb of th ght sd of s o accodng to 5 ths sbstttng nto : ' d f d d f Snc s ncssa th ntgal can onl b o f ' f d d f hs s calld as th El-agang dffntal qaton

25 OUDAMEA DEIIIOS I COIUUM MECHAICS DIEREIA EQUAIO SYSEM O EASICIY AD IS BOUDARY EEMES PROBEM ndamntal dfntons n contnm mchancs Modl: smplfd appomaton of alt whch bhavs smlal as th amnd phnomna In od to solv poblms n th stngth of matals ctan modls a qd as: gomtcal matal- mchancal load constants Gomtcal modls accodng to th dmnsons can b: D: patcl modl all th gomtcal dmnsons a nglctd D: f two dmnsons can b nglctd compad to on Classcal bam-tss lmnts and ln lmnts sd n fnt lmnt mthod D: f on dmnson can b nglctd compad to th two oths Plats and mmbans D: on of ts dmnsons can b nglctd Althogh ths statmnt dos not alwas ncld th complt gomt snc som pats can b stll smplfd n th mchancal pont of vw Onl thos pats mst b gnod whch sgnfcantl ncas th comptaton bt lss lvantl th pcs of th slt Contnm modl: A contnm modl can b dvdd p to fnt o nfnt lmnts and dscbd b contnos and contnosl dvatv fnctons h ponts of th contnm bod can b appontd b a poston vcto j n a gvn coodnat sstm g : Contnm bod and an nfntsmal lmnt István Oldal SZIE wwwtanonvtah

26 6 nt Elmnt Mthod Infntsmal lmnt: an nfntsmall abtal small lmnt of a contnm bod dpndng on th modl t can b an nfntsmal mass o volm Rgd bod: th lngth btwn two abtal chosn ponts of a gd bod s alwas constant ndpndntl th magntd of th load Elastc bod: th bod s capabl to dfom lastcall h lngth btwn ts ponts changs dpndng on th appld load na lastc matal modl: th latonshp btwn th load and dfomaton s lna on-lna lastc matal modl: th latonshp btwn th load and dfomaton s non-lna Plastc matal modl: th sbjct mans dfomd aft th moval of th load and dos not gan ts ognal fom Sval plastc modls st dpndng on th domnanc of lna non-lna lastc o plastc popts g : Matal modls Isotopc matal: th bhavo of th matal dos not dpnd on th dcton; all popts a th sam ndpndntl an abtal dcton Dsplacmnt vcto: th dffnc vcto btwn P and P' ponts h ponts psnt an abta pont of an lastc bod bfo- and aft applng a load ths th ognal ndfomd and dfomd stats P v j w P P P Dsplacmnt fld: dsplacmnt vcto of all ponts of th bod n th fncton of th poston vcto v j w wwwtanonvtah István Oldal SZIE

27 ondamntal of lastct 7 g : Dsplacmnt vcto Small dsplacmnt: th dsplacmnt of th ponts of th bod s lvantl small compad to th gomtcal dmnsons of th bod Knmatc bonda condtons: th gvn o admssbl dsplacmnts of th bod Dnamc bonda condtons: th gvn o admssbl load of th bod Dfomaton: th popotonal dsplacmnt of th ponts of th bod latd to a nt lngth Stan: th gadnt of lngth of vcto oson of angl: th angl gadnt of ppndcla as g b th toson of angl s alwas smmtc h gd bod moton s not tan nto accont g a István Oldal SZIE wwwtanonvtah

28 8 nt Elmnt Mthod P P = + P a b g : h gd bod moton and dfomaton n th - plan Dfomaton vcto: ths vcto dscbs th dsplacmnt of a gvn nt vcto Dfnng t b j nt vctos td: a a j j 5 a j 6 wh h popt of th vcto coodnats: Spcfc stans: popts wthot dmnsons th lngth ncass th lngth dcass Angl toson: th dmnson s n adan th angl dcass th angl ncass Dfomaton stat: th sm of th dfomaton vctos latd to all dctons n a gvn pont Possbl dscpton: nfntsmal nt cb wth dfomaton vctos dfomaton tnso Moh ccl wwwtanonvtah István Oldal SZIE

29 ondamntal of lastct 9 j nso: lna homognos vcto-vcto fncton Dscpton s possbl wth dadc fom o a mat dfnd n a gvn coodnat sstm Dfomaton tnso: It dscbs th dfomaton stat of an pont of an lastc bod b assgnng th dfomaton vcto of a gvn dcton to an abta dcton Dscpton s possbl wth th vctos n mat o dadc fom n a gvn coodnat sstm In dadc fom: a a j a 7 In mat fom: g 5: Dfomaton stat wth dfomaton vcto coodnats 8 h dfomaton vcto coodnats of a n th colmns h a dfomaton vcto latd to an abtal chosn n dcton s dfnd as: n a n n 9 Dfomaton tnso fld: th dfomaton fld of all ponts of th bod n th fncton of poston vcto István Oldal SZIE wwwtanonvtah

30 nt Elmnt Mthod Stss: h ntnst of th ntnal foc sstm dstbtd on th ntnal fac of th bod Dmnson: Pa m Stss vcto: th stss s dfnd b a stss vcto h stss vcto of a da n sfac latd to an abtal chosn n dcton s dfnd as: d n da On a gvn sfac th nomal coodnat of th stss vcto s namd as nomal stss and dnotd b: n cas of tnson n cas of compsson h coodnat of th stss vcto whch s paalll wth th sfac s namd as sha stss and dnotd as: h stss vctos dfnd b j nt vctos on a gvn sfac: j j j wh Stss stat: th sm of th stss vctos latd to all dctons n a gvn pont Possbl dscpton: nfntsmal nt cb wth stss vctos stss tnso Moh ccl wwwtanonvtah István Oldal SZIE

31 ondamntal of lastct Stss tnso: It dscbs th stss stat of an pont of an lastc bod b assgnng th stss vcto of a gvn dcton to an abta dcton Dscpton s possbl wth th vctos n mat o dadc fom n a gvn coodnat sstm In dadc fom: j 5 In mat fom: 6 stss vcto dfnd to an n dcton: n n 7 n Stss tnso: th stss fld of all ponts of th bod n th fncton of poston vcto g 6: Stss stat psntd on an nfntsmal cb 8 h lmnt nd of th stss- and stan tnso can b sd n a vsd od not onl István Oldal SZIE wwwtanonvtah

32 nt Elmnt Mthod as t was psntd al h wo of a foc: a foc actd along a d dsplacmnt cas ot an d nfntsmal wo s th gomtc dscpton of th scala mltplcaton on th g 7; th foc s mltpld b th foc dctd componnt of th dsplacmnt h wo cad ot along a fnt dsplacmnt s th sm of th nfntsmal wos W d 9 d +d d dw= d =cos =d g 7: Wo of a foc Intnal ng: dfomaton ng th ng of th ntnal focs U V dv na cas h ntnal ng can b dvd fom th dobl podct of th stss and dfomaton tnso: U V dv Hamlton opato: abla opato s a vcto whch coodnats a spcal ods to ct th patal dffntatons of th gvn dctons In a Dscats coodnat sstm: j wwwtanonvtah István Oldal SZIE

33 ondamntal of lastct In clndcal pola coodnat sstm: R R R Dffntal qaton sstm and bonda lmnt poblm of Elastct Eqlbm qatons h qlbm qatons dscb th latonshp btwn th q dstbtd foc sstm actng on a volm and th stss fld tnso dv q j a b g 8: oad cas of an nfntsmal bod If an abtal chosn nfntsmal bod nsd of a bod s n stad stat thn th tnal g 8a and ntnal g 8b focs a n qlbm B nvstgatng th focs along th as g 9a t s clal obvos: f no tnal focs a actng pon th bod thn th ntnal focs stsss hav qal magntd and oppost snss on th pop sds of th bod h chang s casd b th tnal dstbtd foc sstm actng on th volm Stss s an ntnal foc dstbtd on a sfac ths t mst b calclatd onto th nfntsmal cb appas on th dd sfac of th cb and t tns to b a foc sstm actng on a volm f t s dvdd b d sd lngth Smlal sha stss mst b dvdd b d whl mst b dvdd b d sd lngth hn all focs actng pon th nfntsmal bod along th as a n qlbm: d d d d d d q István Oldal SZIE wwwtanonvtah

34 nt Elmnt Mthod + d + q + d a b d g 9: oad cas of an nfntsmal bod along th as h gadnt of th stsss can b dscbd b th patal dvatv of th gvn dcton: d d d whch a sbstttd nto w obtan: q 5 Analogosl to th al th oth two dctons: q 6 q 7 h 5-7 qatons a th so calld qlbm qatons n Dscats coodnat sstm In od to dfn gnall th qlbm qatons lt s consd a V volm nsd of a bod smlal to g wwwtanonvtah István Oldal SZIE

35 ondamntal of lastct 5 h nfntsmal foc actng on a dv volm of th nfntsmal bod: d qdv g : V volm nsd a bod wth a foc sstm actng on sfac and volm h nfntsmal foc actng on a da sfac and calclatd fom th stss vcto: n d da nda n h V ntnal bod s n qlbm ths th sm of th focs actng on th sfac and th volm a o: qdv nda 8 V A A Accodng to th Gass-Ostogads ntgal-tansfomaton thom: nda dv V Sbstttng Gass-Ostogads nto 8: qdv dv V V Sttng th spaatd pats of th qaton nto on ntgal: István Oldal SZIE wwwtanonvtah

36 6 nt Elmnt Mthod q dv 9 V Snc th V volm s abtal chosn ths th 9 qaton s onl vald f th ntgal s o hs s th qlbm qaton of lastct q Gomtc qatons h gomtc nmatc qatons dfn th latonshp btwn th dsplacmnt fld and th dfomaton tnso fld On g 8 th dfomaton of an nfntsmal cb s psntd n th plan of a Dscats coodnat sstm d Q dv dv dv Q d d = + dv P P d d d d t s nglct th gd-bod moton and lt s nvstgat th latv dsplacmnt btwn pont P andq B plottng P and P ' ponts on ach oth th gadnt of PQ lngth s th QQ ' vcto whch s dnotd b d d dv j dw nfntsmal dsplacmnt vcto It has two coodnats n a plan naml d and dv Both coodnatd can b bon p nto two pats: d d d dv dv dv d : fom th stan of d sd n th fncton of d : fom th stan of d sd n th fncton of ths g : h gomtc ntptaton of dfomaton d d and d d wwwtanonvtah István Oldal SZIE

37 ondamntal of lastct 7 dv : fom th stan of d sd n th fncton of dv : fom th stan of d sd n th fncton of ths v v v dv d and v v v dv d Accodng to g th stans a: d d dv d Whl th toson of angl s: dv d dv d actan actan d d d d B th s of th patal dvatvs latd to th dsplacmnt vcto: v v hs calclaton can b cad ot on all plans whch slt th gomtc qatons n a Dscats coodnat sstm: v w v v w w h gomtc qatons can b dfnd n a gnal fom t s nvstgat th poston of two ponts on an lastc bod bfo and aft applng an tnal load on t h dstanc btwn th two ponts n th ndfomd stat s d d d j d Accodng to th dfnton of dfomaton th gadnt of dsplacmnt btwn th two ponts has to b amnd and dscbd István Oldal SZIE wwwtanonvtah

38 8 nt Elmnt Mthod g : Dsplacmnt and dfomaton Q Q = Q P d d d P P P g : Dsplacmnt and dfomaton vctos h dffnc of th two ponts s dfnd b th latv dsplacmnt of P and Q ponts: Q P P hs th dsplacmnt of Q : P t s appomat th dsplacmnt fncton n th clos nvonmnt of P b applng a alo-ss on P pont: P d d d d P P P P P d wwwtanonvtah István Oldal SZIE

39 ondamntal of lastct 9 om and can b dvd that n th clos nvonmnt of P th dffnc and th dvatv a appomatl qal In cas of small dsplacmnt th hgh dvatvs can b nglctd: d d d d P P P ang nto accont d d d j d d d qlbms and th gop tho btwn th scala and dadc podct a b c a bc dsplacmnt fld s: th nfntsmal gadnt of th d P P P d j d d j d P P P d B th s of th Hamlton opato: d 5 Wh th dvatv tnso of th dsplacmnt fld whch can b dvdd to a smmtc and ant-smmtc sw-smmtc tnsos h smmtc pat dscbs th dfomaton of th nfntsmal bod whl th antsmmtc dscbs th otaton of th nfntsmal bod hs th dfomaton tnso dvd fom th dsplacmnt fld s dscbd as: 6 Eqaton 6 s th so-calld gomtc qaton h dntcal scala qatons of th tnso fom a dscbd n a Dscats coodnat sstm as t was mntond al n and qatons h oth tp of gomtc qatons s th to so-calld Sant-Vnant compatblt qaton: h compatblt s also latd to th nghbo nfntsmal lmnts snc th matal s contnos and th dsplacmnt of th nghbo lmnts hav to b dntcal as wll István Oldal SZIE wwwtanonvtah

40 nt Elmnt Mthod Consttton qatons matal qatons h constttonal- o matal qatons dtmn th latonshp btwn th stss and stan stat h bhavo of th matal on g can b dscbd as lna and th Hoo law s stabl to dscb to phnomna In cas of sngl as stss stat th smpl Hoo law can dfn th latonshp btwn th stan and th stss: E wh E Yong-modls lastct modls s th coffcnt btwn th stss and stan In cas of tnson o compsson th stss has onl on pncpl dcton ths componnt bt stan appas n two dctons as t s sn on g h s postv longaton n th matal along th as of tnson bt n th sam tm t contacts ppndclal h latonshp btwn th longaton and th contacton s dscbd b th dmnsonlss Posson-coffcnt: In cas of mlt-as stss stat th latonshp btwn th stss and stan stat can b onl dscbd wth a tnso qaton th so-calld gnal Hoo law h law has two sotopc fom to lna lastc matals: G E 7 E 8 G wh G : sha lastc modls whch can b calclatd as: G E : nt mat g : Stans Posson-coffcnt E : th fst scala nvaant of th tnsos th sm of th man ow? h scala qatons wth spct to 7 matal qatons: wwwtanonvtah István Oldal SZIE

41 ondamntal of lastct G G G G G G Bonda condtons p A p A In cas of an lastct poblm two tps of bonda condtons can b dfnd: Knmatc bonda condtons: th admssbl dsplacmnts constants on sfac It stands fo th solton that: Dnamc bonda condtons: th admssbl p load on A p sfac th nloadd sfacs a ncldd as wll snc th hav nown load whch qal to o It stands fo th solton that: p p o n p g 5: Bonda condtons Oth bonda condtons can b dfnd as wll bt ths two a th most common A István Oldal SZIE wwwtanonvtah

42 nt Elmnt Mthod 5 Bonda lmnt mthod h bonda lmnt poblm of lastct s consstd th dffntal qatons of lastct and th bonda condtons: q qlbm qatons gomtc qatons G E constttv qatons nmatc bonda condtons A n p dnamc bonda condtons A p Wth ths dfnton t s povd that th bonda lmnt poblm has solton stnc cta and onl on solton st nct cta wwwtanonvtah István Oldal SZIE

43 EERGY HEOREM O EASICIY CACUUS O VARIA- IO IIE EEME MEHOD DEERMIAIO O SI- ESS EQUAIO I CASE O CO-PAAR ESED EEME Appomat fnctons h appomat solton of an lastct poblm can b obtand b th appomaton of dsplacmnt o ntnal focs stsss B th s of th lastct qatons th dsplacmnt- dfomaton o stss fld of a bod can b dtmnd ndpndntl fom th wa of appoach Knmatcall admssbl dsplacmnt fld A dsplacmnt fld s nmatcall admssbl f: Satsfs th nmatc bonda condtons g 5 Contnosl dffntabl th gomtc qatons a satsfd A g : Knmatcall admssbl dsplacmnt fld of a fd bam h nmatcall admssbl dfomaton fld can b dvd fom : h nmatcall admssbl stss fld can b dvd fom th dsplacmnt fld b th s of th consttton qaton matal qaton gnal Hoo law: G E Snc an lastct poblm can onl hav on solton whl can hav nfnt solton ths gnall t dos not satsf th qlbm qatons and th dnamc bonda condtons Statcall admssbl stss fld Stss fld s statcall admssbl f: István Oldal SZIE wwwtanonvtah

44 nt Elmnt Mthod Satsfs th dnamc bonda condtons g 5 n p Satsfs th qlbm qatons: q h nmatcall admssbl dfomaton fld can b dvd fom ths stss fld b th s of th consttton qaton: E hs dfomaton fld and th dvd nmatcall admssbl dsplacmnt fld gnall do not satsf th gomtc qa- G tons and th nmatc bonda condtons Ap Pncpl of vtal ng Vtal dsplacmnt: small abta admssbl dsplacmnt of th appld constants whch can b dvd fom th dffnc of a nmatcall admssbl dsplacmnt fld and th vald dsplacmnt fld A : ; g : Knmatcall admssbl and vtal dsplacmnt flds Pncpl of vtal wo: f an dllcall lastc sstm bod s dsplacd fom ts qlbm stat n cas of lastct th qlbm s dfnd b th load and constans thn th vtal wo of th tnal focs qal to th vtal chang of ntnal ng: W U Wo don b th focs on volm and sfac: W q dv p da V A p h vtal ntnal ng: wwwtanonvtah István Oldal SZIE

45 ndamntals of fnt lmnt mthod 5 U dv V V dv V dv In th fomla consttton qaton has bn alad sd to dscb th latonshp btwn th stss and dfomaton stat s th thom of vtal wo Sbstttng and qatons: V dv q dv p da V A p Pncpl of mnmm potntal ng h potntal ng of a bod s th dffnc of th ntnal dfomaton ng and th wo don b th tnal focs: U W 5 V dv V q dv h ntnal dfomaton ng: A p p da 6 U V dv Wo don b th tnal focs: W q dv p da V A p t s dtmn th potntal ng b sng a nmatcall admssbl dsplacmnt fld: U W 7 Wo don b th tnal focs focs on volm and sfac on an lastc bod n cas of a nmatcall admssbl dsplacmnt fld: W V q dv p da q dv p da Ap V A p V q dv q dv V p da A p A p p da István Oldal SZIE wwwtanonvtah

46 6 nt Elmnt Mthod q dv p da q dv p da W W 8 V A V A p p h nmatcall admssbl dfomaton fld: 9 Intnal ng stod n an lastc bod d to dfomaton n cas of a nmatcall admssbl dsplacmnt fld b applng th consttton qaton : U V dv dv V V dv V dv V dv V dv dv dv dv U U U V V V h potntal ng dvd fom a nmatcall admssbl dsplacmnt fld b th s of 7 8 and : UW U U U W W U W U W U wh th potntal ng of th vald dsplacmnt solton s: U W h fst vaaton of potntal ng: U W h scond vaaton of potntal ng: U wwwtanonvtah István Oldal SZIE

47 ndamntals of fnt lmnt mthod 7 h fst vaaton of potntal ng s o accodng to th thom of vtal wo W U : h scond vaaton of potntal ng s an ng qantt ths t s vald to an abta : 5 hn th dffnc of a nmatcall admssbl and a vald dsplacmnt fld s: 6 h 6 fomla s th pncpl of mnmm total potntal ng: among all nmatcall admssbl dsplacmnt flds th potntal ng s mnmal n cas of th vald dsplacmnt fld Pncpl of agang vaaton h vaaton fom of th pncpl of mnmm total potntal ng s th pncpl of agang vaaton B sng th vaaton appoach th total potntal ng s a fnctonal dpndng on th dsplacmnt fld: U W Wh th nmatc bonda condton n vaaton fom s: A h condton of th tma s: U W 7 In cas of lastc bods ths pncpl s qal wth th pncpl of vtal wo If th fst vaaton s o thn th fnctonal can b staton mnmm o mamm In o cas th scond vaaton can b th postv o o val ths t can b staton o stabl mnmm István Oldal SZIE wwwtanonvtah

48 8 nt Elmnt Mthod wwwtanonvtah István Oldal SZIE g : Kntc ampl of potntal ng h vaatons of potntal ng dscb th stablt condtons of a ntc poblm on g 5 nt lmnt modl basd on dsplacmnt mthod h most wdl spad fnt lmnt mthod s basd on th moton mthod; th commcal pogams mostl appl ths basc mthod h fndamntals of th mthod a th followngs: th bod mst b dvdd nto lmnts and thn nmatcall admssbl dsplacmnt flds mst b consdd on th lmnts b appomat fnctons Aft that b applng th gomtc and consttton qaton alongsd wth th bonda condtons a lna algbac qaton sstm s catd h solton of ths qaton sstm s th appomat dsplacmnt fld h stss fld calclatd fom th dsplacmnt fld wll patclal satsf th qlbm qatons In th dscpton vctos colmn mats wll b sd nstad of tnsos 5 Intodcton of vcto flds Vcto of stss componnts colmn mat: th vcto ncldng th stss tnso componnts s dscbd n a spatal sstm as: whl n cas of coplana sstm:

49 ndamntals of fnt lmnt mthod 9 István Oldal SZIE wwwtanonvtah Stan vcto colmn mat: vcto ncldng th stss tnso componnts s dscbd n a spatal sstm as: whl n cas of co-plana sstm: If th dsplacmnt mthod s sd thn th gomtc and consttton qatons a also qd hs qatons hav to b fomlatd to vcto qatons t s dfn th scala componnts of th gomtc qaton n a Dscats coodnat sstm: v w v w v w and sbsttt thm nto th dfomaton vcto t s convt thm nto a podct fom: w v w w v v w v hs th dfomaton vcto s dvd as podct of dsplacmnt vcto and ncldng th dffntal ls dffntal opato mat h pop lmnts a sbstttd nto th stss vcto b th s of th consttton qaton E G :

50 5 nt Elmnt Mthod wwwtanonvtah István Oldal SZIE G G G G G G G G G G G G G G G hn lt s convt thm nto podct fom: G G G G G G G G G G G G C G G G G G G G G G G G G hs th stss vcto s dvd as a podct of th dfomaton vcto and th C mat whch nclds th matal constants Intodcng th vcto flds both th gomtc qaton: 8 and th consttton qaton:

51 ndamntals of fnt lmnt mthod 5 C 9 a obtand as sngl podcts Sbstttng 8 nto 9: C ths th dsplacmnt fld s th nnown fncton whl th stss and dfomaton can b dctl calclatd 5 Elastct poblm and th mthod of solton h fnt lmnt mthod s psntd on an lastct poblm h gnal lastct poblm s th followng: p V q P A p A g : Elastct poblm Accodng to g th followng data a gvn: h gomt of th bod h matal constants of th bod loads Constants Dmandd fnctons: Stps to solton: stl th bod s dvdd to fnt domans so-calld lmnts Spcal ponts nods a appontd on ths lmnts h lmnts cov th total volm of th bod and th gomtc psntaton s a msh h sngl lmnts a connctd to ach oth b th nods h dsplacmnt fld s appomatd lmnt b lmnt gnall wth polnomals whch a ft to th nods h dsplacmnt flds of th nab lmnts a ft to ach oth thogh th nods and th dscb a contnos fncton on th bod h appomat stss- and dfomaton fld can b dvd fom th dsplacmnt fld b th s of th gomtc- and consttton qaton hn b th applng th pncpl of agang vaaton a lna algbac qaton sstm can b dvd wth spct to th nods hs s th so-calld stffnss qaton h algbac ss- István Oldal SZIE wwwtanonvtah

52 5 nt Elmnt Mthod tm of qaton s solvabl f a load o dsplacmnt paamt dvd fom th nmatc o dnamc bonda condtons s spcfd to ach nods on th sfac hs th nnown vals a th dsplacmnts of th nods B solvng th sstm of qaton th appomat nodal dsplacmnt fld s obtand ths th appomat stss- and dfomaton flds can b calclatd as wll 5 nt lmnt appomat dsplacmnt fld h bod s dvdd to abta shapd and sd fnt domans fnt lmnts atall t s tan nto accont that th bass fnctons hav to ft to th lmnt g 5: Dsctaton fnt lmnt h dsplacmnt fld of lmnt s appomatd b a contnosl dffntabl fncton h tp of th fncton s dtmnd and accodng to ths fncton th dmandd nmbs of nods ponts n cas of lna fncton ponts n cas of qadatc fncton a appontd on th lmnt hn th dsplacmnt fld s dscbd b th nodal dsplacmnt h dsplacmnt of lmnt nod on lmnt: v w h dsplacmnt vcto of lmnt dvd fom th dsplacmnt of j n nods: wwwtanonvtah István Oldal SZIE

53 ndamntals of fnt lmnt mthod 5 v w j n n vn wn whl ths vcto consst n dsplacmnt vcto fld of lmnt s dvd fom th ntpolaton of nodal dsplacmnt vcto: wh nmb of lmnts h s th appomat mat mat of th ntpolaton fnctons hs mat s blt p b blocs and ach bloc nclds th ntpolaton fncton of ach nod h dsplacmnt of th lmnt can b dvd fom th nodal dsplacmnt of wth spct to lmnt: v w h a th ntpolaton fnctons Dfnton of th nds: fncton dfns th dsplacmnt along dcton of lmnt latd to an Wh th lmnts of abta locaton d to th dsplacmnt of dcton of nod whl th oth componnts of th nodal dsplacmnt vcto of lmnt a o h fnctons mst to satsf th followng condtons: h fnctons mst b contnosl dffntabl E th fncton mst povd nt val of dsplacmnt n nod th fncton mst qal to o n th oth nods j n h mat has n lmnts all th blocs latd to th nods j n has to hav th sam s n: j n B th appomaton of th lmnt s dsplacmnt fld th dfomaton fld can b obtand b sbstttng nto 8: Intodcng th appomat mat: B nodal-dfomaton mat as a podct of th dffntal opato and István Oldal SZIE wwwtanonvtah

54 5 nt Elmnt Mthod B h stss fld of th lmnt: C CB h potntal ng of th lmnt accodng to 6: V dv q dv p da V Ap Rwtng th fomla b fomng th scala and dobl scala podcts nto mat podcts th constants of th ntnal ng a placd and ntodcng thm as vctos nstad of tnsos: V dv q dv p da V Ap Sbstttng and spaatng th constants ot of th ntgals: K q p B CB dv V t s ntodc th stffnss mat: V B CB dv q dv p da V And th nodal load vctos wth spct to th volm and sfac focs: V q dv A p p da 5 Ap : q p hs th potntal ng of th lmnt s: K wwwtanonvtah István Oldal SZIE

55 ndamntals of fnt lmnt mthod 55 h ng thoms can onl b appld on th whol bod; th a not vald on ndvdal lmnts If th bod has Q nmb of lmnts th potntal ng of th bod s dvd fom th sm of all lmnts potntal ng Q U KU U Accodng to th pncpl of agang vaaton th fst vaaton of th potntal ng s o: U KU U KU Sttng th qaton w dvd th stffnss qaton: KU 6 wh: K : s th stffnss mat of th bod U : s th nodal dsplacmnt vcto of th bod : s th nodal foc vcto of th bod 6 qaton s lna sstm of qaton whch povds th solton of th lastct poblm h lmnts n th qaton a a fomlatd accodng to a smpl statc poblm n cas of thmal stsss lastc constants th stffnss mat has mo lmnts whl n cas of dnamc poblm vn oth pats a addd as wll 6 Dfnton and solton of stffnss mat n cas of co-plana tnsd tss lmnt 6 Stffnss mat of D tnsd tss lmnt Gnal popt of th tnsd-compssd stcts tss lmnts that th lmnt a onl loadd aall t s a coodnat sstm f to th as of a tss lmnt On g a psntd as th nodal loads of lmnt wth lngth 6 j j j j j g 6: wo nods on a co-plana lmnt István Oldal SZIE wwwtanonvtah

56 56 nt Elmnt Mthod In nod th dsplacmnt s h tss lmnt: whl n nod j th dsplacmnt s j j 7 s appomatd b a lna fncton: a a 8 h dsplacmnt fld povds th dsplacmnt vals n th nods of th lmnt: a a a a j Sttng th constants and sbstttng nto 8: j hn sbstttng ths qaton nto 7 qaton: j omng nto mat podct: v v j v j wh s th appomaton mat of lmnt and s th nodal dsplacmnt vcto h appomaton mat s blt p fom two blocs wth th ntpolaton fnctons of nod and j : j j wwwtanonvtah István Oldal SZIE

57 ndamntals of fnt lmnt mthod 57 István Oldal SZIE wwwtanonvtah hs ntpolaton fnctons satsf th qd condtons contnos povds nt val n ts own nod dsappas n oth nods and shown on g 7 j j j g 7: Intpolaton fnctons In cas of tss lmnts th onl dfomaton s th longaton ths th gomtc qaton: j j B v v d d d d B nodal-dfomaton mat has constant lmnts whch slts constant stan n th tss In cas of sngl-stss stat th smpl Hoo law can b appld n od to calclat th stss: CB C h consttton mat: E E C h stffnss mat of th lmnt: V Ad CB B dv CB B K d A E E E E Ad E E K AE K 9 hn th stffnss qaton of th lmnt:

58 58 nt Elmnt Mthod K wh v v s th nodal dsplacmnt vcto of th lmnt j j j j s th nodal load vcto of th lmnt In gnal cas th local coodnat sstms fd to tss lmnts a dffnt ths th stffnss qaton mst b tansfomd nto global so calld absolt coodnat sstm n od to smma th stffnss mats of th complt bod v v = g 8: Vcto n otatd coodnat sstm h vcto coodnats show on g 8 a calclatd n a coodnat sstm otatd b angl as follows: ' cos v sn v ' sn v cos In mat fom: ' cos ' v ' sn sn cos v wh s th tansfomaton mat h mat nclds two vctos; can b dscbd b two blocs wh on bloc lats to on vcto: and vctos wwwtanonvtah István Oldal SZIE

59 ndamntals of fnt lmnt mthod 59 cos sn sn cos cos sn sn cos t s dtmn th stffnss mat n a coodnat sstm otatd b angl! In od to ca ot ths calclaton w hav to dtmn th tansfomd qaton as wll: K ' ' ' Accodng to: ' ' smla to ths: ' Sbstttng ths fom nto : K ' ' t s mltpl th qaton wth fom th lft sd: K ' ' E K ' ' b th s of w obtan th followng: K' K s asmmtc ths thn: K' K 5 t s calclat th stffnss mat of a co-plana tss dfnd as 5 n a global coodnat sstm wth spct to fomla 9 latd to th s of th stffnss mat n local coodnat sstms: K cos AE cos sn sn cos cos sn sn István Oldal SZIE wwwtanonvtah

60 6 nt Elmnt Mthod K' K cos AE cos sn cos cos sn cos sn sn cos sn sn cos cos sn cos cos sn cos sn sn cos sn sn 6 In cas of a stct all stffnss mats of th tsss mst b tansfomd nto a global coodnat sstm and th smmad Aft thn th stffnss qaton can b appld on th stct whch povds th solton as dsplacmnts and focs n all nods 6 Eampl 9 ába: sss h stct on g 9 nclds th tsss h gvn data a: m o 5 A A A A mm E GPa Dtmn th focs and dsplacmnts n th nods! ngths of th tsss: tg mm mm cos wwwtanonvtah István Oldal SZIE

61 ndamntals of fnt lmnt mthod 6 István Oldal SZIE wwwtanonvtah h stffnss mat of tss n th local whch s dntcal wth th absolt accodng to 9: mm AE K h nodal blocs pp nd s th nmb of th lmnt; low nd s th nmb of th two nods h bloc dscbs th latonshp btwn nods: K K K K K h stffnss mat of tss n th local coodnat sstm: mm AE K ss s ppndcla n th absolt coodnat sstm ths ts coodnats hav to b calclatd n th absolt coodnat sstm accodng to 6: sn sn cos sn sn cos sn cos cos sn cos cos sn sn cos sn sn cos sn cos cos sn cos cos AE K K wh o 9 mm AE K h nodal blocs: K K K K K

62 6 nt Elmnt Mthod h stffnss mat of tss n th local coodnat sstm: K AE mm ss s otatd b angl n th absolt coodnat sstm ths ts coodnats hav to b calclatd n th absolt coodnat sstm: K K cos AE cos sn cos cos sn cos sn sn cos sn sn cos cos sn cos cos sn cos sn sn cos sn sn o wh 5 K mm h nodal blocs: K K K K K h stffnss mat s smmad b addng th dntcal dscbng blocs of th nab nods togth ths th stffnss mat of th stct s: K K K K K K K K K K K K K wwwtanonvtah István Oldal SZIE

63 ndamntals of fnt lmnt mthod mm KU h stffnss qaton of th stct: v 68v v B sbstttng th nown foc and dsplacmnt bonda condtons: v h podct s th solton of a lna sstm of qatons wth s nnown vals: 6857mm 5986mm v 659mm István Oldal SZIE wwwtanonvtah

64 AAYSIS O WO-DIMESIOA RUSSES USIG IIE EEME MEHOD BASED PROGRAM SYSEM wo-dmnsonal ba stcts In sval chapts of th mchancs w mt stcts whch consst of statc bas h man popts a that th a loadd onl at th two nds and thn onl aal focs consqntl dawn o pssd Sch stcts a calld tsss In ths chapt w dal wth two-dmnsonal tsss onl hs stcts a dfnd so that: As of th bas l n a common plan h bas a connctd n an dal plan jont h bas gomtc as ntsct at on pont h stct a lnd to th gond b dal jont constants h tnal focs can act n th nods and th lns of acton of th focs a n th plan of th bas Dng th amnaton of th tsss w sall loo fo th answ to th followng qstons: magntd and dcton of th acton focs magntd and dcton of focs sltng n bas th focs and stsss gnatd n bas th sltng dsplacmnts of ach pont of th stct and th dfomaton of ach ba Mo stcts whch gnall contan bndng bas smpl sppotd and cantlv bams fam stct cvd bas tc ma b tstd fo th stablt of th stct and dnamc bhavo th ctcal focs of compssd bas and natal fqncs W dal ths poblms n chapts 5-8 and th nstablt of compssd bas n th chapt 9- In dtmnng th actons focs mpotant ss s whth th bam s tnall statcall dtmnd o ndtmnd hs s nflnc th sd mthods In dtmnng th ba focs an mpotant ss whth th bam s ntnall statcall dtmnd o ndtmnd h calclaton of dsplacmnts and dfomatons a v smpl fo both ntnall and tnall dtmnd stcts In ths cas w s gomtc appoach o solv complcatd stcts and statcall ndtmnat stcts w s pncpls of ng Castglano and Btt s thom As w shall s t s lvant whth th stct tnall o ntnall ndtmnd th pocd wll not b affctd whn w s th fnt lmnt mthod-basd solton nt lmnts fo modlng bams Gnall th a two tps of lmnt avalabl fo modlng of bams n EA fnt lmnt analss pogams o modlng of tsss RUSS lmnts and fo th bnt shad twstd bas BEAM lmnts ma b sd In both cass th fnt-lmnt two-dmnsonal and chaactd b a sngl ln wwwtanonvtah István Mohaos ÓE

65 Analss of two-dmnsonal tsss 65 h RUSS lmnt popts h RUSS lmnts a gopd accodng to s thm fo two- o th-dmnsonal modlng W dstngsh two tps of lmnt RUSSD s g and RUSSD s g h RUSSD lmnts a two-nods naal lmnts wth two dsplacmnt dgs of fdom n both nods h lmnt local coodnat sstm -as s dfnd b a vcto that stats at th fst nod and ponts towads th scond nod h -as s paalll th global coodnat sstm XY plan and ppndcla to th -as Y X g RUSSD lmnts h lna statc analss qs mo constants to spcf th al th-dmnsonal lmnt popts In ths cas that s th coss-sctonal aa of th bam hs s not sd onl to calclat lastc popts of lmnts bt also t s ndd fo dtmnaton th ta wght W wll also nd th matal popts of th bas In ths cas t s sffcnt to dtmn th lastc modls h calclatons of th own wght of stcts qs to dtmn th matal dnst W can pfom bclng and hat tansf analss sng RUSSD lmnts h BEAMD s a two-nod naal lmnt too o stctal analss s dgs of fdom th tanslatons and th otatons a consdd p nod h and as of th lmnt coodnat sstm sam as dscbd abov and a thd nod s qd to assgn th lmnt ontaton István Mohaos ÓE wwwtanonvtah

