NUMERICAL PREDICTION OF RC DEEP BEAMS BEHAVIOR

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1 NUMERICAL PREDICTION OF RC DEEP BEAMS BEHAVIOR Mohammad Rza SALAMY 1 and Shigki UNJOH 2 ABSTRACT: Sts o rinord onrt bams ar tstd to valat analytial rslts o nonlinar init lmnt as ll as to invstigat snsitivity o FEM rslts to som important isss sh as mthod o soltion, msh disrtization, onrt onstittiv modl and also rak modls. Sin spimns shar span to dpth ratio a/d is rathr lo (1.5), thy ar lassiid in dp bam atgory and most probably thir bhaviors ar dominatd by shar rsistant mhanism. It is hovr obsrvd that almost all o th bams aild in shar mod in xprimnt. On th othr hand analytial loaddltion rspons as also dominatd by brittl ailr and sddn drop atr th pak load ithot any tra o yilding in longitdinal rinormnts. Th rslts hovr shod onsidrabl snsitivity to svral paramtrs addrssd abov and ill nd mor onrn to proprly slt thos mthods, modls and paramtrs sh as soltion prodr, ratr nrgy and shar rtntion ator. Thos paramtrs and mthods ar xamind and valatd by tst rslts hr in th most aptabl on is sltd or invstigation o this stdy. KEYWORDS: RC dp bam, shar ailr, ratr nrgy, init lmnt, nonlinar analysis. 1. INTRODUCTION This invstigation is a part o a largr stdy on ndrgrond RC strtrs bhavior ndr sismi xitation. It is ond, hovr, that in sh strtrs, mmbrs ar likly to bhav similar to RC dp bams sin hight to span ratio ill b small. Dring arthqak it is vidnt that a shar or is prodd in strtr s irmrn lads to shar dormation o ntir strtr assoiatd ith arth prssr in latral and vrtial dirtion. Sh loading rslts larg stion in rsisting mmbrs hih in dpth to span ratio ill b likly similar to dp bams dinition as mntiond bor. This is th main objtiv o th stdy to invstigat bhavior o thos mmbrs xprimntally and analytially or nginring pratial prpos and improvmnt o rrnt dsign ods daling ith dsign o ndrgrond strtrs. Th xprimntal and odiid as ll as som parts o nmrial invstigation ar prsntd lshr [1, 2]. In ordr to prorm nonlinar init lmnt analysis on RC mmbrs, propr modl or onrt matrial and rakd onrt shold b arlly sltd. So ar a id rang o matrial modls hav bn proposd by rsarhrs dring rnt dads. Th obstal in any analysis is ho to slt th modl its bst to th givn problm in among larg nmbr o availabl modls. It is notd, hovr, thr is no onsnss on hat modl rslts th bst. In othr ords ah modl sits prhaps only partilar problm. In original stdy, smard rak modl is xamind by mans o th ixd rak and th rotating rak approah. It is ond that th rotating rak modl rslts losr to xprimnt rspons that that o th ixd rak modl. Th lattr approah srs disrpany in shar rtntion ator dinition hih is lt as a hoi to b didd by analyzr. Th 1 Dr. Eng., Snior rsarh nginr, Earthqak nginring rsarh tam, Pbli Works Rsarh Institt, Japan 2 Dr. Eng., Tam ladr, Earthqak nginring rsarh tam, Pbli Works Rsarh Institt, Japan 1

2 problm manists itsl hn shar ailr dominats mod o ailr hih in signiiant snsitivity to th hoi o shar rtntion ator is rsltd. Althogh snsitivity o th rslts to shar rtntion ator is blivd to b ngligibl in lxral mod o ailr [] bt it is vidnt hr that variation o this ator may hang th rslts signiiantly. In this papr only th rslts o analyss by mans o th rotating rak modl ar prsntd. Fratr mhanis onpt is tilizd in ordr to liminat snsitivity o rspons to msh disrtization hih is sally obsrvd in simlation o highly nonlinar matrial sh as onrt and othr typ o brittl matrials. 