Fiber Beam Element Model for the Collapse Simulation of Concrete Structures under Fire

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1 Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM. COMPUAIONA MECHANICS ISCM7, Jul 3-August, 7, Bijing, China 7 singhua Unirsit Prss & Sringr Fibr Bam Elmnt Modl for th Collas Simulation of Concrt Structurs undr Fir S.C. Chn, X. Z. u*, A. Z. Rn, J.J. Jiang Dartmnt of Ciil Enginring, singhua Unirsit, Bijing, 84 China luinhng@63.nt Abstract In ordr to anal and simulat th collas of rinforcd concrt (RC) lmnts undr fir, a nol numrical modl basd on th fibr bam modl is roosd in this ar. B diiding th cross sction of bam lmnt into man small concrt and stl fibrs and assigning diffrnt matrials to ach fibr, this modl can considr th non-uniform tmratur distribution across th sction and simulat th bhaior of cracking or crushing for concrt and ilding for stl. h licit tangntial stiffnss matri is dducd for roosd fibr bam with otal agrangian () dscrition, and th incrmntal quilibrium quations ar also stablishd. Finall, th roosd fibr modl is alidatd b comaring with arious rimntal rsults. K words: fibr modl, bam lmnt, fir, collas, nonlinarit INRODUCION Fir is on of th most dangrous disastrs that can dstro buildings. With th occurring of local damag or n comlt collas of buildings undr fir, mor and mor attntions ar aid to th scurit of structurs undr fir [, ]. As a rsult, man thortical and rimntal rsarchs ha bn conductd on th bhaiors of arious structural lmnts, substructurs and n whol structurs undr fir [3]. raditional rsarchs of structur undr fir ar basd on singl lmnt tst in th standard furnac. In th 98's and th 99's, rformanc basd conction for fir dsign is adotd b BS595 Part 8 and EC4[4] rsctil in UK and Euro, which mans th saft of a crtain structur in fir can b dtrmind with th fir rsistant calculation. In China, th currnt cods including th fir saft cod for building dsign and th cod for fir rotction dsign of tall buildings ar still basd on singl lmnt tst in standard fir. With th imroing rquirmnts on disastr rsistanc of building structurs, th rformanc-basd fir rsistant dsign mthod has drawn grat attntion [5], which rquirs th whol structural analsis for ral fir situation. Du to th trm larg workload in th coml building simulations with solid finit lmnts, it is ncssar to dlo a suitabl finit lmnt modl to stud th failur rocss of th structural lmnts osd to fir. Now, fibr bam modl is widl usd in rinforcd concrt (RC) structurs. Hnc, a nol numrical modl basd on th fibr bam modl is roosd in this ar to simulat th collas of rinforcd concrt bam and column undr fir. h modl of fibr bam dlod hr is a two nodal lmnt with si dgrs of frdom at ach nod and th cross sction of bam lmnt is diidd into man small concrt and stl fibrs. h non-uniform stiffnss of cross sction along th bam, which is causd b th arious kinds of loads and th dgradation of matrials at latd tmraturs, is considrd with th 3-oint Gauss intgration along th ais of ach fibr. Gomtric nonlinarit of larg dislacmnt is also modld with otal agrangian () dscrition, and th licit tangntial stiffnss matri is dducd for roosd fibr bam with th considration of larg dislacmnts. B aling th incrmntal thrmo lastic-lastic constituti modl, th incrmntal quilibrium quations ar also stablishd for th lmnt of fibr bam. Finall, th analing rsults ar comard with th data from fir rimnts. h rsults show that th rdictions of th modl roosd in this ar agr wll with th rsults of th tsts and can b usd to anal and simulat th collas of rinforcd concrt lmnts undr fir.