66 66 nt Elmnt Mthod Y Y X g RUSSD lmnt h ncssa al constants and matal popts a th sam as gvn as fo RUSSD lmnts h RUSSD lmnts can also b sd n stablt and thmal analss poblms Bam Elmnt s popts h BEAMD lmnt s a two-nod naal lmnt bt nl th tss lmnts at both two-nods th a th-dgs of fdom two dsplacmnts and a otaton so ths s stabl fo two-dmnsonal modlng of bnt bas h BEAMD lmnt two-nod naal lmnt also bt nl th tss lmnts at both two-nod a s-dgs of fdom th dsplacmnt and th otaton hs lmnts stabl modlng th-dmnsonal ba stcts Mo dtald dscpton of ths lmnts s n chapt -6 Std solton h fnt lmnt std pocd: poblm analss cat a gomt fo gnat a fnt lmnt msh dfn popts of fnt lmnts lmnt tp al constant matal popts dtmn bonda condtons and loads 5 solv th modl 6 valaton of th slts At both nds sppotd tsss a loadd at two nods h focs a - ach s g h bas a stl pps wwwtanonvtah István Mohaos ÓE

67 m Analss of two-dmnsonal tsss 67 o b dtmnd: dflcton of th stct stsss gnatd n th bas = m g h tstd tsss h fnt lmnt pogams sall contan blt-n D gomtc modl gaphcs p- and postpocsso hs w can ppa th gomtc modl n ts s g g Gomtc modl n th fnt lmnt pogam hs blt-n gomtc modls do not alwas off o th convnnc of modn CAD sstms Oftn w hav to anal stng modls In ths cas th data chang pocd wth oth CAD sstms can b convnnt and ffcnt b an avalabl standad fl fomat sch as SA IGS DX tc s g 5 István Mohaos ÓE wwwtanonvtah

68 68 nt Elmnt Mthod g 5 Impot gomtc modl Do not fogt n ths cas th gomtc modl onl hlps to cat a fnt lmnt msh It dos not compl wth th ls of a tchncal dawng and has no lvanc to th al shap of th stct It s t n ths cs bcas th mm damt pps appa onl lns s g hs w hav to tansfom smplf and tnd th tchncal docmntaton bfo th fnt lmnt analss hs s shown n g 6 whch shows th mpotd gomtc modl h on pc chod bas a dvdd at nods bcas t hlps th fnt lmnt msh gnaton It shold b mmbd that w hav to choos a nt sstm fo fnt lmnt modlng If th SI s slctd on dawng nt wll b a mt dng th data chang of gomtc modls g 6 h mpotd gomtc modl wwwtanonvtah István Mohaos ÓE

69 Analss of two-dmnsonal tsss 69 It s also shown that th lmnts l n th XY plan In th nt stp w dtmn th lmnt gop s g 7 W hav clafd that w s lna bhavo RUSSD lmnts g 7 Dtmnaton of lmnt gop It s also ncssa to dtmn th matal popts of fnt lmnts It s sffcnt to spcf th val of th modls of lastct fo th tss lmnt s g 8 Mang s to s th slctd nt sstm In ths cas t s th SI sstm wh th dmnsons a dtmnd n mts and th modls of lastct n Pa /m g 8 Dtmnaton of matal popts t tas s dtmn th al constants of th lmnts s g9 A compl fnt lmnt modls contan vaos tps of lmnts so w hav to also dtmn th assocatd lmnt gop As pvosl dscbd th al constant s onl th coss-sctonal aa fo RUSSD lmnts Do not fogt w hav to s th slctd nt sstm n ths cas too István Mohaos ÓE wwwtanonvtah

70 7 nt Elmnt Mthod g 9 Ral constants dtmnaton Aft dfnng th th msh popts ma follow th fnt lmnt msh gnaton h EM pogams off sval mthods fo ths s g Bcas th ba focs do not chang along th lngth of th bas t s sffcnt to b placd on lmnt n ach objcts g Paamtc msh gnaton Bcas th fnt lmnt msh catd ach gomt objct spaatl t s ncssa to mg th nods n ach nd of th bas s g h dndant nods a movd fom th fnt lmnt modl g Mg of th nd of bas In th nt stp th bonda condtons shold b gvn In ths cas ths a two dsplacmnt constans on th nds of th tsss h lft sd two dgs of fdom a fd and dctons and th oth nd onl th dcton s fd s g wwwtanonvtah István Mohaos ÓE

71 Analss of two-dmnsonal tsss 7 g Spcf dsplacmnt constants nall t shold b gvn th loads whch shown n g two concntatd foc s g h dcton of focs mst b gvn n th global coodnat sstm so th downwad focs a ngatv sgn g Dfnng th concntatd focs B th fnt lmnt modl s blt h calclaton follows s g g Rn a lna statc analss Aft th sccssfll solvng th dspla and valaton of th slts follows h dsplang stsss gnatd n bas s g 5 can b don n sval was h stsss a ntptd on th lmnt and n th lmnt local coodnat sstm h s a possblt that th slts dspla on dfomd shap h dfomaton s not al of cos th pogam gnats a spcfc scal facto so that data can b valatd István Mohaos ÓE wwwtanonvtah

72 7 nt Elmnt Mthod g 5 Dspla stsss h slts s g 6 mst b valatd h ngatv sgn ndcat compssv stss otc that th bas w staght can b ntptd no bndng momnt gnatd n thm g 6 Stsss on dfomd shap O am was to amn th dflcton s g 7 wwwtanonvtah István Mohaos ÓE

73 Analss of two-dmnsonal tsss 7 g 7 Dfomd shap h dfomatons can b b-dctonal dsplacmnt of nods h dflcton s th dsplacmnt n th global coodnat sstm s g 8 h ngatv sgn of slts psnt a downwad dsplacmnt h val of th scal accodng to SI nt sstm g 8 Dsplacmnt n dcton István Mohaos ÓE wwwtanonvtah

74 7 nt Elmnt Mthod It s possbl to dspla th act nmcal slts at nods focs gnatd n bas and dsplacmnt componnts g :9 to : Bcas th tss lmnts a loadd onl b tnson-compsson stsss so th tabl ncld onl ths stsss ntptd n th lmnt local coodnat sstm g 9 g Stss componnt lst h dsplacmnts of nods a ntptd n th global coodnat sstm s g g h dsplacmnts of nods wwwtanonvtah István Mohaos ÓE

75 Analss of two-dmnsonal tsss 75 Rmas Dng th solton w hav not dalt wth bclng of th compssd bas If ths s a al poblm t shold hav to vf wth solton a fnt lmnt poblm o wth an analtc mthod Dng th soltons th ta wght ~859 was nglctd bcas ths od of magntd small than th tnal load Both poblms a pland n lat chapts whch dal wth BEAM lmnts thmo th stctal jont was not amnd h oth spcald aas of stctal dsgn dal wth ths poblms István Mohaos ÓE wwwtanonvtah

76 5 WO-DIMESIOA BE BARS VARIAIO PROBEM SIESS EQUAIOS AD SOVIG HEM BY IIE EE- ME MEHOD 5 wo-dmnsonal bnt bam lmnt vaaton std Eamn th two-dmnsonal staght bam shown n fg 5 h loads a q dstbtd load concntatd foc and M concntatd momnt Dng th solton w s th El-Bnoll s bam tho Accodng to ths tho th coss-scton of th bam mans nomal to th bam ntal as so w do not accont fo th sha dfomaton hs th total potntal can b wttn as a fnctonal ad on v dsplacmnt fncton g 5 h tstd bam h dsplacmnt fncton of th bnt bam nown as dffntal qaton of th lastc cv: Mh v" 5 I E Also nown as th bnt bam stan ng: Mh U d 5 E I Solvng th dffntal qaton of th lastc cv fo M h and sbstttng ths n th qaton U v E I v" d 5 Dfn th total potntal ng ndd th wo of tnal focs whch conssts of th mmbs; Wo of th concntatd focs a ppndcla to th bam: v 5 wwwtanonvtah István Mohaos ÓE

77 5 wo-dmnsonal bnt bas vaaton poblm 77 wo of th concntatd bndng momnts: M j v' j 55 wo of th dstbtd loads ppndcla to th bam: b a v qd 56 So th total potntal: v E I v" d vqd v M jv' j 57 b a h cton of fst vaaton of 57 potntal lads to th basc qaton and to th natal bonda condtons h appomat solton of th tas s th dct mnmaton of th total potntal ng So fnd th mnmm of th potntal v and th cospondng v fncton hs mnmaton poblm s solvd b sng th Rt mthod whn th nnown v fncton s loong as th followng fom: n v a 58 wh th shap fncton whch satsfs th nmatcal bonda condtons th dsplacmnts at th sppots a and th angla dsplacmnts at th stan a ' Wth ths sbsttton th potntal v bcam a mltvaabl fncton fo a a a n hs fncton has a mnmm whn: 59 a Snc th Rt-mthod s an appomaton pocd th solton accac dpnds on how man mmbs of th shap fnctons o smpl tas nogh a sngl tag so th abov qaton dpnds on th a onl nvaat Mat fomlaton and solton of th qaton sstm lads to th bas qaton of th fnt lmnt mthod: K 5 5 Solvng th poblm sng fnt lmnt mthod h poblm shown n g 5 s a two-dmnsonal od stct h stct s ovloadd b two concntatd foc ls n plan wth th cantlv bam Compsson and István Mohaos ÓE wwwtanonvtah

78 78 nt Elmnt Mthod bndng gnatd n th bam of th stct and onl bndng stss gnatd n th bam so ths poblm can not b solvd b sng RUSS lmnts psntd n th chapt g 5 A two-dmnsonal od stct Both bam of th stct a 6 bo scton h popts of coss-sctons a stl standads: A = 869 cm I = 8 cm h two focs a ach Dng th solton w s bnt bam lmnts accodng to El- Bnoll s bam tho W hav sn that th fnt lmnt solton mans th solton of an qatons sstm: K 5 st w hav to dvlop th lmnt stffnss mat thn assmbl th stffnss mat of total stct 5 h lmnt stffnss mat h pvos qatons sstm wttn to sngl lmnt: v v 66 M M 5 wwwtanonvtah István Mohaos ÓE

79 5 wo-dmnsonal bnt bas vaaton poblm 79 h phscal ntptaton of colmns of stffnss mat s focs and momnt ncssa to ns th on nt dsplacmnt and th bonda condtons Usng ths w can asl podc th stffnss mat n cas of sng bam lmnts t on mmb of th vcto on nt and all oth s o In ths cas th lmnt of stffnss mat blongs to = and accodng to th gnal pocd: M 66 M 5 Solton of th qaton sstm: = 5 = M = = = 5 M = 6 h phscal contnt of ths cas s llstatd n g 5 g 5 h phscal ntptaton of th fst colmn of th stffnss mat Basd on th fg th ndvdal bam stssd b p compsson so sng th Hoo's law: l E E 55 A h val of dl s on nt so that aft aangmnt: István Mohaos ÓE wwwtanonvtah

80 8 nt Elmnt Mthod AE 56 o satsf th bonda condtons stll ncssa that: 57 : 58 h oth mmbs of th fst colmn of th stffnss mat a o W ma act smlal wth th scond colmn of th stffnss mat In ths cas th qaton sstm: M 66 M 59 Solton of th qaton sstm: = 5 = M = = = 5 M = 6 h phscal contnt of ths cas s llstatd n fg 5 hs stat s podcd spposton of cantlv bams In th fst cas s g 5 b th nd of th bam s loadd b concntatd foc and n th oth cas s g 5 c loadd b concntatd bndng momnt g 5 h phscal ntptaton of th scond colmn of th stffnss mat wwwtanonvtah István Mohaos ÓE

81 5 wo-dmnsonal bnt bas vaaton poblm 8 hs cass a wll nown n th stngth of matals so w can wt t sng qatons whch fom solton of th dffntal qaton of th lastc cv: M v 5 IE IE and th angla dsplacmnts: M 5 IE IE h solton of th mltvaabl qaton sstm: IE 5 M 6IE 5 thmo nsng th qlbm condtons: 55 5 and momnts to th nd pont: 6IE IE 6IE M M M M M K 6 56 h fst and foth mmbs n th scond colmn of th stffnss mat a o Elmnts n th thd colmn of th stffnss mat s dtmnd smlal so that th n v vcto s on nt and all oth mmb M M h solton of th qaton sstm s: István Mohaos ÓE wwwtanonvtah

82 8 nt Elmnt Mthod = 58 = M = = = 5 M = 6 h phscal contnt of ths cas s llstatd n fg 55 g 55 h phscal ntptaton of th thd colmn of th stffnss mat h dsplacmnts psntd n fg 55 s podcd spposton of two dsplacmnts n ths cas too so: M v 59 IE IE and th angla dsplacmnts: M 5 IE IE h solton of th mltvaabl qaton sstm s: 6IE 5 M IE 5 thmo nsng th qlbm condtons: 5 5 and momnts to th nd pont: IE 6IE IE M M M M M K 6 5 wwwtanonvtah István Mohaos ÓE

83 5 wo-dmnsonal bnt bas vaaton poblm 8 István Mohaos ÓE wwwtanonvtah h fst and foth mmbs n th thd colmn of th stffnss mat s o h mmbs n th -6 th colmn of stffnss mat a dfnd smlal Evntall th nt lmnt stffnss mat: E I E 6I E I E 6I E 6I E I E 6I E I A E A E E I E 6I E I E 6I E 6I E I E 6I E I A E A E 55 It shold b notd that n gnall th lmnt stffnss mat a gnatd b: V dv B C B K 56 qaton whn C s th mat of matal popts and B s th mat of dfomatonstan hs solton fond n thd chapt h abov psntd solton wold b dffclt n cas of sng mo compl lmnts It s onl fo ndstandng of th concpt of stffnss mat h stffnss popts of th lmnt w dtmnd onl n th lmnt local coodnat sstm In th global coodnat sstm ths stffnss vals chang dpndng on th poston of lmnts Elmnts popts n th global coodnat sstm a podcd sng th tansfomaton mat whch was psntd n chapt 5 qaton Howv n ths cas th tansfomaton mat s of od 66 accodng to dg of fdom of bam lmnt cos sn sn cos cos sn sn cos 57 h mmbs of tansfomaton mat can b asl calclatd b nown lmnt nodal coodnats:

84 8 nt Elmnt Mthod cos 58 sn 59 5 So th stffnss mat of st lmnt n global coodnat sstm: 5 h nt stct stffnss mat h s of th stffnss mat of nt stct s qal to th nmb of dgs of fdom of th whol stct So now th stffnss mat of nt sstm s of od 99 bcas th sstm composd of two lmnts wth nods ach wth dgs of fdom In th whol stffnss mat th lmnta stffnss of th common nods a addd togth so: K wwwtanonvtah István Mohaos ÓE

85 5 wo-dmnsonal bnt bas vaaton poblm 85 5 h complt qatons sstm and th solton v v M R R R 5 Dng th solton th dsplacmnt locatons at th sppots a sppd So w can dlt ows and colmns of th stffnss mat n ths placs In o cas ths s th fst th ows and colmns hs w gt th condnsd stffnss mat and th qaton sstm to solv: v v 5 Sbstttng th data solvng th qatons w obtan: 7 m v 79 m 76 ad U 55 7 m v 87 m 7 ad h acton focs can b calclatd b th nown slts om th qatons of nt sstm n ths cas ths a th fst th lns: R 5 6 v 56 R 5v 6 István Mohaos ÓE wwwtanonvtah

86 86 nt Elmnt Mthod MR 5v 6 8 m 5 Rmas h pogam sstms basd on fnt lmnt mthod can handl not onl El-Bnoll's bams In sch a cas th sha facto of th scton mst b dtmnd It shold b notd ths sha facto can onl b labl sd n th cas of lna statc analss wwwtanonvtah István Mohaos ÓE

87 6 AAYSIS O WO-DIMESIOA BE BARS USIG I- IE EEME MEHOD BASED PROGRAM SYSEM 6 Plana bam stcts As dscssd n chapt two-dmnsonal tsss a onl a pat of th ba stcts whch w hav to anal In most cass th bndng gnatd n bams can not b nglctd hs staton ass whn th bas of th bam stct a loadd not onl b aal focs bt vn b bndng momnts Sch cass sall a: A smpl and mlt-sppotd bams cantlv bams Cvd bas If th ta wght of th bam stct can not b nglctd A two-dmnsonal fam stct tc hs chapt dals wth ths bam stcts h chapts 7-8 dal wth thdmnsonal bnt and twstd bams and th chapts 9- dal wth bclng of th compssd bas Oth than as dscbd n chapt th a som mo qstons to b answd: h magntd and dcton of th focs and momnts gnatd n sppots Magntd and dcton of th aal and sha focs bndng and toq momnts n ach ba h and τ stsss whch chaactd of th plana-stssd stat Dsplacmnts of ach pont of th stct and dfomaton of ach bam hs stcts ma b tstng fo th stablt of th stct and dnamc bhavo th ctcal focs of compssd bas and natal fqncs W dal wth ths poblms n th chapts 5- h pvos chapt has mntond th tnall and ntnall dtmnaton and ndtmnaton stcts W wll s that t s lvant n ths cas too 6 h sd fnt lmnts n modlng h chapt clafd that pogam sstm basd on th fnt lmnt mthod s two tps of lmnt fo modlng bam stcts h RUSS lmnt fo modlng stct loadd aal focs onl and th BEAM lmnt fo modlng loadd aal and sha focs bndng and toq momnts In both cass th fnt lmnts a plana so that s chaactd b a sngl staght ln h popts of th RUSS lmnts hav alad dscbd n th pvosl chapts 6 Popts of th BEAM lmnt h BEAM lmnts can b dvdd nto two gops h BEAMD lmnts fo modls chaactd b plana-stssd stat sch as gnall th plana stcts wth smmtcal coss-scton bas loadd th plan of th stct onl h BEAMD lmnts a sd fo th-dmnsonal modlng hs s sall th th-dmnsonal constctons o István Mohaos ÓE wwwtanonvtah

88 88 nt Elmnt Mthod two-dmnsonal constctons loadd ppndcla to own plan o two-dmnsonal constcton consstng of asmmtcal coss-scton bas h chapts 7-8 wll dal wth BEAMD lmnts h BEAMD lmnts a two-nod naal lmnts hav th dg of fdom n both nods two dsplacmnt and a otatonal dgs of fdom h local coodnat sstm of th lmnt s shown n fg 6 h coodnat sstm -as pontng fom th fst to th scond nod th -as paalll to th global coodnat sstm XY plan and ppndcla to -as th as s ppndcla to and as and cat a ght-handd Catsan coodnat sstm g 6 h BEAMDD lmnt h lna statc analss a qd th al constants of BEAMD lmnts In ths cas t mans coss-sctonal aa th momnt of nta dpth of th scton and sha facto h val of sha facto dpnds on th shap of th scton W wll also nd th matal popts of th ba In ths cas th lastc modls and th Poson s ato dtmnaton s sffcnt bcas th a plana-stssd stat n all ponts of th BEAMD lmnts If th ta wght of th stct mst b consdd as a load th matal dnst dtmnaton s ndd h BEAMD lmnts a stabl bclng and thmodnamc analss hs qs addtonal al constants and matal popts 6 h sha dfomaton h sha dfomaton s sall nglctd It s possbl smpl to ta ths nto accont sng fnt lmnt modl fo mo accat slts h sha dfomaton s ddcd fom wo of ntnal focs h wo of sha focs of th two-dmnsonal bam: wwwtanonvtah István Mohaos ÓE

89 6 Analss of two-dmnsonal bnt bas 89 V S GI b Wnt ddd Sot of th qaton th sha shap facto s: A S fs da I A b h wo of th sha focs n constant coss-scton bam: W nt f s l V d GA hs s th shap facto and ths nvs sng n th fnt lmnt solton as sha facto h sha facto vals of som oftn sd coss scton shown n fg 6 h coss scton f s h Sha facto Rctangl 6/5= 5/6=8 Ccl /9= 9/=9 hn walld pp 5 g 6 Sha facto of sctons In th tchncal pactc w oftn s scton wh th tnsond chods and th shad wb a spaabl s g 6 In ths cas th appomat val of sha facto: István Mohaos ÓE wwwtanonvtah

90 9 nt Elmnt Mthod h coss scton f s h Sha facto A/A Wb A Wb /A 6 h std solton g 6 h smplfd dfnton of sha facto h solton of th fnt lmnt stds w follow th followng pocd: Analss of th poblm Caton of th gomt modl Dfn th popts of fnt lmnts lmnt tps al constants matal popts Dfn th bonda condtons and loads Rn th analss Evalaton of th slts h opn fam s shown n fg 6 loadd on mad pont h foc ls n plan of th stct h bas a cold bndd bo sctons W hav to dtmn th acton focs th stss gnatd n th bas th dflctons and th bndng- toq momnts and sha foc dagams g 6 h coss scton h fnt lmnt pogams sall contan blt-n D gomtc modlng gaphcs pand postpocsso hs w can ppa th gomtc modl n ts s g 65 wwwtanonvtah István Mohaos ÓE

91 6 Analss of two-dmnsonal bnt bas 9 g 65 Gomtc modl n th fnt lmnt pogam hs blt-n gomtc modls do not alwas off o th convnnc of modn CAD sstms Oftn w hav to anal stng modls In ths cas th data chang pocd can b convnnt and ffcnt wth oth CAD sstms b an avalabl standad fl fomat sch as SA IGS DX tc s fg 66 István Mohaos ÓE wwwtanonvtah

92 9 nt Elmnt Mthod g 66 Impot gomtc modl fom anoth gomtc modl Do not fogt n ths cas th gomtc modl onl hlps to cat a fnt lmnt msh It dos not compl wth th ls of tchncal dawng and has no lvanc to al shap of th stct It s t n ths cs bcas th mm bo sctons appas onl lns s g 67 hs w hav to tansfom smplf and tnd th tchncal docmntaton bfo fnt lmnt analss hs s shown n g 67 whch shows th mpotd gomtc modl wwwtanonvtah István Mohaos ÓE

93 6 Analss of two-dmnsonal bnt bas 9 g 67 h mpotd gomtc modl It s also shown that th lmnts l n th XY plan h nt stp s to dtmn lmnt gop s g 68 István Mohaos ÓE wwwtanonvtah

94 9 nt Elmnt Mthod g 68 Dtmnaton of lmnt gop W hav clafd that w s lna bhavo BAEMD lmnts s g 69 g 69 Slct th BEAMD lmnts and dtmnaton of ths popts t tas s to dtmn th al constants of lmnts s g 6 wwwtanonvtah István Mohaos ÓE

95 6 Analss of two-dmnsonal bnt bas 95 g 6 Ral constants dfnton As pvosl dscbd w hav to dfn al constants of BEAMD lmnts th cosssctonal aa of th bas th ntal momnt I dp of th scton and th sha facto s g 6 Mang s s th slctd nt sstm what s n ths cas th SI sstm g 6 Ral constants dfnton ds to b pland n th foth and ffth al constants End-las cod h ndlas cod fo ach and s spcfd b a s dgt nmb wth combnatons of and h s dgt cod cosponds n od to th s dgs of fdom at ach nd of th bam lmnts o ampl nd las cod fo a BEAMD lmnt psnt a condton n whch th momnt abot as s o and focs n - and dcton a to b calclatd h dg of fdom fs to th lmnt local coodnat sstm s g 6 h svnth and ghth al constants s onl n thmal analss so w do not dal wth thm now Stll th dfnton of matal popts s g 6 g 6 Dfnton of matal popts It s sffcnt to spcf th val of th modls of lastct and Posson s coffcnt fo th bam lmnts as shown n fg 6 and fg 6 István Mohaos ÓE wwwtanonvtah

96 96 nt Elmnt Mthod g 6 Dfnton of th lastc modls g 6 Dfnton of th Poson s coffcnt If ncssa w can dfn mo matal popts Aft dfnng popts of th fnt lmnt msh ma follow th fnt lmnt msh gnaton h EM pogams off sval mthods fo ths now w slct th atomatc msh s fg 65 h s of th lmnts s dtmnd b qd pcson of th slts th avalabl capablts of th compt and th avalabl tm ow w choos m avag lmnt s g 65 Atomatc msh gnaton h fnt lmnt msh and th nmbd nods shown n th g 66 wwwtanonvtah István Mohaos ÓE

97 6 Analss of two-dmnsonal bnt bas 97 g 66 h fnt lmnt msh It vsbl n th fg that catd an ndpndnt nod at ach th ndpont of th bam lmnts Bcas th fnt lmnt msh catd ach gomt objct spaatl t s ncssa to mg th nods n ach nd of th bas s g 67 g 67 Mg of th nd of bas In th nt stp th bonda condtons shold b gvn In ths cas ths a two s dsplacmnts on th sppots W f two dgs of fdom of th stct n and dctons at th both sppot s fg 68 g 68 Dsplacmnt constants István Mohaos ÓE wwwtanonvtah

98 98 nt Elmnt Mthod nall t shold b gvn th loads th 5 concntatd foc s g 69 h dcton of focs mst b gvn n th global coodnat sstm so th downwad focs a ngatv sgn g 69 Dfnng th load h compltd fnt lmnt modl s psntd n g 6 g 6 h compltd fnt lmnt modl ollows th nnng lna statc analss s g 6 wwwtanonvtah István Mohaos ÓE

99 6 Analss of two-dmnsonal bnt bas 99 g 6 Rn lna statc analss Aft th sccssfl solvng follows th dspla and valaton of slts h dsplang stsss gnatd n bas s g 6 can b don n sval was h stsss a ntptd on th lmnt and n th lmnt coodnat sstm l cas of th RUSS lmnts g 6 Dspla stsss h slts a shown n g 6 h dfomaton s not al of cos th pogam gnats a spcfc scal facto so that data can b valatd otc that th bas a bnt d to bndng momnts g 6 Stsss on dfomd shap István Mohaos ÓE wwwtanonvtah

100 nt Elmnt Mthod It s possbl to dspla stss componnts s g 6 h ngatv sgn of th stss ndcat compssv stss g 6 Dspla stss componnts W amn th dflctons dcton dsplacmnts n nt stp s g 65 g 65 Dspla th dflcton h g 66 shows th slts h ngatv sgns ndcat downwad dsplacmnts wwwtanonvtah István Mohaos ÓE

101 6 Analss of two-dmnsonal bnt bas g 66 h dflctons W can dspla th momnt and sha foc dagams n bam lmnts s g 67 g 67 Dspla th bndng momnt dagam h bndng momnt dagam shown n fg 68 h s not nmcal val n dagam vn so sfl bcas t hlps to dtmn th mnmal stssd locatons István Mohaos ÓE wwwtanonvtah

102 nt Elmnt Mthod g 68 Bndng momnt dagam It s possbl to dspla th acton focs and momnts gnatd n sppots s g 69 g 69 Dspla th actons focs It s possbl to lst th foc and momnts componnts gnatd n lmnts s g 6 wwwtanonvtah István Mohaos ÓE

103 6 Analss of two-dmnsonal bnt bas g 6 st th foc and momnts componnts h lstng of th nodal focs and momnts a shown n th fg 6 g 6 h nodal focs and momnts h lstng of th stss componnt shows fg 6 István Mohaos ÓE wwwtanonvtah

104 nt Elmnt Mthod g 6 h stss componnt lst h nmcal slts tabls can b appa ncomplt som componnt s As pland b th BEAMD lmnts h sha focs ppndcla to plan of stct bndng momnts n ths plan and toq dos not st n ths cas 6 Rmas Dng th soltons w do not dal wth bclng of th compssd bas If ths s a al poblm on shold b to vf wth solton a fnt lmnt poblm o wth an analtc mthod Dng th soltons th ta wght was nglctd Both poblms a pland n lat chapts thmo th stctal jont was not tstd h oth spcald aas of stctal dsgn dal wth ths poblms wwwtanonvtah István Mohaos ÓE

105 7 APPICAIO HE PRICIPE O HE MIIMUM POE- IA EERGY I IED O HREE-DIMESIOA BE BAR EEMES RIZ MEHOD AD IIE EEME MEHOD 7 h-dmnsonal bnt bas vaatonal poblm h chapt 5 dals wth th analss of two-dmnsonal bnt bas ods of ths lmnts hav th dgs of fdom two dsplacmnts and a otaton In ths scton w anal th-dmnsonal bam lmnts tnson of th pvos chapts h poston of th lmnt n th local lmnt coodnat sstm and th sd notatons a shown n g 7 g 7 Elmnt poston n th local lmnt coodnat sstm h fg also shows that dgs of fdom of nods a tndd wth dsplacmnt n th - plan otaton n th - plan and otaton aond -as toson In addton as pvosl dscbd btwn th angl dsplacmnt twst of th bam and th toq s a lna conncton so th wo of th toq: W M t 7 Mt 7 I G ths: p Mt W 7 G I p In th pvos chapt w saw that th basc fomla of th fnt lmnt mthod s th followng lna qaton: István Mohaos ÓE wwwtanonvtah

106 6 nt Elmnt Mthod wwwtanonvtah István Mohaos ÓE K 7 whch n ths cas fo th two nods dg of fdom lmnts: X Y Z S X Y Z S M M M M M M w v w v 75 W can dscb th dfomaton of th lmnt b ntpolaton polnomals l that sn n chapt hs wa dsplacmnt of a pont: h ntpolaton fnctons satsf th bonda condtons and dffntabl Accodng to th El-Bnoll's bam tho w appomat th dsplacmnt n dcton and otaton aond as wth lna ntpolaton fnctons:

107 7 Applcaton th pncpl of th mnmm potntal ng 7 István Mohaos ÓE wwwtanonvtah h bndng of th lmnt s appomatd wth cbc fnctons: 7 l hs fnctons can b obtand analtc solvng th dffntal qaton of th lastc cv of th bnt bam h total potntal ng th dffnc btwn stan ng and th wo of tnal focs s mnmal n th qlbm poston of th gd bod th fst vaaton s o U h applcaton of ths thom s basd on th amnaton of th stan ng changs so th stan ng blongs to ach load cass hav to b pscbd h aal dsplacmnts blong to longatons: ' 77 hs th potntal ng fo constant coss-sctonal ba: d EA d EA U ' 78 h j-th mmb of lmnt stffnss mat n cas = and =7: d EA d EA j j j ' ' ' 79 Dfomaton n cas toson aond -as:

108 8 nt Elmnt Mthod wwwtanonvtah István Mohaos ÓE ' 7 hs th potntal ng fo constant coss-sctonal ba: p p d GI d GI U ' 7 h j th mmb of lmnt stffnss mat n cas = and =: d GI d GI j p p j j ' ' ' 7 h potntal ng of bnt bam s th fncton of th otaton th sha dfomaton s nglctd accodng to El-Bnoll s tho In cas bndng n th plan: " 7 hs th potntal ng fo constant coss-sctonal ba: d EI d EI U " 7 h j th mmb of lmnt stffnss mat n cas = =6 =8 and =: d EI d EI j j j " " " 75 Eas to s that n cas bndng n th plan I mst b sd nstad of I hs th j th mmb of lmnt stffnss mat n cas = =5 =9 and =: d EI d EI j j j " " " 76 hs th lmnt stffnss mat:

109 7 Applcaton th pncpl of th mnmm potntal ng 9 István Mohaos ÓE wwwtanonvtah EI EI EI EI EI EI EI EI GI GI EI EI EI EI EI EI EI EI AE AE EI EI EI EI EI EI EI EI GI GI EI EI EI EI EI EI EI EI AE AE K p p p p h lmnt stffnss mat n global coodnat sstm can b podcd sng tansfomaton mat as wll as dscbd n chapt 5 In ths cas th tansfomaton mat s of od h notatons a shown n fg 7: c s s c s c s s s c c c c s s c s c s s s c c c c s s c s c s s s c c c c s s c s c s s s c c c wh: -c cos -s sn h lmnt stffnss mat n global coodnat sstm: K K 77

110 nt Elmnt Mthod g 7 h lmnt poston n global coodnat sstm 7 Solvng th poblm sng fnt lmnt mthod An otdoo nfomaton boad s placd on a hold s g 7 h boad wght s 5 g A foc actng on th boad ppndcla to ts plan g wnd pss =5 = g 7 h hold W plac th fnt lmnt modl n - plan s fg 7 so w hav to dvlop th tansfomd stffnss mat of th bam onl wwwtanonvtah István Mohaos ÓE

111 7 Applcaton th pncpl of th mnmm potntal ng = 6 =5 g 7 h placd hold n th global coodnat sstm h lmnt stffnss mat b th followng notaton: th stffnss mat of ths lmnt n global coodnat sstm: István Mohaos ÓE wwwtanonvtah

112 nt Elmnt Mthod Combn th stffnss mat of th two lmnts so that n th common nods th stffnss dos add p hs th sstm dscbng qatons: R R R MR MR MR v w v 8 w 9-78 Dng th solton th dsplacmnt locatons at th sppots a sp So w can dlt ows and colmns of th stffnss mat n ths placs In o cas ths s th fst s ows and colmns hs w gt th condnsd stffnss mat and th qaton sstm to solv Sbstttng th data and solvng th qatons sstm obtand th dsplacmnts: wwwtanonvtah István Mohaos ÓE

113 7 Applcaton th pncpl of th mnmm potntal ng U v w m v w h acton focs can b calclatd b th nown slts om th qatons of nt sstm whch a n ths cas th fst s lns: R R v R w M R m 7 M R w m M R v -5 m Rmas W dd not dal wth not ccl o ng coss sctons h popts of ths coss-sctons can b dtmnd onl b appomaton mthods th Bdt s fomla fo thn closd scton Wb s fomla fo thn-wall opn coss-scton Also dd not dal wth asmmtc sctons cold bndd o olld U sctons h sha cnt and cnt of gavt of thos sctons dos not concd so th bndng combns wth toson sall h sha dfomaton was nglctd bcas w sd El-Bnoll bam tho István Mohaos ÓE wwwtanonvtah

114 8 AAYSIS O HREE-DIMESIOA BE BARS USIG I- IE EEME MEHOD BASED PROGRAM SYSEM 8 h-dmnsonal bam stcts In cas of two-dmnsonal bnt ba stcts dscssd n chapt 6 th dflctons ma b gnatd n plan of stct In ngnng pactc sng th-dmnsonal modls a qd man of th cass Sch cass sall a: wo dmnsonal constcton wth asmmtcal coss scton bams wo dmnsonal constcton wth loads ppndcla to plan of stct h gnal th-dmnsonal bam stcts hs chapt dals wth ths stcts h chapt 9- dals wth bclng of th compssd bas Bcas th bclng of th compsson chods and th sha bclng of th wb shts q dffnt calclatons so w do not dal wth ths Qstons to b answd h magntd and dcton of th acton focs and momnts gnatd n sppots Magntd and dcton of th aal and sha focs bndng and toq momnts n ach ba h and τ stsss whch chaactd of th stssd stat Dsplacmnts of ach pont of th stct and dfomaton of ach bam hs stcts ma b tstng fo th stablt of th stct and dnamc bhavo th ctcal focs of compssd bas and natal fqncs W dal wth ths poblms lat h pvos chapt has mntond th tnall and ntnall dtmnaton and ndtmnaton stcts W wll s that t s lvant n ths cas too 8 h sd fnt lmnts n modlng h chapt clafd that pogam sstm basd on th fnt lmnt mthod s two tps of lmnt fo modlng bam stcts h RUSS lmnt fo modlng stct loadd aal focs onl and BEAM lmnt fo modlng loadd aal and sha focs bndng and toq momnts Both RUSS and BEAM lmnts can b two- o th-dmnsonal In all cass th fnt lmnts a chaactd b a sngl staght ln h popts of th RUSS and BEAMD lmnts alad dscbd n th pvosl chapts 8 h popts of th BEAMD lmnts h popts of th BEAMD lmnts hav alad wttn n chapt 6 h BEAMD chaactd b th dmnsonal stssd stat and t s gnal thdmnsonal ba stcts o dsplacd ppndcla to th own plan nd th loads h BEAMD lmnts a two o th-nod naal lmnt hav s dgs of fdom th tanslatons and th otatons p ach nd nod h thd nod ponts towads wwwtanonvtah István Mohaos ÓE

