PRACTICAL FINITE ELEMENT PROCEDURE FOR ACHIEVING MESH OBJECTIVITY IN LOCAL BUCKLING ANALYSIS OF STEEL STRUCTURES BY BEAM ELEMENTS

Transcription

1 Advancd Stl Constuction ol. 5, o., pp. 4-6 (9) 4 PRACTICA FIITE EEMET PROCEDURE FOR ACHIEIG MESH OBJECTIITY I OCA BUCKIG AAYSIS OF STEE STRUCTURES BY BEAM EEMETS Eiki Yamaguchi Pofsso, Dpatmnt of Civil Engining, Kyushu Institut of Tchnology, Tobata, Kitakyushu, Japan yamaguch@civil.kyutch.ac.jp Rcivd: May 7; Rvisd: 4 Mach 8; Accptd: 7 Mach 8 ABSTRACT: Sinc th nonlina finit lmnt analysis of a stl stuctu by shll/solid lmnts is xpnsiv, ffot has bn mad to conduct th local buckling analysis of a stl stuctu by bam lmnts. To this nd, th stuctual dtioation du to local buckling of a stl mmb is implmntd in constitutiv lationship. Th appoach invitably lads to th constitutiv lationship of softning typ, which howv dos not adily yild msh objctiv sult. Th psnt study poposs a finit lmnt pocdu to ovcom th poblm: in a local-buckling zon, avag stat vaiabls instad of local stat vaiabls a usd. Th ffctivnss of th poposd pocdu is vifid by solving xampl poblms. Moov, th applicability of a simpl tilina typ of constitutiv lationship associatd with th poposd bam-lmnt analysis is invstigatd in compaison with shll-lmnt analysis by ABAQUS. Kywods: Stl stuctu; local buckling; bam lmnt; msh objctiv; softning-typ constitutiv lationship. ITRODUCTIO Stuctual stl has bn usd xclusivly as a thin-walld mmb in its civil and building application. On of its majo failu mods is a local buckling, which may dtioat th mmb stngth considably. Thfo, th phnomnon has bn studid fo many yas. vthlss, a numb of stl stuctus undwnt local buckling in th 995 Hyogo-kn ambu Eathquak also known as Kob Eathquak, which was th vy fist tim stl stuctus in svic w damagd so badly in Japan, supising many Japans sachs who had confidnc in th safty of stl stuctus. Quit a fw xpimntal sachs w thn conductd in Japan []. Analytical appoach was also takn and th local buckling has bn simulatd succssfully not only by xpimnt but also by -dimnsional finit lmnt analysis using shll/solid lmnts [-]. Along this lin, ffot has bn mad also to analyz local-buckling bhavio by bam lmnts fo th duction of computational cost. To this nd, th stuctual dtioation du to local buckling is implmntd in constitutiv lationship, which is invitably of a softning typ [4-6]. In fact, th duction of computational cost has bn a challnging subjct in vaious ngining filds. Fo buckling analysis, fo xampl, th wok of ittl is notd [7]. Th softning typ of constitutiv lationship has bn usd fo th cacking analysis of conct. A poblm associatd with this appoach is dpndnc on finit lmnt msh: numical sult by th simpl application of a softning-typ constitutiv lationship would not convg as th msh bcoms fin. In oth wods, msh objctiv sult is not obtaind. To solv this poblm, nonlina factu mchanics has bn xplod and, to b spcific, it has bn solvd by contolling th slop of a softning banch of a constitutiv lationship [8-]. Howv, th issu of msh objctivity has not bn thooughly undstood in conjunction with th local buckling of stl stuctus. Th Hong Kong Institut of Stl Constuction