2 Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM. HE EEMEN MODE OF FIBER BEAM h fibr bam lmnt consists of two nods and si dgrs of frdom at ach nod, as shown in Fig.. h cross sction of bam lmnt is diidd into man small concrt and stl fibrs and th following assumtions ar stablishd: ) Gomtric nonlinarit du to larg dislacmnt is considrd but th strain of matrial is still tratd as small strain situation. ) Plan-in-lan assumtion is adotd to constraint th dformation of diffrnt fibrs in th sam sction. 3) For ach fibr onl th longitudinal strss is considrd. 4) h matrial of ach fibr ma b diffrnt, and th 3-oint Gauss intgration is considrd along th ais of ach fibr.3) Each fibr within th cross-sction can ha a diffrnt tmratur, but this is uniform along th fibr. h dformation of th fibr bam lmnt is basd on th otal agrangian () mthod, and th dislacmnts at an oint on th rfrnc ais btwn two nd nods can b rssd as follows: { u } [ N]{} q () u ar th dislacmnts at an oint. N is th sha function matri gin b Bath [6]. q is th nodal dislacmnt ctor. h dislacmnts of an oint A on an cross-sction can b rssd as whr { } [ u,, w, θ ] [ ] {} Fig.: h fibr bam lmnt u u sinθ sinθ (-a) + + cosθ cosθ + + cosθ cosθ cosθ sinθ (-b) w w cosθ sinθ (-c) whr, ar th coordinats of oint A. It can b sn from Fig.3 and Fig.4 that dw d sinθ w and sinθ d d (3-a) thn cosθ ( w ) and cosθ ( ) (3-b) if sin θ θ and cos θ (3-c) thn th dislacmnts of an oint A can b rssd as u u + (4-a) w + ( ) + ( w ) θ (4-b) w w θ (4-c) + ( w ) ( ) θ θ (4-d) cosθ A o w w θ cosθ o A Fig.: Elmnt cross-sction msh d u A o u dw θ d w w o A Fig.3: Dformations in lan For ach fibr onl th longitudinal strss is considrd, th stain of oint A can b rssd as u u w ε [( ) ( ) ( ) ] ε +ε (5) and u ε B q (6-a) u w ε [( ) ( ) ( ) + + ] (6-b) whr ε rrsnts th small linar-dislacmnt strains and ε rrsnts th non-linar dislacmnt stains. d u A o u d o θ A d Fig.4: Dformations in lan

3 thn Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM. ε B dq (7) d [ dε q B B + B B + B B ] dq q [ B ] dq B dq (8) whr B A B B C C A B B D D, thn B B C B D, B B C B D, 3 A / l, B 6 / l 6 / l, C 3 / l 4 / l +, D 3 / l + / l, E / l, F / l dε can b rssd as dε dε + dε B dq + B dq B dq (9) On th othr hand, th total thrmo lastic-lastic strain incrmnt in ach fibr can b rssd as { d ε} { } dε +{ } whr { } ε +{ dε } +{ dε } +{ } d, dε incrmntal lastic strain comonnts, { ε } cr dε () P d, incrmntal strain comonnts du to tmratur dndnt matrial rortis, { dε } incrmntal thrmal strain comonnts, { dε P } incrmntal lastic strain comonnts. Whn th isotroic hardning schm is considrd, th thrmo lastic lastic constituti quation can b rssd b [7] { σ} d ] D ({ dε} -{ } [ α d - [ D ] { } σ d -{ } cr { } [ D ] [ D ] σ dε - S F d ) () whr [ D ] is thrmo lastic lastic constituti matri, α is th thrmal ansion cofficint, S { σ } [ D ]{ σ } + H F,{ σ },[ D ] [ D ] -[ D ] and [ D ] [ ] σ Bcaus onl th longitudinal strain is considrd for ach fibr in th lmnt, thn [ D ] E, [ D ] Gσ / S, σ σ / 3, S 4σ (3G + H ) / 9, S σ ( + H / 3 ) / 3, H /(/ E / E ) G whr t σ is th aial strss of th fibr at th gauss oint, [ ] D { σ } { } σ D ] / S [ Et is th tangntial stiffnss of th fibr at th gauss oint D If E E - Gσ / S,and E - D D σ E σ E S 3ES () thn dσ E ( dε dε dε cr ) Edε (3) whr strain, dε dε α d -, [ ] [ ]{ } dε dε dε cr is th total incrmntal thrmo mchanical strain, E E σ d + E σ 3ES F d is th incrmntal thrmal strain. dε is th total incrmntal Finall, th tangntial stiffnss matri of th fibr bam can b dducd with th dformation quations and th thrmo lastic lastic constituti quation. h incrmntal strain nrg in th lmnt can b rssd as U / { } V dε { dσ} d / { q } B DBd{ q } + σd { q} Dd { ε } d { q} D{ dε cr} d { dε } Dd { ε }d db B B + / (4) h work don on th lmnt b th trn forc is