115 8 Analss of th-dmnsonal bnt bas 5 -as n th lmnt local coodnat sstm It o an ontaton angl as al constant s qd onl fo dtmn th lmnt ontaton h lmnt coodnat sstm shown n fg 8 h coodnat sstm -as pontng fom th fst to th scond nod th -as ppndcla to as and cntal pncpal as of coss scton as ppndcla to - plan and cat a ght-handd Catsan coodnat sstm g 8 BEAMDD lmnt local coodnat sstm h lna statc analss qs som al constant mang as shown g 8: h coss-sctonal aa Momnt of nta abot th lmnt Y as Momnt of nta abot th lmnt Z as Dpth of th bam Wdth of th bam Rlatonshp btwn th nds of th connctd lmnts nd las cod two sts of data osonal constant J s also 8 Sha facto n th lmnt as s also 8 Sha facto n th lmnt as s also 8 Ontaton angl of th coss scton onl f th ontaton dos not dfn b th thd nod Constant fo mamm sha stss calclaton s also 8 dstanc of th scton cntod latv to th nodal pont n ach nod of th bam total s data dstanc of th sha cnt latv to th scton cntod at ach nod of th bam total fo data dstanc of th pont wh stsss a to b calclatd at ach nod of th bam total fo data Cntodal podct f nta of th lmnt coss scton István Mohaos ÓE wwwtanonvtah

116 6 nt Elmnt Mthod Usall th can b dfnd tapd bam popts and mo al constant fo thmal analss also hs popts a not dalt n ths chapt In sall w can spcf oftn sd coss-sctons n ngnng pactc sch as ctangla a sqa hol ccl ng I sctons b gomtcal dmnsons In ths cas th oth sctonal popts wll b calclatd b th pogam g 8 BEAMD lmnts popts W also nd th matal popts of th lmnts In ths cas s sffcnt to spcf th val of th modls of lastct Posson s coffcnt and dnst of th bam lmnts If ncssa w can dfn mo matal popts fo th bclng o hat tansf analss h ntptaton of th bndng momnts and sha focs shown n g 8 Ms Ms Vs Vs Vt Vt Mt Mt g 8 ocs and momnts n BEAMD lmnts wwwtanonvtah István Mohaos ÓE

117 8 Analss of th-dmnsonal bnt bas 7 8 h spcal popts of BEAMD lmnts h sha dfomaton s sall nglctd h chapt 6 has shown that how ths can b tan nto accont Also n ths chapt w hav popts of sval common sd sctons W hav dalt wth th smplfd dfnton of th sha facto s g 8 hs concpt wll also sd n ths chapt h coss scton f s h Sha facto A/A Wb A Wb /A g 8 h smplfd dfnton of Sha facto h calclatons wll b ndd to dtmn a shap facto C to fo calclat th mamm stss τ coms fom toson In cas of ccla and thn-walld ng scton th mamm stss τ gnatd on pmt of th ccl so: ma I P In cas of non-ccla coss scton th mamm stss τ dpnds on th scton shap In sch cass w can s onl appomat pocds sch as Constantn Wb appomat mthod: ma I W C to K W wh: I W : Wb's cntodal podct of nta K W : Wb's pola scton modls C to : th shap facto Ccla coss scton of cos I W =I P K W =K P and C to = An opn coss-scton s fg 85 wh h>> v w can appl th splttng and so: I W v h K W I v W ma Cto v ma István Mohaos ÓE wwwtanonvtah

118 8 nt Elmnt Mthod h h v v v h g 85 Splttng of opn coss-sctons h η s a facto to cocton o of splttng g 86 h η facto of som coss-scton wwwtanonvtah István Mohaos ÓE

119 8 Analss of th-dmnsonal bnt bas 9 8 h std solton h std s a fam s g 87 assmbld b U standad stl h actal lv load s 5 dstbtd focs h hoontal load s th 6% of th lv load what s gnatd fom movmnt on lv load h sppots a fom ach nd to mm Hav to dtmn th acton focs stsss gnatd n bams th dflctons and bndng momnt dagams 5 h followd pocd: g 87 h tstd fam Std analss Cat a gomt modl Dfn th popts of fnt lmnts lmnt tp al constant matal popts Dfn bonda condtons and loads Rn th analss Evalaton of th slts István Mohaos ÓE wwwtanonvtah

120 nt Elmnt Mthod h coss-sctonal popts of th sd olld bas U and gomtc dmnsons shown n g 88 hs tchncal data a avalabl n standads and dsgn ads tabls g 88 h sd U scton W nd som data what a not ncldd n tabls st w hav to dfn th sha facto n lmnt and as W s th smplfd calclaton shown n g 8 Usng th Zavs s thom th sha stss n a pont of th coss scton s: S si ' wh: :sha foc S : statcal momnt of an aa otsd a pont abot scton as I : Momnt of nta abot th lmnt s: wdth of th scton at th pont wwwtanonvtah István Mohaos ÓE

121 8 Analss of th-dmnsonal bnt bas Of cos th stss dstbton dpnds on th latv poston of th scton and sha foc so that t shold b dtmnd abot th lmnt and as spaatl h sd U coss scton popts nown and shown n th g 89 g 89 Sha stss dstbton n th coss scton h posd aa of th sha can b dtmnd b gaphc dtng so th sha shap facto can b calclatd: S S f f A A nít A A nít In addton w nd th coss-sctonal modls of toson whch can b dtmnd b Wb's mthod s g 85: I W v h cm o th dtmnaton th lagst sha stss τ casd b toson: C to v ma 9 cm Aft dtmnng th ncssa data w can bgn th compt-add analss h gomtcal modl s v smpl so w can cat t n th own gaphcs dto of fnt lmnt pogam h stctal modl s catd n th XY plan bt th loads and th dfomatons wll b th-dmnsonal In th fg 8 th dw lns psnt th ntal as of bams István Mohaos ÓE wwwtanonvtah

122 nt Elmnt Mthod g 8 Daw ln n th fnt lmnt pogam h ontaton of BEAMD lmnt can b dfnd b th thd nod W also nd a gomtc pont pont Snc th lns psnt th ntal as of bams so th thd nod mst l n XY plan too It s sffcnt to ta onl on pont bcas th ntal as of all bas ls n a common plan h dfnton a gomtc pont s shown n th g 8 g 8 Plac a gomtc pont pont h compltd gomtc modl shown n g 8 wwwtanonvtah István Mohaos ÓE

123 8 Analss of th-dmnsonal bnt bas g 8 h gomtc modl In th nt stp w dtmn th lmnt gop W hav clafd that w a s lna bhavo BAEMD lmnts s g 8 g 8 Dtmnaton of lmnt gop Dng th dtmnaton th al constant s fg 8 w s th SI nt sstm th lna dmnson mst b dfnd n m and th wght mst b n g István Mohaos ÓE wwwtanonvtah

124 nt Elmnt Mthod g 8 h al constant dfnton It's also ncssa to spcf th matal popts s g 85 In ths cas t s sffcnt to nt th vals of th modls of lastct and Posson s coffcnt If ncssa w can dfn mo matal popts g th dnst to calclat ta wght of th stct g 85 Spcf th matal popts Dng th fnt lmnt msh gnaton sam s bt dffnt nmb of lmnts can b catd on ach ba s g 86 wwwtanonvtah István Mohaos ÓE

125 8 Analss of th-dmnsonal bnt bas 5 g 86 h fnt lmnt msh gnaton Bcas th fnt lmnt msh catd ach gomt objct spaatl th nds of th bams a not n conncton s g 87 g 87 h fnt lmnt msh o cat conncton btwn bas ncssa to mg th nods n ach nd of th bas s g 88 István Mohaos ÓE wwwtanonvtah

126 6 nt Elmnt Mthod 88 g Mg nods on th nd of th ba h compltd fnt lmnt msh shown n g 89 g 89 h fnal fnt lmnt msh h dsplacmnt constants a placd on th fnt lmnt msh h constants assmd gd bt th ba can dfomd fl btwn th two sppots An ampl of plac dsplacmnt constans shown n g 8 wwwtanonvtah István Mohaos ÓE

127 8 Analss of th-dmnsonal bnt bas 7 g 8 Spcf th dsplacmnt constants t stp dfn th dstbtd load on th hoontal ba whch shown n th g 87 In o cas th spcfd foc ffct n all nods of th lmnt h dfnton loads shown n g 8 g 8 Dfnton th dstbtd loads It s advsabl to chc what focs hav bn catd W can s lstng commands fo ths s g 8 István Mohaos ÓE wwwtanonvtah

128 8 nt Elmnt Mthod g 8 st focs h compltd fnt lmnt modl s psntd n fg 8 wwwtanonvtah István Mohaos ÓE

129 8 Analss of th-dmnsonal bnt bas 9 g 8 h comltd fnt lmnt modl h nnng lna statc analss follows s g 8 g 8 Rn lna statc analss h gnatd stss slts can b dsplad on dfomd shap s g 85 István Mohaos ÓE wwwtanonvtah

130 nt Elmnt Mthod h slts a shown n g 86 g 85 Dspla stss slts g 86 h qvalnt stsss W can s lstng commands to dspla nmcal slts s g 87 wwwtanonvtah István Mohaos ÓE

131 8 Analss of th-dmnsonal bnt bas g 87 Dspla stss componnts Eamn th bndng momnt gnatd n th stct s g 88 g 88 Dspla th momnt dagams fo bams Snc th bas a cvd n two dctons so w amn bndng momnts casd b vtcal and hoontal loads spaatl s g 8:9 István Mohaos ÓE wwwtanonvtah

132 nt Elmnt Mthod g 89 Ms and Mt bndng momnt dagams Eamn th dflctons th dsplacmnts n Y dcton s g 8 h slts shown n g 8 g 8 Dspla th dflctons wwwtanonvtah István Mohaos ÓE

133 8 Analss of th-dmnsonal bnt bas g 8 h dflctons Dspla th nmcal dsplacmnts slt also possbl s g 8: István Mohaos ÓE wwwtanonvtah

134 nt Elmnt Mthod g 8: st of dsplacmnts componnts 8 Rmas Dng th soltons a not dalt wth bclng of th compssd bas If ths s a al poblm w shold hav to vf wth solton a fnt lmnt poblm o wth an analtc calclaton Dng th soltons th ta wght was nglctd Both poblms a pland n lat chapts thmo th stctal jont was not tstd h oth spcald aas of stctal dsgn dal wth ths poblms wwwtanonvtah István Mohaos ÓE

135 9 DYAMICS O BEAM SRUCURES MASS MARIX AU- RA REQUECY AAYSIS 9 Etndng of th fnt lmnt mthod Sch as th hstocal sv also showd th fnt lmnt mthod "nvnton" dos not assocat to a dat It shold not spa abot nvnton ath to tal abot pogss o dvlopmnt hs dvlopmnt s bgn n 9-5s and stll contns toda Aft th fst sccssfl solton n fld of tho of lastct asd th possblt that oth phscal poblms can b solvd sng th fnt lmnt mthod hs toda w can gt fnt lmnt soltons n flds of hat tansf lctomagntc adaton fld flow fatg and oscllatng sstms analss h mathmatcal soltons a sd n ths aas slghtl dff fom thos dscbd at tho of lastct h dvlopmnt of th fnt lmnt solton of th stct-dnamcs analss bgan n th96s whn th lmnt mass mat has bn dtmnd 9 nt lmnt fomlaton of th lastc bods' natal oscllaton h pvos chapts dalt wth lastc bods n balanc W sd th fll potntal s pscbd as a fncton of th dsplacmnt : : dv qdv pda 9 V V A h potntal psnts th qalt of th lastc stan ng and th wo of tnal focs th statc qlbm D 'Alambt aangng th wton nd law and wot th followng fom: ma so th "ma" s no long momntm t s th foc of nta Accodng to th d'alambt s pncpl th tnal focs and th foc of nta act on th bod a balancd hs s calld th ntc qlbm ollowng th pncpl w complmnt th abov potntal wth th wo of foc of nta and so w gt th potntal whch dscbs th sstm of ntc qlbm stat: v : dv qdv pda üdv 9 V V A V wh: -ü-th tm of th scond dvatv of th dsplacmnt vcto acclaton - -Dnst of th matal István Mohaos ÓE wwwtanonvtah

136 6 nt Elmnt Mthod Dng th fnt lmnt solton w follow th mthod whch s psntd n a pvos chapt wth th addton that th th coodnats of th fncton dscbng th moton of th bod a spplmntd b th foth coodnat whch s th tm: t 9 W ntpolat ths fncton b th pvosl psntd shap fnctons: t t 9 hs th acclaton: ü t ü t 95 Wth ths spplmntng th potntal wo of th ntal focs on an lmnt: üdv üdv dvü M ü 96 V V V h M s th consstnt mass mat of th lmnt whch contans th ntal popts h fll potntal can b wttn n a mat fom: U KU U U MÜ 97 h qaton whch s satsfacto of th mn mm condton MÜ KU t 98 a lna dffntal qaton sstm h ght sd of th qaton contans constant and tm vaabl focs p-loadng and ctng focs h ngnng pactc th s v mch std whn tnal focs do not act hn of th most common ngnng pactc oscllaton poblm dtmnng of th ctcal angla vloct of otatng shafts h ctcal angla vloct appomatl qal to th smallst angla natal fqnc of th shat hs th qaton sstm of an ndampd vbaton sstm wthot tnal focs bcoms smpl: MÜ KU 99 h bod dos hamonc oscllaton so th solton of dffntalqatons: U Asn t 9 wh: wwwtanonvtah István Mohaos ÓE

137 9 Dnamcs of bam stcts mass mat natal fqnc analss 7 - A th ampltd vcto of th nods - th natal fqnc - phas angl sbsttt U and th tm of th scond dvatv of th U n th basc qaton: M KA 9 w obtan a homognos algbac qatons sstm Sach th gnvals of A and th assocatd natal fqnc h abov qatons hav solton dffnt th tval solton f th dtmnant of th coffcnt mat s o : dt M K 9 Snc th M and K matcs n th qaton sstm accodng to dgs of fdom of fnt lmnt modl th matcs a of od nn so th qatons hav n oots fo h dgs of fdom of fnt lmnt modls sd n pactc can b fw hndd to sval mllon It nd not dtmn so man gnval and natal fqnc th fst fw val hav lvanc n th pactc 9 atal fqnc calclaton of two-dmnsonal ba stcts sng fnt lmnt mthod S a shaft bangs at th two nds wth a fast pll at an ntmdat pont shown n g 9 Dtmnd th shaft ctcal angla spd g 9 h shaft h damt of th shaft s mm mad of sold stl h pll wghs g h shaft lngth mm = 5 mm and =5 mm István Mohaos ÓE wwwtanonvtah

138 8 nt Elmnt Mthod W hav to dtmn th M mass mat and th K stffnss mat fo solton th M KA qaton 9 Dtmnaton of th lmnt mass mat As w hav sn fo dtmnaton of th lmnt mass mat th ntpolaton fnctons a sd st plac th lmnt n an "s" coodnat sstm whch s ndpndnt of th lngth and concds wth th lmnt as h lmnt locaton n th global coodnat sstm s shown n fg 9 a and th lmnt locaton n th local "s" coodnat sstm shown n fg 9 b g 9 h lmnt local coodnat sstm h lmnt mass mat can b dtmnd basd on th followng: M Ad 9 h aal dsplacmnts a ntpolatd: / / 9 wth shap fnctons lnal th dsplacmnts ppndcla to bam a ntpolatd: wwwtanonvtah István Mohaos ÓE

139 9 Dnamcs of bam stcts mass mat natal fqnc analss / 8 / 8 95 wth shap fnctons cbcal o th lna mmbs: ln 96 If th coss-scton of th bam lmnt and dnst s constant thn th assocatd mass mat s: M A ln lnd A ln 6 97 o th cbcal mmbs: öb and th assocatd mass mat: 56 5 M A öböbd A 99 öb 5 56 hs mat s pandd wth th mat assocatd lna mmbs ths w gt th total lmnt mass mat: István Mohaos ÓE wwwtanonvtah

140 nt Elmnt Mthod M A Elmnt stffnss mat h stffnss mat of bam lmnt s also dvd fom th abov ntpolaton fnctons h aal latv longaton of th lmnt: d d d s 9 ds d ds h aal dsplacmnts of th lmnt ma b ds J th stan-dsplacmnt vcto: d B ' 5 J ln ' 5 9 and th lmnt stffnss mat s: K ln BEB Ads AE BEB AJd 9 Smlal th dsplacmnts and angla dsplacmnts ppndcla to lmnt as a appomatd cbc ntpolaton: ' 5 " 5 5 B öb 9 J 5' 5 6" 5 5 and th stffnss mat: K öb BI EB ds 6 6 I E 6 6 BI EB Jd wwwtanonvtah István Mohaos ÓE

141 9 Dnamcs of bam stcts mass mat natal fqnc analss István Mohaos ÓE wwwtanonvtah h lmnt stffnss mat can b obtand b th combnaton of th two stffnss mat: E I E 6I E I E 6I E 6I E I E 6I E I AE AE E I E 6I E I E 6I E 6I E I E 6I E I AE AE K 9 h sstm total mass and stffnss mat h mass pont shown n g 9 has not mass mat w ta nto accont th mass n th mass mat of th nt sstm as an nta on a nod h mass mat of th nt sstm shown n g 9: A M and stffnss mat of th nt sstm:

142 nt Elmnt Mthod wwwtanonvtah István Mohaos ÓE E I E 6I E I E 6I E 6I E I E 6I E I AE AE E I E 6I E I E I E 6I E 6I E I E 6I E 6I E I E 6I E 6I E I E I E 6I E I AE AE AE AE E I E 6I E I E 6I E 6I E I E 6I E I AE AE K If th as of th lmnt s not paalll to th global coodnat sstm X as thn th stffnss and mass matcs of th lmnt mst b tansfomd fst sng th tansfomaton mat dscbd n pvos chapts h qatons can now b smplfd so that th dsplacmnts and angla dsplacmnts locatons at th bangs a sppd So w can dlt ows and colmns of th qaton sstm n ths placs In o cas ths s th - and 7-9 ows and colmns hs th matcs n th sstm of qatons: A A A m A 56 m A M * * E I E I E 6I E 6I E 6I E 6I E I E I AE AE K h poblm to b solvd: dt * * K M

143 9 Dnamcs of bam stcts mass mat natal fqnc analss István Mohaos ÓE wwwtanonvtah qatons whch fom w gt th: solton Ral oots of : Rmas In pactc w can s th smplfd mass mat: M A whch psss that th mass of th lmnt a dvdd nto two qal pats and plac ths to th two nds of th lmnt hs cosponds to th analtcal calclaton whn w dc th mass of th ba to ts ndpont In th cas of fnt lmnt soltons sffcntl accat slts can b obtand sng ths pocd f th ba s dvdd sffcntl man fnt lmnts

144 DYAMIC AAYSIS O HREE-DIMESIOA BARS DE- ERMIAIO O AURA REQUECY USIG PROGRAM SYSEM BASED O IIE EEME MEHOD Intodcton h ngnng wos a all oscllatng sstms Bldngs stcts vhcls machn pats vbatons ca ot ach hs a nglctd sall d to th hgh fqnc and small ampltd hs do not dstb th fnctonalt of th machn Howv th a man cass n whch ths vbatons can not o shold not b gnod Evon nows dsast of th acoma Bdg h oscllaton of th bdg was focd b th wnd Bt n o dal lvs w can fnd ampls of th mpotanc of oscllatng sstms h whls of o cas a balancd h nbalancd whls cas ncomfotabl dvng and malfncton n bangs shafts and ts Bt th stat of shoc absobs a glal chcd not onl bcas of th convnnc bt also bcas t s latd to saft In fld of manfactng pocss th a sval ampls to th vbaton of machns and machn pats can not b gnod Bt th a som ngnng applcatons wh th vbatons shold not b dampd o avodd bt on th conta shold stngthn thm Consd fo ampl vbaton fds and scns machns a sd n th fld of matals handlng o vbaton compacton machns a sd n th fld of bldng ndst Popts of th sd fnt lmnts h popts of BEAMD lmnts sd fo th-dmnsonal modlng a dscbd n chapts 8 Howv w wll s a nw lmnt h fnt lmnt modlng pogams s a - dmnsonal MASS lmnt mass o nta hs lmnt has onl on nod n ths nod of th lmnt accmlats th total mass and momnt of nta h MASS lmnt has mass nta n X Y and Z dcton and th momnt of nta s ntptd aond th thas hs ntptaton allows m that n th cas of D poblms gno som ffcts h std dscpton In th mchancal ngnng pactc th most common tass a th amnaton th bndng and tosonal vbaton of th otatng shaft In ths std w anal a otatng shaft wth two flangs shown n g wwwtanonvtah István Mohaos ÓE

145 Dnamc analss of th-dmnsonal bas 5 g std shaft hs poblm has bn also dscssd n th sbjcts of mchancal ngnng stds n machn dsgn shafts and coplngs n mchancs dnamcs hs sbjcts showd that th bndng and tosonal vbatons a gnatd n shafts It also clafd that ths vbatons can b dangos f th otaton angla vloct of th shaft qal to th fst angla natal fqnc of th sstm h angla natal fqncs of bndng vbatons n th stct shown n g a calclatd b Dnl's smplfd fomla: m m wh: α - th angla fqnc η - dflcton of th shaft casd b a nt adal foc Accodng to th nown fomlas: α 8 a 8 a m m 8 IE 8 IE hs th angla natal fqnc of th shaft s: α=7676 /s whch s qal to n=686 pm h tosonal vbatons fom th chaactstc qatons of mlt-dg-of-fdom sstm a: István Mohaos ÓE wwwtanonvtah

146 6 nt Elmnt Mthod c wh: - Θ th momnt of nta of th dss aond Y as - c th tosonal spng constant: a c I G p wh: -I p pola momnt of nta -G - modls of gdt On ths bass th tosonal natal fqnc s: α =587 /s n=69 pm h fnt lmnt solton of th tas Stct shown n g s a v smpl gomtc modl shaft can b chaactd b a sngl ln W daw t as th spaatd ln to hlp gnatng of th fnt lmnt msh So w can plac th MASS lmnts on th nd of th sctons on gomtcal - ponts h dawn sctons shown n g g Catng a gomtc modl Aft th caton of th gomtc modl follows th dscbng th popts of fnt lmnt msh st w slct th ndd lmnt tp lmnt gop s g whch s n o cas th BEAMD lmnt wwwtanonvtah István Mohaos ÓE

147 Dnamc analss of th-dmnsonal bas 7 g Slct th lmnt tp In th nt stp w dfn th qd matal popts th lastc modls and th modls of gdt s g g Dfn th matal popts nall th al constants a dfnd h g 5 shows an ampl of smplfd pocds fo dfnton th al constant b gomtcal dmnsons h "" sgn ndcat that th coss-scton s ccla István Mohaos ÓE wwwtanonvtah

148 8 nt Elmnt Mthod g 5 Dfnton th al constant If w hav dfnd all popts of th fnt lmnts thn w can cat th fnt lmnt msh s g 6 W cat - lmnt n ach scton h scton of BEAMD lmnts s a ccl ths dfnton of th thd nod s not qd g 6 Cat th fnt lmnt msh W hav to dfn th popts of th two dss o ths nd w dfn a nw lmnt gop alad dscbd abov th MASS ntal lmnt s g 7 g 7 Dfn th MASS lmnt o ths lmnt tp dos not blong to an matal popt sch as sffcnt fo dfnton th al constant hs constant of th fst dsc shown n g8 h momnts of nta of th ds aond X and Z as can b gnod so th vals shall b wwwtanonvtah István Mohaos ÓE

149 Dnamc analss of th-dmnsonal bas 9 g 8 Ral constant of fst ds h MASS lmnt s placd on a sngl nod n th fnt lmnt msh h caton of th MASS lmnt shown n g 9 g 9 Cat a dsc as fnt lmnt W hav to dfn th al constant of th scond dsc s g g Ral constant of scond ds h caton of th scond dsc s smla to th pvos on jst on anoth pont of th gomtc modl h fnt lmnt msh has fv ndpndnt pats th th shaft scton and th two mass W hav to mg th common nods to jon ths ndpndnt pats s g István Mohaos ÓE wwwtanonvtah

150 5 nt Elmnt Mthod g Mg th common nods In th nt stp w dtmn th bonda condtons shown n g as bangs hs s smla to th pvos ampls t can b dfnd fng th th dsplacmnt dg of fdom at both nds of th shaft s g g Dfn dsplacmnt constans hs th catd fnt lmnt modl shown n g wwwtanonvtah István Mohaos ÓE

151 Dnamc analss of th-dmnsonal bas 5 g h complt fnt lmnt modl wth th nod nmbng Bfo th solvng t s possbl to st nmb of th calclatd natal fqnc s g It s appopat to st calclat mo hamonos bcas w pct two-wa bndng and tosonal vbatons In ths std w wll calclat th fst natal fqncs g h natal fqnc analss sttngs Aft th sttng follows th solton s g 5 István Mohaos ÓE wwwtanonvtah

152 5 nt Elmnt Mthod g 5 Rn fqnc analss Aft a sccssfl n th slts can b dsplad h calclatd fst ght natal angla fqncs a shown n g 6 g 6 h calclatd natal angla fqncs In th lst th fst natal angla fqnc s -5 /s whch s nglgbl small n th ngnng pactc hs s consstnt wth th land n mchancs h fst natal fqnc of th mlt-dg-of-fdom sstms s o W obsv that th - and -5 natal fqncs a th sam at w wll s that ths two oscllaton gnatd n X and Z dctons h 6 natal fqnc has not pa hs s th tosonal oscllaton of th shaft btwn th two dss h fnt lmnt pogams can dspla gaphcall th mod shaps as th dfomd shap of th shaft s g 7 wwwtanonvtah István Mohaos ÓE

153 Dnamc analss of th-dmnsonal bas 5 g 7 Dspla th mod shaps h fnt-lmnt pogams off a scal facto to dspla th dfomd shap W ovd ths scal facto and s 5 to do compaabl mod shaps s g 8 g 8 h and 6 mod shaps In th fg w obsv that onl on nod blongs to th fst mod shap Also obsvd that n cas 6 mod shap th s not vsbl dfomaton bcas th twstng aond th Y as s not vsbl n ths psntaton h dsplacmnts blong to and mod shaps a shown n g 9 István Mohaos ÓE wwwtanonvtah

154 5 nt Elmnt Mthod g 9 Vals of and mod shap h tabl contans v small magntd dsplacmnts hs a not al vals onl gnatd dng th solv as calclaton os h mod shap 6th s shown n g g h 6 mod shap h tabl contans onl otaton slts aond th Y as It s also shown that th tosonal oscllaton can onl b btwn th two dss 5 Rmas In ngnng pactc th tosonal vbaton analss sall a sd onl a long flbl shafts flbl coplngs h bndng oscllaton of otatng shaft wth ccl o pp coss scton ma also b amnd sng BEAMD lmnts wwwtanonvtah István Mohaos ÓE

155 IRODUCIO O PAE PROBEMS SUBJEC APPICA- IO O PAE SRESS PAE SRAI AD REVOUIO SYMMERIC AXISYMMERIC MODES Basc tps of plan poblms In th cas of plan poblms w hav two-dmnsonal o two-vaabl poblms; th basc qatons of lastct can b sgnfcantl smplfd compad to spatal poblms h a two majo catgos of plan poblms []: plan stss a thn stct wth constant thcnss nd n-plan loadng ga plan stan a long stct wth constant coss scton nd constant loads along th lngth gb W not that th gnald plan stss stat blongs also to th two-vaabl poblms f w lat th mchancal qantts to th avag vals g Dmonstaton of plan stss a and plan stan b stats o plan poblms th dsplacmnt vcto fld s th fncton of and onl: v Consqntl vn th stan and stss flds dpnd pon and : In th followngs w dvlop th latonshp among th fom mchancal qantts Eqlbm qaton dsplacmnt and dfomaton h qlbm qaton psnts th ntnal qlbm of a dffntal plan lmnt Basd on g t s possbl to pss th qlbm of th focs n dctons and as []: Andás Séns BME wwwtanonvtah

156 56 nt Elmnt Mthod d d d d d d d q dd d d d d d d d q dd wh s th nomal s th sha stss q and q a th componnts of dnst vcto of volm focs h smplfcaton of Eq lads to th followng qatons: q q g Eqlbm of a dffntal plan lmnt h qlbm qaton can b fomlatd also n vcto fom []: q 5 wh q = q s th dnst vcto of volm focs s th Hamltonan dffntal opato vcto opato n two dmnsons: j 6 In od to stablsh th latonshp btwn th stan and dsplacmnt flds w nvstgat th dsplacmnt and dfomaton of som ponts of th dffntal plan lmnt dpctd n g h nomal and sha stans n dcton of dstanc AB and n dcton of dstanc AD of th lmnt a: A' B' AB A' B' d A' D' AD A' D' d 7 AB d AD d wwwtanonvtah Andás Séns BME

157 Intodcton to plan poblms sbjct 57 B th hlp of th fg w can wt th followng: v A' B' d [ d ] d d 8 fom whch w obtan: v 9 h psson abov s applcabl to calclat th nomal stan n dcton n th cas of th so-calld lag dsplacmnt Aft all wthn th scop of lastct n most of th cass w obtan asonabl accat slts b th lnaaton of th psson abov h nomal stan n dcton s dvd smlal glctng th hgh od tms w obtan th lnad fomla: v Utlng g w calclat th angl dnotd b : v / d d / d g Dsplacmnt and dfomaton of a dffntal plan lmnt Andás Séns BME wwwtanonvtah

158 58 nt Elmnt Mthod Assmng that th a onl small angls w can wt: v Basd on Eq7 w obtan: v W obtan th so-calld stan-dsplacmnt qaton b smmang Eqs and n tnsoal fom h stan-dsplacmnt qaton s vald also fo spatal poblms []: wh th ccl mans dadc podct Constttv qatons h matal bhavo n oth wods th stss-stan latonshp of a homognos lna lastc sotopc bod s gvn b Hoo s law []: G I E G I E 5 wh s Posson s ato E s th modls of lastct G = E/+ s th sha modls E s th dntt tnso I and I a th fst scala nvaants spctvl Plan stss stat h stss componnts nd plan stss stat a: and 6 th nomal stss ppndcla to th - plan and th sha stsss actng on th plan wth otwad nomal n dcton a o h stss and stan tnsos hav th followng foms: / / 7 wwwtanonvtah Andás Séns BME

159 Intodcton to plan poblms sbjct 59 om th fst of Eq5 w obtan: E 8 E E E h nomal stan n dcton s: E E 9 E W not that althogh s not ncldd n th qatons t can alwas b calclatd b sng th stans n th oth two dctons Usng th fom qatons w can pss vn th stsss: E E E An altnatv fomlaton of th stss-stan latonshp s that w collct th componnts n vctos: As a slt th latonshp s stablshd thogh a mat: C wh C s th constttv mat On th bas of Eqs- nd plan stss stat mat C bcoms: E C st h nvs and th dtmnant of th mat s: Andás Séns BME wwwtanonvtah

160 6 nt Elmnt Mthod wwwtanonvtah Andás Séns BME E C st dt E C st h latt fom of th stss-stan latonshp s appld n fnt lmnt calclatons Plan stan stat Und plan stan stat th condton s: = th nomal stan ppndcla to th - plan s o In ths cas th stss and stan tnsos a: / / 5 Accodng to Hoo s law w obtan: E E 6 E Dvlopng th stss-stan latonshp fom C w gt: E C stn 7 and: E C stn dt E C stn 8

161 Intodcton to plan poblms sbjct 6 Andás Séns BME wwwtanonvtah Basc qatons of plan lastct h nmb of nnowns n cas of plan poblms s alwas ght: and v Und plan stss nd plan stan componnt can alwas b calclatd b th hlp of th componnts n dctons and Compatblt qaton h combnaton of Eqs and lads to th so-calld compatblt qaton []: 9 h qaton abov s qall t fo plan stss and plan stan stats It s possbl to fomlat th compatblt qaton n tms of stsss t s pss Eq9 n tms of stsss fo plan stss stat b tlng Eq9: G E W pss th md dvatv of th sha stss fom Eq: q q h combnaton of th two fom qatons slts n: q q wh: In a smla wa w can dvlop th followng qaton fo plan stan stat: q q It can b sn that f th s no volm foc thn th compatblt qaton has th sam fom nd plan stss as that nd plan stan In that cas whn th foc fld s

162 6 nt Elmnt Mthod consvatv thn a potntal fncton U sts of whch gadnt gvs th componnts of th dnst vcto of volm foc : q U and q U 5 A s stss fncton h qlbm and th compatblt qatons can b dcd to on qaton b ntodcng th A s stss fncton t = b th A s stss fncton whch s dfnd n th followng wa []: U U 6 ang thm bac nto th qlbm qatons gvn b Eq t s sn that th qatons a dntcall satsfd h stss fncton can b dvd fo v stss fld whch satsfs th qlbm qatons and th bod foc fld s consvatv In tms of th stsss th compatblt qaton gvn b Eq bcoms: U 7 wh: 8 s calld th bhamonc opato Eq7 s th govnng fld qaton fo plan stss poblms n whch th bod focs a consvatv If a fncton = s fond sch that t satsfs Eq7 and th pop pscbd bonda condtons thn t psnts th solton of th poblm h cospondng stsss and stans ma b dtmnd fom Eqs6 and 9 spctvl If th bod focs a constant o f U s a hamonc fncton thn th govnng qaton s: 9 whch s a patal dffntal qaton calld bhamonc qaton av s qaton ow lt s fomlat th govnng qatons n tms of dsplacmnt fld fo plan stss stat! h combnaton of Eqs and 9 povds th followngs []: v E E v G wwwtanonvtah Andás Séns BME

163 Intodcton to plan poblms sbjct 6 Andás Séns BME wwwtanonvtah Aft a smpl aangmnt w obtan: v E v E v E Sbsttton of th abov stsss nto th qlbm qaton gvn b Eq gvs th av s qaton: q v E G q v E v G W can dvlop av s qaton fo plan stan stat n a smla wa th slt s: q v E G q v E v G Und plan stss stat th fst scala nvaant of th stss tnso s: I Bonda val poblms It can b shown that fo plats nd smmtcall dstbtd tnal focs wth spct to th plan = th act solton satsfng all of th qlbm and compatblt qatons s []: 5 wh: 6 whch satsfs 7

164 6 nt Elmnt Mthod h scond tm n Eq5 howv dpnds on and ma b nglctd fo thn plats n whch cas w hav: 8 hat s fo thn plats soltons b Eq8 v closl appomat th stss dstbtons b Eq5 t s smma what nd of qmnts shold b mt of plan stss stat! h actal lastc bod mst b a thn plat th two sfacs of th plat mst b f fom load th tnal focs can hav onl and componnts th tnal focs shold b dstbtd smmtcall wth spct to th and as h govnng qaton sstm of plan poblms s a sstm of patal dffntal qatons qlbm qaton stan-dsplacmnt qaton and matal law wth cospondng bonda condtons h dnamc bonda condton s th latonshp btwn th stss tnso and th vcto of tnal load at ctan ponts of th latal bonda cv: n p 9 wh p s tacton vcto of th cospondng bonda sfac n s th otwad nomal of th bonda sfac o th otwad nomal of a ctan pat of t whch s paalll to th - plan h nmatc bonda condton psnts th mposd dsplacmnt of a pont o ctan ponts: b 5 wh b s th mposd dsplacmnts vcto and a th coodnats of th actal pont h sstm of govnng patal dffntal qatons togth wth lvant dnamc and nmatc bonda condtons blt a bonda val poblm W not that closd fom soltons of th govnng patal dffntal qatons of plan poblms wth pscbd bonda condtons whch occ n lastct poblms a v dffclt to obtan dctl and th a fqntl mpossbl to achv Bcas of ths fact th nvs and sm-nvs mthods ma b attmptd n th solton of ctan poblms [] In th nvs mthod w slct a spcfc solton whch satsfs th govnng qatons and thn sach fo th bonda condtons whch can b satsfd b ths solton w hav th solton fst and thn as what poblm t can solv In th sm-nvs mthod w assm a patal solton to a gvn poblm A patal solton conssts of an assmd fom fo ach dpndnt vaabl n tms of nown and nnown fnctons h assmd patal solton s thn sbstttd nto th ognal st of govnng qatons As a slt ths qatons wll b dcd to a st of smplfd dffntal qatons whch govn th manng nnown fnctons hs smplfd st of qatons togth wth pop bonda condtons s thn solvd b dct mthods wwwtanonvtah Andás Séns BME

