BEAM ELEMENT FOR TRUSS BEAM WITH ELASTOPLASTIC-BUCKLING BEHAVIOR
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1 BEAM ELEMEN FOR RUSS BEAM WIH ELASOPLASIC-BUCKLING BEHAVIOR S. Mooi Dar o Bi Eviro oo Isi o coog Jaa ooi@vg.ic.ac.j Absrac: I is ar w wi sggs s ba wic abs s o aaz rss bas ivovig asoasic bcig bavior o cord brs wio cosrcig discr od. I is i-sracs or idig ad bcig bavior ar cosidrd i sac o (M i M j N. Eac co o sracs corrsods o idig ad bcig srg o ac cord br i a rss ba. I is assd a oa oda disac ca b xrssd i or o addiiv dcoosiio o asic asic ad bcig coos. Frror w dscrib a i is ossib o vaa c o bcig bavior o a cord br as isoroic soig bavior or o a co o sracs i sac o (M i M j N. Fia w wi xai vaidi o or ba rog a rica xa.. INRODUCION W cosidr a rss ba as sow i Figr (a. is rss ba bogs o Warr rss. Usa srcra dsigrs do o dirc aaz sc g srcrs as ar coosd wi a rss bas sic sc aaica ods av c or rdo dgr br ad ar vr cos. a civ coios od i wic a rss ba is racd wi a sig ba sow i Figr (b is sa sd w aazig daic bavior o sc a g srcr. coios od cod o aid o asic rob i w dvod a rss ba wic abd s o sia asoasic rob (Mooi a.(000b. Howvr cord brs bc ad ca o carr a axia orc as soo as id i corssio. I is ar w wi dscrib cosis ad covi rss ba o cosidr asoasic bcig bavior o cord brs i coios od. A irs w xai a asoasic bcig bavior o a cord br ca b aroxia rrsd as asic bavior wi soig dr assio a is sdrss raio is ss(mooi a.(000a. Nx w ora rss ba basd o rodaics aroac. Fia w sow vaidi o rs b coariso wi rss b a discr od o wic cord brs ar dividd b sadard ba s. L H N Q M i i Q N δ M j j (a Discr od Figr wo s o rss od (b Eciv coios od
2 . BASIC EQUAIONS FOR CHORD MEMBER I is assd a Hoz r rg ψ or cord br wic cosis o a asic sraig bar ad a rc asoasic roar srig i Figr (b ca b wri b ( ( ψ ( wr ad ar a oa src ad oa roaio ad ar asic coos o a src ad a roaio ad ad ar a asic siss o a bar ad a asic siss o a roar srig. sb-six as va o - br. I is ar i is asd a is awas osiiv ad a cord br dos o bc i is br ids. Sbsiig i Casis-D iqai; 0 ψ ro Eq.( givs ( 0 Γ ( wr Γ is cad asic dissiaio r ad is xrssio a variabs o ad ar rodaic orc o ad rsciv. id cio a roar srig; ( ( 0 ca b rwri i or ( ( 0 wr ( ( ( (3 Hr rici o axi asic dissiaio is irodcd o sci os-bcig bavior. So cosidr Lagragia; Γ L wr is a asic cosis arar. Diriaio o Lagragia wi rsc o or givs ( s qaios ar cad voio qaios. Frror i id codiio o Eq.(3-a is aciv oowig K-cr coar codiios s b saisid 0 0 ad 0 (5 Now cosidr cas a br s sdrss raio is c ss. I is cas assio a is aroxia qa o is saisid. id cio o Eq.(3 ca b rwri i or ( ( 0 wr ( ( ( ( ( (6 (a rss (b Easoasic bcig od Figr Sig cord br / / Aaica arg / A B /
3 b W irodc a civ asic-bcig coo wic is did b or o addiiv dcoosiio. b b b b (7 Fro ird oadig codiio o Eq.(5-c cssar codiio o sais scod aw o rodaics is giv b > 0 ( EXEND O RUSS BEAM 3. Basic Eqaios or rss Ba Hoz r rg Ψ or wo o a rss ba ca b giv b s o ac br s Hoz r rg ψ. is qaio ca r rwri i or a ( Ψ ψ (9 a { ( } Ψ (0 wr ( sigiis rasos is a asic siss arix or a civ coios od ad is civ asic coo o oda raiv disac. Ad w ass a oa coo ca b rwri b or o addiiv dcoosiio. wr b i j δ ( ad b ar asic ad bcig coos o oda raiv disac. b b is irodcd. Frror assio o ( ad ( a a a b a ( b ( b ( I is cas Casis-D iqai ca b rwri as Ψ 0. is oda orcs; N M M. sbsiig i is iqai ro Eq.(0 givs i j a a { } b [ ( ] [ ( ] 0 (3 For is iqai qaio o b r or a vas o givig b ( or ir coicis s b zro b [ ] 0 a ( ird qaio rrss dissiaio r. B coariso wi Eq.(-c i is drsood a b bo ad ca b rrsd b wic is did b (as sow i Figr 3. b (5 Ad ro Eq.(3 id cio or a rss ba od ca b xrssd i or ( ( 0 (6 xrssio o Eq.(5 ad Eq.(6 as a bo asic ad bcig coo ra ar