2. TESTED RC DEEP BEAMS In this stion tstd bams to valat analytial simlation ar xplaind and gomtris and matrial proprtis o spimns ar prsntd. Totally nin spimns ar tstd dring th yar 24 in Pbli Works Rsarh Institt. Th bams hav shar span to dpth ratio a/d=1.5 and tiv dpth siz rom 4 mm to 14 mm. Th longitdinal tnsil rinormnt ratio is kpt almost onstant in abot 2% or all spimns hil latral rinormnt (stirrps) ratio varis by.% and.4% in shar span. Gomtry haratristi and matrial proprtis ar givn in Tabl 1. In this Tabl, ρ, ρ st and y ar stirrps ratio, longitdinal tnsil rinormnts ratio and thir yild strsss rsptivly. All spimns, ith or ithot stirrps in shar span, hav a minimm latral rinormnt in mid-span and ot o span. Othr paramtrs o th tabl ar L,, a, d, H and b as spimn s lngth, distan btn loading points, shar span, tiv dpth, bam hight as ll as stion idth rsptivly. Maximm load apaity is dnotd by P. All spimns ar sbjtd to or points monotoni stati load ondition. Tabl 1. Gomtri and matrial Proprtis o spimns Bam Gomtry siz (mm) ρ st y ρ P MPa L a d H b % MPa % KN B B B B B B B B B MATERIAL CONSTITUTIVE MODELS FOR FINITE ELEMENT ANALYSIS Matrial modls or onrt and stl ar prsntd hr hras snsitivity to matrial modls; thogh stdid in original stdy, is not prsntd in this papr. Stl rinormnts ar modld as an lasti prt plasti matrial ith no hardning atr th yild point. This modl is shon in Figr 1. Th onstittiv bhavior o onrt is rprsntd by a smard rak modl, hih in th damagd matrial still ontinm basd on matrial modls xplaind in this stion. Morovr, th modl is assmd in ratr matrial ramork ith a haratristi lngth paramtr to liminat msh siz t hr in th ratr nrgy in ithr tnsion or omprssion is st onstant. This assmption aomplishs an objtiv analysis hih in th rslt is irrlvant to th init lmnt msh siz. Th lngth paramtr h is a ntion o lmnt siz and stimatd by A hr A is lmnt ara. 2

3 .1 CONCRETE IN TENSION An mpirial xponntial ratr modl drivd by Hordijk [4] is sd or onrt in tnsion dpitd in Figr 2 and xprssd by olloing qation σ = xp 2 ( 1+ 1 ) xp( 2 ) t (1) hr is th rak opning; is th rak opning at th omplt rlas o strss hih is a ntion o Mod I ratr nrgy G and dind by Eqation 2; and σ is th normal strss in rak I and t is th tnsil strngth o onrt in on dimnsion systm or tiv tnsil strngth in to dimnsion systm. Vals o th onstants ar 1 = and 2 =6.9. I G = (2) t Tnsil strngth as ll as ratr nrgy as a ntion o onrt maximm aggrgat siz and omprssiv strngth ar dtrmind by mans o JSCE rommndation [5]..2 CONCRETE IN COMPRESSION Conrt in omprssion is modld throgh a modiid paraboli rlationship btn strss and strain to tak into aont ratr nrgy o onrt in omprssion [6]. Elasti bhavior p to 1 is assmd in this modl. Figr 2 along ith th olloing qations ar rprsnting this modl. 2 ( ) 2 σ = 2 (1 ( ) ) i i < < () (4) E = 4 Consqnt to th lngth paramtr assoiation, ltimat strain ill b a ntion o omprssiv ratr nrgy, lngth paramtr h, and also as blo. σ y E T = σ r G 11 = 1.5 (5) h 48 σ t y E S y G I h 1 G h Figr 1. Rinormnt modl Figr 2. Conrt tnsil (lt) and omprssion (right) modls

4 Sin onrt is assmd to b linar lasti p to 1 thror pr-pak nrgy ill b takn into aont by a orrtion ator 11 in Eq.5. Constant val or G =5 N/mm is adoptd in analyss 48 throghot. Frthrmor, th onpt o Modiid Comprssion Fild Thory [7] is assoiatd in analyss by mans o onrt omprssiv strngth sotning d to th latral tnsil raks. 4. ANALYTICAL INVESTIGATION Analytial shm and init lmnt msh disrtization is shon in Figr. It is notd hovr that FE msh disrtization shon in this igr is jst a shmati vi hr in th nmbr lmnt and othr haratristis o th modl dir or ah spimn. For spimns ith stirrps in shar span, latral rinormnt ill b xtndd to ntir