4 V - dq { } Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM. dq (5) So th total incrmntal otntial nrg can b rssd as U + V (6) Uon substituting quation: B DBd { dq} U and V into th rssion for and aling th ariation rocss lads to th following { q} - B D{ dε }d - B D{ d ε }d -{ } cr dq (7) It can b rssd as u F (8) whr K K + Kσ is th lastic-lastic stiffnss matri, Kσ is th gomtr stiffnss matri and D K { } { } K B Bd ( B + B ) D (B + B )d EB d B + E d B B + EB d B + E d B B K σ d B σ d d B σ, { F} F + FC + FM is th incrmntal thrmo mchanical load matri. E Eσ F F (B + B ) Edε d (B + B ) E ( α d σ d + d ) d E 3 ES F C (B + B ) Edε cd and FM { dq } Whn th local quilibrium quation is transformd into global coordinats through th transformation matri, th structural quilibrium quations can b obtaind. { q} F K (9) and th incrmntal dislacmnts can b obtaind b soling ths quations. MAERIA MODE Du to th trm nonlinarit of concrt and stl undr latd tmratur, th tmratur dndnt matrial modls ar rquird in ordr to rform numrical analsis of structur undr fir. Bcaus onl longitud dformation of ach fibr is considrd in th fibr bam modl, th uniaial matrial modls ar adotd. h modls usd hr for bam is in accordanc with EC4 art.[4], which has riousl bn usd b Huang [8] in concrt slabs analsis. th tmratur-dndnt strss-strain rlationshis for concrt in comrssion ar shown in Fig.5. Concrt is assumd to ha no rsidual strngth in ithr comrssion or tnsion onc it has crushd. h tmratur-dndnt strss-strain rlationshis for stl ar shown in Fig.7. A bilinar modl is adot to rsnt th tnsil softning of crackd concrt (Fig.6 [9]), which can mak th numrical solution rocss mor stabl. Strss ratio(σc/fc( )) Strain Strss f t aft a.5~. 7 ε cra Strain ε cu Strss ratio Strain(.) Fig.5: Strss-strain rlationshi of concrt at latd tmraturs Fig.6: Strss-strain rlationshi of concrt in tnsion Fig.7: Strss-strain rlationshi of stl at latd tmraturs