165 Intodcton to plan poblms sbjct 65 5 Eampls fo plan stss 5 Dtmnaton of th tacton on th bondas of a sqa shap plat o th sqa shap plat shown n g w now th A s stss fncton n th - coodnat sstm []: p 5 a 6 wh p s th ntnst of th dstbtd ln load h bod foc s nglgbl; w assm that th plat s n plan stss stat g Sqa shap plat nd plan stss What nd of sstm of focs loads th bonda cvs of th plat? st w podc th stss fld: p p 5 a a p a h tacton vctos can b calclatd b th hlp of th dfnton of dnamc bonda condton and th localaton of t nto th bonda cvs hfo w nd th otwad nomal of ach bonda cv: bonda cv constant coodnat otwad nomal n = - =a = -j = a j Andás Séns BME wwwtanonvtah

166 66 nt Elmnt Mthod wwwtanonvtah Andás Séns BME thmo w nd Eqs9 and 5 W obtan th tacton vctos b applng th fom qatons: a p p 5 a p a a p a a a a a a p j p p a p a a a a a a j p h sstm of focs actng on th bonda cvs can b obtand b plottng th componnts of th vctos abov along th cospondng bonda cv g5 dmonstats th fncton plots wh g5a dpcts th loads n th nomal dcton ppndclal to th bonda cv g5b psnts th tangntal wth spct to th bonda cv stss dstbtons g5 omal a and tangntal b loads on th bonda cvs of a sqa plat nd plan stss stat

167 Intodcton to plan poblms sbjct 67 5 Analss of a tangntall loadd plat o th plat shown n g6 wth dmnsons of h th bod foc s nglgbl w can assm plan stss stat h fom of th A s stss fncton fo th load shown n g 6 s []: pt 5 h h h h g6 hn plat loadd b tangntall dstbtd foc nd plan stss stat Is th gvn fncton an act solton of th poblm abov? A fncton s th act solton of th poblm f t satsfs th govnng patal dffntal qaton of plan poblms and th dnamc bonda condtons Basd on th gvn fncton t s sn that Eq9 s satsfd n ths cas snc th govnng qaton s a foth od patal dffntal qaton whl th fnctons contans to a mamm th thd pow of t s nvstgat th dnamc bonda condtons! Smlal to th fom ampl w calclat th stss fld fst: pt 55 h h pt h h Basd on th stsss th loads on th bonda cvs a: = : p t 56 h h = h: pt = -h: Andás Séns BME wwwtanonvtah

168 68 nt Elmnt Mthod nall ndpndntl of Eq56 w fomlat th dnamc bonda condtons b th hlp of g 6 In accodanc wth th dnamc bonda condton dfnton th stss componnts actng on th actal bonda cv shold b qal to th cospondng nomal o tangntal componnts of th tacton vcto hat mans: = : 57 = h: pt = -h: Compang th bonda condtons to th bonda loads t s sn that on condton s not satsfd naml th sha stss on th bonda at = s not o on of th condtons s volatd vthlss th a two ponts wh n accodanc wth th fomla: p t h h h h h h 58 wth soltons of = /h and = h at two ponts th dnamc bonda condton s satsfd As a fnal wod th gvn fncton s not th act solton of th poblm n g6 bcas on of th dnamc bonda condtons s volatd Aft all t s accptabl snc togth wth Eq9 th gvn fncton satsfs ght fom th total tn condtons It shold b hghlghtd that th bonda at = s a fd bonda whch nvolvs nmatc bonda condton that s wh w dd not nvstgatd ths bonda cv n th ampl 6 h govnng qaton of plan poblms sng pola coodnats h soltons of man lastct poblms a convnntl fomlatd n tms of clndcal coodnats On th bas of g7 w hav th fnctonal latons []: cos sn 59 actan wwwtanonvtah Andás Séns BME

169 Intodcton to plan poblms sbjct 69 g7 Paamts of a pola coodnat sstm h dvatvs of th pola coodnats wth spct to and sng th last of Eq59 a: cos sn 6 sn cos Agan th dvatvs wth spct to and can b fomlatd basd on th chan l: sn cos 6 cos sn o dv th govnng qatons n tms of pola coodnats w ncopoat th stss tansfomaton pssons [] h nomal and sha stsss a tansfomd to a coodnat sstm gvn b otaton abot as b an angl : n n mn m n 6 wh: n n cos sn sn cos whch lads to: m 6 cos sn sn 6 Andás Séns BME wwwtanonvtah

170 7 nt Elmnt Mthod wwwtanonvtah Andás Séns BME sn cos sn sn cos cos sn h stan componnts can b tansfomd smlal ang Eq6 bac nto th qlbm qatons gvn b Eq moov b assmng that th a also bod focs w hav []: q 65 q wh th fom s th qaton n th adal th latt s th qaton n th tangntal dcton B a smla tchnq th stan-dsplacmnt qatons ma b tansfomd nto: 66 wh and a th adal and tangntal dsplacmnts Elmnatng th dsplacmnt componnts w obtan th compatblt qaton: 67 In th cas of Hoo s law th s no nd to pfom th tansfomaton d to th fact that th pola coodnat sstm s an othogonal sstm hfo g n Eq fng to plan stss stat w hav to sbsttt b and b: E E E 68 E E E h fomlaton ncopoatng plan stan stat basd on Eq6 lads to: E E E 69 E E E

171 Intodcton to plan poblms sbjct 7 Andás Séns BME wwwtanonvtah h fst scala nvaant of th stan tnso plan dlataton nd plan stan stat s: I 7 Sbstttng th stss and stan componnts nto th qlbm qaton gvn b Eq65 plan stan and ncopoatng th fst scala nvaant w obtan th av s qaton n tms of pola coodnats []: I q G G 7 q G G I wh 7 s th otaton abot as s th amé-constant: E 7 h govnng qaton of plan poblms n tms of pola coodnats can b fomlatd b sng th Hamlton opato Basd on Eqs8 and 6 w gt: 7 h stsss ma b obtand b sng th dffntal qotnts gvn b Eq6 and th tansfomaton pssons gvn b Eq6: 75 h last th fomla a qall vald nd plan stss and plan stan stats h qlbm qatons stan-dsplacmnt latonshp can also b fomlatd b sng nfntsmal lmnts n pola coodnat sstm []

172 7 nt Elmnt Mthod 7 Asmmtc plan poblms h s of pola coodnats s patclal convnnt n th solton of volton smmtc o n oth wods asmmtc poblms In ths cas dsplacmnt fld stsss a ndpndnt of th angl coodnat consqntl th dvats wth spct to vansh vwh In accodanc wth Eq7 th govnng qaton of plan poblms bcoms: d d d d d d d d 76 B ntodcng a nw ndpndnt vaabl ths qaton can b dcd to a dffntal qaton wth constant coffcnts: 77 As a slt Eq76 bcoms: d d d d d d 78 fo whch th gnal solton s: A B C D 79 ang bac w hav: A ln B C ln D 8 wh A B C and D a constants h stsss basd on Eq75 a: 8 ang th solton fncton bac w s that: C C Aln A B ln A A B 8 7 Sold ccla clnd and thc-walld tb t s s som ampls fo th applcaton of th qatons and fomla abov []! o a sold ccla clnd th stsss at = can not b nfntl hgh thfo: A C 8 wwwtanonvtah Andás Séns BME

173 Intodcton to plan poblms sbjct 7 h stsss n a sold ccla clnd a: B 8 hs s th solton of a ccla clnd loadd b tnal pss wth magntd of B on th ot sfac In th cas of a hollow ccla clnd o a thc-walld tb g8a t s not sffcnt to nvstgat onl th dnamc bonda condtons w nd to mpos also nmatc bonda condtons g8 Hollow ccla clnd wth mposd dsplacmnt at th nn bonda a thc-walld otatng ds b h stan componnts b sng Eq66 bcom: d d 85 Usng th stss-stan latonshp gvn b Eq68 w obtan th qatons blow: d d K K K K 86 wh: K E K 87 fo plan stss and K K 88 E fo plan stan t w pss th stan componnts: Andás Séns BME wwwtanonvtah

174 7 nt Elmnt Mthod d d C C K Aln A B K Aln A B 89 C C K Aln A B K Aln A B Intgatng th fom qaton w gt: C C K A ln A B K A ln A B H 9 wh H s an ntgaton constant Dvdng th fomla abov b and qatng t to th scond of Eq89 gvs th followng: A H 9 Snc th qaton mst b satsfd fo all vals of n th gon w mst consd th tval solton: A H 9 h manng constants B and C a to b dtmnd fom th bonda condtons mposd on th nn and ot bonda sfacs hfo th gnal solton s: C KB K K 9 h poblm of hollow ccla clnd can also b solvd b av s qaton If th dsplacmnt fld s ndpndnt of coodnat thn = fom Eqs7-7 w obtan: d d d d 9 fo whch th gnal solton s: c c 95 It s sn that t s mathmatcall dntcal to 9 o a ccla clnd wth fd ot sfac and wth ntnal pss th nmatc bonda condtons a: b 96 Basd on th solton fncton th constants a: wwwtanonvtah Andás Séns BME

175 Intodcton to plan poblms sbjct 75 c b c 97 b b b and th solton s: b 98 b h stan componnts a to b dtmnd b Eq85 th stsss b Eq68 7 Rotatng dss If th thcnss of th ccla clnd s small thn t s sad to b a ds g8b If th ds otats thn th s a bod foc n th fnc coodnat sstm h qlbm qaton n th adal dcton s Eq65 bcoms []: d d q and q 99 wh s th angla vloct of th ds s th dnst of th ds matal Raangng th qaton w obtan: d d hs qaton can b satsfd b ntodcng th stss fncton n accodanc wth th followng: d d h stan componnts hav alad bn dvd fo a hollow ccla clnd lmnatng fom Eq85 w obtan: d d Assmng plan stss stat and tlng Eq68 w hav: d E E d E d E d Andás Séns BME wwwtanonvtah

176 76 nt Elmnt Mthod ang t bac nto Eq lds th followng: d d d d w hav a scond od dffntal qaton fo th stss fncton whch nvolvs th followng solton: 8 A B 5 h stss componnts basd on Eq a: 8 A B 8 A B 6 wh A and B a ntgaton constants whch can b dtmnd b th bonda condtons o calclat th dsplacmnt fld w ncopoat Eq85 fom whch w hav: d d E E 8E A B 7 and th ntgaton of t lds: E E 8E A B 8 h basc qatons of th otatng ds a thn: A B C 9 A B C c wh: a b 8 8 C C wwwtanonvtah Andás Séns BME

177 Intodcton to plan poblms sbjct 77 a A b B c E E 8E t s solv an ampl sng th qatons abov! h lastc ds shown n g9 s fd to th shaft wth an ovlap of [] g9 Rotatng ds on a gd shaft Gvn: b = m = m h = m = - m = 78 g/m E = GPa = a How lag can b th mamm angla vloct f w want th ds not to gt loos? b Calclat th contact pss btwn th shaft and ds whn th stct dos not otat! o pont a fst w fomlat th bonda condtons A nmatc bonda condton s that h adal dsplacmnt on th nn sfac of th ds mst b qal to th val of ovlap: b ab b c b b h ot sfac of th ds s f to load thfo n accodanc wth th dnamc bonda condton th adal stss ppndcla to th ot sfac s o: A B C If th ds gts loos thn a f sfac s catd that s wh th adal stss shold b qal to o : b A B C b b h sstm of qatons contans th nnowns: A B and snc a and b a not ndpndnt of A and B W now sbtact Eq fom Eq and w obtan: Andás Séns BME wwwtanonvtah

178 78 nt Elmnt Mthod B C b B Cb 5 b h bac sbsttton nto Eq gvs: A C b 6 consqntl: a C b E b E C b 7 ang th constants bac nto th nmatc bonda condton qaton lds: C E b b C E b b 8E b 8 Incopoatng th constant C and aangng th sltng qaton th mamm angla vloct bcoms: 885 ad / s 9 ma In tms of th angla vloct th constants can b dtmnd: 8 A 8 Pa B 995 C 9 95 / m C 9 6 / m a 5 b 59 7 m c 9 / m o pont b w fnd ot that f th ds dos not otat thn = and ths wa: C = C = c = Und ths ccmstancs th adal dsplacmnt on th nn sfac mst b qal to th val of ovlap: b ab b c b b h ot sfac of th ds s stll f to load : A B C h solton s: 6 A 5 Pa B 689 wwwtanonvtah Andás Séns BME

179 Intodcton to plan poblms sbjct 79 6 a 556 b m h dstbton of th adal and tangntal stsss nd two dffnt condtons a dmonstatd n g g Dstbton of th adal and tangntal stsss n th ds stct whn th stct otats a and whn th s no otaton b 8 Bblogaph [] P Ch Cho cholas J Pagano Elastct nso dadc and ngnng appoachs D Van ostand Compan Inc 967 Pncton w Js oonto ondon [] S moshno J God ho of lastct McGaw-Hll Boo Compan Inc 95 w Yo oonto ondon [] Jósf Uj cts and pactcs of th sbjct Elastct and EM Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng Dpatmnt of Appld Mchancs 998/999 atmn smst Bdapst n Hngaan Andás Séns BME wwwtanonvtah

180 MODEIG O PAE SRESS SAE USIG EM SO- WARE SYSEMS MODEIG AAYSIS O PROBEM EVAUA- IO nt lmnt solton of plan poblms In th applcaton of th fnt lmnt mthod w dvd th plan doman of th whol stct nto dsct lmnts as t s llstatd n g [] g h basc concpt of th fnt lmnt mthod n th cas of plan lmnts In th E mthod w appl th mnmm pncpl of th total potntal ng to dvlop th fnt lmnt qlbm qatons o a sngl plan lmnt th total potntal ng s []: U W V Ap V n dv pda qdv wh s th vcto of stss componnts s th vcto stan componnts spctvl: moov = + vj s th dsplacmnt vcto fld p s th dnst vcto of sfac focs q s th dnst vcto of volm o bod focs s th vcto of concntatd focs actng on th plan lmnt wth coodnats of pont of acton and A p s that pat of th bonda cvs whch s loadd b sfacs focs V s th volm of th lmnt spctvl W povd th dsplacmnt vcto fld b ntpolaton: wwwtanonvtah Andás Séns BME

181 Modlng of plan stss stat sng EM softwa sstms 8 wh s th mat of ntpolaton fnctons ts dmnson dpnds on th dgs of fdom of th plan lmnt s vcto of nodal dsplacmnts Rfng to th basc qatons of lastct th latonshp btwn th stan and dsplacmnt flds n mat fom s: wh s th mat of dffntal opatos t can b obtand b Eqs and : 5 h combnaton of th latt latons gvs: B 6 wh B s th stan-dsplacmnt mat h stss fld can b obtand b: C CB 7 wh C s th constttv mat ts calclaton has alad bn mad n scton fo plan stss and plan stan stats h stan ng fo a sngl fnt lmnt s: U V dv B C B vdd K 8 wh K s th lmnt stffnss mat K B C BdV V B C Bvdd 9 ts dmnson dpnds on th dgs of fdom of th lmnt o plan lmnts th dffntal volm s wttn n th fom of: dv vda vdd wh v s th thcnss of lmnt h wo of tnal focs actng on th lmnt b th hlp of Eq bcoms: W Ap pda V qdv n pda qdv c Ap V Andás Séns BME wwwtanonvtah

182 8 nt Elmnt Mthod wh c s th vcto of concntatd focs actng n th nods of lmnt hs th total potntal ng can b wttn as: K wh: pda qdv c b p c Ap V s th vcto focs actng on th lmnt W can fomlat th qlbm qaton n th lmnt lvl b mans of th mnmm pncpl of th total potntal ng: K h assmbl of lmnt stffnss matcs vcto of nodal dsplacmnts and focs lads to th stctal qlbm qaton: K U wh K s stctal stffnss matu s th stctal vcto of nodal dsplacmnts s th stctal vcto of nodal focs spctvl hat s th fnt lmnt qlbm qaton s a sstm of algbac qatons fo whch th soltons a th vals of nodal dsplacmnts In tms of th nodal dsplacmnts w can calclat th nodal focs and stsss o th solton of plan poblms th st man tps of plan lmnts In th sql w vw th smplst lmnt tps na th nod tangl lmnt h lna tangl lmnt n tangl [] whch s oftn calld th tangl mmban lmnt o constant stan tangl CS lmnt s dpctd n g At ach nod th a two dgs of fdom Consqntl th dgs of fdom a qal to s fo th whol lmnt h aow n th cnt pont of th lmnt fs to th ontaton of th lmnt fo ach lmnt w hav a dcton whch mans how th nods a followd b ach oth Intpolaton of th dsplacmnt fld W collct th nodal coodnats and dsplacmnt componnts n vctos: 5 v v v wwwtanonvtah Andás Séns BME

183 Modlng of plan stss stat sng EM softwa sstms 8 Andás Séns BME wwwtanonvtah g na tangl lmnt odal coodnats and dsplacmnts h tangl aa can b pssd as a dtmnant: A 6 h and v componnts of th dsplacmnt fld a fomlatd as th lna fncton of and : a a a 7 b b b v wh a a a b b and b a nnown constants h vcto of stan componnts s: 8 wh sng Eqs and w hav: a b v a b v 9 h nodal dsplacmnts mst b obtand f w ta bac th nodal coodnats nto th and v fnctons gvn b Eq7 : a a a b b b v

184 8 nt Elmnt Mthod wwwtanonvtah Andás Séns BME a a a b b b v a a a b b b v h solton of th sstm of qatons abov slts n: A a A a A a A v v v b A v v v b A v v v b wh: Sbstttng th solton abov bac nto th componnts of dsplacmnt fld Eq7 w obtan: ] [ A ] [ v v v A v Consdng th fact that fo th tangl lmnt w hav th ntpolaton fnctons s Eq w can wt that: v v v v v In accodanc wth Eq th ntpolaton fnctons can b dvd n th followng fom: A = 5

185 Modlng of plan stss stat sng EM softwa sstms 85 th mat of ntpolaton fnctons bcoms: Basd on th laton of 6 h paamt lns of th ntpolaton fncton a shown n g whch mpls th followng popts: at th nods : at th mdponts of th tangl sds : / / / / / / at th cntod : / / / t s sn that at v pont: fnall:! j!! j da A 7 j! A g Paamt lns of th ntpolaton fnctons of lna tangl lmnt Calclaton of th stffnss mat Accodng to Eq9 th dfnton of th lmnt stffnss mat s: K B C Bvdd 8 wh th pvosl mntond stan-dsplacmnt mat sng Eqs5 and 6 bcoms: Andás Séns BME wwwtanonvtah

186 86 nt Elmnt Mthod wwwtanonvtah Andás Séns BME A B 9 hs fomlaton mpls that th lmnts of mat B a ndpndnt of th and vaabls th dpnd onl on th nodal coodnats hfo th stffnss mat can b wttn as: BV C B BvA C B K wh A s th lmnt aa V = A v s th lmnt volm spctvl As a consqnc th stffnss mat of th lna tangl lmnt can b comptd n a latvl smpl wa and n closd fom Dfnton of th loads Bod foc o volm foc t th vcto of bod focs b qal to: q q q fom whch w hav: A A b vda q q q q q q vda q q qvdd Utlng th spcal popts of th ntpolaton fnctons gvn b Eq7 g f = j = and = w hav:

187 Modlng of plan stss stat sng EM softwa sstms 87 da A A Consqntl w hav: b Av q q q q q q As an planaton th bod foc actng on th lmnt g th whol wght and th sltng sltant foc s dvdd nto th qal pats and pt nto th nods h bod foc can b ognatd fom gavtaton o acclaton nta foc Dstbtd foc along lmnt dgs o th calclaton of foc vctos as a slt of ln loads along lmnt dgs w shold ta th - dg of th lmnt shown n g nto consdaton W dfn a dmnsonlss paamt along th lmnt dg h ac lngth along th lmnt dg s thn: s l and l d 5 ds g na tangl lmnt wth ln load n dcton along lmnt dg - h lnal dstbtd load n dcton can b dscbd b th followng fncton: p p p 6 Smlal th dsplacmnt fncton n dcton along lmnt dg - can b wttn as: 7 h wo of th dstbtd load s gnall th ntgaton of th load fncton mltpld b dsplacmnt fncton btwn th cospondng nods: Andás Séns BME wwwtanonvtah

188 88 nt Elmnt Mthod wwwtanonvtah Andás Séns BME l l p p p p p p v l vd l p p vd l p p vd l p vds p W 8 hat s th foc vcto fom a lnal dstbtd ln load bcoms: p p p p p v l 9 If th ln load s constant along th lmnt dg thn p = p = p whch mpls: p p p v l h fom of th vcto of focs n th fnt lmnt qaton s smla n th cas of a lnal dstbtd load n dcton Concntatd focs Concntatd focs can act onl at nods h foc vcto can b smpl fomlatd basd on th nods: c h total foc vcto s th sm of th vctos dtald n th pvos ponts : c p b W dmonstat th solton of th fnt lmnt qaton and th constcton of th stffnss mat and foc vcto thogh an ampl Eampl fo th lna tangl lmnt plan stss stat h modl shown n g5a s loadd b dstbtd focs Calclat th nodal dsplacmnts and focs n that cas whn w blt-p th plat sng two lna tangl lmnts! Calclat th stan and stss componnts []!

189 Modlng of plan stss stat sng EM softwa sstms 89 g5 Plan modl loadd b dstbtd focs a fnt lmnt modl mad b two lna tangl lmnts b Gvn: p = MPa E = 5 GPa a = mm c = mm p = 6 MPa = 5 b = mm v = 5 mm In th cos of th comptaton w calclat th dstancs n [mm] and th foc n [] ollowng g5b w s that th modl s constctd b two tangl lmnts h nodal coodnats a: nod [mm] [mm] h so-calld lmnt-nod tabl s: lmnt nods h fnt lmnt qlbm qaton to b solvd s: KU wh: v v v v U Andás Séns BME wwwtanonvtah

190 9 nt Elmnt Mthod s th stctal vcto of nodal dsplacmnts Bcas of th bonda condtons v = v = = = w hav: U v v 5 In od to calclat th stffnss mat w nd th constttv mat fo plan stss stat s scton : C C st 6 E MPa 6 h coffcnts of th ntpolaton fnctons fo th fst lmnt a: mm mm 7 mm mm mm mm and fo th scond lmnt spctvl: mm mm 8 mm mm mm mm h tangl aas a: A mm A 5 mm 9 Mat B fo th fst lmnt s: B A mm 5 wwwtanonvtah Andás Séns BME

191 Modlng of plan stss stat sng EM softwa sstms 9 o th scond lmnt t s: B A mm 5 Basd on Eq th lmnt stffnss matcs a: K B C B V / 5 57 / 6 5 / / 6 5 5/ 5 / / / 8/ 5 mm 5 K B C B V / / / 5 5/ mm wh V = A v = 5 = 5 mm and V = A v = 55 = 75 mm a th lmnt volms o th constcton of th stctal stffnss mat w complt th lmnt matcs wth mpt ows and colmns cospondng to th mssng dgs of fdom On th bas of g5 and th lmnt-nod tabl t s sn that th fst lmnt nclds onl nods and Consqntl thos ows and colmns whch blong to nod shold b flld p wth os: K Andás Séns BME wwwtanonvtah

192 9 nt Elmnt Mthod In contast fo th scond lmnt th ows and colmns cospondng wth th fst nod mst b compltd b th placmnt of os: K h stctal stffnss mat s calclatd as th sm of th two fom matcs: K K K h foc vcto latd to th dstbtd load s calclatd b Eq9: lv p p p 56 lv p p p wh l m and l = mm a th lmnt dg lngths btwn th nods ndcatd n th sbscpt B compltng th lmnt vctos wth os at th postons of th pop dgs of fdom w gt th stctal foc vctos: lv p p l l v p p 5 5 l W consd th acton focs as concntatd focs at th constand nods: 57 wwwtanonvtah Andás Séns BME

193 Modlng of plan stss stat sng EM softwa sstms 9 c 58 ang t nto accont that at nod th s no tnal foc and that th sfacs a fctonlss : = = w hav: c 59 h stctal foc vcto s: p p c 6 h fnt lmnt qlbm qaton s KU w hav: / 5 57 / 6 5 / / / 5 / / 5 5/ / 5 / 5 5 / v 5 v 5 h nodal dsplacmnts can b dtmnd fom th sstm of qatons constctd b th st 6 th 7 th and 8 th componnt qatons of th mat qaton: v 6 5/ 6v 5v 5 5 v 5v / 6v 5 h qatons abov n fact w obtand b th condnsaton of Eq6 Whn w pfom th mat condnsaton onl thos componnt qatons man whch contan nnowns wth spct to th dsplacmnts onl On th ght hand sd n th foc vcto th a no nnowns h soltons a: mm v 997 mm mm v 6 mm Andás Séns BME wwwtanonvtah

194 9 nt Elmnt Mthod ang th nodal dsplacmnts bac nto th nd d th and 5 th ows of th mat qaton w can dtmn th nodal focs: 5 5 / v v 5 5v 5 / v 5 / v 5v 5v h soltons a: Usng Eq9 w calclat now th stan componnts: v v a b a b 66 o th fst lmnt w obtan: a A 67 b v v v A a 9 A b v v v v 8 55 A h vcto of stan componnts fo th fst lmnt s: wwwtanonvtah Andás Séns BME

195 Modlng of plan stss stat sng EM softwa sstms 95 o th scond lmnt w can wt: a A 5 69 b v v v v v A 5 a A b v v v v A 5 fom whch w hav: Snc th plat s nd plan stss stat w can wt basd on Eq9 that: h nomal and sha stans a accodngl constants wthn th ndvdal lmnts w fd to ths fact n th ntodcton of th tangl lmnt Incopoatng th constttv mat w can dtmn th stss componnts too basd on Eq6: st C 7 hs qaton gvs th stsss of th lmnts whch s n gnal fd to as lmnt stss n th commcal fnt lmnt pacags o th fst lmnt w hav: MPa MPa Andás Séns BME wwwtanonvtah

196 96 nt Elmnt Mthod MPa Smlal fo th scond lmnt th stsss a: MPa MPa MPa Consdng th stsss t s possbl to podc nodal stss solton B comptng th avag stsss n th mtal nods w obtan th so-calld nodal stss o avag stss solton: od : 6 MPa 88 MPa 87 MPa 75 od : 65 MPa 7865 MPa 8786 MPa od : 658 MPa 5 MPa 5685 MPa od : 65 MPa 7865 MPa wwwtanonvtah Andás Séns BME

197 Modlng of plan stss stat sng EM softwa sstms MPa h poblm psntd n scton was vfd b th fnt lmnt cod ASYS sltng n th sam slts h solton wth th abov appld low msh solton s natall v naccat Qadatc s nod tangl lmnt h mo advancd vson of th lna tangl lmnt s th s nod qadatc tangl lmnt n whch th a addtonal nods n th mdponts of th lmnt sds [5] Bcas of th addtonal nods w nd dsplacmnt fnctons ncldng s nnowns whch a: a a a a a a5 76 v b b b b b b5 h calclaton of th stffnss mat and foc vcto can b pfomd n th sam fashon as t was don n th lna tangl lmnt Wthn th ndvdal lmnts th stan and stss componnts va lnal As a consqnc sng dntcal msh solton th qadatc tangl lmnt povds a btt appomaton of th poblm than th lna on 5 Isopaamtc fo nod qadlatal h sopaamtc qadlatal s g6a s on of th most mpotant fnt lmnt tp fo plan poblms [5] An lmnt s calld sopaamtc f w fomlat th local gomt and dsplacmnt fld b th sam st of fnctons 5 Intpolaton of th gomt o th sa of smplct w map th qadlatal lmnt to a gla sqa nto th - natal coodnat sstm as t s shown n g 6b W gv th fnctons of th and coodnats of lmnt dgs n th followng fom: 77 wh: 78 Andás Séns BME wwwtanonvtah

198 98 nt Elmnt Mthod g6 Isopaamtc qadlatal n th global a and natal b coodnat sstms D to th fact that w hav fo nods th ntpolaton fncton ma contan to a mamm fo nnowns: a a a a P A 79 wh A s th vcto of coffcnts P s th vcto of bass polnomals spctvl: A a a a a P 8 h fncton gvn b Eq79 mst satsf th followng condtons: a a a a 8 a a a a a a a a a a a a In mat fom t s: M A 8 wh: M 8 wwwtanonvtah Andás Séns BME

199 Modlng of plan stss stat sng EM softwa sstms 99 Andás Séns BME wwwtanonvtah hn th coffcnts can b dtmnd b sng Eq8: M A and: M P A P 8 h soltons fo th coffcnts a: a a 85 a a ang thm bac nto Eq79 w gt: 86 h ntpolaton polnomals on th bas of Eq86 a: 87 Pfomng th sam comptaton fo coodnat w obtan th sam ntpolaton fnctons h th dmnsonal plot of th ntpolaton fnctons psnts ln sfacs of whch val n th locaton of th th nod s qal to nt whl n th locaton of th oth nods t s qal to o as t s dmonstatd n g7

200 nt Elmnt Mthod wwwtanonvtah Andás Séns BME g7 Intpolaton fnctons of th sopaamtc qadlatal lmnt h smma of th gomt s gvn b th fomla blow: R 88 wh: R 89 s th vcto of nodal coodnats and: h compact fom of th ntpolaton fnctons s: 9 wh and a th con nod coodnats accodng to g6b

201 Modlng of plan stss stat sng EM softwa sstms Andás Séns BME wwwtanonvtah 5 Intpolaton of th dsplacmnt fld h dsplacmnt vcto fld of th sopaamtc qadlatal lmnt can b wttn as: v 9 wh: 9 v v v v v moov th mat of ntpolaton fnctons and th vcto of nodal dsplacmnts a: 9 v v v v h dsplacmnt fld mst slt n th nodal dsplacmnts f w sbsttt th coodnats of th pop nods bac t mst satsf th followng condtons: 9 Mathmatcall ths s th sam st of condtons fo th dsplacmnts as that fomlatd n th cas of th gomtcal paamts Consqntl th comptaton lads to th sam ntpolaton fnctons as thos gvn b Eq87 h qadlatal lmnt s calld sopaamtc lmnt bcas of th fact that th sam ntpolaton fnctons a appld fo th dsplacmnt fld and local gomt 5 Calclaton of stan componnts Jacob mat and Jacob dtmnant h vcto of stan componnts sng Eq s th followng: B v v 95 wh s th patal dvatv of wth spct to v s th patal dvatv of v wth spct to Moov:

202 nt Elmnt Mthod wwwtanonvtah Andás Séns BME B 96 Appantl mat B contans th fst dvatvs of th ntpolaton fnctons wth spct to and t can b laboatd basd on Eq87 that th ntpolaton fnctons a nown n tms of and W f to th chan l of dffntaton: 97 Utlng Eq77 th local gomt and th fst dvatv of th fnctons wth spct to and a: 98 Wtng t n mat fom w hav: J 99 wh J s th so-calld Jacob mat:

203 Modlng of plan stss stat sng EM softwa sstms J J J J J h Jacob dtmnant s: J J JJ J h dvatvs wth spct to and a povdd b th hlp of th nvs Jacob mat: J J J J J J J fthmo: J om whch w obtan th followngs: J J J J J J Wth th ad of th fom mat B bcoms: J J B J J J J J J J J J J J J J J J 5 J J J J J J J J J J J J J J J J Andás Séns BME wwwtanonvtah

204 nt Elmnt Mthod W nd th dvatvs of th ntpolatons fnctons and th lmnts of th Jacob mat whch a: spctvl and: J J J J 6 7 Basd on th fom qatons w can fomlat th Jacob mat th n th followng fom: J J J J J 8 5 h mpotanc of th Jacob dtmnant ampl Calclat th lmnts of th Jacob mat fo th qadlatal shown n g8! h nodal coodnats a: a a a a a 9 wwwtanonvtah Andás Séns BME

205 Modlng of plan stss stat sng EM softwa sstms 5 g8 Isopaamtc qadlatal lmnt wth cssv dstoton h lmnts of th Jacob mat basd on Eq7 a: J J J J a a a a a 6 a a a a 6 6 a a a a a 6 6 a a a a 6 fom whch th Jacob dtmnant s: J JJ JJ a h Jacob dtmnant s f g = - and = o = and = - hs cas s sad to b cssv dstoton t mans that w hav dgnat lmnt If J = thn th nvs Jacob mat dos not st at th pont nd consdaton Moov th paamt lns ntsct ach oth otsd th doman of th qadlatal hat s wh th sm of th nn angls of qadlatal mst b lss than 8 n oth wods th qadlatal can not b concav 55 Calclaton of th stss fld h vcto of stss componnts can b obtand fom Eq7: Andás Séns BME wwwtanonvtah

206 6 nt Elmnt Mthod C CB whc st C fo plan stss and stn C C fo plan stan s scton 56 Calclaton of th stffnss mat h stffnss mat fo plan poblms s calclatd b Eq9: K B C Bvdd In th cas of th sopaamtc qadlatal th lmnts of mat B contans th dvatvs of th ntpolaton fnctons Consqntl fo th stffnss mat calclaton th tansfomaton of sfac ntgals mst b pfomd h vctos and paamts whch a qd fo th analss a shown n g9 h angs of paamts c and c a: c c g9 ansfomaton of sfac ntgal n th sopaamtc qadlatal h dffntal vctos wttn b lowcas ltts can b fomlatd b tlng Eq : d d d d J d d d d d d J 5 onst and smlal: onst wwwtanonvtah Andás Séns BME

207 Modlng of plan stss stat sng EM softwa sstms 7 d d J d d d d J 6 onst h dfnton of th lmnta aa s: da d d j abs Jd Jd JJ JJ dd dd 7 J d J d ths lds: da dd Jdd and: dd Jdd 8 h stffnss mat bcoms: A K A B C Bvdd B C BvJdd 9 th stffnss mat can b comptd b th hlp of an aa ntgal o th calclaton w can appl analtcal o nmcal mthod h commcal fnt lmnt pacags n gnal mplmnt th Gassan qadat to pfom th ntgaton hs mthod wll b psntd n scton 6 57 Calclaton of th foc vcto Dstbtd load along th lmnt dg h foc vcto sltng fom th dstbtd load along lmnt dg - shown n g can b dfnd as: v p p ds wh: s l and ds l d wh l s th lmnt dg lngth btwn nods and Andás Séns BME wwwtanonvtah

208 8 nt Elmnt Mthod wwwtanonvtah Andás Séns BME g Dstbtd load along th lmnt dg of an sopaamtc qadlatal lmnt Moov w now that along dg - = and s g6 W can wt aft all that: d p p p p p p p p vl d l p p v ds p v l p o fth calclaton w mst valat th ntpolaton fnctons along th paamt ln fo whch = -: hs lds:

209 Modlng of plan stss stat sng EM softwa sstms 9 Andás Séns BME wwwtanonvtah p p p d p d p p p p d p d p B tang th slts bac nto th foc vcto w obtan: p p p p p vl 5 h sltant of th nfoml dstbtd load s dvdd nto two pats and smlal to th bam and lna tangl lmnts pt nto th nods of lmnt dg h calclaton can b mad also n th cas of lnal dstbtd load; natall t slts n a dffnt foc vcto Bod foc h foc vcto calclatd fom th bod foc s: qjdd v b 6 fo whch w nd agan th valaton of sfac ntgal Smlal to th stffnss mat th Gassan qadat wll b appld to valat th ntgal Concntatd loads o plan poblms th a concntatd focs actng n th nods and th a no momnts h and componnts of th concntatd focs a collctd n a vcto: c 7 h total vcto of focs s th sm of vctos psntd n th last th ponts: c b p 8 6 mcal ntgaton th Gass l o th calclaton of th lmnt stffnss mat and th bod foc vcto of sopaamtc qadlatals th a nmcal ntgaton schms mplmntd n th fnt lmnt pacags Commonl th Gass l s appld bcas t ss mnmal nmb of sampl ponts and t s latvl accat [6]

210 nt Elmnt Mthod 6 On dmnsonal Gass l h man am s th appomat bt latvl accat calclaton of th aa nd th cv shown b g sng th on dmnsonal l g Sampl ponts of th on dmnsonal Gass l h appomat aa nd th cv s calclatd b: p d w 9 h sampl o ntgaton pont coodnats and th ntgaton wghts w a lstd n tabl h on dmnsonal l povds th act solton fo a polnomal p to th od of p- p w / / / 5 / 5 = - = - 6/5 / 7 6/5 / 7 / / 5/9 8/9 5/9 5/ 6 / 6 5/ 6 / 6 w = w w = w abl Paamts of th on dmnsonal Gass l t s solv an ampl fo th applcaton of th Gass l! Calclat th act val of th ntgal: wwwtanonvtah Andás Séns BME

211 Modlng of plan stss stat sng EM softwa sstms I d as wll as ts appomat val sng on two and th ntgaton ponts! Eact solton: ln ln ln 986 I Gass l p = t = - If = thn = on th oth hand f = thn = - consqntl: I d d and: h appomat val of th ntgal s: I w hat mans an o of 99% compad to th act solton Gass l p = In ths cas: I w w 999 h val of th ntgal dffs wth 7 % fom th act solton Gass l p = I w w w h o of appomaton s onl 5% 6 wo dmnsonal Gass l h two dmnsonal Gass l mas t possbl to valat th appomat val of sfac ntgals h ntgal s appomatd b th psson blow: b d f dd f Jdd j a c n n w w f J 6 j j j Andás Séns BME wwwtanonvtah

212 nt Elmnt Mthod wh w and w j a th ntgaton wghts and j a th ntgaton pont coodnats moov th angs a - - j spctvl Dpndng on th nmb of ntgaton ponts w can dfn dffnt Gassan qadats as t s dmonstatd n g g Intgaton ponts of th and Gassan qadats o th qadat th s onl a sngl ntgaton pont fo th w hav tc h paamts of th and Gassan qadats a smmad n abl a / b / 5 w w j = = -a = -a = -b = -b 5/9 5/9 = = a = -a = b = -b 5/9 5/9 w = = a = a = b = b 5/9 5/9 = -a = a = -b = b 5/9 5/9 5 = 5 = -b 8/9 5/9 w = w = 6 = b 6 = 5/9 8/9 w = w = 7 = 7 = b 8/9 5/9 8 = -b 8 = 5/9 8/9 9 = 9 = 8/9 8/9 abl Paamts of th Gassan qadats Eampl fo th applcaton of Gassan qadat Calclat th appomat val of th ntgal: wwwtanonvtah Andás Séns BME

213 Modlng of plan stss stat sng EM softwa sstms I da 7 A sng th Gass qadat fo th doman of paalllogam dpctd n g! Compa th slt to that of th act ntgaton []! g Eampl fo th applcaton of Gassan qadat h nodal coodnats a: = = = = = = = = 8 Basd on th appomat psson of th Gassan qadat w can wt: I da A j w w j f J 9 j j h calclaton qs th lmnts of th Jacob mat W nd th nodal coodnats and also th dvatvs of th ntpolaton fnctons s Eq: J J J J h Jacob mat s: Andás Séns BME wwwtanonvtah

214 nt Elmnt Mthod J J J J J and th Jacob dtmnant s: J const h and paamts tlng th ntpolatd fom gvn b Eq77 a: h fncton f s: f 5 5 W can calclat th appomat val of th ntgal basd on th fg and tabl abov: I [ f a a f a a f a a f a a] I h act val of th ntgal s: f dd dd It s shown appantl that th Gassan qadat povds th act val n ths cas Most of th commcal fnt lmnt pacags mplmnts qadat 7 Eampl fo th sopaamtc qadlatal Solv th ampl psntd n scton sng on sopaamtc qadlatal lmnt! h data a th sam as thos gvn n th lna tangl lmnt Appl a sngl fnt lmnt b followng g [] Dtmn th nodal dsplacmnts and th actons! wwwtanonvtah Andás Séns BME

215 Modlng of plan stss stat sng EM softwa sstms 5 g Eampl fo th applcaton of th sopaamtc qadlatal lmnt h fnt lmnt qlbm qaton to b solvd s: KU 7 Snc w hav onl a sngl lmnt n ths cas Eq7 cosponds to th qlbm qaton n th lmnt lvl: K 8 wh: v v v v U 9 s th vcto of nodal dsplacmnts D to th bonda condtons v = v = = = w hav: U v v 5 Smlal to th lna tangl lmnt w hav a sstm of qatons ncldng fo nnowns h lmnt stffnss mat s calclatd b th Gassan qadat w can wt that: K B C BvJd d v j p p j j w w B C B J 5 j j h lmnts of th Jacob mat a qall qd: Andás Séns BME wwwtanonvtah

216 6 nt Elmnt Mthod J J J J Constctng th Jacob mat w hav: 5 5 J J J 5 J 5 5 J 5 and th Jacob dtmnant s: 5 75 J 5 It can b sn that f thn J > fo ach cas consqntl th lmnt s not dgnat whch s obvosl sn basd on g5 h nvs Jacob mat s: J J 5 J 55 J J J 5 5 As a nt stp w calclat mat B s Eq96 wh fng to Eq5 w hav: wwwtanonvtah Andás Séns BME

217 Modlng of plan stss stat sng EM softwa sstms 7 Andás Séns BME wwwtanonvtah J J J 56 J J J J J J J J J J J J J J J J J J 5 J J J Mat B bcoms: 5 5 B 57 W calclat th lmnt [] of th stffnss mat b th Gassan qadat o that lt s calclat th followng: 6 B C B 58 and: d d J f v d d v d Jd B C B v K 59

218 8 nt Elmnt Mthod wh v = 5 mm s th thcnss of th plat W ca ot th calclaton n th was b sng th Gassan qadats and th act ntgaton spctvl I Gassan qadat: K v { f a a J a a f a a J a a f a a J a a f a a J a a} mm 6 II Gassan qadat: v f b b J b b f b b J b b f b b J b b f b b J b b K v f b J b f b J b f b J b f b J b v f J mm 6 III Eact ntgaton: 5 K mm Calclatng all of th componnts of th lmnt stffnss mat w obtan: K mm 6 h vcto of focs can b constctd n a smla wa to that shown n th tangl lmnt: lv p p p l v p p 5 5 p 6 W consd th actons as concntatd focs n th nmatcall constand nods: wwwtanonvtah Andás Séns BME

219 Modlng of plan stss stat sng EM softwa sstms 9 c 65 Consdng th fact that th sfacs a fctonlss and that at nod th s no tnal foc w hav = = = = whch lads to: c 66 h stctal foc vcto bcoms: p p c 67 h constcton of th fnt lmnt qlbm qaton slts n: v v 5 68 In th stffnss mat w lmnat thos ows and colmns fo whch th cospondng dsplacmnt componnt s a pscbd h constand val hs wa w obtan th socalld condnsd stffnss mat whch s sd to pand th sstm of qatons of whch soltons a th nodal dsplacmnts: v v h soltons a: mm v 999 mm mm v 5 mm hn th actons a calclatd b th nd d th and 5 th componnt qatons of Eq 68 h soltons a: Andás Séns BME wwwtanonvtah

220 nt Elmnt Mthod h stan and stss componnts of th lmnt can b pssd n paamtc fom as th fncton of and fom Eqs95 and 5 ang th coodnats of th cospondng nod bac th stan and stss componnts can b calclatd h ampl abov was vfd b th fnt lmnt cod ASYS 8 Qadatc sopaamtc qadlatal h advancd vson of th lna qadlatal s th qadatc qadlatal n whch th cvs of th lmnt sds as wll as th dsplacmnts a appomatd b a scond od fncton of th and coodnats [7] On ach lmnt dg w povd a mdsd nod as t s shown n g5 mplng 8 nods and 8 nnown coffcnts n th appomat fncton of g th coodnat: a a a a a a5 a6 a7 7 Usng th nodal condtons w can dv th ntpolaton fnctons of th qadatc lmnt n a smla wa to that shown n th fo nod qadlatal h ntpolaton fnctons bcom: g5 Qadatc sopaamtc qadlatal h ntpolaton fnctons can b fomlatd also n compact fom: wwwtanonvtah Andás Séns BME

221 Modlng of plan stss stat sng EM softwa sstms = = 6 8 wh and a th coodnats of th nods g6 shows th fncton plot of th ntpolaton fnctons 5 and 8 g6 Intpolaton fnctons of th qadatc sopaamtc qadlatal h val of cospondng to th th nod s qal to nt n th oth nods t s o h calclaton of th stffnss mat and th vcto of focs can b mad n th sam wa as that shown n th qadlatal wth staght dgs h Jacob dtmnant and th Gassan qadat s qall qd 9 Bblogaph [] Gábo Vöös nt lmnt analss Mchancal Engnng Modlng MSc fomaton Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng Dpatmnt of Appld Mchancs lct not 8/9 atmn smst [] Jósf Uj cts and pactcs of th sbjct mcal mthods n mchancs PhD fomaton Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng Dpatmnt of Appld Mchancs / spng smst n Hngaan [] Sngs S Rao h fnt lmnt mthod n ngnng foth dton Elsv Scnc & chnolog Boos [] Edogan Madnc Ibahm Gvn h fnt lmnt mthod and applcatons n ngnng sng ASYS Spng Scnc+Bsnss Mda Inc 6 w Yo USA [5] Im Bojtá Gög Molná amás ag Applcaton of th fnt lmnt mthod to plan poblms chncal Boo Pblsh 988 Bdapst n Hngaan Andás Séns BME wwwtanonvtah

222 nt Elmnt Mthod [6] Ádám Kovács Jósf Uj ndamntals of fnt lmnt mthod Unvst Pss 7 Bdapst n Hngaan [7] Klas-Jügn Bath nt lmnt pocds Pntc Hall Upp Saddl Rv 996 w Js 758 wwwtanonvtah Andás Séns BME

223 MODEIG O AXISYMMERIC SAE BY EM SOWARE SYSEMS MODEIG AAYSIS O PROBEM EVAUAIO nt lmnt solton of asmmtc poblms o asmmtc poblms both th gomt and th load a ndpndnt of th angl coodnat An ampl s shown n g g hc-walld tb nd ntnal pss a asmmtc modl of th tb b and th smplfd fnt lmnt poblm c Plan poblms a dfnd n plan as th mdan scton of an actal bod; mathmatcall th can b solvd as two-vaabl poblms h lmnt tps of asmmtc poblms a actall ng shap lmnts hat s wh th s no concntatd foc n sch poblms cpt fo th cas whn th foc concds wth th as of smmt A ln load wth constant ntnst on th ot sfac of th modl dfnd b a ads of loos as a concntatd foc o asmmtc poblms th dsplacmnt fld has th followng fom []: w h stan-dsplacmnt qaton s: wh s th Hamlton opato n clndcal coodnat sstm CCS It can b dvd b th hlp of Eq6 Basd on g7 th adal and tangntal nt bass vctos bcom []: cos sn j sn cos j t Andás Séns BME wwwtanonvtah

224 nt Elmnt Mthod wwwtanonvtah Andás Séns BME Opato nabla n th -- coodnat sstm s: j Utlng Eq6 and sbstttng t nto Eq lads to: t 5 h stan componnts n CCS can b wttn as [] s Eq66: t w w 6 In vcto fom: t 7 h vcto of stan componnts s wttn n th followng fom: 8 wh basd on Eq6 th mat of dffntal opatos s compltd wth an addtonal lmnt compad to th plan stss o plan stan stats: 9 h vcto of stss componnts s: t Indpndntl of th coodnat sstm w hav Hoo s law n th fom blow: E G I

225 Modlng of asmmtc stat b EM softwa sstms 5 Andás Séns BME wwwtanonvtah t t / / fom whch w hav: ] [ t t E E ] [ t t t t E E ] [ t t E E E Accodngl th constttv mat basd on C s []: E C h calclaton of th lmnt stffnss mat s possbl thogh th followng dfnton []: V BdV C B K 5 wh th dmnson of mat B dpnds on th dgs of fdom of th lmnt h vcto of focs can b dtmnd n th sam wa as t was shown fo plan poblms h doman of asmmtc bods can b mshd b ng shap lmnts Elmnts can b dfnd n th mdan scton n plan In th fnt lmnt softwas th sam lmnt tps a avalabl as thos fo plan poblms; howv th asmmtc bhavo shold b st In th cos of th fnt lmnt analss th sam ntpolaton fnctons a appld as thos psntd fo plan stss and plan stan stats In most of th fnt lmnt cods th plan modl shold b ppad n th - plan wh s th as of volton s gc In th sql w vw th applcaton of th lna tangl and th sopaamtc qadlatal lmnts

226 6 nt Elmnt Mthod Asmmtc lna tangl lmnt h stps of th fnt lmnt dsctaton sng lna tangl lmnt hav alad bn psntd n scton Som modfcaton s qd consdng th asmmtc applcaton of th tangl lmnt In th dsplacmnt fld w chang th and paamts to and spctvl []: w 6 wh th dsplacmnt componnts can b povdd b changng th coodnat to and coodnat to n Eq spctvl: 7 w w w w moov th mat of ntpolaton fnctons and th vcto of nodal dsplacmnts bcom: 8 w w w h calclaton of th stan componnts s mad n a smla fashon to that psntd n plan poblms: B 9 wh th stan-dsplacmnt mat sng Eqs9 and 8 s: wwwtanonvtah Andás Séns BME

227 Modlng of asmmtc stat b EM softwa sstms 7 Andás Séns BME wwwtanonvtah B wh n th scond ow th tm / appas Consdng th asmmtc nat of th poblm w can wt that: dd B C B BdA C B K A h vcto of focs conssts of th dffnt tms vn n asmmtc poblms o a dstbtd load th fomla s: ds p p wh p s th vcto of psss n th adal and aal dctons: p p p In th cas of bod foc th foc vcto bcoms: dd q b wh: q q q 5 s th dnst vcto of volm focs nall th vcto of concntatd focs s:

228 8 nt Elmnt Mthod c 6 h total foc vcto s th sm th followng th vctos: 7 p b c h poblm solton nvolvs th composton of th lmnt and stctal stffnss matcs W calclat fst th nodal dsplacmnts fom th stctal qaton thn th actons and stan and stss componnts spctvl t s s an ampl fo th applcaton of th lmnt Eampl fo th applcaton of asmmtc tangl lmnt g shows a hollow ds wth tangla coss scton nd ntnal pss h angla vloct of th ds s = 5 ad/s Consd also th own wght of th ds! Calclat th nodal dsplacmnts and actons! g nt lmnt modl of a hollow ds wth tangla coss scton Gvn: p = KPa E = GPa d = 6 m D = 8 m g = 98 m/s = h = m Solv th poblm sng a sngl asmmtc tangla lmnt []! h dstancs a gvn n [m] th foc s gvn n [] h nodal coodnats a: nod [m] [m] Snc w hav onl a sngl lmnt th lmnt qlbm qaton s th sam as th stctal qaton: wwwtanonvtah Andás Séns BME

229 Modlng of asmmtc stat b EM softwa sstms 9 K 8 wh: w w w 9 Bcas of th bonda condtons onl fo nnowns man : w h constttv mat basd on Eq s: E C Pa h coffcnts of th ntpolaton fnctons sng Eq and g a: m m m m m m h aa of th tangl s: A m h ntpolaton fnctons can b calclatd as: A whch lds: Andás Séns BME wwwtanonvtah

230 nt Elmnt Mthod wwwtanonvtah Andás Séns BME 5 Mat bcoms: 6 Accodngl th stan-dsplacmnt mat B s: B 7 h stffnss mat s gvn b: m dd B C B K 8 wh all of th lmnts w calclatd b act ntgaton sng th cod Mapl h pp ang of th fst ntgaton s th qaton of th hpotns of th tangl: = - h vcto of focs s constctd as th sm of th vctos h fst on s latd to dstbtd load along lmnt dg - basd on Eq t s: ds p p KPa p p p 9 Obvosl th ads s = m constant along lmnt dg - fthmo th coodnat of ntgaton s ladng to:

231 Modlng of asmmtc stat b EM softwa sstms Andás Séns BME wwwtanonvtah 6 6 d pd p h foc vctos latd to th volton and own wght qs vcto q whch s calclatd sng g and : g q q q Aft ths w calclat b sng Eq: b dd g qdd w hav: b nall th nnown actons a collctd n vcto c Consdng th bonda condtons w obtan: c hs th fnt lmnt qlbm qaton bcoms: w 5

232 nt Elmnt Mthod h solton can b obtand b th st d 5 th and 6 th componnt qatons h oth possblt s th applcaton of th mat qaton sng th condnsd stffnss mat whch has alad bn psntd n scton h soltons a: m m 675 m w m 6 h actons tlng th nd and th componnt qatons of th fnt lmnt qlbm qaton a: h ampl was vfd b th fnt lmnt cod ASYS W not that smlal to th ampls of scton w consdd th actons n th vcto of tnal focs h tm / appang n th scond ow of mat B can cas tobl n th cos of ntgaton f on of th lmnt dgs ls on th as of volton wh = o avod ths poblm a local coodnat sstm s ntodcd fo ach lmnt o th ntgaton s mad b appoachng to o b constctng a hol wth v small damt [] Asmmtc sopaamtc qadlatal lmnt h sopaamtc qadlatal lmnt fo plan poblms has bn psntd n scton h lmnt s applcabl to solv asmmtc poblms too h fnctons of th local and coodnats of lmnt dgs a []: 8 wh n Eq77 coodnat was changd to coodnat was changd to Consqntl th sam ntpolaton fnctons can b sd: 9 h dsplacmnt s fomlatd n th sal wa: w 5 wh: wwwtanonvtah Andás Séns BME

233 Modlng of asmmtc stat b EM softwa sstms Andás Séns BME wwwtanonvtah 5 w w w w w wth that th mat of ntpolaton fnctons and th vcto of nodal dsplacmnts a spctvl: 5 w w w w 5 h wll-nown stan-dsplacmnt mat s sd to calclat th stan componnts as: B 5 wh: B 55 As t s shown w nd th dvatvs of th ntpolaton fnctons wth spct to and D to th fact that th fnctons a nown n tms of th natal coodnats and w nd agan th Jacob mat and ts dtmnant fng to Eq w hav []: J J J 56 J J J

234 nt Elmnt Mthod wwwtanonvtah Andás Séns BME wh: J 57 J J J Mat B can b podcd n a smla wa as t was shown b Eq5 cpt fo th fact that w mst consd th tm / appang n th scond ow of th mat h calclaton of th nw tms s possbl ncopoatng Eqs8-9 Coodnat n tms of ξ and η paamts s gvn b Eq8 h fomla of th stffnss mat s: d BJd C B K 58 o povd th vcto of focs w nd th vctos th fst on s: Jd p p Jd p p 59 dpndng on th fact that whch on of th lmnt dgs s loadd b th ln load moov th scond and thd vctos a: d qjd b 6 c 6 nall th total foc vcto s: c b p 6 In th sql w psnt an ampl fo th applcaton of th lmnt

235 Modlng of asmmtc stat b EM softwa sstms 5 5 Eampl fo th applcaton of asmmtc sopaamtc qadlatal lmnt Solv th poblm of th otatng ds of whch analtcal solton has bn psntd n scton 6 sng two sopaamtc qadlatal lmnts! h fnt lmnt modl of th ds s shown n g a h angla vloct of th ds s = 885 ad/s vf f th ds gts loos! b Calclat th stsss n that cas whn th s no volton : = bt th s an ovlap of = - m! Gvn: b = m = m h = m = 78 g/m E = GPa = g A smpl fnt lmnt modl of a otatng ds W gv th dstancs n [m] and th foc n [] h nodal coodnats a: h lmnt-nod tabl bcoms: nod [m] [m] nod [m] [m] 5 6 lmnt nod 5 6 In th nowldg of th bonda condtons th stctal vcto of nodal dsplacmnts s: U 6 w w w 5 w5 6 w6 h constttv mat sng Eq s: Andás Séns BME wwwtanonvtah

236 6 nt Elmnt Mthod C Pa h lmnts of th Jacob mat mst b podcd fo both lmnts basd on Eq57: J J J J 5 65 and: J J J J h lmnts of th Jacob mat and so th dtmnant s constant and dntcal fo both lmnts: J J J 9 67 Contnng th calclaton w compt th dvatvs of ntpolaton fnctons wth spct to and n accodanc wth Eq56 D to th dntcal Jacob dtmnants of th lmnts th dvatvs of th ntpolaton fnctons wll b dntcal too hfo w can omt th spscpts of th lmnts of Jacob mat: wwwtanonvtah Andás Séns BME

237 Modlng of asmmtc stat b EM softwa sstms 7 J J J J J J J J J J J J J J J J J 5 5 J J J 5 5 J J J 5 5 J Coodnat shold b gvn fo both lmnts spaatl basd on Eq8: wh w consdd also th lmnt ontaton th local nmbng of th nods of lmnt As a nt stp w povd th stan-dsplacmnt mat fo ach lmnt sng Eq55 h lmnts of matcs a th fnctons of and whch a tml complcatd thfo w do not gv thm h h lmnt stffnss matcs can b calclatd sng th B matcs: Andás Séns BME wwwtanonvtah

238 8 nt Elmnt Mthod K 6 B 5857 C B Jdd m 7 K 8 B 79 C B Jdd m As th nod nmbng dos not cospond to th lmnt ontatons w nd to aang th lmnt stffnss matcs n accodanc wth th nmals of dgs of fdom t th vcto of nodal dsplacmnts b qal to: w w w w 7 w w 5 w5 6 w6 Cospondng to th fom th ognal lmnt stffnss matcs a aangd as: wwwtanonvtah Andás Séns BME

239 Modlng of asmmtc stat b EM softwa sstms 9 Andás Séns BME wwwtanonvtah K 7 Basd on th nods of th scond lmnt th aangmnt s mad as: K 7 ow w can constct th stctal stffnss mat h mtal nods a th thd and foth ons Accodngl th combnaton of th two matcs slts n: K 75 W not that th fnt lmnt cods povd th stctal stffnss mat sng th lmnt-nod tabl h nmcal vals can b obtand sng Eq7 h foc vcto conssts of th vctos of bod and concntatd focs h dnst vcto of th bod foc s:

240 nt Elmnt Mthod q q and: q q 76 fom whch w hav: b b q q Jdd Jdd Smlal to th stffnss matcs th aangmnt s qd also n th foc vctos accodng to th local nod nmbng: b 686 b b b b7 686 b7 b8 5 b and b8 5 b 78 b b b b b5 5 b b6 b6 h stctal foc vcto s calclatd as th sm th two fom vctos: b b b5 b6 b b b b7 b8 b b b5 b6 79 b b b7 b h vcto contanng th acton s: 8 c h stctal foc vcto s: b c 8 nall th stctal qaton s: KU 8 wwwtanonvtah Andás Séns BME

241 Modlng of asmmtc stat b EM softwa sstms h sstm of qatons conssts of twlv qatons om th st and d - th qatons w dtmn th nodal dsplacmnts h soltons a: 68 m w w 9 m 8 68 m w 65 m w m m w 5 m w 98 m It s sn that f th ds otats wth mamal angla vloct thn n accodanc wth th fnt lmnt modl w do not ach th ovlap val of - m calclatd fom th analtcal modl th ds wll not gt loos hs dsagmnt can b pland b th coas msh of th fnt lmnt modl whch conssts of onl two lmnts h dfomd shap of th stct compad to th ognal stat s shown n g Basd on th dsplacmnt soltons w constct th nodal dsplacmnt vctos of th lmnts: w w w 8 w 5 w5 6 w6 w In th fom two vctos w followd th ognal od of th local nod nmbng bcas mat B was constctd n accodanc wth ths fact h vctos of stan componnts fo both lmnts a calclatd sng mat B : 6 B B 85 h vcto of stss componnts a: C C 86 g Dfomd shap of th fnt lmnt modl of otatng ds Andás Séns BME wwwtanonvtah

242 nt Elmnt Mthod h slts a smmad n abls and In th tabls w lstd th nodal soltons Elmnt soltons a possbl to calclat onl at mtal nods and b avagng th nodal solton Accodng to abl t s sn that th dnamc bonda condtons a volatd conctl spang th adal stss at nods 5 and 6 s not o h ason fo that s th low solton of th msh and th lna ntpolaton On th conta th tangntal stss ags qt wll at th nn and ot bondas wth th slts psntd n ga h ampl was vfd b th cod ASYS lmnt nod [ - ] t [ - ] [ - ] [ - ] abl Stan componnts n th otatng ds n th cas of = 885 ad/s lmnt nod [MPa] t [MPa] [MPa] [MPa] abl Stsss n th otatng ds n th cas of = 885 ad/s In that cas whn th s no otaton th stctal vcto of nodal dsplacmnts bcoms: U w w w w w h stffnss mat mans th sam th vcto of focs s: 88 c h soltons a: m w w 9 m m w 56 m w m wwwtanonvtah Andás Séns BME

243 Modlng of asmmtc stat b EM softwa sstms 8 m w m w 7 m abl contans th stsss n th ds whn th s no otaton Compad to th slts of th analtcal solton th dffncs a qt lag whch can b pland agan b th low solton fnt lmnt msh and th lna ntpolaton lmnt nod [MPa] t [MPa] [MPa] [MPa] abl Stsss n th ds n th cas of = 6 6 Bblogaph [] Jósf Uj cts and pactcs of th sbjct mcal mthods n mchancs PhD fomaton Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng Dpatmnt of Appld Mchancs / spng smst n Hngaan [] Davd V Htton ndamntals of fnt lmnt analss st dton McGaw- Hll w Yo [] OC Znwc R alo h fnt lmnt mthod ffth dton Volm : h bass Bttwoth-Hnmann Ofod Acland Boston Johanns-bg Mlbon w Dlh [] Edogan Madnc Ibahm Gvn h fnt lmnt mthod and applcatons n ngnng sng ASYS Spng Scnc+Bsnss Mda Inc 6 w Yo USA Andás Séns BME wwwtanonvtah

244 MODEIG O HI-WAED SHES AD PAES I- RODUCIO O HE HEORY O SHE IIE EEME MODES Plat and shll thos Plan stcts a calld plats f th thcnss of stct s sgnfcantl lss than th oth dmnsons moov f th stct s loadd ppndclal to ts plan h plat can b bondd along ts sds b an optonal gomtcal objct; th nmatc bonda condtons can b vaos pont-sppotd gdl o lastcall sppotd along th sds smpl sppotd tc [] h plat can b consdd as th tnson of a bam n two dmnsons bcas both mpls th domnanc of th bndng load and most commonl th load s ntodcd tansvsl vthlss th a sgnfcant dffncs too snc g th fl of th bam can b th staght o cvd on th oth hand th mdplan of a plat s alwas flat If th mdplan of th plat s cvd thn t s no long plat bt a shll [] In th sql w ovvw th most mpotant dtals of th tho of plats and shlls h basc qatons of Kchhoff plat tho h Kchhoff plat tho s oftn calld th tho of thn plats W not that f th plat s latvl thc thn th tansvs sha dfomaton can b consdd too h lvant plat solton s povdd b th Mndln plat tho [] Dsplacmnt fld Basd on g w nvstgat th dsplacmnt of a pont of th mdplan of an lastc flat plat [] h dsplacmnt fld can b captd b th componnts: th tansvs dsplacmnt along and th otatons abot and : w wh = s th otaton abot as = s th otaton abot as and w = w s th tansvs dsplacmnt gdsplacmnt of a pont n th mdplan of a flat plat wwwtanonvtah Andás Séns BME

245 Modlng of thn-walld shlls and plats 5 Stan componnts Assmng small stans w can calclat th stan componnts b sng th standsplacmnt qaton dfnd n scton b Eq []: v v w v w w w wh fo th sa of smplct - th dvatvs wth spct to and a ndcatd n th sbscpt In th sql w assm that th coss scton plans man flat and th otwad nomal of ach coss scton s ppndcla to th coss scton plan aft th dfomaton hs assmpton s calld Kchhoff-ov hpothss [] om th latt t follows that n th plans ppndcla to th mdplan of th plat th sha stans a qal to o: w and w Utlng th fom w obtan fom Eq that: w w w h stan componnts bcom: 5 w w w Consqntl n th mdplan ponts = Accodng to th Kchhoff plat tho nd th assmpton of small stans th componnts of th dsplacmnt and stan fld can b dfnd b w Stss fld focs and momnts n th mdplan Assmng plan stss stat w pss th stss componnts b Eqs8 and 5: E w w A E 6 E w w B E Andás Séns BME wwwtanonvtah

246 6 nt Elmnt Mthod E E w C wh E = E/- A B and C a constants Smlal to th tho of bams sbjctd to bndng th stss dstbtons a gvn b lna fnctons along th thcnss dcton as t s shown b g g Dstbton of th stsss along th thcnss dcton of a dffntal plat lmnt h stss copls n th mdplan of th plat a calclatd b ntgatng th stsss ov th thcnss []: M t / t / d A d I E w w 7 t / t / M t / t / d B d I E w w t / t / t / t / M d C d IE w t / t / t / t / M d C d IE w t / t / wh M s th bndng momnt along as M s th bndng momnt along as M and M a th twstng momnts Moov: t I 8 wwwtanonvtah Andás Séns BME

247 Modlng of thn-walld shlls and plats 7 whch s smlal to bams th scond od momnt of nta of th coss scton h stss copls n th mdplan of th plat a dmonstatd n ga h latonshps btwn stsss and stss copls bndng and twstng momnts basd on Eqs6 and 7 a: M 9 I I I M M o th qlbm of a dffntal plat lmnt tansvs sha focs a qd ansvs sha focs a shown b gb and th can b calclatd sng th followng fomla []: t / Q d t / t / Q d t / g Stss copls n th mdplan of a thn dffntal plat lmnt a and ts qlbm n th cas of tansvs sha focs and dstbtd load b h qlbm and govnng qaton of thn plats h homognos qlbm qaton wth spct to th stss fld has alad bn ntodcd n scton []: of whch fst componnt qaton s: Intgatng th qaton wth spct to lds: Ad Cd Andás Séns BME wwwtanonvtah

248 8 nt Elmnt Mthod and: A C fnall: 5 t w ntgat Eq5 wthn th angs of t/ and t/: t / t t / t / d d d 6 / t / t / wh s an ntgaton constant A possbl solton fo whch satsfs vn th dnamc bonda condtons s []: 7 t In fact Eq7 gvs th dffnc btwn th aa nd a ctangl and a paabola whch s / of th total aa Accodngl f t s mltpld b two thn mathmatcall w obtan th aa nd th paabola that s fom Eq6 w hav: t / t / t / d d Q 8 t / whch s not ls than th sha foc along as gvn b Eq ang Eqs6 9 and 8 bac nto th qlbm qaton w obtan: M M Q 9 h scond componnt qaton and th cospondng qlbm qaton n tms of th stss copls and tansvs sha foc a: M M Q wwwtanonvtah Andás Séns BME

249 Modlng of thn-walld shlls and plats 9 and: M M om th thd componnt qaton of Eq w obtan th followng: W ntgat Eq wthn th angs of t/ and t/ wth spct to : t / t / and: d t / t / d t / t / d t / t / t / t / d d d t / t / Basd on Eq th fst two tms a th sha focs Q and Q th thd on s n accodanc wth th dnamc bonda condton th ntnst of th dstbtd load p ppndclal to th mdplan : Q Q p 5 Smmang th qlbm qatons w hav: M M Q 6 M M Q Q Q p o dv th plat qaton w aang th fst two qatons: Q M M 7 Q M M ang thm bac nto th thd of Eq6 w obtan th followng: Andás Séns BME wwwtanonvtah

250 5 nt Elmnt Mthod M M M p 8 B th hlp of Eq7 w hav: I E w w w w w p 9 whch aft a smpl aangmnt hav th fom of [5]: w w w p I E o: p I E w Consqntl th govnng qaton s a foth od patal dffntal qaton wth th pop nmatc and dnamc bonda condtons hat mans that th poblm of plats sbjctd to bndng s a bonda val poblm nt lmnt qatons of thn plats o th fnt lmnt solton of th poblm of thn plats sbjctd to bndng w collct th stan and stss fld componnts nto vctos and w assm plan stss stat [6]: Basd on Eq5 th stan componnts can b wttn as: wh s th vcto of cvats: w w w Incopoatng th matal law w fomlat th vcto of stss componnts as: st C 5 wwwtanonvtah Andás Séns BME

251 Modlng of thn-walld shlls and plats 5 h stan componnts can b obtand b a two-vaabl fncton w th fnt lmnt ntpolaton of th w fncton dpnds on th lmnt tp and th chosn dgs of fdom bt t can alwas b fomlatd n th fom blow: w A 6 wh A s th vcto of nnown coffcnts s th vcto of bass polnomals h vcto of nodal dsplacmnts s: M A 7 whch fo ampl n th cas of a tangl lmnt wth th nods bcoms: w 8 w w In Eq7 mat M can b calclatd basd on th appomat w fncton and Eq h and paamts a th otatons abot th as and n th cospondng nods wh = om Eq7 w hav: A M 9 Gnall spang th vcto of stan componnts can b dtmnd sng th standsplacmnt mat: B wh Eq can b fomlatd tlng Eqs5 7 and 9 as follows: RA RM wh mat R stablshs th latonshp btwn th vcto of stan componnts and th vcto of nnown coffcnts ts dmnson s lmnt dpndnt Consqntl w hav: B RM K ollowng th dfnton b Eq9 w fomlat th lmnt stffnss mat as: B C BdV V st Andás Séns BME wwwtanonvtah

252 5 nt Elmnt Mthod h dmnson of th lmnt stffnss mat dpnds on th nmb of nods and th nmb of nodal dgs of fdom Smlal to th plan mmban lmnt th vcto of focs s composd as th sm of sval tms h most common s th dstbtd sfac load and concntatd foc B fomlatng th wo of tnal focs w dv th foc vcto latd to th dstbtd load: Ap W p w da p wh p s th ntnst of th dstbtd load ppndclal to th mdplan of th plat w s th appomat fncton of th dflcton sfac accodng to Eq6 h vcto p can b dtmnd basd on th vcto of nodal dsplacmnts In th cas of concntatd loads consdng g a tangla shap plat lmnt wth th nods at ach nod th can b a foc ppndclal to th plat sfac and vn concntatd momnts actng abot th and as spctvl: c M M M M M M 5 hs th vcto of focs bcoms: 6 p c Evntall th fnt lmnt qlbm qaton fo a sngl lmnt and fo th whol stct s: K KU 7 Smlal to th plan mmban lmnts th s lag nmb of plat bndng lmnts hs lmnts wll b vwd n scton 5 Basc qatons of th tchncal tho of thn shlls In that cas whn th mdplan of a thn-walld stct s not flat bt cvd thn w tal abot shlls h analtcal nvstgaton of shlls qs consdabl complcatd mathmatcal comptatons hfo n th sql onl th most mpotant qatons wll b psntd Gomtcal qatons D to th fact that th mdsfac of shlls s cvd w nd to ntodc cvlna coodnat sstms as t s shown b g wwwtanonvtah Andás Séns BME