2 5 Eiki Yamaguchi M o θ p y M o θ x S o v S o v y Figu. Bnding Poblm of Bam In th psnt study, th msh objctivity is fist lookd into, which shows that just lik in th conct cacking analysis, sult dos not convg with th finmnt of msh in th analysis of a stl stuctu by bam lmnts coupld with th simpl application of th softning-typ constitutiv lationship. Th objctiv of th psnt study is thn to popos a finit lmnt pocdu to yild msh objctiv sult. Th ffctivnss of th poposd pocdu is dmonstatd by solving xampl poblms. onlina analysis of a two-dimnsional bam is conductd to this nd.. COETIOA FORMUATIO FOR OIEAR BEAMS Th convntional finit lmnt pocdu fo th bnding bhavio of a two-dimnsional bam is bifly dscibd in this sction fo th sak of slf-sufficincy. onlinaity is in constitutiv lationship. Assuming that displacmnt is small, th Bnoulli-Eul bam thoy fo th bam shown in Figu lads to th following govning quations: Equilibium quations: M '' p () y Cuvatu-displacmnt lationship: v" () Momnt-cuvatu lationship: M C () Bounday conditions: S S o v v, M M o v' at x (4) S S o v v, M M o v' at x (5) wh M, p y,, v, S and a bnding momnt, distibutd load in th y-diction, cuvatu, displacmnt in th y-diction, sha foc and otational angl, spctivly. A quantity with th top ba such as S is th pscibd on. Th top dot such as that in M and th pim such as that in M ' psnt th at and th divativ with spct to x, spctivly. C is bnding igidity and is qual to EI (th poduct of th Young Modulus and th momnt of intia) at th initial stag of dfomation. As dfomation incass, C may vay, initiating nonlina bhavio.

3 Pactical Finit Elmnt Pocdu fo Achiving Msh Objctivity in 6 ocal Buckling Analysis of Stl Stuctus by Bam Elmnts θ θ Figu. Standad -nod Fou-d.o.f. Bam Elmnt Th application of th wightd sidual mthod to ths quations yilds ' ", x y Mw Sw w dx p dx Mw H (6) wh w is th wight function. Using a standad -nod fou-d.o.f. bam lmnt shown in Figu [] and th Galkin mthod, th displacmnt and th wight a disctizd as T v U (7) T W W W W w W (8) wh,,, (9) is th lngth of a bam lmnt and is th local coodinat attachd to ach lmnt, taking at od and at od. Th disctization by this bam lmnt givs th following xpssion fo ach lmnt: H T F I F EX W () wh BM d F I () y M S d p, EX F () d d B ()

4 7 Eiki Yamaguchi Th assmblag of all th lmnt contibutions and th abitainss of th wight function yild th following disctizd govning quations: F I F EX (4) Bcaus of th nonlinaity of Eq., Eq. 4 is a st of nonlina quations. To obtain th solution, Eq. 4 is linaizd as K( U ( m) ( m) ( m) EX I ) U F F ( U ) (5) Th cofficint matix K, also known as th stiffnss matix, has th following xpssion at an lmnt lvl: 6 6 T C 4 6 K Β CΒ d (6) 6 sym. 4 (m) Eq. 5 is solvd fo U and U is updatd. Th pocdu is patd until convgnc is obtaind. Th supscipt m in th panthsis shows th numb of itation. In ality, sha dfomation contibuts to dflction. Th Timoshnko bam thoy can b usd to includ th ffct of sha dfomation. In this thoy, th dflction is xpssd as th sum of thos du to bnding and sha dfomation: b s (7) b s S (8) GkA wh th supscipts b and s stand fo th contibutions du to bnding and sha dfomation, spctivly, G is th sha modulus, A is a coss-sctional aa and k is th coction facto that accounts fo th diffnc fom actual sha stss distibution ov a coss sction. Th valu of k dpnds on th coss sction of a bam, and fo a thin-walld mmb th atio of th coss-sctional aa of a wb ov th whol coss-sctional aa A can b assignd to k []. Th cofficint matix K fo th Timoshnko bam thoy is thn givn by 6 6 C (4 ) 6 ( ) K (9) ( ) 6 sym. (4 ) wh

