An Explicit Method for Geometrically Nonlinear Dynamic Analysis of Spatial Beams

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1 An Eplii Meho or eomerilly Nonliner ynmi Anlysis o pil Bems hu hng Hung Wen Yi Lin umio uii n Kuo Mo Hsio Asr An eplii meho se on he enrl ierene meho or nonliner rnsien ynmi nlysis o spil ems wih inie roions using oroionl ol Lgrngin inie elemen ormulion is presene. The kinemis o he em elemen is esrie in he urren elemen oorine sysem onsrue he urren onigurion o he em elemen. The em elemen hs wo noes wih si egrees o reeom per noe. Three roion prmeers reerre o he urren elemen oorines re eine o eermine he orienion o elemen ross seion. A roion veor is use o represen he inie roion o se oorine sysem rigily ie o eh noe o he isreize sruure. Noe h he vlues o nol roion veors re rese o zero urren onigurion. The elemen eormion nol ores n ineri nol ores re sysemilly erive y onsisen linerizion o he ully geomerilly nonliner em heory he 'Alemer priniple n he virul work priniple in he urren elemen oorines. The snr enrl ierene meho is pplie o he inremenl isplemen n roionl veor n heir ime erivives. The orienion o he en ross seion o he em elemen is upe y he inremenl nol roion veor. A Numeril emple is presene o emonsre he ury n eiieny o he propose meho. Ine Terms oroionl ol Lgrngin ormulion ynmis Eplii ime inegrion eomeril nonlineriy. T I. INTROUTION HE implii mehos se on he Newmrk ire inegrion meho hve een eensively employe in nonliner rnsien ynmi nlyses o em sruures unergoing lrge isplemens n inie roions (e.g. [- 4]). In [] oroionl ol Lgrngin inie elemen ormulion or he nonliner ynmi nlysis o spil elsi Euler em using onsisen linerizion o he geomerilly non-liner em heory ws presene. The snr Newmrk meho ws pplie o he inremenl isplemen n roionl veors n heir ime erivives. The ormulion ws proven o e very eeive Mnusrip reeive erury hu hng Hung is wih eprmen o Mehnil Engineering Nionl hio Tung Universiy 00 T Hsueh Ro Hsinhu 00 Tiwn (e-mil: 8@homil.om). Wen Yi Lin is wih eprmen o Mehnil Engineering e Lin Insiue o Tehnology Alley 80 hing Yun Ro Tuheng Tiwn (e-mil: wylin@li.eu.w). umio uii is wih eprmen o Mehnil Engineering iu Universiyiu 50-9 Jpn Kuo Mo Hsio is wih eprmen o Mehnil Engineering Nionl hio Tung Universiy 00 T Hsueh Ro Hsinhu 00 Tiwn (phone: ; : ; e-mil: kmhsio@mil.nu.eu.w). y numeril emples suie in []. However he ppliion o he eplii meho in he nonliner ynmi nlysis o hree-imensionl ems wih inie roions hs een rher limie (e.g. [5 6]). The oe o his pper is o presen n eplii meho se on he enrl ierene meho or nonliner rnsien ynmi nlysis o spil ems wih inie roions using oroionl ol Lgrngin inie elemen ormulion. The elemen esripion is se on he oroionl ol Lgrngin ormulion esrie previously in [ 7 8]. In his ormulion eh emen is ssoie wih resin oorine sysem onsrue he urren onigurion o he em elemen. The elemen oorine sysem is us lol oorine sysem upe urren onigurion o he em elemen no moving oorine sysem. Thus he veloiy n elerion eine in he elemen oorine sysem re solue veloiy n elerion. or he purpose o reing rirrily lrge roion o noe in spe he orienion o he noe is esrie y se oorine sysem rigily ie o eh noe o he isreize sruure. A nol roion veor [7] is use o represen he inie roion o he se oorine sysem. In his suy he vlues o nol roion veors re rese o zero urren onigurion hus he vlues o he irs n seon ime erivive o he nol roion veor re equl he vlues o he spil nol ngulr veloiy