253 Modlng of thn-walld shlls and plats 5 g Coodnat lns and nt bass vctos of th mdsfac of a shll h two-paamt psntaton of th mdsfac of shlls can b fomlatd n th fom of a vcto qaton []: R R q q 8 wh: X X q q Y Y q q Z Z q q 9 a th global coodnats R s th poston vcto of a pont n th q and q a th gnal o cvlna coodnats of th sfac s g If th paamts ta on th vals q = constant and q = constant w obtan th q and q coodnat lns h tangnt nt vctos and th ac lngths ds of th coodnat lns a: R R Hdq H q H ds 5 wh: H R = 5 a th so-calld amé paamts [] o mtc coffcnts [] In th followngs w assm that th local coodnat as a mtall ppndcla at ach pont and th cvlna sstm s othogonal = h otwad nt nomal vcto of th mdsfac bcoms: n 5 h tad of nt othogonal vctos [ n] dtmns an othogonal cvlna coodnat sstm at an actal pont P h cvat and th toson of coodnat lns a gvn b th nt fomla [7]: Andás Séns BME wwwtanonvtah

254 5 nt Elmnt Mthod R R R n n n R = 5 S H q H R R n n R S S H H wh R and R a th ad of cvat If R = thn th q and q lns a th lns of pncpal cvats on th mdsfac moov th dctons of th nt bass vctos and a th pncpal dctons h cvat of th mdsfac s a tnso qantt If th dctons of vctos and a not th pncpal dctons thn th angl whch dtmns th pncpal dctons can b obtand b: ' / R tg 5 / R / R ' ' h vals of th pncpal cvats a [7]: R cos sn sn 55 R R R ' ' ' R sn cos sn R R R R ' ' ' In th followngs w nvstgat th spcal cas whn th dctons of nt bass vctos concd wth th pncpal dctons h dvatvs of th nt bass vctos of th coodnat sstm on th mdsfac a []: H H H R j j j n j j j H H H j n j j j j = 56 R Pont P * s locatd on a sfac paalll to th mdsfac and th dstanc of pont P * fom pont P s gvn b coodnat masd along th nomal vcto n Basd on g th poston vcto of pont P * s: j R * R n 57 h nt vctos a ndpndnt of coodnat v: * 58 h dvatv of th poston vcto of pont P* can b wttn as: wwwtanonvtah Andás Séns BME

255 Modlng of thn-walld shlls and plats 55 * R R n H 59 R Consd th followngs: H * * H and ds ds = 6 R R whch a th amé paamts and ac lngths wth spct to pont P* Stss sltants and copls qlbm qatons g5 shows th stsss on th bonda plans of a dffntal shll lmnt whl g6 psnts th stss sltants and copls ntnal focs and momnts on th mdsfac of th dffntal shll lmnt wth dmnsons of ds ds g5 Stss componnts on th bonda plans of a dffntal shll lmnt g6 Intnal focs and momnts n th mdsfac of a dffntal shll lmnt W mst consd th latonshp btwn th ac lngths ds and ds * gvn b Eq6 whn w stablsh th latonshp btwn th stsss actng on th dffntal shll lmnt Andás Séns BME wwwtanonvtah

256 56 nt Elmnt Mthod wth thcnss t and th ntnal focs momnts on th mdsfac of th shll lmnt h stss sltants and stss copls actng on th cv wth otwad nomal a: t / R t / t / d d Q R d R t / R t / t / t / t / t / M d M d M 6 R t / wh s th n-plan nomal foc and a th n-plan sha focs Q s th tansvs sha foc M s th bndng momnt M and M a th twstng momnts spctvl It mst b tan nto consdaton that althogh th cpoct law of sha stsss mpls = n th qatons abov and M M whch can b pland b th fact that th ad of cvats a n gnal not qal to ach oth : R R h dvlopmnt of qlbm qatons stablshng th qlbm btwn tnal loads and ntnal focs and momnts n th shll stct s also v complcatd hfo w psnt onl th sltng qatons h qlbm qatons n th cas of stss sltants a [7]: Q H H H H HH p R Q H H H H HH p R H Q HQ HH p R R 6 wh p and p a th tangntall dstbtd loads along dctons and p s th dstbtd load ppndclal to th shll mdsfac h qlbm qatons n th cas of stss copls and momnt of stss sltants a: H M H M M H M H H H Q 6 H M H M M H M H H H Q M R M R 6 wh n th sbscpt th comma and th nmb f to th dffntaton wth spct to th cospondng coodnat wwwtanonvtah Andás Séns BME

257 Modlng of thn-walld shlls and plats 57 Dsplacmnt fld stan componnts Basd on g7 th vcto of dsplacmnts and otatons n a pont P on th shll mdsfac can b wttn as: v wn n 65 g7 Dsplacmnt of a pont on th mdsfac of a thn shll In accodanc wth th nmatc hpothss of th shll tho th componnts of vcto n a pont P * ot of th mdsfac a [7]: * * v w w 66 * v th ln of matal ponts whch s ppndcla to th shll mdsfac mans ppndcla dng th dfomaton h qatons dscbng th n-plan stans and changs n cvat a [7]: H v w 67 H H R H H v w H H R H H H v H H H H H H H H H H H H Andás Séns BME wwwtanonvtah

258 58 nt Elmnt Mthod H H H H H H R R wh and a th n-plan stans n th dctons of q and q coodnat lns s th sha stan latd to th chang of angl btwn nt vctos and dng th dfomaton and a th changs n cvats n th dctons of q and q paamts s th twstng cvat h sha stans latd to th nt nomal and nt vctos a [7]: w R H v w R H 68 W assm that dng th dfomaton of shll an actal ln of matal ponts man ppndcla to th cvd shap of shll mdsfac accodngl th sha stans gvn b 68 a qal to o h nmatc hpothss of shll tho togth wth th on mntond bfo s calld th Kchhoff ov hpothss Und thss assmptons w hav: w R H R H v w 69 In oth wods th addtonal tansvs sha dfomaton s nglctd smlal to Kchhoff s tho of thn plats h otaton abot as can b fomlatd b th followng psson [7]: [ H v H ] 7 H H vthlss n most of th cass th otaton abot s nglgbl; thfo t s not consdd n th qatons Appomatons wthn th tchncal tho of thn shlls h shll s consdd to b thn f th thcnss s latvl small compad to th small ads of cvat v []: R 7 Consqntl th amé paamts and th ac lngths on th mdsfac and ot of th mdsfac a appomatl qal whch lads to: H * H and: ds ds = 7 * Accodngl Eq6 can b smplfd sgnfcantl: wwwtanonvtah Andás Séns BME

259 Modlng of thn-walld shlls and plats 59 t / d d Q d 7 t / t / t / t / t / t / M d M d M t / t / t / It s sn that n ths cas th tansvs sha focs and tosonal momnts a qal to ach oth whch volats th qlbm qatons gvn b Eq 6 hs appomaton s pmttd wthn th tchncal tho of thn shlls 5 Majo stps n th fnt lmnt modlng of shlls In th cos of th fnt lmnt dsctaton of shlls smlal to th plan and plat poblms w pocd th ntpolaton of th gomt and th dsplacmnt fld [7] h vcto of dsplacmnt and otaton componnts n a pont locatd on th shll mdsfac s: v w 7 h componnts of ths vctos a not ndpndnt of ach oth om Eq67 w calclat th n-plan stans and th changs n cvat: 75 W collct th n-plan focs and momnts nto a vcto: 76 M M M M ansvs sha focs Q Q a not consdd n th calclaton of th dfomaton nall th vctos of th sfac loads and concntatd focs and momnts a: p p p p 77 Q M M M Andás Séns BME wwwtanonvtah

260 6 nt Elmnt Mthod wh p contans th dstbtd loads n th dctons of coodnat lns q and q and also th dstbtd load ppndclal to th shll mdsfac and M contan th concntatd focs and momnts actng n th nods Usng th vctos gvn b Eqs75-77 th total potntal ng s fomlatd as: A M HH dqdq phh dqdq A S M ds 78 W assm that th matal of th thn shll s lna lastc homognos and sotopc hn th vcto of n-plan focs and vcto of momnts can b calclatd as follows: st t st tc M C 79 wh th constttv mat assmng plan stss stat s: E C st 8 Accodngl Eq78 bcoms: A t C st A t C ph H dq dq st H H dq dq S ds M 8 Utlng th dfnton of th lmnt stffnss mat and th vcto of nodal focs w can dv th psson blow: K 8 fom whch th fnt lmnt qlbm qaton fo a sngl lmnt th fst of Eq7 can b dvd As a nt stp w smma th potntal ng of ach lmnt: U KU U 8 and fnall applng th mnmm pncpl of th total potntal ng w obtan th stctal qlbm qaton: wwwtanonvtah Andás Séns BME

261 Modlng of thn-walld shlls and plats 6 KU 8 o th fnt lmnt modlng of shlls th s v lag nmb of lmnt tps ot onl th flat shll lmnts whch gv mo accat slt nd hgh msh solton bt also th cvd g clndcal shll lmnt and dobl-cvd shll lmnt tps a avalabl whch appomat btt both th gomt and th dsplacmnt fld sng th sam lmnt nmb h dffnt plat and shll lmnts a dscssd n sctons Bblogaph [] Im Bojtá Gábo Vöös Applcaton of th fnt lmnt mthod to plat and shll stcts chncal Boo Pblsh 986 Bdapst n Hngaan [] Robt D Coo nt lmnt modlng fo stss analss John Wl and sons Inc 995 w Yo Chchst Bsban oonto Sngapo [] Gábo Vöös cts and pactcs of th sbjct Elastct and fnt lmnt mthod manscpt Appld Mchancs Modl Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng Dpatmnt of Appld Mchancs Bdapst n Hngaan [] P Ch Cho cholas J Pagano Elastct nso dadc and ngnng appoachs D Van ostand Compan Inc 967 Pncton w Js oonto ondon [5] S moshno S Wonows-Kg ho of plats and shlls chncal Boo Pblsh 966 Bdapst n hngaan [6] Sngs S Rao h fnt lmnt mthod n ngnng foth dton Elsv Scnc & chnolog Boos [7] Edad Vntsl hodo Kathamm hn plats and shlls ho analss and applcatons Macl D Inc w Yo Basl Andás Séns BME wwwtanonvtah

262 5 MODEIG O I-PAE HI-WAED SHES UDER I- PAE AD RASVERSE OAD BY IIE EEEM MEHOD BASED SOWARE SYSEMS 5 Plat lmnts sbjctd to bndng lat plat lmnts a stabl to dtmn th ntnal focs stss sltants and stss copls n plat shap stcts h plat lmnt s th tnson of bam lmnts so that bndng sha and toson ta plac n two othogonal plans nvolvng som ntactons Smlal to th plan mmban lmnts th tangl and qadlatal shap lmnts a avalabl fo th modlng of shlls h applcaton of gnal tangl shap lmnts s asonabl whn th shap of th stct s gla tangla o smla to th tangl In ths scton w ovvw pmal th plat lmnts sbjctd to tansvs load In that cas whn th plat s loadd n-plan and also tansvsl w can solv th poblm b combnng th plan mmban and plat bndng lmnts W hav alad sn b Eq that d to nglctng th tansvs sha focs th otatons n an actal pont of th plat a: w and w 5 h cvats latd to th bndng dfomaton a: w w and 5 w o thn plats w assm plan stss stat : 5 In th cos of th ntodcton of Kchhoff plat tho w hav obsvd that th dflcton sfac s gvn b a two-vaabl w fncton wth that both th cvats and stan componnts can b calclatd o plat bndng poblms ths w fncton mst b podcd b ntpolaton polnomals and thn w can povd th lmnt stffnss mat and foc vcto In th followngs w gv th dtals of fw lmnt tps fo plat bndng 5 angla plat bndng lmnt o och tangl lmnt In th cos of th fnt lmnt dsctaton of plat shap stcts w appomat th tansvs dflcton b a thd od polnomal n tms of th and coodnats []: w a a 5 a a a a a5 a6 a7 8 wwwtanonvtah Andás Séns BME

263 5 Modlng of n-plan thn-walld shlls 6 hs appomaton was on of th fst tangla fnt lmnts whch was pblshd b och [] h lmnt s shown n g5 h dflcton sfac n vcto fom s: w A 55 wh A s th vcto nnown coffcnts s th vcto of bass polnomals spctvl: a a a a a5 a6 a7 a8 A a 56 g5 h nn dgs of fdom och tangla plat lmnt h nnown coffcnts can b calclatd basd on th nodal condtons aml th dsplacmnt fncton mst gv bac th actal nodal dsplacmnt f w sbsttt th nodal coodnats of th sam nod hfo n Eq5 th nmb of tms s alwas qal to th nmb of dgs of fdom o th och tangl lmnt th ghth tm contans th sm of and Actall n vcto A th s nn nnown coffcnts ollowng g5 th vcto of nodal dgs of fdom fo a sngl lmnt s: w 57 w w wh w s th tansvs dsplacmnt ppndclal to th mdplan and a th otatons abot as and spctvl Accodngl th och plat lmnt has nn dgs of fdom h nodal condtons fo th calclaton of th coffcnts a []: w w w w 58 Andás Séns BME wwwtanonvtah

264 6 nt Elmnt Mthod wwwtanonvtah Andás Séns BME w w w w w w w w W nd th dvatvs of th w fncton wth spct to and to calclat both th coffcnts and th stan componnts w can wt sng Eq 5: 7 6 a a a a a w a a a a a w a a a w a a a w a a w h sbsttton of th dvatvs abov nto Eq57 lads to th followng sstm of qaton dcd to mat fom []: M A 5 wh: M 5

265 5 Modlng of n-plan thn-walld shlls 65 h coffcnts of th ntpolaton fncton a th soltons of th sstm of qaton gvn b Eq5: A M 5 h pssons of th coffcnts a tml complcatd; hnc th a not dtald h h vcto of stan componnts basd on Eqs5 and 55 a: w w RA 5 w wh mat R s: 6 R 6 5 ang bac Eq5 nto Eq5 th stan-dsplacmnt mat can b dvd: RA RM B 55 o thn plats w assm plan stss stat consqntl w can wt: st st C C B 56 wh matc st lmnt stffnss mat s: fs to plan stss stat Accodng to Eq th dfnton of th K B C BdV 57 V st Incopoatng Eq55 w obtan: t / st K M R C RddA M 58 t / A h mddl tm n th psson abov s []: Andás Séns BME wwwtanonvtah

266 66 nt Elmnt Mthod wwwtanonvtah Andás Séns BME 6 6 } 8 8 { 6 6 / / da E I da Rd C R A A t t sf 59 wh I = t / and E = E/- o calclat th stffnss mat th nvs of mat M s qd Snc t s v complcatd t s not dtald h In Eq59 t s possbl to smplf th componnts b th sfac ntgal tansfomatons gvn blow []: ] [ A dd da A 5 A dd da A A dd da A 6 A dd da A 6 A dd da A A dd da A 6 A dd da A

267 5 Modlng of n-plan thn-walld shlls 67 Andás Séns BME wwwtanonvtah 6 A dd da A A dd da A A dd da A w h A s th tangl aa and = a th nodal coodnats spctvl In most of th cass th foc vcto s composd b two tms h foc vcto latd to th dstbtd foc can b dvd b pssng th wo of tnal foc: p A da pw W p 5 h calclaton of p s dffclt w nd th nvs of mat M and th smplfcaton of sfac ntgals spctvl h concntatd focs and momnts a collctd n a vcto n accodanc wth th nodal dgs of fdom: c M M M M M M 5 wh s th concntatd foc ppndclal to th mdplan of plat M and M a th concntatd momnts actng n th and dctons In th sql w psnt a dtald ampl 5 Eampl fo th applcaton of th och tangl plat lmnt Dtmn th dsplacmnt and th actons of th blt-n plat dpctd n g5 []! g5 angl shap blt-n plat loadd b concntatd foc

268 68 nt Elmnt Mthod Gvn: E = GPa = t = 5 mm = a = mm b = 75 mm h nodal coodnats a: nod -b/ a b/ In th sql th dstancs a calclatd n [mm] th foc s gvn n [] Bcas of th nmatc constants blt-n nods th vcto of nodal dsplacmnts bcoms: w 5 o th calclaton of stffnss mat w nd th constttv mat whch s: C C st E GPa Utlng th nodal coodnats w calclat mat M basd on Eq5: M h dtmnant of mat M s 86 th mat s not sngla ts nvs sts h stffnss mat s obtand b calclatng mat R s Eq5 and comptng th sfac ntgals: K M RddA M t / st R C A t / wwwtanonvtah Andás Séns BME

269 5 Modlng of n-plan thn-walld shlls mm 56 In Eq56 onl th ndpndnt componnts a ndcatd th ason fo that s th stffnss mat s alwas smmtc h foc vcto basd on th concntatd loads s: c M M M M 57 h condnsd stffnss mat and th sltng mat qaton fo th calclaton of nodal dsplacmnts s th followng: w h nodal soltons a: w 76 mm 76 ad 77 ad 59 It s sn that althogh th poblm s smmtc wth spct to as fo both th gomt and load th dfomaton of th tangl lmnt s not smmtc ang th nodal dsplacmnts bac to ognal qaton w can dtmn th actons: 555 M 785 mm M mm 5 Usng Eq55 th vctos of stan and stss componnts a: RM C st h stan and stss componnts can b obtand at an pont of th tangl lmnt b tang bac th coodnats n [mm] h ampl abov was vfd b a fnt lmnt Andás Séns BME wwwtanonvtah

270 7 nt Elmnt Mthod cod dvlopd n Matlab [] and w obtand th sam slts In gnal th accac of th och plat lmnt s not satsfacto and vn th convgnc of th slts s bad o dc th dfcncs of th och tangl th so-calld dcd tangl lmnt was dvlopd wh aa coodnats a ntodcd [] Apat fom th och tangla plat lmnt th a sval mo lmnt tps g: Adn o Cowp tangl lmnt Adn- Clogh-Mlosh Bogn-o-Scmt ctangl lmnt tc [] In th sql w psnt som ctangl shap plat lmnts 5 Incompatbl ctangla shap plat lmnt g5 psnts on of th fst ctangl shap lmnts n a global local and natal coodnat sstm g5 Incompatbl ctangl shap plat lmnt n a global a local b and natal c coodnat sstm h dmnsonlss local and coodnats a: and: d d d d 5 a b a b h followng dffntal qotnts a also qd: d d d d 5 d a d d b d wwwtanonvtah Andás Séns BME

271 5 Modlng of n-plan thn-walld shlls 7 Accodng to g5b w consd th dgs of fdom n th local coodnat sstm at ach nod whch a th dsplacmnt w ppndclal to th mdplan of plat and th otatons abot th and as spctvl h vcto of nodal dsplacmnts fo a sngl lmnt s: w 5 w w w totall th lmnt has dgs of fdom wh th otatons can b dtmnd b mans of Eq5 and th Kchhoff-ov hpothss h dsplacmnt n dcton and th otatons a not ndpndnt of ach oth and th can onl b to a mamm of nnown paamts n th ntpolaton fncton Along th lmnt dgs th psson of w shold b a thd od fncton and accodngl th dvatv n th nomal dcton shold va lnal [] A complt thd od fncton contans tms bt n accodanc wth th nmb of nodal paamts w nd two addtonal tms n th ntpolatd fncton W can choos fom th th possblts blow: and o: and o: and 55 An of th abov possblts s chosn w obtan a cbc chang n th dvatvs n th nomal dcton nstad of th pctd lna on [] hfo ths lmnt s not compatbl n oth wods t s ncompatbl Choosng th fst altnatv w hav: w a a a a a a a 7 8 a a a a a h nodal condtons fo th dtmnaton of th nnown coffcnts a: w w w b w 57 a w w w b w a w b w a w w w w w b w a ang bac th coffcnts nto th fncton gvn b Eq56 moov b tlng th fact that th dsplacmnt fncton can b fomlatd as th podct of ntpolaton fnctons and nodal paamts t s possbl to obtan: Andás Séns BME wwwtanonvtah

272 7 nt Elmnt Mthod w w 7 8 w w w 58 fom whch w obtan th mathmatcal fom of th ntpolaton fnctons: 59 b a 5 b a 6 7 b 8 a 9 b a and: w 5 wh: wwwtanonvtah Andás Séns BME

273 5 Modlng of n-plan thn-walld shlls 7 s th vcto of ntpolaton polnomals As a nt stp w pss th vcto of stan componnts sng Eq5: w w w wh: 5 5 wh and a vctos contanng th scond od dvatvs of th ntpolaton fnctons b Eq5 wth spct to th cospondng sbscpt Hnc th vcto of stan componnts and th vcto of stss componnts bcom: 5 st st C C h vcto of bndng and twstng momnts can b gvn n vcto fom; th a calclatd basd on Eqs7 and 5: M M M M IE 55 ang th pvosl calclatd vctos bac nto th total potntal ng w obtan: V st dv pda C dv pw da 56 Ap V Ap W tansfom th volm ntgal ov th lmnt b ntgaton wth spct to th paamts and Moov w assm that n th scond tm th ntnst of th dstbtd load s constant Consqntl w can wt: K 57 p st t ab C dd p abdd wh th lmnt stffnss mat s: Andás Séns BME wwwtanonvtah

274 7 nt Elmnt Mthod t st K ab C dd 58 and th foc vcto fom th nfoml dstbtd load s: p p abdd pab b a 6 6 b 6 a 6 b 6 a 6 b 6 a 6 59 that s smlal to th bam lmnt sbjctd to bndng th dstbtd load s psntd b concntatd focs and momnts at th nods fng to th dsctaton pocd It s also ncssa to consd that th can b concntatd loads n th nods v: c and: M M M M M M M M c p h applcaton of th mnmm pncpl lds th lmnt qlbm qaton: K 55 whch can b sd onl f th stct conssts of a sngl lmnt o mlt-lmnt stcts w obtan th stctal qaton b smmng th potntal ngs of th lmnts: KU 55 t s solv an ampl fo th ncompatbl ctangl shap lmnt! 55 Eampl fo th applcaton of th ncompatbl ctangl shap lmnt Calclat th nodal dsplacmnts and th actons of th blt-n plat shown n g5! wwwtanonvtah Andás Séns BME

275 5 Modlng of n-plan thn-walld shlls 75 g5 Eampl fo th applcaton of ncompatbl plat lmnt Gvn: E = GPa = t = mm = 5 a = 6 mm b = mm In th sql th dstancs a sbstttd n [m] th foc s gvn n [] h nodal coodnats a: nod a a b b Consdng th nmatc constants n th constcton of th vcto of nodal dsplacmnt w obtan: w w 55 c h foc vcto consdng th tnal foc and th actons s: 555 M M M M o plan stss stat th constttv mat s: 6 E C st 6 GPa t w calclat mat whch s qd fo th stffnss mat: Andás Séns BME wwwtanonvtah

276 76 nt Elmnt Mthod h dmnson of th stffnss mat s ; thfo t s not dtald h Instad of th stffnss mat w gv th sltng fnt lmnt qlbm qaton sstm fom Eq55: 7w 5 8α- 87β-96 5w 58 6α - 69β w w 885 M 87w 6 69w 87 M 96w w w w w w 7-75w w w 65-7w 5-5 wwwtanonvtah Andás Séns BME

277 5 Modlng of n-plan thn-walld shlls 77 9w 7 557w w w w w 59 M 69w 87 87w 6 M Calclatng th nodal dsplacmnts fom th th 5 th 6 th 7 th 8 th and 9 th qatons of Eq556 w hav: w 655 m 9 ad 59 ad 559 w 8 m 65 ad ad om th st nd d and th th th qatons of Eq556 w can dtmn th actons: 5 M 7 m M 79 m 56 M m M m h bndng and twstng momnts can b obtand fom Eq55 th stsss can b dtmnd fom Eq5 b tang bac th nodal coodnats Eampl 55 was vfd b th fnt lmnt cod ASYS and w obtand th sam slts 56 Compatbl ctangla shap plat lmnt In that cas whn w want to dvlop a compatbl plat lmnt th ntpolaton fncton gvn b Eq56 has to b modfd n accodanc wth th followngs []: w a a a a a a a a a 9 a a a 5 a 6 a 7 a 8 a 5 56 Howv ths fomlaton mpls 6 nnown nodal paamts hat s at ach nod w mst consd th md dvatv w h vcto of nodal dsplacmnt bcoms: w w w w w w w w 56 h condtons fo th dtmnaton of th nnown paamts a: w w w b w a w ab w 56 Andás Séns BME wwwtanonvtah

278 78 nt Elmnt Mthod w w w b w a w ab w w b w a w ab w w w w w w b w a w ab w h dflcton sfac s appomatd b sng 6 ntpolaton fnctons: w w 9 w w w 5 w w w 6 w 56 h ntpolaton fnctons can b wttn b th hlp of th Hmtan polnomals whch a psntd n th bam fnt lmnts s g55 []: f f f 565 f wth that th 6 ntpolaton fnctons bcom: f f 9 f f 566 b f f b f f a f f a f f a b f f a b f f f f f f 5 b f f b f f 6 a f f a f f 7 5 a b f f a b f f 8 6 wwwtanonvtah Andás Séns BME

279 5 Modlng of n-plan thn-walld shlls 79 g55 ncton plot of th Hmtan ntpolaton polnomals B sng th ntpolaton polnomals th stffnss mat can b blt-p b th sam mthodolog as that psntd n th ncompatbl plat lmnt h onl dffnc s that w obtan a mat wth dmnson of 66 Assmng a constant dstbtd foc th lvant tm n th foc vcto s: p p abdd pab b a ab b 6 a 6 ab 6 b 6 a 6 ab 6 b 6 a 6 ab smlal to th plan bam lmnt sbjctd to bndng th dstbtd load s psntd b concntatd focs and momnts n th nods As sal w hav to consd th cas of concntatd loads th lvant vcto tm s: c M M M M M M M M M M M M 568 Eampl 55 was solvd b sng th compatbl plat lmnt too In ths cas th nodal dsplacmnts a: w 658 m 6 ad 65 ad ad w 569 m w 5 m ad 8 ad ad w 5 m h actons a gvn blow: M m M 8 m M 8 m 57 Andás Séns BME wwwtanonvtah

280 8 nt Elmnt Mthod 5 M 7 m M 55 m M 5 m It s sn that th dffnc btwn th slts of th two soltons s not sgnfcant 57 Plats nd n-plan and tansvs load If th plat s loadd b n-plan and tansvs focs smltanosl thn w hav to podc an lmnt b havng both n-plan and bndng load-cang capablt t mans th spposton of plan mmban and plat bndng lmnts hs poblm can b solvd basd on sctons and 5 n a latvl smpl wa st w collct th cospondng nodal dsplacmnts nto a vcto Scond w cat th stffnss mat of th combnd lmnt b placng th stffnss mat componnts cospondng to th mmban and bndng dfomaton nto th ght postons h vcto of focs s obtand b a smla combnaton of th lmnt vctos hs tchnq s stabl to modl n-plan plat stcts too Howv f w connct th lmnts b contanng an angl dffng fom 8 among th sfacs thn t s possbl to appomat cvd sfacs In oth wods th combnd mmban-plat lmnt s stabl to modl spatal shlls and shll stcts too Snc n th modlng of plan and spatal shlls smla stps a qd ths sss wll b dtald n scton 6 58 Bblogaph [] Sngs S Rao h fnt lmnt mthod n ngnng foth dton Elsv Scnc & chnolog Boos [] Im Bojtá Gábo Vöös Applcaton of th fnt lmnt mthod to plat and shll stcts chncal Boo Pblsh 986 Bdapst n Hngaan [] Al Ogl Edg ods of angla Kchhoff Bndng hn Plat EM Adn och BCIZ Ag 6 Updatd 5 Sp 6 [] OC Znwc R alo h fnt lmnt mthod ffth dton Volm : h bass Bttwoth-Hnmann Ofod Acland Boston Johannsbg Mlbon w Dlh wwwtanonvtah Andás Séns BME

281 6 MODEIG O SPAIA HI-WAED SHES BY IIE EEME MEHOD-BASED SOWARE SYSEMS 6 Smpl flat shll lmnts h stffnss mat of flat shll lmnts a asl calclatd sng th stffnss matcs of th mmban and plat bndng lmnts Accodngl t s possbl to dv th dffnt vson of flat shll fnt lmnts b combnng th avalabl tangl and ctangl shap lmnts [] h appomaton of a cvd sfac b flat shll lmnts s shown b g6 hs nd of appomaton s anoth soc of o apat fom th dsplacmnt fld ntpolaton B ncasng th nmb of lmnts w can dcas th gomtcal naccacs h applcaton of flat shll lmnts s jstfd whn th advantag of th hgh od lmnts naml th lag lmnt s can not b plotd In th sql w dmonstat th combnaton of th lna mmban tangl and th och plat bndng lmnts g6 angla shap flat shll lmnt n th global and local coodnat sstms 6 Spposton of th lna tangl and och bndng plat lmnts h lmnt mntond abov s not confom bcas of th dscontnt of dsplacmnts at th lmnt bondas [] Howv d to ts smplct w s ths combnaton to dmonstat th applcaton of flat shll lmnts h lna tangla mmban lmnt s g has two dgs of fdom at ach nod th stffnss mat n th local lmnt coodnat sstm s: Andás Séns BME wwwtanonvtah

282 8 nt Elmnt Mthod wwwtanonvtah Andás Séns BME ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 6 6 m m m m m m m m m m K 6 wh th sbmatcs m j ~ cospond to th stffnss mat componnts assocatd wth nods and j h tld ov th mat ndcats th local coodnat sstm; th spscpt m fs to th mmban acton h fnt lmnt qaton s: ~ ~ ~ m m m K 6 wh th vcto of nodal dsplacmnts and concntatd focs of th mmban lmnt a: 6 ~ ~ ~ ~ ~ ~ ~ v v v m 6 6 ~ m c wh s th dsplacmnt n th local v s th dsplacmnt n th local dcton In th dsplacmnt vcto w f to th local paamts b sng th tld In th foc vcto w dntf th local paamts b lowcas and n th sbscpt of componnts A dstncton l that was not ncssa ntl now whch can b pland b th fact that n all of th pvos ampls th local and global coodnat sstms concdd At ach nod of th och tangla plat lmnt s g5 th a th dgs of fdom; thfo th stffnss mat has nn ows and nn colmns: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 6 6 b b b b b b b b b b K 6 wh th spscpt b ndcats bndng acton In th local coodnat sstm th dsplacmnt and concntatd foc vctos of th och tangla plat lmnt a:

283 6 Modlng of spatal thn-walld shlls 8 Andás Séns BME wwwtanonvtah 9 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ w w w h 65 9 ~ h c M M M M M M h dgs of fdom of th combnd lmnt a shown b g6 h mmban and bndng stffnss matcs hav to b combnd n accodanc wth th followng obsvatons []: a fo small dsplacmnt s th mmban and bndng stffnsss a ncopld ndpndnt b th n-plan otaton n th local - plan s not ncssa fo a sngl lmnt howv and ts conjgat momnt M hav to b consdd n th analss b ncldng th appopat nmb of os to obtan th lmnt stffnss mat fo th ppos of assmblng sval lmnts o assmblng th flat shll lmnt wth dffnt tp of lmnts g6 Combnaton of th lna mmban tangl lmnt and th och tangla plat bndng lmnt h nodal dsplacmnt vcto of th combnd lmnt n th local coodnat sstm s: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 8 w v w v w v b m 66 h vcto of concntatd focs bcoms: ~ 8 b m c M M M M M M M M M 67

284 8 nt Elmnt Mthod Accodngl th stffnss mat s shown blow []: ~ mb K = 88 ~ m ~ m ~ m ~ b ~ b ~ b ~ m ~ m ~ m ~ b ~ b b ~ m ~ m ~ m ~ b ~ b ~ b 68 h stffnss mat abov s vald n th local coodnat sstm W hghlght agan that th tld ov th matcs and vctos fs to th local sstm In th analss of thdmnsonal stcts n whch dffnt fnt lmnts hav dffnt ontatons t s ncssa to tansfom th local stffnss matcs to a common st of global coodnats In th qantts of global coodnat sstm th s no tld ndcatd h tansfomaton of th lmnt stffnss mat s gvn b th psson blow: mb ~ mb K K wh s th tansfomaton mat wth dmnson of 8 8: wwwtanonvtah Andás Séns BME

285 6 Modlng of spatal thn-walld shlls 85 Andás Séns BME wwwtanonvtah 6 and: 6 Mat contans th nt bass vctos of th local coodnat sstm and s g6 n th fom of colmn vctos fomlatd n th global coodnat sstm: Z Z Z Y Y Y X X X 6 Evntall mat contans th dcton cosns of th angls btwn th local and global as h dfnton of th dcton cosns fo an optonal A vcto basd on g6 s []: cos A A A A l 6 cos A A A A m cos A A A A n

286 86 nt Elmnt Mthod g6 Dcton cosns of vcto A Snc th bass vctos and a nt vctos t s not as to s that th componnts a vntall th dcton cosns o constct th stctal stffnss mat th local qantts hav to b tansfomd nto th global sstm h vcto of nodal dsplacmnt and vcto of focs n th global sstm a: mb mb ~ 6 m b m b ~ o shll stcts th most common load tp s th constant pss ppndclal to th shll sfac n dcton of th local as It s a asonabl assmpton that th s mmban stss stat nd ths assmptons th foc vcto s: []: A p ~ mb p 65 wh A s th tangla aa If th pss on th shll sfac s not constant bt ts chang ov th lmnt aa s nsgnfcant thn w can stll s th vcto abov bt w s th pss avagd b th nodal loads nstad of p: p p p p 66 nall w smma th fnt lmnt qatons In th local sstm th qantts ndcatd b th tld a sd : ~ K mb ~ mb ~ mb 67 ansfomng Eq67 nto th global XYZ sstm b sng th tansfomaton mat w hav: wwwtanonvtah Andás Séns BME

287 6 Modlng of spatal thn-walld shlls 87 K mb mb mb 68 wh th qantts n th global coodnat sstm a calclatd basd on Eqs69 and 6 o th whol stct th fnt lmnt qaton s: K mb U mb mb 69 of whch soltons a th componnts of vcto U m+b whch a th nodal dsplacmnts n th global coodnat sstm om that w can calclat th global lmnt dsplacmnt vctos m+b and thn w can tansfom thm nto th local sstm b th followng psson: m b m b ~ 6 h tansfomaton mat s othogonal thfo w can wt that: E v: and m b m b ~ 6 Usng th local dsplacmnts n th nods w can calclat th mmban and bndng stsss A sgnfcant advantag of th flat shll lmnts s a novl softwa can b asl constctd b combnng th softwas of th stng mmban and plat lmnts whch can b sd fo ngnng calclatons [] hs comptaton qs onl th nowldg of mat h accac of th slts dpnds on th lmnt s Hgh msh solton s ncssa wh th cvat of th sfac s lag o th chang n stsss s pctd to b mo sgnfcant h pctd o of th calclaton s hgh n th vcnt of th sds notchs and th conncton of dffnt sfacs t s solv an ampl to ndstand th applcaton of th mthod! 6 Eampl fo th combnaton of th lna tangl and och tangl lmnts Solv th shll poblm gvn n g6! Calclat th nodal dsplacmnts actons n th local coodnat sstm and tansfom th slts nto th global coodnat sstm! Andás Séns BME wwwtanonvtah

288 88 nt Elmnt Mthod g6 lat tangla shll lmnt n th local and global coodnat sstms a applcaton ampl fo th flat shll lmnt b Gvn: a = 8 m b = 5 m t = mm E = GPa = = 6 = 8 p = /m h dstancs a sbstttd n [m] th foc s ntptd n [] h nodal coodnats n th local coodnat sstm a: nod -b/ a b/ W gv th nodal coodnats also n th global coodnat sstm W not that th global coodnats dpnd on how th lmnt s blt-n th actal stct nod X [m] Y [m] Z [m] h vctos of nodal dsplacmnts and concntatd focs n th local coodnat sstm a: mb 8 ~ ~ ~ ~ ~ ~ ~ ~ ~ v~ ~ ~ v ~ 6 ~ mb c 8 wwwtanonvtah Andás Séns BME

