Uncertainty in non-linear long-term behavior and buckling of. shallow concrete-filled steel tubular arches

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1 CCM th July, Cambridg, England Unrtainty in non-linar long-trm bhavior and bukling of shallow onrt-filld stl tubular arhs *X. Shi¹, W. Gao¹, Y.L. Pi¹ 1 Shool of Civil and Environmnt Enginring, Th Univrsity of Nw South Wals, Sydny, NSW 5, Australia. *Corrsponding author: shixu1985@hotmail.om Abstrat:This papr prsnts a thortial analysis for th non-dtrministi tim-dpndnt non-linar bhavior of shallow onrt-filld stl tubular (CFST arhs with intrval paramtrs undr a sustaind uniform load. Th hang rangs of th shrinkag strain and rp offiints of onrt or ar drivd from xprimntal rsults. Th virtual work mthod is usd to stablish th diffrntial quations of quilibrium for th tim-dpndnt bhavior and bukling analyss of shallow CFST arhs, and th ag-adjustd fftiv modulus mthod is adoptd to modl th rp bhavior of th onrt or. Analytial solutions of th intrval tim-dpndnt displamnts and intrnal fors of shallow CFST arhs ar drivd. Th lowr and uppr bounds of strutural rsponss ar dtrmind. Comparisons of th intrval analytial solutions with th intrval finit lmnt rsults show that th analytial solutions of th prsnt study ar aurat. Kywords: CFST arhs, intrval analysis, rp, shrinkag 1. ntrodution Appliations of onrt filld stl tubular (CFST arhs ar inrasing in nginring struturs, partiularly in bridg onstrutions. Th viso-lasti ffts of rp and shrinkag of onrt or ar invitabl in th long trm for CFST arhs. Whn a CFST arh is subjtd to a sustaind load, th rp of th onrt or will lad to th inras of its dformations with tim and th dformations may b signifiant, whil th shrinkag strain may also dvlop vn whn th arh is not subjtd to any load [B.C. Chn ]. Hn, an invstigation of signifiant ffts of rp and shrinkag of th onrt or on th tim-dpndnt strutural bhavior of CFST arhs is muh ndd. Howvr, it is notd that th rp offiint obtaind from tsts vary signifiantly from on xprimnt to anothr. Vry diffrnt prditions of th tim-dpndnt bhavior of CFST olumns hav bn rportd in diffrnt studis. This shows that th unrtaintis of rp and shrinkag of th onrt or do xist. To prdit th long-trm bhavior of CFST olumns rasonably, ths unrtaintis hav to b onsidrd. n this papr, intrvals ar adoptd to rprsnt th unrtaintis. This papr, thrfor,

2 aims to invstigat th intrval tim-dpndnt non-linar bhavior and bukling of shallow CFST arhs undr a uniform radial load, to driv analytial solutions for thir tim-dpndnt non-linar dformations, intrnal fors and bukling, and to dtrmin thir strutural lif tim prior to th bukling. To invstigat th ffts of th rp and shrinkag of onrt or, it is important to us an ffiint and aurat mthod to dsrib th rp and shrinkag of onrt. [Pi t al. (, Pi t al. (7] t is known that a numbr of mthods hav bn proposd and usd for th rp and shrinkag of th onrt. Among ths mthods, th ag-adjustd fftiv modulus mthod rommndd by AC Committ-9 and Australia dsign od for th onrt struturs AS36 ar ommonly onsidrd to b ffiint and aurat in valuating th tim-dpndnt bhavior of th onrt and it ould onvnintly b inorporatd into th strutural analysis [AC Committ 9 198]. Algbrai formulas usd in this mthod an b fftiv and pratiabl in modling rp and shrink-ag of onrt or, so th ag-adjustd fftiv modulus mthod is usd in this invstigation.. ntrval nonlinar lasti analysis of long-trm bhavior of shallow CFST arhs To prdit th long-trm prforman, intrval onstitutiv modl onsidring rp and shrinkag of th CFST olumn nds to b stablishd. Th basi assumptions adoptd for th intrval long-trm linar lasti analysis of CFST olumns in this papr ar: (1 dformations of CFST arh ar lasti and satisfy th Eulr-Brnoulli hypothsis, i.. th ross-stion rmains plan and prpndiular to th arh axis during dformation; ( th dimnsions of th ross-stion ar muh smallr than th lngth and radius of th arh so that th arhs ar suffiintly slndr; and (3 th ohsion and adhsion of two diffrnt matrial omponnts ar fully bondd. To aount for th non-linarity rsultd from rp and shrinkag of th onrt or, th drivation of th diffrntial quations of quilibrium for shallow CFST arhs nd to onsidr non-linar longitudinal normal strain-displamnt rlationship and th non-linar longitudinal normal strain of an arbitrary point in th ross-stion of shallow CFST arhs an thn b xprssd as [Pi t al ] 1 yv w v ( v (1 R Aording to th third assumption, th dformations of ah omponnt should b ompatibl with ah othr, so thir mmbran strains and also th strains at th intrfa ar idntial. Howvr, du to diffrnt Young s moduli and th ffts of rp and shrinkag in onrt or, th strss s in th stl tub and th strss σ in th onrt or ar diffrnt and thy ar givn by and s E ( E ( u ( sh s s sh s E (3