n elerion []. The elemen eormion n ineri nol ores re sysemilly erive y using he 'Alemer priniple n he virul work priniple. A numeril proeure o eplii meho se on he enrl ierene meho is propose here or he soluion o he nonliner equions o moion. A numeril emple is presene n ompre wih he resuls oine using he Newmrk meho o emonsre he ury n eiieny o he propose meho. II. INITE ELEMENT ORMULATION The elemen evelope here hs wo noes wih si egrees o reeom per noe. The kinemis o he em elemen n he oroionl ol Lgrngin inie elemen ormulion propose in [ 7 8] re pe n employe here. In he ollowing only rie esripion o he em elemen is given. A. Bsi Assumpions The ollowing ssumpions re me in erivion o he em elemen ehvior: () The em is prismi n slener n he Euler-Bernoulli hypohesis is vli. () The ross seion o he em is ouly symmeri. () The uni eension o he enroi is o he em elemen is uniorm. (4) The ross seion o he em elemen oes no eorm in

2 is own plne n srins wihin his ross seion n e neglee. B. oorine ysems In orer o esrie he sysem we eine hree ses o righ hne rengulr resin oorine sysems:. A ie glol se o oorines X i (i = ) (see ig. ); he nol oorines isplemens roions veloiies n elerions n he equions o moions o he sysem re eine in his oorines.. Elemen ross seion oorines i (i = ) (see ig. ); se o elemen ross seion oorines is ssoie wih eh ross seion o he em elemen. The origin o his oorine sysem is rigily ie o he enroi o he ross seion. The is is hosen o oinie wih he norml o he unwrppe ross seion n he n es re hosen o e he prinipl ireions o he ross seion.. Elemen oorines i (i = ) (see ig. ); se o elemen oorines is ssoie wih eh elemen whih is onsrue he urren onigurion o he em elemen. The origin o his oorine sysem is loe noe n he is is hosen o pss hrough wo en noes o he elemen; he n es re eermine y he meho propose in [7]. Noe h his oorine sysem is us lol oorine sysem no moving oorine sysem. The eormions eormion nol ores ineri nol ores n mss mri o he elemen re eine in erms o hese oorines.. Kinemis o Bem Elemen In his suy only he ouly symmeri ross seion is onsiere. Le Q (ig. ) e n rirry poin in he em elemen n P e he poin orresponing o Q on he enroi is. The posiion veor o poin Q in he uneorme n eorme onigurions reerre o he elemen oorines my e epresse s [8]: r0 e ye ze r r e r e re r p y( ) z( ) () () r v y( ) z( ) r w y( ) z( ) w w w () s s v v v (4) s s s (5) where p p( v v( n w w( re he n oorines o poin P respeively in he eorme onigurion ( n ( re he wis ngle n wis re o he eorme enroi is respeively ( y z) is he in enn wrping union or prismi em o he sme ross seion s is he r lengh o he enroi is n is he uni eension o he enroi is. The orienion o elemen ross seion is eermine y i (i = ) hus i re lle roion prmeers [7 8]. Here he lerl eleions o he enroi is v n w re ssume o e he ui Hermiin polynomils o n he roion ou he enroi is is ssume o e he liner polynomils o. X The relionship mong p given s [8] p 0 ] v n w n my e u [( ) v w (6) where u is he isplemen o noe in he ireion. Noe h ue o he einiion o he elemen oorine sysem he vlue o u is equl o zero. However he vriion n ime erivives o u re no zero. The il isplemens o he enroi is u( p( my e eermine rom he lerl eleions n he uni eension o he enroi is using (6). Mking use o he ssumpion o uniorm uni eension o he enroi is my e lule using (6) n he urren hor lengh o he em elemen.. Elemen Nol ore eor n Mss Mri The elemen eormion nol ores n ineri nol ores re sysemilly erive y onsisen linerizion o he ully geomerilly nonliner em heory he 'Alemer priniple n he virul work priniple in he urren elemen oorines. The virul work priniple requires h q X X s ig.. oorine sysem. P P u P v z Q P y q q (7) w (l00)