5 Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM. NUMERICA EXAPME hr rrsntati fir tsts of structural lmnts ar anald to alidat th fibr modl roosd abo. h first is a four-fac hatd column, which could ro th aial bhaior of th sction bcaus of th smmtric tmratur distribution. hn a thr-fac hatd bam is usd to tstif th bnd bhaior of th bam sction. Finall a thr-fac hatd column subjctd to a constant comrssi forc is anald, th column is bndd bcaus of th unsmmtrical tmratur distribution and comrssd bcaus of th trnal aial rssur, so th could bhaiors of bnding and comrssion could b considrd. Four-fac hatd column h fibrs strss will chang and strss rdistribution will occur across th sction bcaus of th non-uniform tmratur distribution. In ordr to tstif th aial bhaior of th sction, th rdictions of th roosd modl ar comard with th fir tst rsults of a four-fac hatd rinforcd concrt column with aial load, rortd on b i and Irwin[].th gomtric of th column and stl rinforcmnt ar shown in Fig.8. In th tst, th r-loadd aial forc is 67kN, and th fir situation is simulatd b osing th column to hot surrounding air and th tmratur of th air changing follows th ASM fir cur. h ild strss of th rinforcmnt is 44MPa, th comrssi strngthn of concrt is 36.MPa and th masurd fir rsistanc tim of th column is 8 min []. h tmratur distribution on th cross-sction of th column should b dtrmind bfor th structural analsis; th transint hat conduction roblm is sold b th comutr rogram dlod b th authors [], which has bn alidatd b comaring th rdictions with th tsts rsults. h matrial aramtrs ndd in th analsis, including th hat conductiit and scific hat, ar stimatd according to Eurocod 4[4]. h othr aramtrs such as hat transfr cofficint and hat missiit ar 5W/mK and. rsctil. Fig. shows th calculatd and masurd tmratur distributions in concrt cross-sction at arious tims. In th structural analsis, th column is simulatd with on bam lmnt, th cross-sction is diidd into fift concrt fibrs and four stl fibrs and ach fibr has thr Gauss oints, as a rsult thr ar 6 Gauss oints togthr in on bam lmnt. Fig. shows th calculatd and masurd aial dislacmnts at arious tims. From th rsults of th tst, it can b found that th aial dislacmnt incrass during th first min, showing that th column is longatd du to th growing thrmal strain. Subsquntl, with th incrasing of th tmratur, th aial dislacmnt starts to dcras du to th dgradation of th matrials and bcoms ngati at 8 min. th column collass at 8 min. From th rdiction of th modl, it can b found that th largst dislacmnt occurs at 5 min. It is r clos to th masurd alu min. Finall, th dislacmnt incrass raidl at about 5 min which shows that th column collass undr fir. hough th rdiction of th formula roosd b i[] was min, th numrical modl roosd hr ar mor flibl than th mirical formula b i[]. ASM P67kN Φ5mm 35mm tmratur( ) min 8min min rsult from tst rsn 5 5 Hight of sction(mm) Fig.8: Four-fac hatd column in fir Fig.9: mratur from tst and calculation

6 Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM. Aial dislacmnt(mm) rsnt -4 tst analsis from i -6 im(min) Fig.: h dlomnt of aial dislacmnt of th column hr-fac hatd bam In this aml, th roosd modl is tstifid b comaring th its rdictions with th fir tst rsults of a thr-fac hatd four-oint bndd RC bam [4]. h dimnsion of th column and th loading data ar gin in Fig.. h ild strss of th rinforcmnt is 7MPa; th comrssi strngth of concrt is 8.9MPa. h concntratd load P is 4kN, th tmratur of th air in th furnac masurd in th tst [4] is usd to simulat th fir situation in th transint thrmal analsis. Strss(N/mm ) min min 8min 3-5 Fibr location at th sction(mm) Fig.: h dlmnt of th sction strss P P f c f Fig.:hr-fac hatd bam In th structural analsis, in ordr to obtain th dislacmnt at th mid-oint of th column connintl, th column is mshd into four bam lmnts. h cross-sction is diidd into fift concrt fibrs and four stl fibrs and ach fibr has thr Gauss oints, as a rsult thr ar 6 Gauss oints togthr in on bam lmnt. Fig.3 shows th calculatd and masurd dislacmnts of th mid-san of th bam at arious tims. From th rsults of th tst, It can b sn that th mid-san dislacmnt incrass slowl whn th tmratur is blow 4. Aftr that, its dformation acclrats du to th thrmal strain and non-uniform tmratur distribution. Whn th tmratur cds 55, though th ll of non-uniform tmratur rducs, th mid-san dislacmnt incrass raidl and th bam collass bcaus th matrials dgnrat. From th calculation rsults, It can b sn that th rdictions of th modl agr wll with th tst rsults whn th tmratur is low. Whn th tmratur cds 4, th rdictions bcam som smallr than th tst rsults bcaus th tmratur ffct of cracking on th rinforcmnt is not considrd in th numrical modl. But th mid-san dislacmnt incrass raidl and th bam collass aftr 55, which agrs wll with th tst rsults. h modl is alid furthr to th analsis of th sam bam subjctd to diffrnt lls of th trnal loads and Fig.4 shows th numrical rsults. Whn th load ll is r low and P quals kn, th mid-san dislacmnt incrass slowl until th tmratur rachs.with th incras of th load ll, th ultimat tmratur of th bam rducs to 6 Whn P quals 4kN. Aftr that, th ultimat tmratur of th bam still rducd but th sd of rduction bcam much slowr. hs rdictions agr wll with th tst rsults, so th roosd modl can b usd to simulat th collas of rinforcd concrt bams undr fir.