lngth o th mmbr. Aording to onrt rak modl, to approahs hih r xplaind in prding portion ar applid in analysis bt only thos analyss ith th rotating rak modl ar prsntd hr. In th analyss, prt bond is assmd or both smard rinormnts and mbddd bar rinormnts. Displamnt ontrol ith Nton-Raphson soltion thniq is adoptd to solv qilibrim qations. Spimns ar partly modld by FEM d to th symmtri gomtry and sbjtd to th proportional monotoni loads ith 2D lmnts in plan-strss ondition. Analytial rslts by mans o prditd load to xprimnt load ratio or ntir spimns along ith som load-dltion rspons or som rprsntativ spimns ar prsntd hr. Assmbling (omprssiv) Bar Stirrps Stirrps Tnsil Bar 2 Tnsil Bar 1 Figr. Analytial modl and FE msh disrtization (shrink msh) 4.1 PREDICTED ULTIMATE LOADS AND LOAD-DEFLECTION RESPONSE Load-dltion rsponss o thr rprsntativ bams B1, B14 and B16 ar illstratd in Figr 4 and Figr 5 shos prditd load to tst load ratio P max /P xp or ntir spimns. Th rslts gnrally ar in good agrmnt ith xprimnt althogh onsrvativ rsponss hav bn stimatd by FEM or rtain spimns. In spit o RC bams bhavior aild in lxral mod hih old b prditd by FEM qit aratly, simlation o shar ailr otn as many diiltis in trms o th applid onstittiv las and nmrial soltion pross. In many ass, hovr, FE analysis prdits highr load apaity or bams aild in on o shar ailr mod atgoris partilarly thos rqntly or in RC dp bams sh as shar propr or shar omprssiv ailr. In ontrary to analytial rsponss hr, sh ovrstimatd prdition on shar apaity o RC mmbrs mislads dsignrs to ovrstimat ral apaity o th mmbr and om p ith a ak lmnt in shar vntally. Figr 5, on th othr hand, shos an abot 15 prnt in avrag ndrstimatd prdition o ral shar apaity o th mmbrs by th applid init lmnt mthod. All o th prditd shar load apaitis ar loatd insid P max /P xp =1 irl hih mans dsign saty margin is nsrd. This mans that thr has bn ond a good orrlation btn analysis and xprimnt in trms o shar apaity o RC dp bams and an b tilizd or dsign o thos mmbrs in ral prati ith an avrag ration o P analysis /P xp =.85. Rslts o analyss in most ass, as an b sn in Figr 4 as ll, sho stir bhavior in pr-pak rgim bor shar rak tak pla. This iss an b attribtd to modling assmptions sh as 4

5 Total Load (KN) 2 1 Tst Analysis Mid-span Displ. (mm) 45 B1 B14 Total Load (KN) 15 Tst Analysis Mid-span Displ. (mm) Total Load (KN) Tst Analysis B Mid-span Displ. (mm) Figr 4. Analytial rspons vrss tst rslts o rprsntativ bams B1, B14 and B16 1 B1 B1. B18 1 Panalysis/Pxp B16 Panalysis / Pxp B1-1.5 B15 B1-2 B17 B14 B18 B16 B15 B17 B14 B1-2 B1-1 B1. B1 Figr 5. Analysis to tst pak load ratio P analysis /P xp prt bond assmption, ixd onntion btn dirnt matrial or instan loading plats and onrt body as ll as matrial proprtis o onrt bor and atr raking. It is also obsrvd that atr shar raks xtnsion to som part o shar span, stinss is rdd sharply and sho vn sotr bhavior than that o xprimnt in som spimns. Nvrthlss shar rak load on th othr hand is adqatly aptrd by analysis and dpitd by a hang in stinss agr ll ith tst rslts. Shar rak load hovr is a milston in dsign mthodology i shar apaity o th mmbr is th main onrn o dsign. 