5 Pactical Finit Elmnt Pocdu fo Achiving Msh Objctivity in 8 ocal Buckling Analysis of Stl Stuctus by Bam Elmnts P t b (a) Sid iw b (b) Coss Sction Figu. Cantilv Bam M M M My My Mc My κ κ κy κy κy κc (a) Typ I (b) Typ II (c) Typ III κ Figu 4. Bnding Momnt M cuvatu Rlationships C GkA (). UMERICA AAYSIS I: MESH DEPEDECE Th two-dimnsional bhavio of a stl cantilv bam subjctd to a concntatd load at th f nd (Figu ) is mployd as a numical xampl hin. Th bam lngth is m and th coss sction is a squa box consisting of fou thin plats, whos width b is mm and thicknss t is 4 mm. Th Young Modulus E is 6 k/mm and th yild stss y is 5 /mm. Th main objctiv of this sction is to shd light on th difficulty in daling with th constitutiv modl of a softning typ. To this nd, it is sufficint to consid only th bnding bhavio of th bam: no sha dfomation is takn into account in this sction. Fo nonlina bhavio, th simpl constitutiv lationships btwn th bnding momnt M and th cuvatu shown in Figu 4 a assumd. Typ I psnts a typical lasto-plastic bhavio. o dscnding banch (softning banch) xists. Typ II has no hadning banch but has th softning banch, which mbodis th dtioation du to local buckling without undgoing plastic dfomation. Typ III includs both hadning and softning banchs: th lastic bhavio is followd by th lasto-plastic bhavio, which thn lads to th dtioation du to local buckling. Th slops of th hadning and softning banchs a assumd to b EI/ and EI/, spctivly. Fo Typ III, c is assumd to b twic as lag as y, i.. c y. In th hadning and softning banchs, unloading bhavio may tak plac. Th slop of th unloading M lationship is th sam as that of th lastic bhavio, which is illustatd by dottd lins in Figu 4.

6 9 Eiki Yamaguchi 6 6 P (k) 4 lmnts lmnts 4 lmnts 8 lmnts 4 6 (a) Typ I P (k) 4 lmnts lmnts 4 lmnts 8 lmnts 4 6 (b) Typ II 6 P (k) 4 lmnts lmnts 4 lmnts 8 lmnts (c) Typ III Figu 5. oad P Displacmnt v Cuvs of Cantilv Bam Using th bam lmnts dscibd ali, fou finit lmnt mshs a constuctd to s th influnc of th msh on numical sult. Th fou mshs us,, 4 and 8 lmnts, spctivly. All th lmnts in ach msh a qual in lngth. Th numical sults of Typ I a psntd in th fom of th load P displacmnt v lationship at th loading point in Figu 5(a). Evn though dpndnc on th msh is not significant, clos obsvation vals th tndncy of th convgnc: as th lmnt lngth bcoms small, th sult tnds to convg. Th -, 4- and 8-lmnt mshs yild pactically th sam spons. Fo instanc, at v=6 mm, P is qual to 599 k in th cas of th -lmnt msh whil it is just about 597 k in th oth th cass. Msh objctiv sult is thus obtaind with Typ I. Figu 5(b) shows th sults of Typ II. Th dpndnc on th msh is cognizd mo claly, as th sults by th fou mshs a vy diffnt fom ach oth. Unlik th pvious sult of Typ I, no tndncy of th convgnc is obsvd in this typ of constitutiv modl, which indicats that th sult hin is not msh objctiv. Th P v lationship du to Typ III is psntd in Figu 5(c). This figu claly indicats th dpndnc on th msh. As th msh bcoms fin, th dpndnc ducs in th hadning potion of th P v lationship whil such tndncy is not obsvd in th softning potion. Th poblm associatd with th softning typ of constitutiv lationship is mad appant again by Typ III.