3 ( ) r r q u u (8) { q u u (9) { { u θ u θ I { m m I { m m I { m m q (0) () () () ( ) r r r r y.. z r r (4) where = u u v w enoe he virul { isplemen veors noes { enoe veors o virul spil roion noes { enoe he orresponing virul vriion o roion veors noes { n θ { w v enoe { θ { he orresponing virul vriion o w v ( ) ( ) noes. { { ( ) re generlize elemen inernl nol ore veors onuge o q { ( ) re nol ore veors orresponing o u n m m m m ( ) re generlize nol { momens orresponing o n θ I respeively. n ( ) re eormion nol ore veors n ineri nol ore veors orresponing o ) n r r ( respeively. is he volume o he uneorme em elemen ( = ) re he vriion o he reen srin in (4) orresponing o q. ( = ) re he seon Piol-Kirhho sress. or liner elsi meril E n ( = ) where E is Young s moulus n is he sher moulus. is he ensiy r n r re he vriion n he seon ime erivive o r in () respeively. Noe h he elemen oorine sysem is us lol oorine sysem no moving oorine sysem hus r is he solue elerion. The higher orer erms o nol prmeers in he elemen nol ores re neglee y onsisen seon orer linerizion in his suy. Noe h he vlues o roion prmeers i (i = ) will onverge o zero n heir ime erivives i n i will onverge o onsns wih he erese o he elemen size. Thus he oupling eween roion prmeers n heir ime erivives re no onsiere in his suy. Noe h i re ininiesiml roions ou he i es hus m i re momens ou he i es elemen lol noes respeively n m is veor quniy. re no ininiesiml roions ou he i es hus re no momens ou he i es noes n i m i m is no veor quniy. However he vlues o he roion veors noes re rese o zero he urren onigurion o he sruure in his suy. Thus he vlues o equl o he vlues o he orresponing o n re n he vlues re equl o he vlues o he orresponing ngulr veloiy veors ω n ngulr elerion veors ω noes respeively []. The vlues o m n re hereore equl o he vlues o he orresponing m ω n o veor iion lso pply o he iion o in his suy. ou he i es hus ω respeively so he rules m n i re no ininiesiml spil roions m i re no onvenionl momens. The vlues o m i re no equl o he vlues o orresponing m i euse he vlues o i re no equl o he vlues o orresponing i eorme se [ 7] so he rules o veor iion n no pply o he iion o m θ n θ. Here he glol nol prmeers o he sysem re hosen o e he omponens o nol isplemen veor n nol roion veor reerre o he glol oorines. To ssemle he elemen equions ino he glol equions he elemen nol prmeers n elemen nol ores shoul e onsisen wih he glol nol prmeers n glol nol ores. Thereore q n re hosen o e he elemen nol isplemen veor n he elemen nol ore veor. q n n e rnsorme rom elemen oorine sysem o he glol oorine sysem using he snr proeure o veor rnsormion. The relion eween q n q my e epresse s [ 8] q T q (5) q is rele o eiss eween q hrough he sme relionships h q n q [] i.e.: q T q (6) q The ime erivive o (6) my e epresse y T q T q (7)