mratur( ) 6 5 4 3 Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM.

4: Rlationshi of load and ultimat tmratur 3 hr-fac hatd column In this aml, th rdictions of th roosd modl ar comard with th fir tst rsults of a thr-fac hatd RC column [4].

th comrssi strngth of concrt is 36.MPa. In th tst, th r-loadd forc is 8kN; th masurd tmratur of th air in th furnac is usd to simulat th fir situation in th transint thrmal analsis. P Fig.

7 mratur( ) Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM. Prsnt st 5 5 Mid-san dislacmnt(mm) Fig.3: Dlomnt of th mid-san dislacmnts mratur( ) oad(kn) Fig.4: Rlationshi of load and ultimat tmratur 3 hr-fac hatd column In this aml, th rdictions of th roosd modl ar comard with th fir tst rsults of a thr-fac hatd RC column [4]. h gomtric of th column and th loading data ar gin in Fig.5. h ild strss of th rinforcmnt is 44MPa; th comrssi strngth of concrt is 36.MPa. In th tst, th r-loadd forc is 8kN; th masurd tmratur of th air in th furnac is usd to simulat th fir situation in th transint thrmal analsis. P Fig.5: hr-fac hatd column Fig.6: mratur-distribution at 3min 6min and min h tmratur distribution on th cross-sction of th column is also calculatd with th rogramm dlod b th authors. Fig.6 shows th rsults of th ariation of tmratur-distribution at 3min, 6min and min rsctil. In th structural analsis, in ordr to obtain th dislacmnt at th mid-oint of th column connintl, th column is mshd into two bam lmnts. h cross-sction is also diidd into fift concrt fibrs and four stl fibrs and ach fibr has thr Gauss oints, as a rsult thr ar also 6 Gauss oints togthr in th bam lmnt. Fig.7 shows th calculatd and masurd aial dislacmnts at arious tims. Fig.8 shows th calculatd and masurd latral dislacmnts at arious tims. From th rsults of th tst, It can sn that th aial dislacmnt incrass du to th thrmal strain whn th tmratur is blow 7. Aftr that, th matrials dgnrat in high tmratur. th latral dislacmnt was ositi during th first min, showing that th column bnds to th hatd sid of th column. Subsquntl, th column bnds to th othr sid of th column du to th dgradation of matrials with th incrasing of th tmratur and th latral dislacmnt bcam ngati and th column collasd at 5 min.from th rdiction of th modl, It can sn that th largst aial dislacmnt occurs at about 6 and incrass raidl at about 9 whn th column collass. At th sam tim, th latral dislacmnt also incrass raidl at about 4 min, which shows that th rdictd fir rsistanc of th column is about 4 min.