7 Pactical Finit Elmnt Pocdu fo Achiving Msh Objctivity in ocal Buckling Analysis of Stl Stuctus by Bam Elmnts P P (a) Whol Bam Figu 6. Fixd-fixd Bam (b) Half Bam P (k) 8 4 lmnts lmnts 4 lmnts 8 lmnts Figu 7. oad P Displacmnt v Cuvs of Fixd-fixd Bam (Typ III) Additionally, a statically indtminat stuctu of a stl fixd-fixd bam with a concntatd load at th mid-span (Figu 6(a)) is also analyzd with Typ III. Th total lngth of th bam is m and th coss sction is th sam as that of th cantilv (Figu (b)). Sinc it is sufficint to consid only a half of th bam du to symmty, th analysis of th bam shown in Figu 6(b) is conductd. Th numical sults in th fom of th P v lationship at th loading point a plottd in Figu 7. This figu shows that msh objctiv sult is not obtaind in th statically indtminat stuctu as wll. 4. PROPOSED FORMUATIO FOR SOFTEIG BEHAIOR Th following obsvations a gnally mad in th local buckling phnomnon of a stl thin-walld mmb:. Rgion subjctd to local buckling is limitd. (Th gion is haft calld th local-buckling zon o BZ fo shot.). Stain distibution in BZ dpnds on buckling mod and is vy complicatd in gnal, so that stain in BZ vais fom point to point wildly. It is notwothy that th study on BZ has bn conductd, poviding th following mpiical fomula fo valuating th lngth of BZ of a box-sction mmb []: BZ Min(.7b, a) () wh BZ is th lngth of BZ, a is th distanc btwn two adjacnt diaphagms and b is th width of a flang. Min indicats that th small of th two valus in th panthsis shall b takn.

8 Eiki Yamaguchi Making us of th abov obsvations, th constitutiv lationship btwn M and in BZ is assumd to b contolld by th avag quantity in BZ in th psnt study. To b spcific, th constitutiv lationship of Eq. is placd by M C( ) fo av c (a) M ) fo av c (b) C( av wh av is th avag cuvatu in BZ. Spcifically, av is dfind by av BZ dx BZ BZ d BZ () wh BZ is th lngth of th lmnt locatd in BZ. Th poposd appoach may b viwd as a kind of nonlocal fomulation with BZ bing th chaactistic lngth [4, 5]. Fo th bam lmnt mployd in this study, th cuvatu vais as follows: d v BU T d (4) As Eq. indicats, B is a st of lina functions with spct to. Thfo, vais in a lina fashion within a bam lmnt. Eq. can thn b wittn as BZ BZ av (5) BZ wh / BZ is th valu of valuatd at th middl of th lmnt. If th lngths of th lmnts in BZ a qual to ach oth, th abov quation can b futh simplifid as BZ av (6) wh is th numb of lmnts locatd in BZ. It is also notd that Eq. b lads to M av C av M ( ) d C( av ) d d C( av ) av (7) BZ BZ BZ BZ BZ BZ wh M av is th avag bnding momnt in BZ. Sinc dfomation in BZ is complicatd, xpimnts povid infomation only fo this class of constitutiv lationship in pactic, i.. th lationship only in tms of avag stat vaiabls [5, 6]. In oth wods, C( av ) can b dtmind xpimntally.

9 Pactical Finit Elmnt Pocdu fo Achiving Msh Objctivity in ocal Buckling Analysis of Stl Stuctus by Bam Elmnts It is notd that th finit lmnt pocdu dscibd ali can b usd with this constitutiv lationship of Eq.. 5. UMERICA AAYSIS II: EFFECTIEESS OF PROPOSED APPROACH Th two poblms of th cantilv bam (Figu ) and a half of th fixd-fixd bam (Figu 6(b)) a solvd by th poposd appoach. Sinc th coss sction at th fixd nd dos not dfom and a diaphagm is placd at th coss sction und th concntatd load, BZ can dvlop na th fixd nd in both bams and also na th loading point in th cas of th fixd-fixd bam. Following Eq., it is dcidd that th lngth of ach BZ is 4mm. Th poposd constitutiv lationship of Eq. is applid to ths BZs. Th valus of th paamts in th constitutiv lationship a assumd to b th sam as thos mployd in th pvious analysis, i.. umical analysis I. Howv, th physical maning of th citical valu c fo th judgmnt on th initiation of local buckling has bcom diffnt: it is now in tms of th avag cuvatu av whil it is local cuvatu in th pvious analysis. Outsid BZ no local buckling taks plac so that no stuctual dtioation occus. Th constitutiv lationship outsid BZ is thfo assumd to b givn by Typ I (Figu 4(a)) and th paamts tak th sam valus as thos mployd pviously. Fou finit lmnt mshs a mployd in this analysis. In th coasst msh, svn lmnts a usd fo th whol bam: BZ is modld by on lmnt and th maining zon is modld by th st of th lmnts. Th lmnt msh is find by halving th lmnt lngth, so th fou mshs consist of 7, 4, 8 and 56 lmnts, spctivly, and BZs in th fou mshs a modld by,, 4 and 8 lmnts, spctivly. Th lmnts in BZ a qual in lngth and th lmnts in th maining zon a qual in lngth as wll, but th lmnt lngths in BZ and th maining zon a slightly diffnt fom ach oth. Th sults a shown in Figus 8 and 9. Both figus indicat that th vaiations du to th diffnc in msh a much small than th countpats in th pvious analysis with Typ III (Figus 5(c) and 7). ot only hav th vaiations bn ducd but th convgnc is obsvd as th msh bcoms small, which can b alizd mo claly in Figus 8(b) and 9(b). Th convgnc appas to b lativly slow in th fixd-fixd bam, yt th 8- and 56-lmnt mshs hav yildd indistinguishabl sponss. Th ffctivnss of th poposd appoach is thus vifid. 6 6 P (k) 4 7 lmnts 4 lmnts 8 lmnts 56 lmnts (a) Ovall Rang P (k) 55 7 lmnts 4 lmnts 8 lmnts 56 lmnts (b) Clos-up of Pak Rgion Figu 8. oad P Displacmnt v Cuvs of Cantilv Bam by Poposd Appoach