4 In view o (7) n (5) he relion eween n my e epresse s T (8) where my e lule using (-7) n (4). or onveniene re ivie ino our veors i (i = ) n epresse s I I I I { { m m { m m { m m (9) [ AE ( ) (0) E( I y Iz ) EI y EIz w v ] L L L EI z N v EI y N w ( J EI p ) N EK I N I m u () m A N u { u u N u L AN[ ( v w ) ( v w ) ] L 0 0 I m u m A N u { v N I v v v z N N I N v I N w z z I m u m u A N N I w w w { w y N N I N w I N v y y I m u y m ( I I ) N u z { N I y z ( I I ) N v w I y J N N z A y A I A () I z {( z y ) ( y z ) A ( y z K I ) in whih he rnge o inegrion or he inegrl () is rom 0 o L A is he ross seion re N k (k = ) re shpe unions o elemenry r elemen N (k = ) re k A shpe unions o elemenry em elemen n ( ) ( )/. m i (i = ) re onsisen mss mries n i (i = ) re veors o veloiy oupling erms o elemen ineri nol ores. I In view o () elemen ineri nol ore veor my e epresse y I mq () where m is he elemen mss mri ssemle y he sumries given in (). I he oupling eween roion prmeers i (i = ) n heir ime erivives re no onsiere rom (6) (7) n () one my oin I mq (4) mt q (5) E. Equions o Moion The nonliner equions o moion my e epresse y MQ vi ( (6) P where M is he mss mri Q is he seon ime erivive o he veor o glol nol prmeers Q P( is he eernl nol ore veor is he eormion nol ore veor n vi is he ineri nol ore veor orresponing o he veloiy oupling erms o he elemen ineri nol ore. M vi n re ssemle rom he elemen mss mri elemen nol ore veors whih re lule using (7)- (5) irs in he urren elemen oorines n hen rnsorme rom elemen oorine sysem o glol oorine sysem eore ssemlge using snr proeure. III. NUMERIAL PROEURE An inremenl meho se on he enrl ierene meho (M) is propose here or he soluion o he nonliner equions o moion. The si seps involve in he numeril soluion o (6) re ouline s ollows. Le enoe he ime sep size n ime n n ( n ). Le Q n Q n n Q n enoe he veor o glol nol prmeers veloiy n elerion o he sruure ime ( n ) respeively n n Q Q Q n n n Le n i e (i = = ) enoe he uni veors ssoie wih he i es o elemen ross seion oorines reerre o he glol oorines elemen lol

5 noes n ime n ( n ). Le Q n Q n Q n Q n Q n n e i enoe he known vlues ime n ( n ). The vlues o Q n Qn Q n n Q n ne i my e oine y he ollowing inremenl proeure. () Er he glol inremenl nol roion veor orresponing o eh elemen lol noes ( = ) rom Q n or eh elemen. Er u he glol nol isplemen veor orresponing o eh elemen lol noes ( = ) rom Q or eh elemen. eermine n e i y ppliion n o n e i. Then eermine he urren elemen oorines elemen eormion nol prmeers i n uni eension or eh elemen using n e i u n he meho esrie in [ 7 8]. Then lule he elemen eormion nol ore veors using (8) (0) n ssemle sruurl eormion nol ore n. () Rese he vlues o he elemens in veor Qn orresponing o nol roion veors o zero. () Er he glol nol veloiy veor orresponing o eh elemen rom Q n (or kwr ierene Q n Qn / ) n hen rnsorm hem o he urren elemen oorines using snr proeure. Then lule ime erivive o elemen nol roion prmeers using (6) n (7). Then lule elemen ineri nol ore veor using () n (5). Then rnsorm rom he urren elemen oorines o he glol oorines using snr proeure o ssemle vi n. () Trnsorm he elemen mss mries rom he urren elemen oorines o he glol oorines using snr proeure. Then ssemle he sruurl mss mri M using he elemen mss mries. (e) lule Q vi n using MQ n Pn n n ((6)). () Le Qn Qn Q n Qn Qn Qn n Q n ( Qn Qn ) /. When n Qn Q0 n Q Q n 0 n e oine rom he iniil oniions n Q Q n 0 n e lule using (6); Q is lule y Q Q 0 Q 0 0 Q. I. NUMERIAL TUIE The emple suie is lmpe em suee o enrl eenri onenre lo s shown in ig.. The lo hisory is lso shown in ig.. The geomery o he em n he meril properies re L m m h.750 m Young's moulus E 07 P Poisson s rio 0. n ensiy 7 kg / m. The eenriiy o he onenre lo is e 60 m. Beuse o symmery only one-hl o he em is moele y 5 elemens. igures - 5 show omprison eween he ime hisories o he mi-spn eleions n il roion oine y he presen meho (M) n y he Newmrk meho []. The ime seps s is use or he presen meho n s n 0 s re use or he Newmrk meho. ery goo greemen eween hese wo resuls n e oserve. A Mi ross Lo hisory ig.. lmpe em suee o onenre lo. isplemen (0 - m) Time(0 - s) ig.. Time hisory o he mi-spn eleion ( ). isplemen W (0 - m) X W h X X X U e X ig. 4. Time hisory o he mi-spn eleion ( W ). L ( ( N) 844 Newmrk s Newmrk s M s (s) Newmrk = 0-5 s Newmrk = 0-6 s M = 50-7 s Time(0 - s) B