4.2 CRACK PATTERNS AND STRESS DISTRIBUTION IN REINFORCING BARS As or strss in longitdinal rinormnt, no yilding is signald in ntir analyss and shar thror triggrd ailr vntally. Strss-jmp in hr th rak xists partilarly in th loation hr ot o balan raks xist is aratly aptrd. Uppr longitdinal rinormnts ar gnrally providd to ontrol omprssiv strss o onrt hil strss in onrt gos byond onrt strngth or jst as assmbling bars. In simpl span normal bams thy ar alays ndr omprssion. In dp bams hovr th nbalan tnsil strss in ppr a o th bam rslts rak in onrt and tnsil strss in rinormnt hras in normal as is xptd to b ndr omprssion d to th positiv momnt. Th biggr th bam hight is th svr raks ar xptd to b prodd. Many o th spimns in this stdy shod sh raking in hr th momnt is positiv or zro. Morovr in th loation hr ths ot o balan rak obsrvd larg strss jmp is also obsrvd onsqntly. Figr 6 dpits rak pattrn o analysis vrss tst obsrvation in bam 14. Th igr shos vry arat prdition by analysis hr in almost all important raks ar aptrd. Shar rak xtnsion is ll monitord by appriabl hang in strtr s ovrall stinss. In othr ords it mrgs a basis to valat damag lvl in RC mmbr i dynami analysis is prormd and rslts hang in natral priod o strtr. Analytial rak pattrns ar dran in shado o xprimntal raks (solid lins) or bttr valation. It an b sn hr that ot o balan raks (abov spports 5

6 B14 on ppr sid) ar also aptrd aratly. D to symmtri gomtry o spimn raks o othr part in Fig.6 ar mirrord to ovr ntir lngth o spimn tst rslts hih in sally rial raks dvlop in only on sid o spimn. 5. CONCLUSION Invstigation on RC dp bams bhavior and thir modling ith nonlinar init lmnt analysis is prsntd in this papr. Th matrial modl proposd by Hordijk and Fnstra ar applid or onrt in tnsion and omprssion rsptivly in ratr mhanis ramork to liminat msh dpndny o th rslts. Th applid mthods ith th rotating rak modl hovr old prdit RC dp bams rspons to monotoni stati load ith aptabl aray. In all spimns prditd pak load hav bn lor than tst rslts. This nsrs a saty margin or dsign o RC mmbrs i load apaity o th mmbr is th major onrn o dsign. Prditd rak pattrns ar also valatd by xprimnt obsrvation. Th main indings o this papr ar dran as ollos: 1. Pr-pak rgim old b proprly aptrd by th applid matrial modls and init lmnt analysis sing Nton-Raphson soltion mthod ith displamnt ontrol. For post pak analysis, ar-lngth mthod is sggstd sin th snap-bak instability is possibl to or. 2. Corrlation btn prditd raks by analysis and monitord dring tst pross is sitably adjstd. Almost all important raks inlding thos o tnsil raks on ppr a o th bam nar to th nd rgion ar vry ll prditd. Th prditd rak pattrns thror an b ll tilizd or damag valation o RC mmbrs.. Th rslts onirmd that th applid init lmnt mthod an b proprly sd or rthr paramtri stdy and analyzing n RC mmbrs thogh in rtain ass (sh as totally dirnt loading pattrns or gomtry o spimns) xprimntal vriiation is rommndd. 6. REFERENCES Figr 6. B14 rak pattrns o analysis at ltimat stat vrss xprimnt 1. Kobayashi, H., Unjoh, Sh. and Salamy, M.R., Exprimntal stdy on shar apaity o larg sald dp bams (in Japans), JCI, V.27, No.2, 25, pp Salamy, M.R., Kobayashi, H., Unjoh, Sh., Kosa, K. and Nishioka, T., A od-basd omparativ stdy on RC dp bams bhavior ith shar span to dpth ratio btn.5 and 1.5", 8th Symposim on Dtility Dsign Mthod or Bridgs, Tokyo, Fb. 1-2, 25 pp Kak, H., and Flippo F.C., Finit lmnt analysis o rinord onrt strtrs ndr monotoni loads, Rport No. UCB/SEMM-9/14, Dpartmnt o Civil Enginring, Univrsity o Caliornia Brkly, Nov. 199, pp Hordijk, D.A., Loal approah to atig o onrt, PhD thsis, Dlt Univrsity o Thnology, Japan Soity o Civil Enginrs, Standard spiiations or onrt strtrs, Strtral prorman vriiation (in Japans), Tokyo, Marh, 22, pags 67 and Fnstra, P.H., Comptation aspts o biaxial strss in plain and rinord onrt, PhD thsis, Dlt Univrsity o Thnology, Vhio, F. and Collins, M.P., Th modiid omprssion ild thory or rinord onrt lmnts sbjtd to shar, ACI strtral jornal, V., No.4, 1986, pp