10 Eiki Yamaguchi P (k) lmnts 4 lmnts 8 lmnts 56 lmnts (a) Ovall Rang P (k) 7 lmnts 4 lmnts 8 lmnts 56 lmnts (b) Clos-up of Pak Rgion Figu 9. oad P Displacmnt v Cuvs of Fixd-fixd Bam by Poposd Appoach 6. UMERICA AAYSIS III: APPICABIITY OF TRIIEAR TYPE OF COSTITUTIE REATIOSHIP I TERMS OF BEDIG MOMET M AD CURATURE Th finit lmnt pocdu fo achiving msh objctivity is poposd and its ffctivnss is confimd in th abov. Sinc th objctiv of th psnt study is to popos th msh-objctiv finit lmnt pocdu, th constitutiv lationship usd is on of th simplst softning-typ modls. Yt it may b intsting to xamin th applicability of this simpl tilina typ of constitutiv lationship fom pactical point of viw. Thfo, th compaativ study btwn th poposd bam-lmnt analysis with th tilina typ of constitutiv lationship and th shll-lmnt analysis by ABAQUS [6] is caid out hin. Th ABAQUS analysis taks advantag of symmty, and ach bam is modld by 44 fou-nod shll lmnts. Th matial in th ABAQUS analysis is assumd to b a von Miss typ of lasto-plastic matial and its uniaxial bhavio is bilina with th scond slop bing qual to E/. Th Poisson atio is., and th Young Modulus E and th yild stss y tak th sam valus as thos in th pvious bam-lmnt analysis. 6 P (k) 4 56 lmnts ABAQUS (a) oad P Displacmnt v Cuvs (b) Dfomd Configuation (ABAQUS) Figu. Analysis of Cantilv Bam und Concntatd oad Sinc in th ABAQUS analysis, sha dfomation contibuts to dflction, th Timoshnko bam thoy nds to b incopoatd into th poposd bam-lmnt analysis. This can b don simply by mploying Eq. 9 instad of Eq. 6. Futhmo, ca must b takn in th dtmination of th valus of th bnding igiditis C and th citical cuvatu c, sinc thy a not pu matial poptis but includ th stuctual chaactistics, paticulaly coss-sctional poptis. To this