289 6 Modlng of spatal thn-walld shlls 89 h tms n th foc vcto latd to th dstbtd load can b calclatd basd on th ntgal tansfomaton fomla psntd n scton 5 W can constct th foc vcto of th och tangl fom dstbtd load b fomlatng th wo of th dstbtd load: W ~ pw da ~ 6 Ap p Basd on Eq6 and th ntgal tansfomaton pssons gvn b Eq5 w hav: 7 / ba /8b 669 /ab / ba 6 /ab / 8b 6 / ba 6 ~ b p p /ba p /b 7 / ba /8b 7 /ab / ba 97 /ab / 8b 69 W not that n th och tangl th dstbtd load s dvdd nto th pats and pt nto th nods; howv th dvson s mad n nqal dg as t s sn n th vcto of focs On th oth hand b smmng th focs n dcton and th momnts abot and w obtan: ~ b ~ b ~ b p p p pa pab 65 7 ~ b ~ b ~ b pa pa p A p 5 p 8 5 b ~ b ~ b ~ b pb pab p A p 5 p 8 a whch a th sltant focs n dcton and th sltant momnts abot as and h foc vcto of th 8 dgs of fdom flat shll lmnt n th local coodnat sstm s: Andás Séns BME wwwtanonvtah

290 9 nt Elmnt Mthod wwwtanonvtah Andás Séns BME ~ ~ ~ b m p b m c b m p 66 h stffnss mat of th lna mmban tangl lmnt basd on th calclatons of scton 5 s: m ~ m K 67 wh d to smmt of th mat onl th ndpndnt componnts a ndcatd h stffnss mat of th och tangla lmnt n th local coodnat sstm s:

291 6 Modlng of spatal thn-walld shlls ~ b K h combnaton of th mmban and bndng stffnss matcs basd on Eq68 lad s to: 66 ~ mb K h nt stp s th constcton of th fnt lmnt qaton sng Eq67 h nodal dsplacmnts a calclatd fom th th 5 th 7 th 8 th th th th 6 th and 7 th qatons of th sstm of qatons h soltons a: Andás Séns BME wwwtanonvtah

292 9 nt Elmnt Mthod ~ ~ 6 ad 78 ad 6 ~ 87 m v ~ 68 m ~ ~ 98 ad 9 ad v ~ 77 m ~ 8 ad ~ 99 ad In th nowldg of th dsplacmnts w can dtmn th actons tlng th st nd d 9 th th and 5 th qatons: W not that th slts b Eq6 a th componnts of th vcto of concntatd focs whch s th fst tm n Eq66 Moov th 6 th th and 8 th componnt qatons w not tld h whch s pland b th fact that ths qatons a assocatd to th local otatons abot as and th vals a o n th local sstm h total foc vcto b sng Eq6 and Eq66 bcoms: ~ mb In th sql w tansfom th slts nto th global XYZ coodnat sstm h global poston vctos of th nods basd on th global coodnats a: wwwtanonvtah Andás Séns BME

293 6 Modlng of spatal thn-walld shlls R m R 5 m R 5 m h poston vcto of th ogn of local coodnat sstm can b gvn n th global sstm as: 975 R R R 5 m 6 5 W dtmn th nt bass vctos basd on th poston vctos n th global sstm sng g6: 685 R R R R 875 Smlal th nt vctos and a 959 R R 66 R R 8 97 R R 688 R R 9659 Basd on th nt bass vctos and Eq6 mat bcoms: Wth that t s possbl to constct mat wth dmnson of 88 Usng Eqs6 6 and 6 th nodal dsplacmnts n th global coodnat sstm a: v w 68 ad 85 ad 5 ad Andás Séns BME wwwtanonvtah

294 9 nt Elmnt Mthod 66 m v 75 m w 88 m 5 ad 958 ad 5 ad 765 m v 7 m m w 8 ad 587 ad 7797 ad It s sn that althogh n th local coodnat sstm th otatons abot a o n th global sstm as a slt of th tansfomaton vn otatons abot Z st h nodal focs a th followngs: X 886 Y 77 Z M X 6 m M Y 8859 m M Z 68 m X 9769 Y 575 Z 98 M X 87 m M Y 55 m M Z 7856 m 765 X Y 77 Z 79 M X 99 m M Y 965 m M Z 98 m Accodngl n th global sstm th a bndng momnts abot as Z whch a n fact th pojctons of th momnts abot local and as wth spct to Z h solton mthod s applcabl also fo ctangl shap lmnts 6 Bblogaph [] Sngs S Rao h fnt lmnt mthod n ngnng foth dton Elsv Scnc & chnolog Boos [] Im Bojtá Gábo Vöös Applcaton of th fnt lmnt mthod to plat and shll stcts chncal Boo Pblsh 986 Bdapst n Hngaan [] Klas-Jügn Bath nt lmnt pocds Pntc Hall Upp Saddl Rv 996 w Js 758 [] P Ch Cho cholas J Pagano Elastct nso dadc and ngnng appoachs D Van ostand Compan Inc 967 Pncton w Js oonto ondon wwwtanonvtah Andás Séns BME

295 7 MODEIG O CURVED AD DOUBY-CURVED SHES BY IIE EEME MEHOD BASED SOWARE SYSEMS 7 Cvd shll lmnts Cvd shll lmnts a stabl to modl th mdsfac gomt mo accatl In th cas of ctan sfacs fo ampl th clndcal shll t mans th act dscpton of th ognal sfac o mo complcatd cass smlal to th dsplacmnt fld - th cvats of th sfac a appomatd b ntpolaton fnctons In ths spct sch lmnts blong to th paamtc lmnt tps [] 7 hn-walld clndcal shll lmnt h thn clndcal shll lmnt s psntd n g7 In accodanc wth th basc qatons of th tchncal tho of thn shlls th gomtcal popts of th clndcal shll a th followngs []: q H R 7 q H R R R g7 Paamts of th thn clndcal shll lmnt Apat fom th dsplacmnt componnts v and w w can dv th angl of otatons b applng th basc qatons of th tchncal tho of thn shlls basd on Eqs69 and 7: w 7 w R H Andás Séns BME wwwtanonvtah

296 96 nt Elmnt Mthod wwwtanonvtah Andás Séns BME w v R w H R v Rv R t w calclat th stan componnts sng th paamts of th clndcal shll sfac s Eq67: w R v H H H H 7 w R w R H H H v H v R H v H H H H H w H H H H w v R H H H H v w R R R H H H H H H h gd bod-l moton of th lmnt nvolvs s dgs of fdom whch a gvn b th dsplacmnt vcto fld gvn blow []: sn cos cos R a R a a 7 cos sn cos cos cos sn 6 5 a a R a a a v sn cos cos sn sn cos 6 5 a a R a a a w In mat fom: 75 wh:

297 7 Modlng of cvd and dobl-cvd shlls 97 Andás Séns BME wwwtanonvtah w v 76 sn cos cos sn sn cos cos sn cos cos cos sn sn cos cos R R R R 6 5 a a a a a a wh s th mat of ntpolaton fnctons whch capt th gd bod-l motons s th vcto of nnown coffcnts h dsplacmnt fld of gd bod-l moton and that of th dfomaton togth gv th total dsplacmnt fld whch s: w v 77 wh: w v 78 a a a a a v a a a a a a a a a a a a a w In mat fom w hav: 79 wh s th ntpolaton fnctons mat latd to th dfomaton dsplacmnt fld: 7 s th vcto of nnown coffcnts:

298 98 nt Elmnt Mthod a 7 a 6 a 8 a 7 a 9 a a 8 a a 9 a a a a a a a a 5 a 7 In th psson abov th a nnown coffcnts o dtmn all of thm dsplacmnt paamts a qd lt s choos th followngs: ~ ~ ~ ~ v w ~ ~ ~ ~ w ~ ~ ~ ~ v w ~ w~ v~ w~ ~ ~ ~ ~ w ~ ~ ~ ~ v w w~ 7 at ach nod th a dsplacmnts n th dcton of th bass vctos th a otatons abot and th sth dgs of fdom s chosn to b th md dvatv w h dgs of fdom fo th clndcal shll lmnt basd on Eqs7 and 77 a: v v v w w w 7 w a 9 a cos a 9 a a sn a a a a a 5 a 6 a a 7 v w ar a a a a a6 a R R a a a a a a w a sn a a cos a a 6a a 6 a a 6a 7 9a 9 h condtons fo th dtmnaton of th paamts a = a: / ~ / / ~ / ~ v / v~ v / v v v~ 7 a ~ 7 ~ v / v~ / w / w~ ~ w / w w / w~ w / w~ ~ / ~ / 8 wwwtanonvtah Andás Séns BME

299 7 Modlng of cvd and dobl-cvd shlls 99 ~ ~ / / ~ ~ / / ~ ~ / / w w ~ w ~ / w / w ~ w ~ / w / w h vcto of nodal dsplacmnts s fomlatd smlal to th plat lmnt psntd n scton 5 v []: ~ M A 75 wh vcto A contans th lmnts of and n od : A a a a a a a a a 5 5 a a 6 6 a 7 a 7 a 8 a 8 a 9 a 9 a a a a a a a a 76 h coffcnts of th ntpolaton polnomals a dtmnd b th nvson of M : A M ~ 77 D to th lag lngth th coffcnts a not dtald h Assmng that th angl otaton s appomatl o th stan componnts a fomlatd as follows: w v 78 R R w v w w v R R h stan componnts can b classfd nto two pats: stans sha stans and th cvats spctvl W can wt that: R A R A 79 Andás Séns BME wwwtanonvtah

300 nt Elmnt Mthod wwwtanonvtah Andás Séns BME wh matcs R and R a calclatd basd on th dvatvs of th dsplacmnt fnctons: cos sn sn cos cos sn sn cos cos sn R R R R R R R R R R R R R R R R R R R R R R cos sn 6 R R R R R R R R R R R R R R R R R R R R R R R 7 o th calclaton of th stffnss mat and th foc vcto latd to th dstbtd load w fomlat th total potntal ng of a sngl lmnt W not that th psson blow contans onl th stan ng and th wo of th dstbtd load: A p V da p dv 7 h constttv law of th lna lastc matal s: C 7 wh smlal to th plats w assm plan stss stat st C C Usng Eqs77 and 79 w obtan: B M R A R ~ ~ 7 M R B CB C ~

301 7 Modlng of cvd and dobl-cvd shlls Andás Séns BME wwwtanonvtah fthmo: B M R A R ~ 75 M R B CB C ~ Basd on Eq77 th dsplacmnt fld bcoms: M A ~ 76 wh: 77 Eq7 s latd to th stss sltants whl Eq75 s latd to th stss copls h total potntal ng bcoms: p A V V da p M dv C dv C ~ 78 whch s wttn as: ~ ~ ~ ~ ~ / / / / / / d d R p M d d B R C B t d d R B C B t 79 It s mpotant to not that n Eq79 th tm latd to th concntatd loads s cldd consqntl th vcto of concntatd loads shold b podcd addtonall hs s an as tas basd on th nodal dgs of fdom: ~ c M M M M M M M M M M M M 7 and th compltd total potntal ng bcoms: ~ ~ ~ ~ ~ ~ c p K 7 In Eq7 K ~ s th lmnt stffnss mat n th local coodnat sstm

302 nt Elmnt Mthod ~ K / / t t B C B B C B R d d 7 h foc vcto fom th dstbtd load s: ~ p / / M p R d d 7 nall th wll-nown fnt lmnt qlbm qaton n th local sstm s: ~ K ~ ~ 7 whch s applcabl onl fo a sngl lmnt h global qaton of dvlopd b a pop tansfomaton h stctal qaton s qd whn th a sval lmnts connctd to ach oth whch s mathmatcall th sam as Eq 86 h advantag of th thn clndcal shll lmnt s that th clndcal sfac s captd actl; as a consqnc t povds accat slt vn f th nmb of lmnts s latvl low 7 Asmmtc shll poblms concal shll lmnt h mdsfac of asmmtc shlls s podcd b th otaton of th mdan cv abot a staght as [] An ampl s shown b g7 g7 Asmmtc shll h mdan cvs and th ccla cvs ppndclal to th mdan cvs a pncpal cvat lns of th sfac If th load of th stct s asmmtc thn n ths nd of poblm th dsplacmnt fld s th fncton of ac lngth along th mdan cv onl wwwtanonvtah Andás Séns BME

303 7 Modlng of cvd and dobl-cvd shlls h mdan cv can b modld b staght lns and so w appomat th ognal shll stct b concal shll lmnts Rfng to th basc qatons of th tchncal tho of thn shlls th paamts of th concal shll lmnt shown n g7 a: q s H R 75 q H R cos wh s s th ac lngth s th angl coodnat s th ads fo a pont P s th half angl of nclnaton o calclat th stan componnts w nd to dtmn th s latonshp basd on g7 w hav: d s ssn and sn 76 ds g7 Asmmtc concal shll lmnt and ts nodal paamts Usng Eqs67 69 and 7 of th tchncal tho of thn shlls w can calclat th stan componnts as: 77 s w s s s sn wcos sn s w ss w s h dsplacmnt n th tangntal dcton at pont P s v = d to th asmmt h admssbl gd bod-l moton of th lmnt s a dsplacmnt gvn b d n dcton Z fo whch th dsplacmnt componnts a = -dcos and w = dsn s g7 W consd th dgs of fdom at ach nod ths a: dsplacmnt along th mdan dcton w dsplacmnt ppndclal to th mdan cv and s angl of o- Andás Séns BME wwwtanonvtah

304 nt Elmnt Mthod wwwtanonvtah Andás Séns BME taton abot th as ppndclal to th mdan cv n accodanc wth g7 thfo th lmnt has s dgs of fdom h dsplacmnt n th mdan dcton s ntpolatd b a lna fncton of th ac lngth On th oth hand w appl thd od ntpolaton wth spct to th dsplacmnt n th nomal dcton: s s s s w 78 wh s th vcto of nnown coffcnts: 6 5 a a a a a a 79 h vcto of nodal dsplacmnts s: ~ ~ ~ ~ ~ ~ ~ s s w w 7 h condtons qd fo th dtmnaton of th coffcnts a: ~ s a a s ~ w s a s a s a a s w 6 5 ~ s s s a s a a s ~ s a a s 6 5 ~ w s a s a s a a s w 6 5 ~ s s s a s a a s h soltons fo th coffcnts a modatl complcatd thfo th a not ncldd h h dsplacmnt fnctons can b fomlatd also n th wa psntd blow: ~ ~ s ~ ~ ~ ~ s s w w s w wh = 6 a th ntpolaton fnctons: s s s s s s s s 7

305 7 Modlng of cvd and dobl-cvd shlls 5 Andás Séns BME wwwtanonvtah s s s s s s s s s s s s s 5 s s s s s s s 6 s s s s s s and: w ~ ~ h stan componnts n mat fom a: s B 75 wh th stan-dsplacmnt mat s: s s B cos cos sn cos cos sn W collct also th cvats n mat fom: s H ~ 77 wh: s s s s s s s s H sn sn sn sn 78 Basd on th constttv law th vcto of stss componnts s: CB C ~ 79 CH C ~

306 6 nt Elmnt Mthod wwwtanonvtah Andás Séns BME h vctos of stan componnts and cvats contan onl two lmnts thfo th constttv mat dcs to: E C 75 ang th fom bac nto th total potntal ng smlal to th clndcal shll lmnt w can calclat th lmnt stffnss mat n th local coodnat sstm: l s s ds s H C H t B C B t ds H C H t B C B t K sn ~ 75 In th abov psson t was consdd that s = and s = l and so = ssn h act comptaton of th stffnss mat s qt complcatd and consqntl th fnt lmnt cods mplmnt nmcal mthods g th Gass l psntd n scton s stabl to calclat th mat componnts h foc vcto s composd b two tms h vcto of concntatd focs can b constctd basd on th nodal dgs of fdom: ~ M M n s n s c 75 wh fs to th concntatd foc M s a concntatd momnt abot th sam dcton tan that of s h foc vcto fom th dstbtd load s calclatd basd on th wo of th load: p l n s l n s ds p p ds w p p W ~ ~ ~ 75 accodngl: l n s p ds s p p sn ~ 75 Consdng that lsn = - and assmng that both p s and p n a constants w obtan:

307 7 Modlng of cvd and dobl-cvd shlls 7 Andás Séns BME wwwtanonvtah 7 7 ~ l p l p l p l p l p l p n n s n n s p 755 In th local coodnat sstm th nodal dsplacmnt and actons a calclatd fom th sal: K ~ ~ ~ 756 qaton wh: p c ~ ~ ~ 757 o a fnt lmnt stct w nd th stctal qaton gvn b Eq86 Snc th lmnts a connctd nd a gvn angl th local dsplacmnt coodnats shold b tansfomd nto th global clndcal coodnat sstm wth longtdnal as gvn b Z h tansfomaton can b pfomd basd on g7: w w s s ~ ~ ~ ~ cos sn sn cos 758 Basd on th fom th tansfomaton of th stffnss mat bcoms: K K ~ 759 wh: 76 s an othogonal tansfomaton mat h tansfomd foc vcto s:

308 8 nt Elmnt Mthod ~ 76 o a sngl lmnt th fnt lmnt qaton n th global sstm s: K 76 Moov fo th whol stct w hav: KU 76 In th fnt lmnt ltat th a mo lmnt tps g cvd asmmtc shll lmnt [56] whch opats smlal to th concal shll lmnt 7 hc-walld shll lmnts o th solton of th-dmnsonal poblms w can appl th spatal SOID tp lmnts g7 shows a nod sopaamtc lmnt Isopaamtc psntaton mans that th gomt and th dsplacmnt fld s dscbd b th sam st of ntpolaton fnctons [5]: v w v 76 w g7 Qadatc two and th dmnsonal lmnts h thc-walld shll lmnts a constctd n accodanc wth sopaamtc fomlaton n ths spct w pont ot that th sds ppndclal to th shll mdsfac a staght th ntpolaton n th thcnss dcton s lna h lmnt s dtmnd b th 8 nods of th = mdsfac As t can b sn n g7 th dcton of th nt bass vctos changs fom pont to pont thfo th nodal nmb s ndcatd b sbscpt wwwtanonvtah Andás Séns BME

309 7 Modlng of cvd and dobl-cvd shlls 9 Andás Séns BME wwwtanonvtah h coodnats of th ponts on th mdsfac of th thc-walld shll lmnt a gvn b: n t wh n a th colmn vcto of nomal vctos at th mdsfac nods t s th thcnss n th actal nod a th ntpolaton fnctons spctvl h ntpolaton fnctons a th sam as thos of th qadatc sopaamtc plan mmban lmnt s scton h compact fom of th ntpolaton fncton s: = = 6 8 wh and a th local nodal coodnats On th mdsfac th and coodnat lns a othogonal thfo th bass vctos a calclatd as: R R R R n R R n 767 h nodal dsplacmnt paamts a th v w dsplacmnts and th and angl of otatons In th cas of ght nods t mans that th lmnt has dgs of fdom Vcto n can b fomlatd b sng th otatons and th bass vctos : n ~ ~ 768 whch s th tm captng th tansvs sha dfomaton t cass an ncmnt n and v Accodng to th sopaamtc psntaton th dsplacmnt fld bcoms: ~ ~ ~ ~ ~ 8 8 t w v w v 769 o calclat th stffnss mat w hav to stablsh th stan-dsplacmnt latonshp h dvatvs of th dsplacmnt paamts wth spct to th local coodnats a:

310 nt Elmnt Mthod wwwtanonvtah Andás Séns BME t t t t t t 8 ~ ~ ~ ] [ ] [ ] [ 77 o th oth two componnts w obtan smla qatons h fth comptatons q th Jacob mat and dtmnant [5]: J and: J 77 h lmnts of th Jacob mat can b obtand sng Eq765 Also th dvatvs of th dsplacmnt componnts can b dtmnd n th global coodnat sstm o ampl th dvatvs of th componnt n mat fom a: J J J J J J J J J 77 wh J j - a th lmnts of th nvs Jacob mat Basd on Eq77 w obtan th followng: ~ ~ ~ ~ ~ ~ 8 8 g g G J J J t J J J t J J 77 wh: J J J G 77 and:

311 7 Modlng of cvd and dobl-cvd shlls Andás Séns BME wwwtanonvtah t g t g 775 h dvatvs wth spct to th oth two coodnats a: ~ ~ ~ ~ ~ ~ 8 8 g g G J J J t J J J t J J 776 ~ ~ ~ ~ ~ ~ 8 8 g g G J J J t J J J t J J wh: J J J G 777 J J J G Wttn n mat fom w hav: g G g G g G g G g G g G h dvatvs of th oth two componnts can b povdd smlal Usng th dvatvs w can calclat mat B whch s th latonshp btwn th stan componnts and th nodal dsplacmnt paamts: ~ B ~ 779 wh ~ s th vcto of nodal paamts n th local coodnat sstm h vctos of stan and stss componnts n th global sstm a:

312 nt Elmnt Mthod 78 Hoo s law n th local sstm can b wttn as: ~ C ~ 78 wh C s th constttv mat: E C 78 h mat abov dffs fom th gnal th dmnsonal cas n accodanc wth th followngs h stss nomal to th shll sfac s o d ow d colmn Snc th lmnt s thc-walld t consds also th ffct of tansvs sha dfomaton bt onl n th fom of an avag stss h constant n th lmnts of th 5 th ow 5 th colmn and th 6 th ow 6 th colmn s a sha cocton facto = 5/6 [5] h ason fo that s th al dstbton of th sha stsss s assmd to b paabolc ov th thcnss and t s not constant as consdd n th shll modl h cocton facto s th ato of th stan ngs fom th two dffnt dstbtons Basd on th tansfomaton of local stss and stan componnts w can wt th followngs: ~ ~ 78 ~ ~ wh s th tansfomaton mat fo gnal spatal stss and stan stats h calclaton of s possbl sng th dfntons gvn b Eq6 ang bac Eq78 nto Hoo s law w hav: C 78 h pmltplcaton wth lads to: wwwtanonvtah Andás Séns BME

313 7 Modlng of cvd and dobl-cvd shlls Andás Séns BME wwwtanonvtah C 785 Snc s an othogonal mat w can wt that: E and v: C 786 h tansfomaton mat s []: l n l n n m n m m l m l n n m m l l l n l n n m n m m l m l n n m m l l l n l n n m n m m l m l n n m m l l l n n m m l n m l l n n m m l n m l l n n m m l n m l 787 wh l m and n a th dcton cosns of th nt bass vctos at th actal pont [7]: cos l cos j m cos n 788 cos l cos j m cos n cos l cos j m cos n h tansfomaton mat shold b valatd n th nods moov d to th nmcal ntgaton vn n th ntgaton ponts h stffnss mat n th global coodnat sstm can b calclatd sng Eq57: V V d d BJd C B BdV C B K 789 wh J s th Jacob dtmnant whch can b calclatd sng Eqs765 and 77 o th dtmnaton of th foc vcto w call th dsplacmnt vcto fld n th sal fom: ~ 79 wh s th mat of ntpolaton polnomals As a slt th vctos of bod sfac and ln focs n th global coodnat sstm a: V b V b b d d Jd p dv p ~ 79

314 nt Elmnt Mthod ~ p p da p A A p p Jdd ~ l p ds l S whch can b dtmnd b tansfomaton nto th global sstm n a smla wa to that psntd n scton 6 In th nods concntatd focs ma act th lvant vcto can b obtand n th sam wa as that shown n plat lmnts Bcas of h hgh nmb of nods t s not dtald h h fnt lmnt qlbm qaton s fomd n th sal wa fo a sngl lmnt t s: K 79 wh s th sm of th vctos of bod sfac ln and concntatd focs nall th stctal qaton s: KU A shll-sold tanston lmnt In compl stcts somtms th s th ncsst of th smltanos applcaton of sold and thc-walld shll lmnts hs lmnts can not b connctd dctl bcas th nodal dgs of fdom a not dntcal In ths cass t s asonabl to s a tanston lmnt btwn th sold and shll lmnts [5] A qadatc tanston lmnt s shown n g75 wh th nods -8 a locatd n th sold sd nods - ca locatd n th shll sd of th lmnt g75 A shll-sold tanston lmnt h gomt of th tanston lmnt s captd b th fncton blow: t n 79 wwwtanonvtah Andás Séns BME

315 7 Modlng of cvd and dobl-cvd shlls 5 h ndcs = 8 f to th ntpolaton fncton of th sold lmnt gvn b Eq76 f = 9 thn th actal ntpolaton fnctons of th thc-walld shll lmnts a fd to n accodanc wth Eq765 h composd sstm of fnctons satsfs th followng condtons []: f j j j j f j f j j j f j wh j j and j a th nodal coodnats n th local coodnat sstm Smlal to th thc-walld shll lmnts th dsplacmnt fld s pssd b: v w 8 ~ v~ w~ ~ ~ v w~ ~ t ~ h dgs of fdom n nods -8 a qal to th n nods 9- th a fv dgs of fdom Consqntl th tanston lmnt has 9 dgs of fdom h fth calclatons can b pfomd n smla fashon to that psntd n th thc-walld shll lmnt 76 Bblogaph [] Im Bojtá Gábo Vöös Applcaton of th fnt lmnt mthod to plat and shll stcts chncal Boo Pblsh 986 Bdapst n Hngaan [] Gábo Vöös cts and pactcs of th sbjct Appld mchancs manscpt Bdapst Unvst of chnolog and Economcs aclt of Mchancal Engnng Dpatmnt of Appld Mchancs 978 I smst Bdapst n Hngaan [] Sngs S Rao h fnt lmnt mthod n ngnng foth dton Elsv Scnc & chnolog Boos [] Klas-Jügn Bath nt lmnt pocds Pntc Hall Upp Saddl Rv 996 w Js 758 [5] homas J Hghs h fnt lmnt mthod na statc and dnamc fnt lmnt analss Pntc Hall Inc A dvson of Smon & Schst Englwood Clffs 987 w Js 76 [6] OC Znwc R alo h fnt lmnt mthod ffth dton Volm : Sold mchancs Bttwoth-Hnmann Ofod Acland Boston Johannsbg Mlbon w Dlh Andás Séns BME wwwtanonvtah

316 6 nt Elmnt Mthod [7] P Ch Cho cholas J Pagano Elastct nso dadc and ngnng appoachs D Van ostand Compan Inc 967 Pncton w Js oonto ondon wwwtanonvtah Andás Séns BME

317 8 AAYSIS O D PROBEMS WIH IIE EEME BASED PROGRAM SYSEMS IRODUCIO O D EEMES In sval cass th gomt of a stct o a bod cannot b modld as a ln o sfac In that spcal cas t has to b modld as a bod Bods l ths can onl b appomatd b D lmnts n od to pval nglctng mpotant pats Compl gomt appas n smpl stcts as wll o ampl f a wldd bam stct s modld thn t s stabl to s bam lmnts whch s abl to anal th stss stat of th stct If th stss stat has to b anald n th jonts of a stct thn shll modl mst b appld If th wld coalscncs hav to b amnd as wll thn D modl mst b appld atall th mo pcs modlng whch nvolvs mo nods ncass th amont of calclatons as wll h D lmnts can b hahdons ttahdon lss oftn pntahdon ths lmnts can b dvd fom hahdons whch can b dscbd b lna- qadatc of hgh dg of bass fnctons 8 Hahdon lmnts h hahdon lmnts a mappd to a cb wth nt lngth of two Dpndng on th dg of th appomatng polnomal lmnts wth 8 o nods can b sd as wll C c c b - - C b C - a a C g 8: ocal coodnat sstm of th Hahdon István Oldal SZIE wwwtanonvtah

318 8 nt Elmnt Mthod h local coodnat sstm s dvd fom th global coodnats and th a b c lngths of th cb hs coodnats a: C 8 a C and 8 b C 8 c hn th dvatvs: d d 8 a d d és b 85 d d 86 c 8 Hahdon lmnt wth 8 nods g 8: Hahdon wth 8 nods and th mappd cb If th nvstgatd bod s to b anald b lna hahdons thn lmnts wth 8 nods mst b sd h lmnt s appomatd b: wwwtanonvtah István Oldal SZIE

319 8 Intodcton of D lmnts 9 87 and fomlas wh s th bass fncton of th th nod: 8 8 wh a th local coodnats of th nod h J Jacob-mat dtmns th latonshp btwn th local and global dvatvs of th bass fncton: lo J glob wh 8 J 8 h bass fnctons smlal to qaton a ntpolaton fnctons hs fnctons can b sd to appomat th dsplacmnt of th lmnt ths th s no nd to ntodc nw ntpolaton fnctons bt to appl th 8 bass fnctons n cas of a hahdon lmnt wth 8 nods v v w w h dscbd lmnts a namd as sopaamtc lmnts István Oldal SZIE wwwtanonvtah

320 nt Elmnt Mthod 8 Hahdon lmnt wth nods g 8: Hahdon wth nods and th mappd cb If th nvstgatd bod s to b dscbd b qadatc hahdons thn lmnts wth nods mst b sd h lmnt s appomatd b: and fomlas wh s th bass fncton of th th nod In cas of an lmnt wth nods th nods n th cons and th nods n th mddl of th sds mst b dstngshd In cas th nod s locatd n a con th bass fncton s: 8 8 wh a th local coodnat of con In cas th nod s locatd at th mddl of a sd th bass fncton s: If thn 8 wwwtanonvtah István Oldal SZIE

321 8 Intodcton of D lmnts If thn 85 If thn 86 wh a th local coodnats of nod at th mddl of th sd Smlal to th lmnt wth 8 nods th J Jacob-mat dtmns th latonshp btwn th local and global dvatvs of th bass fncton: lo J glob J wh If 8-86 bass fnctons a sd to appomat th dsplacmnt of a hahdon wth nods thn th lmnts a namd as sopaamtc lmnts as wll 8 Hahdon lmnt wth nods g 8: Hahdon wth nods and th mappd cb István Oldal SZIE wwwtanonvtah

322 nt Elmnt Mthod If th nvstgatd bod s to b dscbd b cbc hahdons thn lmnts wth nods mst b sd h lmnt s appomatd b: and fomlas wh s th bass fncton of th th nod In cas of an lmnt wth nods th nods n th cons and th nods along th sds mst b dstngshd In cas th nod s locatd n a con th bass fncton s: wh a th local coodnats of con h bass fnctons at th pont of th thd lngth of th sd: If thn If thn If thn wh a th local coodnats of nod along th sd Smlal to th al th J Jacob-mat dtmns th latonshp btwn th local and global dvatvs of th bass fncton: J wh lo glob wwwtanonvtah István Oldal SZIE

323 8 Intodcton of D lmnts J If 87-8 bass fnctons a sd to appomat th dsplacmnt of a hahdon wth nods thn th lmnts a namd as sopaamtc lmnts as wll 8 Pntagon lmnts Among th pntagon lmnts mostl th psm and th pamd s sd whch a dscbd as a dgnatd hahdon 8 tahdon lmnts Bass fnctons can b dscbd n two nds of coodnat sstms n cas of ttahdon lmnts: n a coodnat sstm wh th lmnt s mappd as a ttahdon wth nt lngths o n a so-calld volm coodnat sstm sam as n th ASYS Both wa of dscpton wll b ntodcd In cas of th fst dscpton th ogn of th local coodnat sstm s allocatd n on con of th ttahdon g 85 c A b A a g 85: tahdon n th nt lngth coodnat sstm In ths ogn th local coodnats a dscbd b th a b c lngths: A 8 a István Oldal SZIE wwwtanonvtah

324 nt Elmnt Mthod A and b 8 A c 8 h volm coodnat sstm s dfnd b th followng qaton sstm: 8 a th global coodnats of an nn pont a th global coodnats of th cons a th volm coodnats B solvng th 8 qaton sstm th volm coodnats a: a b c d 6V a b c d 6V a b c d 6V a b c 6V d wh a d a constants and V s th volm of th ttahdon B sttng and smplfng th fomlas th sngl coodnats can b calclatd at an nn P pont of th ttahdon f th ognal bod s dvdd nto fo ttahdons wth spct of pont P wwwtanonvtah István Oldal SZIE

325 8 Intodcton of D lmnts 5 P P P P g 86: Dsctaton of ttahdon fo volm coodnats hs th volm coodnats of P latd to ach con can b obtand as th ato of th oppost volm of th amnd small ttahdon and th volm of th ognal ttahdon: V P V V P V V P V V P V 85 István Oldal SZIE wwwtanonvtah

326 6 nt Elmnt Mthod 8 tahdon lmnt wth nods g 87: Map of ttahdon lmnt wth nods to coodnats If th nvstgatd gomt of th bod s to b dscbd b lna ttahdons thn lmnts wth nods mst b sd In a coodnat sstm th lmnt s appomatd b: and wh s th bass fncton latd to th nod: In volm coodnat sstm th shap s appomatd b: wwwtanonvtah István Oldal SZIE

327 8 Intodcton of D lmnts 7 and omlas wh s th bass fncton of nod: tahdon lmnt wth nods g 88: Map of ttahdon lmnt wth nods to coodnats If th nvstgatd gomt of th bod s to b dscbd b qadatc ttahdons thn lmnts wth nods mst b sd In a coodnat sstm th lmnt s appomatd b: István Oldal SZIE wwwtanonvtah

328 8 nt Elmnt Mthod and wh s th bass fncton latd to th nod n th cons: h nods n th mddl of th sds: In volm coodnat sstm th shap s appomatd b: and wh s th bass fncton latd to th nod Bass fnctons n th cons: wwwtanonvtah István Oldal SZIE

329 8 Intodcton of D lmnts 9 In th mddl of th sds: h advantag of th volm coodnat sstm bcoms mo obvos snc th bass fnctons a smpl and smla to ach oth 8 tahdon lmnt wth nods g 89: tahdon lmnt wth nods István Oldal SZIE wwwtanonvtah

330 nt Elmnt Mthod If th nvstgatd gomt of th bod s to b dscbd b cbc ttahdons thn lmnts wth nods mst b sd Snc th volm coodnat dscpton has a smpl fom onl ths mthod wll b psntd In volm coodnat sstm th shap s appomatd b: and wh s th bass fncton latd to th nod Bass fnctons n th cons: Bass fnctons n th mddl of th sds: wwwtanonvtah István Oldal SZIE

331 8 Intodcton of D lmnts 9 5 Mddl of th sfac: Hachc bass fnctons In cas of lmnts wth hgh dg hachc bass fnctons can b sd whch has th ognal dg of th fncton n th cons bt low dg along th sds and th sfacs 8 Dfnton of stffnss mat and nodal loads 8 mcal Gass ntgaton mthod In od to solv a fnt lmnt poblm th lmnts of th stffnss qaton stffnss mat nodal loads mst b dtmnd hs lmnts can b obtand b ntgaton as t s shown n Chapt 5 Most of th tms th ntgals cannot b solvd analtcall ths nmcal ntgaton tchnqs mst b appld In cas of a th dmnsonal fncton wth spct to a V volm b th s of th Gass ntgaton mthod: dv ddd w wjw j V j wh: w w w : wght factos j j : Gass coodnats If w choos to s local coodnats thn: ddd dt J ddd j j WW W dt J j j István Oldal SZIE wwwtanonvtah

332 nt Elmnt Mthod wh: W W W : Gass wght factos j j : Gass local coodnats J : Jacob-mat 8 Dfnton of stffnss mat n cas of D lmnts In cas of sopaamtc lmnts th stffnss mat mst b dfnd as: K V B CB dv wh B K t b dfnd as fncton: B CB hn th stffnss mat of a hahdon lmnt: V j dv dt J ddd WW W dt J j j j 8 Dvaton of nodal loads fom dstbtd foc sstm on volm h nodal loads a dvd fom th dstbtd foc sstm on a volm o sfac Accodng to th load fom th dstbtd foc sstm on a volm: f q q V q dv Intodcng th f q q fncton: Dvng th nodal load fom th dstbtd foc sstm on a volm of a hahdon lmnt: wwwtanonvtah István Oldal SZIE

333 8 Intodcton of D lmnts q V f q j dv f dt J ddd WW W dt J j q f j q j 8 Dvaton of nodal loads fom dstbtd foc sstm on sfacl Accodng to 5 th load fom th dstbtd foc sstm on a sfac: p A p p da a a a j g 8: Dtmnaton of th nomal of a sfac In an abta pont of coodnat th a and a tangnts can b dtmnd as: a és a B nowng ths tangnts th nomal a vcto s: a a a h bass fncton s lmtd to th sfac: István Oldal SZIE wwwtanonvtah