3 whr E is th ag-adjustd fftiv modulus of onrt, of onrt and an b givn by AS36 [AS36 1] sh ( t t 35 t sh is th shrinkag strain (4 whr t is th loading tim, is th shrinkag strain of onrt whn t. E an b alulatd by E E (, t t (5 1 (, t t (, t t whr τ is th ag at loading, χ(, t t is th aging offiint and (, t t is th rp offiint that an b xprssd as ( t t (, t t [ ] ( t t (6 whr χ(, t t is th rp offiint whn t. Th aging offiint an b xprssd as whr with χ(, t t 1 * k1τ χ k τ * ( 1 χ ( t t ( t t (7 (8 1.33,7 k (9 1.33,7 k.16.8 (1 t (11.118,7 1.5 Th diffrntial quations for th long-trm analysis of a CFST arh an b obtaind using a virtual work mthod. Whn th virtual work prinipl is usd for th long-trm quilibrium of th CFST arh, it an b statd as rquiring that th funtional Θ [-NR( dw - dv v dv -Mdv ]- qr dvdq -Θ (1 whr th axial omprssiv for N is givn by N 1 s d A d ( s s [ ( ] A s A AE AE w v sh s A v AE (13 and th bnding momnt is givn by 3

4 in th axial dirtion: in th radial dirtion: v M s ya d s ya d ( E s s E A s A R (14 N (15 M NRv NR qr (16 and lads to th stati boundary onditions for pin-ndd arhs as M at θ ±Θ From Eq.(15, th axial omprssiv for N is onstant along th arh axis. Substituting th onstant axial omprssiv for N and th xprssion for M givn by Eq.(14 into Eq.(16 lads to iv v ( v P (17 whr is a tim-dpndnt dimnsionlss axial for paramtr dfind by and P is a dimnsionlss load dfind by NR NR [, ] (18 E E E E s s s s P qr N (19 N By using th kinmati boundary onditions and th stati boundary onditions and th kinmati boundary onditions, th solutions of Eq.(17 an b obtaind as For Pin-ndd: Th radial displamnt an b xprssd as os( os( P θ Θ 1 P os( θ os( Θ 1 v [ { [( θ ( ]} { [( ( ]} Θ, θ Θ ( os( Θ os( Θ And th axial displamnt an b xprssd as: Lowr bound of axial displamnt: w PPθθ Θ (1 PP [ Θsin( θ θ sin( Θ] Θ Θ (1 ( 3 6 os( θ θ P θ θ θ [1 ] 3 os( Θ 4 Θos ( Θ P os( [ Θsin( os( sin( Θos( Θ Uppr bound of axial displamnt: (1 4

5 w PPθθ Θ (1 PP [ Θsin( θ θ sin( Θ] Θ Θ (1 ( 3 6 os( θ θ P θ θ θ [1 ] 3 os( Θ 4 Θos ( Θ ( P os( [ Θsin( os( sin( Θos( Θ ( For Fixd: Th radial displamnt an b xprssd as Lowr bound of radial displamnt: P ( Θ[os( θ os( Θ ] 1 v Θ (3 { [( θ ( ]} sin( Θ Uppr bound of radial displamnt P ( Θ[os( θ os( Θ ] 1 v Θ (4 { [( θ ( ]} sin( Θ And th axial displamnt an b xprssd as: Lowr bound of axial displamnt: Θsin( θ w [ ] (1 PP θθ ( Θ (1 PP θ 6 sin( Θ P θθ Θ P Θ [ os( θ ] sin( Θ 4 4 sin ( Θ 3os( sin( θ θ os( θ P θ os( θ P [ Θsin( θ os( θ θ sin( Θos( Θ [1 ] os( Θ 4 Θos ( Θ 3 (6 Uppr bound of axial displamnt: Θ Θsin( θ w [ ] (1 PP θθ ( (1 PP θ 6 sin( Θ P θθ Θ P Θ θ sin( 4 Θ 4 sin ( Θ 3os( sin( θ θ os( θ [ os( ] θ θ P θ θ θ [1 ] 3 os( Θ 4 Θos ( Θ P os( [ Θsin( os( sin( Θos( Θ (7 3. ntrval finit lmnt analyss For intrval bukling analysis, bukling load and quilibrium paths ar sought by solving th quilibrium quation at ah load inrmnt. Th quilibrium quation an b xprssd as: (8 KT u P At ah load inrmnt, prturbation mthod was mployd to alulat th global 5

6 stiffnss matrix. n th following, th ffts of th bam axial for and bnding ar inludd. Th rp offiint and shrinkag strain ar onsidrd as intrval paramtrs. Using th first-ordr Taylor xpansion and th intrval arithmti oprations [Rao t al. 1997], th intrval stiffnss offiints of th bam-olumn lmnt an b obtaind as K K (, T i KT i (, (, ( T i KTi (, ( K K (, K Ti (, Ti (, Ti (9 whr i dnots th numbr of lmnts,,,,, [-1, 1]. From Eq. (9, it an b asily obsrvd that th stiffnss is th funtion of th rp offiint and shrinkag strain. Th intrval global stiffnss matrix of th strutur is assmbld using th following quation K n T (, KTi (, i 1 n K Ti KTi T (, i 1 ( ( K E E Thn th intrval stati quation of quilibrium in th intrval finit lmnt systm boms yilds 6 (3 K (, ( U U P (31 T K ( K ( U K E E U (3 n 1 Ti Ti T (, [ ] i 1 To ovrom th omputational diffiultis ausd by th singularity of th tangnt stiffnss matrix at th limit points on th quilibrium path, th intrval inrmntal-itrativ solution is introdud for this study.