11 Pactical Finit Elmnt Pocdu fo Achiving Msh Objctivity in 4 ocal Buckling Analysis of Stl Stuctus by Bam Elmnts nd, th cantilv poblm with th concntatd load (Figu ) is solvd fist by ABAQUS. Rfing to this ABAQUS sult, th bnding igiditis and th citical cuvatu a valuatd by tial and o, and C= EI/. fo th hadning bhavio, C= EI/46.8 fo th softning bhavio and c. 7 y a obtaind. Ths valus a to b usd not only fo th cantilv poblm with th concntatd load, but also fo all th oth poblms. Th sam fou mshs as thos dscibd in th pvious sction a usd again in this compaativ study. As is in th pvious xampls (Figus 8 and 9), th numical sult convgs also hin with th finmnt of th msh: th sponss obtaind by th 8-lmnt msh and th 56-lmnt msh a pactically th sam. Bcaus of th convgnc, only th numical sult by th 56-lmnt msh is psntd in Figu (a) togth with that du to ABAQUS. Th diffnc in th maximum load btwn th two sults is. %. Th post-pak dfomd configuation obtaind by th ABAQUS analysis is givn in Figu (b). Fo th cla psntation, th dfomation is magnifid by th facto of in this figu. Th local buckling is claly obsvd in th compssion flang na th fixd nd. Compad to this ABAQUS sult, th BZ lngth of 4mm mployd in th poposd bam-lmnt analysis appas justifiabl. Rplacing th concntatd load with unifomly distibutd load, th cantilv bam (Figu (a)) is analyzd again. All th valus of th paamts in th constitutiv lationship of th poposd appoach including th valus of C and c main th sam as thos in th cantilv-bam poblm with th concntatd load. Th numical sult obtaind is psntd in Figu (b) togth with that du to ABAQUS. Bcaus of th convgnc, only th numical sult by th 56-lmnt msh is givn. Th two sults a in good agmnt with ach oth: th diffnc in th maximum load is.56 %. Th dfomd configuation in th ABAQUS analysis is psntd in Figu (c). Th dfomation is magnifid by th facto of. Th local buckling is claly obsvd in th compssion flang na th fixd nd, and th BZ lngth of 4 mm appas good also in this poblm. q (a) Cantilv Bam und Distibutd oad q (k/m) lmnts ABAQUS (b) oad P displacmnt v Cuvs (c) Dfomd Configuation (ABAQUS) Figu. Analysis of Cantilv Bam und Distibutd oad

12 5 Eiki Yamaguchi P (k) lmnts ABAQUS (a) oad P Displacmnt v Cuvs (b) Dfomd Configuation (ABAQUS) Figu. Analysis of Fixd-fixd Bam A half of th fixd-fixd bam (Figu 5(b)) is thn analyzd. All th valus of th paamts in th constitutiv lationship a kpt th sam as thos in th pcding two xampls. Th numical sult is psntd in Figu (a) togth with that du to ABAQUS. Bcaus of th convgnc, only th numical sult by th 56-lmnt msh is givn. Th poposd appoach ags wll with th ABAQUS sult: th diffnc in th maximum load is 4. %. Figu (b) shows that local buckling occus in th compssion flangs na th fixd nd and th middl sction of th bam wh th concntatd load is applid, which matchs th sult of th poposd bam-lmnt analysis: BZs in ths two gions undgo th softning bhavio. Th occunc of th local buckling in th multipl locations is attibutd to th fact that th bam is statically indtminat. Th lngth of 4 mm appas good fo both BZs in th analysis of th fixd-fixd bam. Basd on th sults obtaind in this sction, it may b statd that th tilina typ of constitutiv lationship is pomising in th pactical analysis of a stl bam undgoing local buckling in spit of its xtm simplicity. 7. COCUDIG REMARKS ocal buckling of a stl stuctu can b simulatd wll by shll-lmnt analysis. vthlss, such an analysis quis much computational cost and is yt to b pactical. Hnc, ffot has bn mad to implmnt th stuctual dtioation du to local buckling in th constitutiv lationship so that a bam lmnt can b usd fo th local buckling analysis of stl stuctus. Howv, such a constitutiv lationship has a softning banch invitably, and as dmonstatd in this study, simpl application of th softning-typ constitutiv lationship dos not lad to msh objctiv sult. Against this backgound, th psnt study has poposd a finit lmnt pocdu that uss th avag stat vaiabls in th local-buckling zon to contol th stuctual dtioation du to local buckling. Th ffctivnss of th poposd finit lmnt pocdu has bn confimd by solving xampl poblms: th msh objctivity is shown to b achivd. It is notd that th application of th undlying concpt of th poposd pocdu is not stictd to th constitutiv lationship mployd in th psnt study: it can b applid staightfowadly to any constitutiv lationships.