334 nt Elmnt Mthod a a hn th tangnts a: j j And th foc vcto on an nfntsmal sfac s dfnd as: a a dd pa dd p da p d A p f p Sbstttng ths nto 5: Ap p da Intodcng p p f fncton: p a pa dd Dvng th nodal load fom th dstbtd foc sstm on a sfac of a hahdon lmnt: p f dd WW j f j q j q wwwtanonvtah István Oldal SZIE

335 9 AAYSIS O D PROBEMS WIH IIE EEME BASED PROGRAM SYSEMS APPICAIO O D EEMES 9 Caton of gomtc modl h gomtc modls of th-dmnsonal bods a dntcal wth th ognal D bods f D lmnts a sd dng th caton Wth th capact of th modn compts vn th most compl stcts lna matal law statc poblm can b solvd n asonabl tm Althogh n man cass th complt dtald analss s lvant h comptatonal tm can b dcasd and n cas of non-lna poblms th modfcaton of th ognal shap of th bod can b an ssntal condton h mpotanc of ths qstons wll b psntd n th followng chapt h a two was to mpot th gomt of a bod nto a fnt lmnt pogam: w th s to constct th modl wth th basc dsgn modl of th fnt lmnt pogam o w s a commcal dsgn pogam to cat and mpot th gomt of a bod 9 Edtng th ognal gomt Dng th caton of th gomtc modl th p dgtalaton of th gomt s not sffcnt spcal sfacs mst b catd wh: oads and constants can b appld h msh can b modfd Rslts can b plottd th on ths otns can b onl ndtan f th appontd sfacs st and a avalabl to f n th pogam In g 9 an ampl s shown abot th dfnton of sfacs On th ognal gomt th sfac of th clnd s a compact doman bt th load s appld on onl on ndvdal sfac hat sfac has to b spaatd snc t wll b tatd as a fnc sfac n cas of dfnng foc pss o dflcton g 9: Caton of sfac on a clnd István Oldal SZIE wwwtanonvtah

336 6 nt Elmnt Mthod 9 Modlng sds and cons D modls ncld th gomt of sds and chamfd o ondd cons On th oth hand t s commndd to dcas th comptatonal tm b slctng th ssntall mpotant pats fo modlng and nglctng th ons whch a lss mpotant latd to th analss As a gnal ngnng l of thmb f w sspct pa stsss n a ctan aa thn a mo dtald modl has to b sd whl n cas of an nloadd aa th sam modl s lvant In cas of g 9 w assm that th loads and constants a locatd at th nds of th shaft Usng ondd cons sgnd as d n ths patcla cas onl ncass th complt of th modl wthot addng mo nfomaton In conta th ondd gn pat on th modl has vald nflnc on th pa stsss ths nglctng t wold cas gat nlablt n th calclaton In cas of ths poblm th gn ondd ads s gvn and appas n th modl whl th cons sgnd wth d colo s nglctd and also not psntd on th modl g 9: Modlng sds and cons 9 Modlng nloadd pats Dng th modlng th nloadd pats can b nglctd If t s nown that a dcoaton of nfomaton boad wll b boltd to th canshaft n g 9 thn ts nflnc can b nglctd and th handl can b sppd n th modl wwwtanonvtah István Oldal SZIE

337 9 Applcaton of D lmnts 7 a b g 9: Modlng nloadd pats 9 Modlng smmtc pats In cas of modlng a machn lmnt th smmt tslf can b tld f th load s smmtc as wll It s sffcnt to s th half of th gomt n cas of sngl smmt whl th on-foth of th gomt n cas of dobl smmt whl th pop constants hav to b appld on th ntsctd sfacs n od to modl th nglctd pats g 9 hs mthod s coct n cas of stngth calclaton and analss whl n cas of stablt and gnfqnc can onl b sd wth som stctons a sngl smmt b dobl smmt g 9: Utlaton of smmt popts 9 Caton of fnt lmnt modl h gomt of th bods a dsctd to fnt lmnts accodng to mthod and psntd lmnt tps n Chapt 8 h dsctaton of th bod s calld mshng h mshng s cad ot b th softwa althogh som paamts hav to b spcfd o th dfalt paamts modfd h two most mpotant paamts a th tp and th s of th lmnts István Oldal SZIE wwwtanonvtah

338 8 nt Elmnt Mthod 9 Dfnng th msh In cas of classc commcal softwa th lmnt tp has to b chosn whch s followd b th s o th nmbs of th lmnts Aft ths stps coms th conct mshng Modn softwa s abl to msh th gomt wthot sttng an paamts In ths spcal cas th pogam ss a dfalt sttng whch s appopat fo a ogh stmaton althogh ths onl gvs th s a ln on th slts In most cass w can onl gv captal cdt to th msh fasbl and appopat f th sttngs a wll chosn In th followng chapt w shall nvstgat th nflnc of th lmnt tp and s 9 Inflnc of lmnt s Eal a canshaft was amnd nd dffnt sttngs wth faton appld on on nd and concntatd foc appld on th oth nd In g 95 th mshng was cad ot wth th dfalt sttng As t s sn th applcaton of ttahdons slts a coas msh wth mamal dcd stss of 8MPa g 95: EM msh wth dfalt sttngs and calclatd dcd stss MPa t s dfn th avag lmnt s to 5mm whch slts fn msh whl th dcd stss ncass to 55MPa g 96: Calclatd dcd stss n cas of 5 mm avag lmnt s MPa wwwtanonvtah István Oldal SZIE

339 9 Applcaton of D lmnts 9 It appas v vsbl that th ctcal pat s locatd at th tanston of th damts at th ondd con If w a alad awa of th ctcal sgmnts th fth fnmnt of th msh on th complt gomt s lvant t s dc th lmnt s bt onl n th ctcal sgmnt and nvstgat th addtonal nflnc t s dc th s th lmnt s to mm n th ctcal sgmnt whl th avag lmnt s mans 5mm n oth sgmnts g 97: 5 mm of avag lmnt s mm of dcd lmnt s n th ctcal sgmnt and th calclatd dcd stss MPa In g 97 t s sn that fth fnmnt n th msh sltd addtonal MPa of stss ncmnt n th calclaton As long as th slts a so snstv to th msh th coct solton s not vn clos ths lt s dc th lmnt s to mm n th ctcal sgmnt g 97: 5 mm of avag lmnt s mm of dcd lmnt s n th ctcal sgmnt and th calclatd dcd stss MPa h fth fnmnt n th msh casd no lvant dffnc ths th slt statd convgng h solton s appomatd and fth fnmnt n th msh wll not hav consdabl nflnc on th slt ot that th lmnt s was halvd whch cass powd István Oldal SZIE wwwtanonvtah

340 nt Elmnt Mthod ncmnt n th lmnt nmb In th dmnsons t ncass th od of magntd n th ctcal sgmnt wth closl on As a valdaton lt s consd a mch fn msh wth 5 mm of lmnt s n th ctcal sgmnt In g 98 t s sn that ths msh cass lss than 5 % dffnc n th slt g 98: 5 mm of avag lmnt s 5 mm of dcd lmnt s n th ctcal sgmnt and th calclatd dcd stss MPa 9 Inflnc of lmnt tp t s nvstgat th nflnc of th ttahdon lmnts wth nods lna appomaton compad to th al psntd ttahdon lmnts wth nods qadatc appomaton V ll th dcas of th od of appomat fnctons wll hav ngatv nflnc on th slts h lmnt tp mans ttahdon onl th nods n th lmnt a dcd h mshs a not psntd dntcal wth g 95-8 onl th dcd stsss a compad n cas of dntcal msh lmnt s and dffnt o nods tahdon wth nods tahdon wth nods g 99: Calclatd dcd stss MPa n cas of dffnt ttahdons lmnt s s dfalt wwwtanonvtah István Oldal SZIE

341 9 Applcaton of D lmnts In g 99 t s sn that th ttahdon wth nods cannot appomat popl th clndcal gomt snc th sd of ctan lmnts a a plan d to th lna appomat fnctons n contast wth th nods lmnt whch can modl th sds as cvs as wll tahdon wth nods tahdon wth nods g 9: Calclatd dcd stss MPa n cas of dffnt ttahdons avag lmnt s s 5 mm h ondng ads was st to mm ths th mamm stss dd not chang sgnfcantl compad to th dfalt sttng In th nt stp th ondng ads s st to mm whch wll v ll nhanc th accac of th slt tahdon wth nods tahdon wth nods g 9: Calclatd dcd stss MPa n cas of dffnt ttahdons avag lmnt s s 5 mm lmnt s n th ctcal sgmnt s mm In g 9 t s sn that th slts a hghl fnd f w s small lmnts than th ondng ads Whl th vald slt was alad appomatd wth th nods lmnt th solton gvn b th nods lmnt had lvant dffnc t s fn th msh whch was alad appopat fo th lmnts wth nods and obsv th nflnc on th lmnts wth nods István Oldal SZIE wwwtanonvtah

342 nt Elmnt Mthod tahdon wth nods tahdon wth nods g 9: Calclatd dcd stss MPa n cas of dffnt ttahdons avag lmnt s s 5 mm lmnt s n th ctcal sgmnt s mm tahdon wth nods tahdon wth nods g 9: Calclatd dcd stss MPa n cas of dffnt ttahdons avag lmnt s s 5 mm lmnt s n th ctcal sgmnt s 5 mm In g 9 and 9 t s sn that th lmnts wth nods do not convg to th vald solton vn f th msh f v fn In cas of ttahdon lmnts wth nods w obtan an accptabl solton wth fw lmnts than th od of magntd of two W can daw th followng conclsons: th s of lmnts wth nods has to b avodd f th modld bod nclds cvd gomt hs mans pactcall most cass 9 Bonda condtons Anoth most mpotant pat of EM modlng s th pop sttngs of th bonda condtons W cannot ma falts b sng appomat fnctons wth hgh od o a v fn msh hat mght lvantl ncas th comptatonal tm bt th solton wll b ltmatl vald h falt of bonda condtons sttngs wll appa ndpndntl fom th fn msh Man cass th o d to th wong bonda condtons s ncasd b th fn msh wwwtanonvtah István Oldal SZIE

343 9 Applcaton of D lmnts 9 oads Ral bods a sbjctd to dstbtd loads on th sfac o volm oads can b concntatd n a pont o dstbtd along a ln f th modlng dmnson s low h applcaton of ths loads s a falt n th D modlng whch popotonall ncass th o n th solton b th fnmnt of th msh In g 9 a cb s plottd wth mm of sd lngth and of concntatd foc s appld on th mddl of ts pp plan h oth low plan of th cb s fd t s amn th stsss as a fncton of lmnt s If th foc s actd on th total plan as a pss: A g 9: Cb loadd on th mddl of ts pp plan 5MPa mm mm hn nomal stss appas n th total coss scton If w ma th followng falt b applng a concntatd foc n th mddl of th pp plan nstad of a pss thn w obtan dffnt dcd stsss as a fncton of lmnt nmbs g 95: Calclatd stsss MPa n a cb loadd n a pont wth concntatd foc lmnt s mm István Oldal SZIE wwwtanonvtah

344 nt Elmnt Mthod g 96: Calclatd stsss MPa n a cb loadd n a pont wth concntatd foc lmnt s 5 mm g 97: Calclatd stsss MPa n a cb loadd n a pont wth concntatd foc lmnt s mm g 98: Calclatd stsss MPa n a cb loadd n a pont wth concntatd foc lmnt s mm W can obsv that th dcas of th lmnt s constantl and ncas of th lmnt nmb ncass th dcd stss g 99 B th dcas of th lmnt s w wwwtanonvtah István Oldal SZIE

345 Eq Stss [MPa] 9 Applcaton of D lmnts 5 stp b stp appomat th thotcal concntatd load whch slts nfnt stss W obtan smla slt f w appl dstbtd load along a ln In cas of D modlng w can onl appl dstbtd loads on th sfacs and th volms W do not analss fth th poblm bt th smla poblm appas f a ln- o shll lmnt s dctl connctd to a D bod of lm g 99: Rdcd stss as a fncton of lmnt nmb 9 Constants W hav to pa spcal attnton to th constants f w modl D bods Snc th constants a nfntl gd th mght slt npctd and nalstc stsss and dfomaton n th calclatons Althogh ths poblm dos not appa alwas so dctl as t was dmonstatd wth th concntatd foc n th al scton D to ths fact t can cas poblms snc t s had to notc t s amn a bam fd n on nd and loadd wth a concntatd foc n th oth nd g 9 István Oldal SZIE wwwtanonvtah

346 6 nt Elmnt Mthod g 9: omal stsss MPa n a fd bam hahdon lmnts wth mm Calclatd stss fom p bndng: l 6 mm 875MPa K mm h calclatd stss n th bod s smmad n a tabl as a fncton of lmnt tp lmnt nmb and nmb of nods: wwwtanonvtah István Oldal SZIE

347 9 Applcaton of D lmnts 7 p S mb of nods tahdon wth nods tahdon wth nods stctd tahdon wth nods Hahdon wth 8 nods Hahdon wth nods mb of Ma nomal lmnt stss [MPa] df Ma dcd stss [MPa] df df df Cold not msh 5 6 df B obsvng th slts w can dv that th solton dos not convg bt th ncas of th o s not as sgnfcant as t was wth th concntatd foc hs falt cass nlablt dng th valdaton of th slts snc th calclatd hgh stsss a nalstc d to th gd faton Evn hgh stsss a sltd f th nmatc constant s dfnd onl on a sgmnt nstad of th total sfac h nalstc stsss can b dcasd b coasng th msh n th aa of th dal constants hs s onl a mgnc solton f w do not hav th possblt to modl contact o alstc constants István Oldal SZIE wwwtanonvtah

348 PRICIPES ABOU MODEIG ACCURACY AD APPI- CABIIY COMPARISO O DIERE IIE EEME MOD- ES AAYSIS O RESUS Modlng bams Mltpl modls can b sd n a fnt lmnt sstm n cas of bams wth constant coss sctonal aa Shot bams can b modld as D bam D bod whl thn-walld stcts vn as shlls In th followngs w a gong to nvstgat that dpndng on th condtons whch modl s applcabl In od to compa th slts smpl poblms wll b solvd b th s of dffnt modls Analss of a bam wth ccla coss scton t s consd a bam wth ccla coss scton wth 5 mm of ads and mm of lngth All otatons and tanslatons a constand at on sd whl on th oth sd a foc wth th magntd of s appld t s dtmn th mamm stss n th bam wo modls wll b appld to solv th poblm Applng ln lmnts th bam s modld b ts ntal as g a In ths cas on nd of th bam s fd and th oth nd s loadd b a sngl foc B sng D lmnts th gomtc modl s a clnd whch s loadd b a dstbtd foc sstm at on nd At th oth nd no faton s appld bcas th constant of dfomaton wold cas addtonal stsss bsd th bndng stss Instad th sam dstbtd load s appld wth oppost dcton at th oth nd whl th aal dsplacmnt s pscbd to o g b a b g : Gomtc modl of th bam h most mpotant advantag of th D modlng latd to th EM s th dcd comptaton h lmnt tslf s fa smpl than th D lmnts and bsd th smla accac lss lmnt s ndd n th modlng h mshd modls a shown n g wwwtanonvtah István Oldal SZIE

349 Compason of dffnt fnt lmnt modls 9 In cas of bam lmnts o nods a sffcnt; whl 8 and 9 nods a qd to cat an appopatl pcs modl wth D lmnts h nmb of D nods can b dcd f th lmnts a longatd aall bt ltmatl fa mo a qd; f not D lmnts a sd h stsss a dtmnd analtcall as wll n od to compa t to th lat nmcal slts In cas of a fd bam wth ccla coss scton th mamm stss calclatd fom th bndng momnt s: M h l mm MPa K d 5 mm wh: : Stss M h : Mamm bndng momnt : Concntatd foc K : Scton modls d : Damt of th coss scton a b g : EM msh wth D and D lmnts a b g : Calclatd nomal stsss n MPa István Oldal SZIE wwwtanonvtah

350 5 nt Elmnt Mthod h stsss a plottd n g h slt latd to th bam modl s appomatl th sam as th analtcal slt h dffnc btwn th analtcal slt and th slt of th D modl s lss than % whch s pactcall accptabl h applcaton of bam lmnts cold b sfl f w wsh to modl lag stcts wth fn msh gnong th s of D lmnts d to th lmt of comptatonal tm o t s smpl bond possblt If th bonda condtons a popl gvn th slt s labl Stll w hav to b awa of th lmt of bam lmnts d constants contact stsss coss scton tanstons cannot b alstcall modld wth t h calclatd stsss cannot b so dscbd n dtals as good as f th gomt of th coss scton s nvolvd n th modl Modlng of thn-walld bams t s consd th bam n th pvos scton as a thn-walld ctangla coss scton wth dmnson of 66 and matal of stl In th modlng now w hav th possblt to s shll lmnts bsd th D and D lmnts It s clal sn n g that th bam s modld and dscbd as a ln lmnt b sfac lmnt c o bod a b c g : Gomtc modl of th bam wth ts constants and loads Smlal to th pvos modls faton s onl appld n cas of ln lmnts In cas of shll and bod modl copl and aal constant a appld snc onl th stsss fom th bndng a dmandd B mshng ach gomtc modls w obtan th fnt lmnt modls g 5 wwwtanonvtah István Oldal SZIE

351 Compason of dffnt fnt lmnt modls 5 a b c g 5: D D and D fnt lmnt modls h modl blt fom ln lmnts s n g 5a nclds lmnts and nods h modl blt fom shll lmnt nclds 5 lmnts and 6 nods n g 5b whch s sgnfcantl mo compad to th ln lmnts In cas of th bod modl n g 5c th msh s spas aall stll lmnts and 97 nods a sd to bld th modl As t was pctd dng th mshng th modls wth hgh od qd popotonall mo comptatonal tm d to th nd of mltpl lmnts and nods h mamm stss calclatd fom th bndng momnt s: M h 6 l a 6 mm 6mm 76 8MPa I a a v 6 mm 5 mm a b c g 6: Calclatd stsss of D D and D fnt lmnt modls n MPa István Oldal SZIE wwwtanonvtah

352 5 nt Elmnt Mthod wh: : Stss M h : Mamm bndng momnt : Concntatd foc I : Scond momnt of aa a : Hght wdth of th coss scton v : hcnss of th coss scton : Dstanc fom th ntal as In g 6a t s sn that th slt of th bam modl compltl cosponds wth th analtcal solton whch s pctd snc th analtcal solton s dvd fom th tho of th bam modl h slt gvn b th shll mod n g 6b shows closl % of ncmnt whl n g 6c ths dffnc s 8% h dffnc can b ddcd fom th nqalt of aal stsss n thn-walld coss sctons and onl hgh odd modls can popl dscb ths phnomnon Modlng of thn-walld opn coss scton bams In th followngs w shall nvstgat th o f smpl bam modl s sd to modl thnwalld opn coss scton bams h most sgnfcant dffnc s casd b th wapng ffct snc most modls a nabl to dscb ths phnomnon t s s smlal to th pvos ampl a bam wth mm of lngth whl th dmnson of th cold fomd U scton s h bam s loadd wth on sngl foc wth of magntd st lt s dtmn th nomal stsss analtcall th sha stsss a nglctd althogh w a awa that th cas addtonal ncmnt n th qvalnt stss h mamm stss calclatd fom th bndng momnt s: M h mm 5mm 8 5MPa I 89mm wh: : omal stss M h : Mamm bndng momnt : Concntatd foc I a a v a v mm : scond momnt of aa a : Hght wdth of th coss scton v : hcnss of th coss scton : Dstanc fom th ntal as Calclaton of mamm nomal stss fom wapng momnt: Sctoal scond momnt of aa of thn-walld opn coss scton: wwwtanonvtah István Oldal SZIE

353 Compason of dffnt fnt lmnt modls 5 I c vs mm 98mm 96mm 69mm wh: v : hcnss of th flangs s : Badth of th flangs h sctoal coodnat fncton: d d d s wh: : a th coodnats of th coss scton conto B C -68 sha níás cnt öéppont A cntod súlpont h sctoal coodnat fncton s calclatd wth spct to th pol B ntgatng th sqa oot of th fncton w obtan th scond momnt of aa of th coss scton wth spct to th pol: 8 98 da v ds v ds A s s I Intodcng: g 7: fncton wth spct to th pol mm -bn mm 9 6 s ds 76 8s ds 5 G I 8GPa 69mm c 777mm 9 6 E I GPa 5 mm István Oldal SZIE wwwtanonvtah

354 5 nt Elmnt Mthod wh: G 8GPa : Sha modls E GPa : Yong-modls Rlatv angla dsplacmnt of th coss scton wth spct to th pol along th as [Csmada: Modllalotás]: M c c sh c ch ch G I wh: M c : toson wth spct to th pol : coodnat of th as of th bam c c : constants h dvatvs: d M c c ch c ch sh d G I At th f nd of th bam: d d ths: c At th fd nd of th bam: M c ths: c G Ic ch l c c B sbstttng th obtand constants th latv angla dsplacmnt of a on-ndfd bam: M c sh G Ic ch l h dvatv of latv angla dsplacmnt: d M c d G I c sh ch l omal stss wth spct to th bmomnt d to th wapng: wwwtanonvtah István Oldal SZIE

355 Compason of dffnt fnt lmnt modls 55 B I wh: B EI In o cas: d d : bmomnt M c sh M c sh B EI G Ic ch l ch l h mamm of th fncton s at th faton l van: B 6667 mm 7 hn th stss n th con of th U scton d to bmomnt n g 7 pont B 76mm s: B 65 MPa At th nd of th U scton n g 7 pont C 68mm : C 86 5MPa hn th sm of stsss casd b bndng momnt and bmomnt s: B B 9 7MPa C C 58MPa omal stsss calclatd b VEM modls t s compa th analtcall obtand stsss to th fnt lmnt modls n cas of ln- shll- and bod lmnts h gomtc modls a dscbd as th as of th bam g 8a mddl plan g 8b o ts complt coss scton g 8c István Oldal SZIE wwwtanonvtah

356 56 nt Elmnt Mthod a b c g 8: Gomtc modl of th bam wth ts constants and loads Smlal to th pvos modls faton s onl appld n cas of ln lmnts whl n cas of shll and bod modls copl and aal constant a sd a b c g 9: D D and D fnt lmnt modls h modl blt fom ln lmnts shown n g 9a nclds lmnts and nods h modl blt fom shll lmnt shown n g 9b nclds 75 lmnts and 6 nods whl n cas of th bod modl n g 5c lmnts and 8 nods a sd to bld th modl In g th calclatd stsss a plottd n MPa It s obvosl sn that smpl bam modl wth ln lmnts tas onl th bndng nto consdaton and nglcts th wwwtanonvtah István Oldal SZIE

357 Compason of dffnt fnt lmnt modls 57 bmomnt not: som commcal softwa a abl to modl wapng wth ln lmnts bt pont of applcaton n th coss scton has to b dfnd b th s h stmatd slts gvn b th shll and bod modls a hgh than th analtcal slts h ason s ognatd to th analtcal dscpton snc th stsss w calclatd wth spct to th mddl plan and consdd constant along th thcnss of th flangs whl th fnt lmnt modls calclatd th chang along th thcnss as wll a b c g : Calclatd stsss n MPa n cas of D D and D fnt lmnt modls If w loo at th stsss n th mddl plan of th shll lmnts g that th slts B 9 7MPa C 58MPa colat wth small o g : Calclatd stsss MPa n th mddl plan n cas of shll István Oldal SZIE wwwtanonvtah

358 58 nt Elmnt Mthod Modlng of thc-walld clnds tbs t s amn a tb wth 6mm of nsd- and mm of otsd damt whl MPa of ntnal pss s actng n t and dtmn how pcsl th phnomnon can b dscb wth dffnt modls Analtcal modl In th thc-walld tbs th dstbton of th longtdnal stsss s assmd constant whl th chang along th ads as a fncton of qadatc hpbol h tb dagams a commonl plottd as a fncton of latv cpocat ads: b wh: : th ads of th tb vaabl : th ntnal ads of th tb b In o cas th nmb of consdng th tnal and ntnal ads s: 6 5 b b b b Accodng to ths nmbs th tb dagam: [Mpa] b C 5 =-p = b =-p =- b g : b dagam h adal stss n th tnal and ntnal wall qals th tnal and ntnal pss B tlng th popotonalt of th stsss th tangntal stsss can b obtand wwwtanonvtah István Oldal SZIE

359 Compason of dffnt fnt lmnt modls 59 as: b 5MPa MPa h longtdnal stss dpnds on th fact whth th tb s closd o opn ths havng a constant val of C o t s obsv th slts gvn b ach fnt lmnt modl! nt lmnt modls A thc-walld pssd tb can b popl dscbd wth th D o D modls Smplfcaton s also possbl n cas of th D modl b tlng th fact that th longtdnal stsss a constant ths onl a shot pat of th ognal tb has to b anald If th smmt s also tld thn onl th half o on-foth of th ognal tb s sffcnt to anal althogh w hav to b awa of pscbng th coct constants at th ct-off pat accodng to th smmt h a two possblts to dscb th tb wth D modls W assm that all coss sctons of th tb a nd th sam plana dfomaton ths th tb can b modld as onl on coss scton H w can also tl th smmt b onl sng th half o onfoth ng of th ognal coss scton cafll pscbng th coct constants at th ctoff pat h oth opton s to tl th as-smmtc gomt and load and slctng a D as-smmtc modl hn t s sffcnt to modl onl a sgmnt of th complt tb In od to compa th slts th modls wll b solvd and psntd fom th mltpl chocs a b c g : Modlng optons of thc-walld tbs loads constants In g a onl a shot sgmnt of th tb s consdd wth th D modl th ctoff pats a sbstttd b constants: no aal tanslaton s avalabl on th ntsctd sfac B MPa of pss s dfnd on th ntnal sfac of th tb In g b th on-foth pat of th coss scton s modld ths w hav to dfn plana dfomaton n th D modl On ln B and C th ppndcla dsplacmnt s nhbtd b tlng th smmtc gomt h MPa of load s appld on ln A István Oldal SZIE wwwtanonvtah

360 6 nt Elmnt Mthod In g c th longtdnal scton of th bam s modld wth D as-smmtc lmnts h constcton of th gomtc modl s cad ot b sttng mm of dstanc th ntnal ads btwn sfac A and th as of otaton Vtcal dsplacmnt s nhbtd on ln B and on th oth addtonal lns bsd t as wll hs s how th tb s modld fthmo MPa of pss s appld on sfac A as a constantl dstbtd foc sstm In g th fnt lmnt modls a shown ach of thm blt fom pdtmnd lmnts h ognal complt fnt lmnt modl blt fom D lmnts nclds 756 lmnts and 695 nods g a h D modl wth plana dfomaton condton s shown n g b whl th msh nclds lmnts and 55 nods h as-smmtc modl n g c s blt fom D lmnts as wll and th msh nclds 887 lmnts and 588 nods Dng th compason t s woth to not that sng lmnts wth th sam nmb and s th nods of th modls can b dcd onffth o vn on-twntth n cas of as-smmtc modlng of th ognal complt D modl whl th sam pcson s obtand Accodng to ths calclatons th bst qng th last comptatonal tm appomat fnt lmnt modl s th as-smmtc scond s th plana dfomaton and th last on s th ognal bod modl a b c g : D D plana dfomaton and D as-smmtc modls angntal stsss calclatd b dffnt modls a shown n g 5 wwwtanonvtah István Oldal SZIE

361 Compason of dffnt fnt lmnt modls 6 a b c g 5: angntal stsss calclatd b D D plana dfomaton and D as-smmtc modls n MPa B compang th slts to ach oth and to th analtcal solton th followng conclsons can b dawn; th bst appomaton s gvn b th as-smmtc modl althogh non of th modls pfomd mo o than 5% compad to th analtcal solton h dffnc can b mphasd btt f th lmnt s of th th modls s dtmnd to ft to 5% of o In ths spcfc cas w hav to tl th dobl smmt of th bod modl a b c g 6: D D plana-dfomaton and th mnmm nmb of lmnt n cas of D assmmtc modl wth spct of 5% of o n th tangntal nomal stsss n MPa h stsss n g 6 w obtand b contnosl modfng th msh ntl t achd th 5% o o n th ang of th thotcal 5 MPa hs coas mshs a shown n g 7 István Oldal SZIE wwwtanonvtah

362 6 nt Elmnt Mthod a b c g 7: D D plana-dfomaton and th mnmm nmb of lmnt n cas of D assmmtc modl wth spct of 5% of o h nmb of lmnts and nods latd to ach modls: Modl tp mb of lmnts mb of nods Bod 87 D plana dfomaton 6 D as-smmtc 6 Accodng to ths slts w can daw th sam conclsons as al: th as-smmt modl povds th most pcs slt wth th last comptaton tm wwwtanonvtah István Oldal SZIE

363 EVAUAIO AD APPICAIO O COMPUAIOA RE- SU I DESIG AD QUAIICAIO REAED MECHAICA EGIEERIG ASKS REAIOSHIP BEWEE IIE EE- ME MEHOD AD SADARDIZED SREGH BASED DESIG Pcson of nt Elmnt Mthod h nt Elmnt mthod s adqat to obtan appomaton slt abot an ngnng poblm h ncssa accac of th appomaton dpnds on th applcaton and podcton of th stct o bod and t dtmns th qantt of th calclaton Consdng th pactc th appopat accac of th slt shold b n th ang of 5% of o althogh som cass dmand vn mo accat solton In man cass not vn th loads a nown pcsl ths ths o appas n th solton ndpndntl fom th mthod of calclaton ow w a gong to nvstgat th accac of calclaton n cas of gvn bonda condtons h accac of th slt cold b asl calclatd f w nw th act solton nfotnatl apat fom som smpl poblms ths act soltons cannot b obtand ths w hav to stmat th o If w now th magntd of th o and t dos not mt th qmnts thn th accac can b stll mpovd Manl th a two mthods to mpov th accac h fst on alad ntodcd n th al chapt s basd on th s dcton of th lmnts whch s calld h-tp appomaton h oth mthod s basd on choosng hgh-od appomatng polnomals thn w a talng abot p-tp appomaton h dcton of th lmnt s and th ncas of polnomal od can b combnd as wll hp-tp appomaton h attanabl accac of a gvn bonda val poblm solvd b fnt lmnt mthod s manl dtmnd b th appld paamts lmnt tp s dng mshng In od to dtmn th accac of th solton w hav to calclat th dffnc btwn th act dsplacmnt fld and th EM dsplacmnt fld whch s calclatd b fnt lmnt mthod h qston s that f th act solton s nnown how can w dtmn VEM o? h poblm can b solvd b nvstgatng th latonshp btwn th dg of fdom of th fnt lmnt modl and th nom of th o h ng nom of th dsplacmnt fncton can b dfnd as follows: U Wh U s th dfomaton ng: U V dv W can tl th gomtc qaton: István Oldal SZIE wwwtanonvtah

364 6 nt Elmnt Mthod and th constttv qaton: C C Wh : mat of dffntal ods C : mat of matal constants hn th dfomaton ng: U dv C dv V V h ng nom of th act solton: U C dv V And th ng nom of th o: C dv V 5 Wth th fnt lmnt mthod w nd a nmatcall admssbl dsplacmnt fld sm of fnctons wth fnt vaabl whch povds ng mnmm hs qmnt satsfs th followng qaton as follows: VEM mn 6 hs: mn 7 Wh accodng to and 7 dpnds on th lmnt s and th od of th appld polnomals ths t contans nnown paamts Dpndng on th choosng th ncas of polnomal od o th dcas of lmnt s th slt gvn b th fnt lmnt mthod convgs dffntl to th act solton wwwtanonvtah István Oldal SZIE

365 Applcaton of fnt lmnt mthod 65 Estmaton of o n cas of h-tp appomaton h h-tp appomaton mans that dng th dsctaton w dc th lmnt s bt w do not va th od of th appomat polnomals t s nvstgat how th o vas as a fncton of lmnt s n cas of a bam wth l lngth! t s dsct th bam to nmb of lmnts wth dntcal lngth h lngth of on lmnt s: l h 8 h appomat solton of act dsplacmnt fld s VEM whch s a pcws fncton hs fncton povds qal vals wth th act solton n th ntpolaton ponts jh jh j 9 VEM h o of appomaton n th th lmnt: VEM h; h If th solton s contnosl dffntabl thn th o fncton as wll Accodng to 9 th o s o n th bondas of th lmnts and ths contnt follows that th o fncton wll hav an tma nsd th lmnt h locaton of th o s dnotd wth g In ths pont = =-h =h g : Eo fncton n th th lmnt VEM lna ths VEM thn István Oldal SZIE wwwtanonvtah

366 66 nt Elmnt Mthod d d h ; h If C thn C h ; h and ma C h h; h t s pand th fncton n th pont nto alo ss sng th agang fom of th mand as wll: If th mamm of th o fncton s locatd n th scond half of th lmnt h h 5 thn: h h h ;h sbstttng and : h h 6 fom 6 tlng and 5: ma h h ma C 7 8 If th mamm of th o s locatd n th fst half of th lmnt thn th alo ss hav to b nvstgatd at h wh th val s o ths w obtan th 7 qaton h dfomaton ng of th bam: wwwtanonvtah István Oldal SZIE

367 Applcaton of fnt lmnt mthod 67 l U AE d 8 h dfomaton ng of th o: n h AE d AE l h d nhaec U h tang nto consdaton that n h l and th smmng th constants th nom of th o s: U Ch 9 In ths fomla th constant s nown and b nowng th solton of th fnt lmnt modl C can b stmatd wth small o h stands as lmnt s In accodanc wth ths slt th o of th solton s popotonal to th lmnt s If w wsh to stmat th o bfo solvng th poblm thn w hav to smma th constants hn th mamm val of th o cannot b calclatd d to not nowng C constant bt th convgnc of th slt wll b vsbl om 9 and 8: In man poblms latd to th pactc th dsplacmnt fnctons a not smooth fnctons hn th latonshp btwn th nom of th o and th nmb of lmnts changs as follows: wh dpnds on th p od of th appomat polnomals and th chaact of th solton mn p Stct condton f not onl th nom of th o bt th o tslf s nvstgatd on th total doman In som cass t s possbl that th nom of th o monotoncall convgs bt th solton s not monotonc hs can b onl notcd f not onl th global bt th local o s nvstgatd In g a thn plat s bnt and th ng and solton convgnc s amnd h plat has mm thcnss and m momnt s appld on t Snc th load of th plat s co-plana t s modld as a plana stss poblm as wll István Oldal SZIE wwwtanonvtah

368 68 nt Elmnt Mthod R M g : Bnt plat h fnt lmnt modl s sd to nvstgat th h-tp convgnc ths th lmnt s vas btwn and 5 mm whl th od of th appomat polnomals sta nchangd h convgnc s amnd wth lna- and qadatc ntpolaton fnctons as wll wwwtanonvtah István Oldal SZIE

369 Applcaton of fnt lmnt mthod 69 g : EM msh and nomal stsss n MPa lmnt s of 5mm István Oldal SZIE wwwtanonvtah

370 omal Stss [MPa] Eng nom 7 nt Elmnt Mthod In th followng tabl th ncssa paamts and slts a smmad n od to amn convgnc Avag lmnt s [mm] dg of fdom Ma Stss [MPa] Mn Stss [MPa] Eng nom [ mj ] p lna appomat fncton In ths cas both th calclatd stss and th ng nom monotoncall convgd b ncasng th dg of fdom of th modl g stss nom g : Rslt n th ccal pont and th nom calclatd on th complt doman In ths cas th o stmaton can b dtmnd accodng to th nom of th o fncton ths t dos not cas local poblm t s amn how th slts chang n cas of qadatc appomat fnctons! h modl was ctd wth th sam paamts and lmnt ss as al g 5 g 5: Modl and bonda condtons wwwtanonvtah István Oldal SZIE