8 Aial dislacmnt(mm Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM rsult from tst rsnt rdiction mratur( ) Fig.7:h ariation of aial dislacmnt atral dislacmnt(mm rsnt rdiction rsult from tst im(min) Fig.8:h ariation of latral dislacmnt of th column Fibr osition(mm) min 8 3min 6 9min 4 min Strain(-4) Fibr osition(mm) min 3min 9min min Strss(N/mm ) Fig.9:Cross-sction of th column Fig.:Variation of th sction strain Fig.:Variation of th sction strss CONCUSIONS A fibr bam modl was dlod in this ar. B aling th incrmntal thrmo lastic lastic constituti law, th roosd modl was usd to anal th collas of rinforcd concrt structural lmnts undr fir. h modl considrd th non-uniform tmratur distribution across th sction and nonlinar bhaior of th matrials. hr rrsntati lmnt fir tsts wr anald to alidat th fibr modl roosd in this ar, and th rdictions of this modl agr wll with th tst rsults, it indicats that th fibr bam roosd hr is caabl of analing and simulating th collas of rinforcd concrt lmnts undr fir with acctabl workload. REFERENCES. Brnard,Monahan P.E. World rad Cntr collas ciil nginring considrations. Practic riodical on structural dsign and construction. August, : u X.Z, Jiang J.J. Dnamic finit lmnt simulation for th collas of world trad cntr ciil nginring journal.34(6), 8-,. (in chins) 3. Bail C. Holistic bhaiour of concrt buildings in fir. th Procdings of th Institution of Ciil Enginrs, Structurs and Buildings 5, August, Issu 3, Eurocod4: Dsign of comosit stl and concrt structurs. Part.: Structural fir dsign, CEN/C5/SC4 N39, Commission of th Euroan Communitis, Brussls,. 5. Matthw A, Johann A.M. Prformanc basd structural fir saft. Journal of rformanc of constructd facilits,6: Bath. Finit Elmnt Procdurs.Prntic-Hall,Nw Jrs, Hsu.R. h finit lmnt mthod in thrmomchanics. Alln&Unwin, Huang Z, Burgss I.W, Plank R.J. Nonlinar analsis of rinforcd concrt slabs subjctd to fir. ACI Structural Journal, 96() :7-35, Hinton.E, Own D.R.J. Finit lmnt softwar for lats and shlls, Pinridg rss,swansa. 984.

9 Proc. Intrnational Smosium on Comutational Mchanics (ISCM7), Yao ZH & Yuan MW (ds.), Bijing: singhua Unirsit Prss & Sringr, Jul 3-August, 7, Bijing, China, 88 & CDROM.. i., Irwin R.J.Mthod to calculat th fir rsistanc of rinforcd concrt columns with rctangular cross sction. ACI Structural Journal 9(),5-6,993.. Chn S.C, Rn A.Z. A fir simulation of rinforcd concrt fram structurs for th thrmal and structural analsis. 5-37, INCIE/ICSED, 6, NEW DEHI, INDIA.. Mohamad J,rro. Numrical modling of th bhaior of concrt structurs in fir. ACI Structural Journal/March-Aril, Dotr J.C, Franssn J.M. Erimntal rsarch on th dtrmination of th main aramtrs affcting th bhaior of rinforcd concrt columns undr fir conditions. Mag. Concrt Rs., 49(79), 7 7, Guo Z.H, Shi X.D. Bhaiour of rinforcd concrt at latd tmratur and its calculation. singhua Unirsit Prss.3.(in Chins). 5. Shi X.D, an.h. Effct of forc-tmratur aths on bhaiors of rinforcd concrt flural mmbrs. Journal of Structural Enginring ,.

A UNIFIED APPROACH FOR FIRE RESISTANCE PREDICTION OF STEEL COLUMNS AND FRAMES

A UNIFIED APPROACH FOR FIRE RESISTANCE PREDICTION OF STEEL COLUMNS AND FRAMES Chu Yang TANG Nanang Tchnological Univrsit, School of Civil and Environmntal Enginring,N-B4-04, Singaor 4449244@ntu.du.sg Kang