13 Pactical Finit Elmnt Pocdu fo Achiving Msh Objctivity in 6 ocal Buckling Analysis of Stl Stuctus by Bam Elmnts On of th simplst constitutiv lationships that includ th influnc of th stuctual dtioation du to local buckling is th on mployd in th psnt numical xampls. At th nd, th psnt study has xplod th applicability of this simpl tilina constitutiv lationship in compaison with th shll-lmnt analysis by ABAQUS. Th sults hav indicatd that this typ of constitutiv lationship is pomising in th pactical analysis of a stl bam undgoing local buckling in spit of its xtm simplicity. Yt much mains to b don as to how th bnding igiditis and th citical cuvatu fo a givn stl mmb a dtmind, which is in fact a subjct of an on-going pojct in th autho s sach goup. REFERECES [] Public Woks Rsach Institut t al., Sismic Dsign fo Highway Bidg Pis, Tchnical Rpot of Joint Rsach, PWRI, Ministy of Constuction, Japan, 997. [] Goto, Y., Wang, Q. and Obata, M., FEM Analysis fo Hysttic Bhavio of Thin-walld Column, Jounal of Stuctual Engining, ASCE, 998, ol. 4, pp. 9-. [] Yamaguchi, E., agamatsu, T. and Kubo, Y., Influnc of Finit Elmnt Msh on Buckling Analysis of Stl Pip-sctiond Bidg Pis, Fifth Wold Congss on Computational Mchanics (WCCM ), Pap o. 8566,. [4] Yamaguchi, E., Ab, K. and Kubo, Y, Analysis of Stl Bidg Pis Undgoing ocal Buckling by Bam Elmnts, Pocdings of 5th Intnational Colloquium on Stability and Ductility of Stl Stuctus, 997, pp [5] Sakimoto, T., Watanab, H. and akashima, K., Hysttic Modls of Stl Box Mmbs with ocal Buckling Damag, Jounal of Stuctual Mchanics and Eathquak Engining, JSCE,, o.647/i-5, pp [6] Watanab, H. and Sakimoto, T., Sismic Rspons Analysis of Conct-filld Stl Box Pis Considd on ocal Buckling, Jounal of Stuctual Mchanics and Eathquak Engining, JSCE,, o. 647/I-5, pp [7] ittl, G.H., Rapid Analysis of Plat Collaps by iv Engy Minimisation", Intnational Jounal of Mchanical Scinc, 974, ol. 9, o., pp [8] Hillbog, A., Mod, M. and Ptson, P.E., Analysis of Cack Fomation and Cack Gowth in Conct by Mans of Factu Mchanics and Finit Elmnts, Cmnt and Conct Rsach, 976, ol. 6, pp [9] Bazant, Z.P. and Oh, B.H., Cack Band Thoy fo Factu of Conct, Matials and Stuctus, RIEM, 98, ol. 6, pp [] Yamaguchi, E. and Chn, W.F., Cacking Modl fo Finit Elmnt Analysis of Conct Matials, Jounal of Engining Mchanics, ASCE, 99, ol. 6, pp [] Cook, R.D., Malkus, D.S. and Plsha, M.E., Concpts and Applications of Finit Elmnt Analysis, d Edition, w Yok, Y, John Wily & Sons, 989. [] ishino, F. and Hasgawa, A., "Elastic Analysis of Stuctus", Gihoudo, 98. [] Usami, T. (dito), Guidlins fo Sismic and Damag Contol Dsign of Stl Bidg, Tokyo, Japan, Gihodo, 6. [4] Bazant, Z.P., onlocal Damag Thoy Basd on Micomchanics of Cack Intactions, Jounal of Engining Mchanics, ASCE, 994, ol., pp [5] Khaloo, A.R. and Taivdilo, S., ocalization Analysis of Rinfocd Conct Mmbs with Softning Bhavio, Jounal of Stuctual Engining,, ol. 8, pp [6] ABAQUS/Standad Us s Manual,.5.7, HKS, 997.