4 roorioa o gradi o id srac. Na associa ow r is saisid i is od. I is od a Eq.( is siiar o xdd Koir s or. For siici ass a cas a br s sdrss raio is c ss. I is cas assio a is aroxia qa o is saisid. id cio o Eq.(6 ca b rwri i or ( ( 0 ( ( ( (7 wr is a civ id srss. Cosq abov assio abs o ra asoasic bcig rob as asic rob wi ardig (soig sciid wi Eq.(7-b. Ad id srac bcos o b so-cad i-srac i (M i M j N sac as Eq.(7-a s b saisid a a. Hr w irodc a civ asic-bcig coo b wic is did b b b ad b b. Sbsiig i ar qaio ro Eq.( ad coarig wi Eq.(7 givs aciv a b 0 0 < or Θ (8 is qaio is asic ow r or rs od. Frror ardig (soig coici ca b cacad b ( 6 (9 id cio asic ow r ad ardig (soig ror ar cariid. H H ξ ξ µ ± Figr 3. Loca coordia ss 3. Cosis ag Siss Marix Sic 0 or aciv sbsiig i i ro Eq.(8 ad Eq.(9 { } 0 aciv b (0 w av siaos qaios rsc o as oows; aciv G or aciv ( wr δ 6 G δ :rocr s da sbo ( gravi o axia orc or ac cord br i od j od 0
5 B sovig is siaos qaio w ca id asic cosisc arar G aciv wr G is ivrs o G. Ad ro oda orc ra cosis ag siss arix b is giv ( ( b G ( aciv aciv wr sigiis sor rodc.. (3 3.3 Nrica iaio o caca oda orc vcor rs cacaio od o a oda orc vcor bogs o Rr Maig Agori or Mi-srac id cio (Sio a.(988. vas o ad i coigraio a ad a ar assd o b ow. Easic rdicor : A icra asic doraio is roz; b ria b b 0 0 ria ria. B sig s vas ria vas ca b cacad b ria ria b ( ria ria 6 ria (5 s ria va o id cio is giv b ria ria ria ria ria ( (6 ria I is grar a zro asic corrcor s b xc sic roar srig is i asic oadig. Orwis i is i asic sa icdig oadig sa. Pasic corrcor : srss b aca asic doraio is raxd. va o oda orc vcor ad civ id srss a ca b rwri i or ria b ria ria ( ( 6 (7 Isrig Eq.(7 io id cio givs oiar siaos qaios wi rsc o ( aciv. w sov b Nwo od. ( ( δ aciv or aciv (8 ( aciv ( ( ( ( Sovig s iar qaios ad daig ac vas. 6 ( ( b b ( ( ( δ ( ( aciv ( δ δ δ (9 ( ( ( ( b ( ( ( ( ( 6 ( ( (30 wr owr idx ( sigiid iraio br. Ra s cacaios i ( orac.
6 . NUMERICAL EXAMPLES Fia w sow wo rica xas i Figr ad Figr 5 o vaid rs od. O xa is sbjcd o oooic oadig ad aor is sbjcd o ccic oadig. rs rss ar cacad wi o o ad aor is do wi od o wic cords ar dividd b 0 asoasic ba s. caica roris o aaica od: rss g; 3.6 Hig; 0 Sdrss raio o cord brs; 0 (aria 0. aria roris: Yog s ods; 06GPa Easic-rc asic aria Yid srss; 35MPa. wo qiibri as ar cos ad is as rs od is vaid. -.0 δ /δ N/N N Prs N δ Discr od Proosd od Mi i Discr od Loadig ar : M j0 δ H i Mj M i /M Discr od Proosd od Discr od Proosd od Loadig ar : M j 0 δ H i Bcig cord (a Load disac crv (b N-M i sac Figr Nrica rss (oooic oadig Yidig cord Bcig cord 0.8 N/N N/N -.0 δ /δ 0 Bcig cord Discr od Proosd od N/N Bcig cord 0 Yidig cord M i /M.5 Bcig cord Bcig cord 0 (a Load disac crv Figr 5 Nrica rss (ccic oadig (b N-M i sac 5. CONCLUSIONS I is ar w dscribd cosis ad covi aaica od o sia rss ba robs icdig wi asoasic bcig bavior o cord brs ad cariid a sc robs rdc o r asoasic rob dr codiio a a cord br s sdrss raio is c ss. Frror vaidi o rs od was xaid rog rica xas. Rrcs: Sio J. C. Kd J. G. Govidj S. No-soo Misrac Pasici ad Viscoasici. Loadig/Uoadig Codiios ad Nrica Agoris Iraioa Jora o Nrica Mods i Egirig 988 Vo Mooi S. Osa. Cosis ad Covi Aaica Mod or Easoasic Bcig Prob o Corssio Mbrs ASCS0 Procdigs 000 Vo.II So Kora. Mooi S. Osa. Aaica Mod or Easo-asic Bavior o rss Girdr Basd o or o Pasici Jora o Srcra ad Cosrcio Egirig (rasacio o Arcicra Isi o Jaa 000 No (i Jaas.
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