7 Th inrmntal-itrativ quilibrium quation at tim t and itrativ stp i an b xprssd as [ K ] U Pu ( P (33 t t t t t T i i r whr th subsript i dnots th itrativ yl, th suprsript t dnots th load stp, t Pu ( is th inrmntal xtrnal fors at tim t and t t Pr is th unbalan fors at tim t t. 4. Modl validation and disussions 4.1 Dtrmination of intrvals for th shrinkag strain and rp offiint th intrval of th shrinkag strain and rp offiint of thir onrt ors an b drivd from othr rsarhrs [Uy.B 1] tst rsults as [43.5, 34] and [.5, 1.7] rsptivly, whih ar usd in this study. t an b xptd that th rsults obtaind by th intrval modls proposd in this papr will ontain ths xprimntal rsults; in othr words, th xprimntal rsults will fall into th intrval bounds produd by th proposd modls. Th rp offiints dtrmind by Uy [Uy.B 1] is adoptd to ompar with th rsults obtaind by th intrval analytial modl dvlopd in this papr, whih ar illustratd in Figs. 1. Similarly, th total shrinkag strain dtrmind by Uy is ompard with th rsults givn by th proposd intrval modl in Fig.. t an b asily sn that th rsults produd by th proposd modl ontain ths xprimntal rsults as prdiatd. Th bounds an b furthr updatd if mor xprimntal rsults availabl. Ths bounds will b usful for th futur xprimntal invstigations and dsign. Fig.1 Comparison of rp offiint 7

8 Fig. Comparison of shrinkag strain Th typial long-trm nonlinar rsponss of pin-ndd and fixd shallow CFST arhs undr a sustaind uniform radial load ar ompard in Figs. 3 with thir linar ountr parts as th variations of th dimnsionlss ntral radial displamnts v / v,15 and th dimnsionlss axial displamnts is wθ/ / wθ /,15 at th quartr point with tim t, whr th Young s moduli of th stl and onrt wr assumd as Es GP and E3 GPa..A irular ross-stion with outr and innr radii: R5 mm and r4 mm was usd in th invstigation. Th span of th arh was L15 m and th ris-to-span ratio of th pin-ndd and fixd CFST arh wr f/l1/1, rsptivly. Figur 3 ntrval radial displamnts 5. Conlusions This papr prsnts a thortial study on th unrtain long-trm and bukling analysis of shallow onrt-filld stl tubular arhs subjtd to a sustaind uniform radial load. An intrval analytial modl basd on th algbraially tratabl ag-adjustd fftiv modulus mthod is proposd to dsrib th tim-dpndnt bhavior of onrt in CFST arhs. Non-linar analytial solutions for th tim-dpndnt displamnts and intrnal ations wr drivd. t has bn found that rp and shrinkag of onrt or hav signifiant ffts on th 8

9 tim-dpndnt non-linar dformations, intrnal fors and bukling bhavior of shallow CFST arhs. An intrval finit lmnt was dvlopd to dsrib th long-trm bhavior and analysis bukling. Th bukling load or bukling tim an b valuatd using this modl. Th rsult is ompard with analytial rsults; it ould b found it shows a good agrmnt.n th futur, th proposd modls will b furthr dvlopd to analyz othr typs of CFST struturs aounting for th unrtaintis in thir matrial and gomtri proprtis. Rfrn AC Committ 9, Prdition of Crp, Shrinkag and Tmpratur Effts in Conrt Struturs, Amrian Conrt nstitut (AC, Dtroit, 198. AS36, Australian Standard: Conrt Struturs, Standard Assoiation of Australia, Sydny, 1 B.C. Chn, Exprimntal study on mhanial bhaviors of onrt-filld stl tubular rib arh undr in-plan loads, Enginring Mhanis Rao SS, Brk L.Analysis of unrtain strutural systms using intrval analysis. AAA J 1997;35(4:77±35. Uy. B. Stati long-trm ffts in short onrt-filld stl box olumns undr sustaind loading. AC Strut. J., 98, Y.-L. Pi, M.A. Bradford, B. Uy, n-plan stability of arhs, ntrnational Journal of Solids and Struturs 39 ( Y.-L. Pi, M.A. Bradford, F. Tin-Loi, Nonlinar analysis and bukling of lastially supportd irular shallow arhs, ntrnational Journal of Solids and Struturs 44 (7 8 (