6 isplemen (r) Time(0 - s) ig. 5. Time hisory o he mi-spn roion ( ).. ONLUION Newmrk s Newmrk s M s An eplii meho se on he enrl ierene meho or nonliner rnsien ynmi nlysis o spil ems wih inie roions using oroionl ol Lgrngin inie elemen ormulion is presene. The glol nol prmeers o he sysem re hosen o e he omponens o nol isplemen veor n nol roion veor. The vlues o he nol roion veors re rese o zero he urren onigurion o he sruure in his suy. The snr enrl ierene meho is pplie o he inremenl isplemen n roionl veor n heir ime erivives. The orienions o he noes re upe y he inremenl nol roionl veors. The elemen oorine sysem onsrue he urren onigurion o he em elemen is us lol ineril oorine sysem no moving oorine sysem hus he irs n seon ime erivives o he posiion veor o he em elemen eine in he elemen oorines re he solue veloiy n elerion. The em elemen evelope hs wo noes wih si egrees o reeom per noe. Three roion prmeers reerre o he urren elemen oorines re eine o eermine he orienion o elemen ross seion. Boh he elemen ineri n eormion nol ores re sysemilly erive y using onsisen seon orer linerizion o he ully geomerilly nonliner em heory he 'Alemer priniple n he virul work priniple. The vlues o roion prmeers will onverge o zero n heir ime erivives will onverge o onsns wih he erese o he elemen size hus he oupling eween roion prmeers n heir ime erivives re no onsiere in his suy. The ormulion is inene or eplii inegrion proeures so siness mries re no evelope. The elemen equions re onsrue irs in he elemen oorine sysem n hen rnsorme o he glol oorine sysem y using snr proeure. The snr enrl ierene meho is pplie o he inremenl isplemen n roionl veor n heir ime erivives. rom he numeril emple suie he ury n eiieny o he propose meho re well emonsre. I is elieve h he oroionl ol Lgrngin ormulion o he em elemen n he numeril proeure o he eplii meho presene here my represen vlule engineering ool or he ynmi nlysis o spil em sruures. REERENE [] A. ron n M. erin A em inie elemen non-liner heory wih inie roion Inern. J. Numer. Mehs. Engrg. ol pp [] J.. imo n L. u-quo On he ynmis in spe o ros unergoing lrge moions- geomerilly e pproh ompu. Mehos Appl. Meh. Engrg. ol pp [] K. M. Hsio J. Y. Lin n W. Y. Lin A onsisen o-roionl inie Elemen ormulion or eomerilly Nonliner ynmi Anlysis o - Bems ompu. Mehos Appl. Meh. Engrg. ol pp. -8. [4] T. N. Le J. M. Bini n M. Hi ynmis o em elemens in oroionl one: omprive suy o eslishe n new ormulion inie Elemens in Anlysis n esign ol. 6 0 pp. 97. [5] T. Belyshko n B. J. Hsieh Nonliner rnsien inie elemen nlysis wih onvee oorines Inern. J. Numer. Mehs. Engrg ol pp [6] T. Belyshko n L. hwer Lrge isplemen rnsien nlysis o spe rmes Inern. J. Numer. Mehs. Engrg. ol. 977 pp [7] K. M. Hsio A o-roionl Tol Lgrngin ormulion or Three imensionl Bem Elemen AIAA Journl ol pp [8] W. Y. Lin n K. M. Hsio o-roionl ormulion or eomeri Nonliner nlysis o ouly ymmeri Thin-Wlle Bems ompu. Mehos Appl. Meh. Engrg. ol pp