Title: Practical reliability-based calculation tool for the post-fire assessment of concrete beams

Transcription

1 Cover page Ttle: Pratal relablty-based alulaton tool for the post-fre assessment of onrete beams Authors (sample names): Ruben Van Cole Robby Caspeele Lu Taerwe PAPER DEADLINE: **JANUARY 15, 015** PAPER LENGTH: **8 PAGES MAXIMUM ** SEND PAPER TO: Venkatesh Kodur CIVIL AND ENVIRONMENTAL ENGINEERING 3580 ENGINEERING BUILDING Mhgan State Unversty East Lansng, MI 4884, USA Tel: Fax: E-mal: Please submt your paper n Mrosoft Word format. We enourage you to read attahed Gudelnes pror to preparng your paper ths wll ensure your paper s onsstent wth the format of the book. Fnal book wll prnt n Blak nk only, please submt pture fles n graysale f possble Do not use olor n artwork. NOTE: Sample gudelnes are shown wth the orret margns. Follow the style from these gudelnes for your page format. Hardopy submsson: Pages an be output on a hgh-grade whte bond paper wth adherene to the spefed margns (8.5 x 11 nh paper. Adjust outsde margns f usng A4 paper). Please number your pages n lght penl or non-photo blue penl at the bottom. Eletron fle submsson: When submttng fnal eletron paper please -- make sure that you submt your paper n Mrosoft word.

2 (FIRST PAGE OF ARTICLE) ABSTRACT If a struture has been able to mantan stablty durng fre exposure, the resdual load-bearng apaty of the strutural elements after fre should be determned when dedng upon the further use of the struture. Sne adequate safety s a prmary requrement for all strutures and sne many unertantes are assoated wth the post-fre assessment, only a relablty-based assessment an be aeptable for real-lfe applatons. In ths ontrbuton an easy-to-use relablty-based tool s presented for the post-fre assessment of the load-bearng apaty of onrete beams. INTRODUCTION Although fre s a very severe load ondton for strutures, onrete elements generally have a good fre resstane and rarely ollapse durng fre [1]. Consequently, after fre exposure the queston of the resdual load-bearng apaty arses: an the struture be used wthout repar or rehabltaton, or should the struture be demolshed or repared? As exposure to elevated temperatures may result n permanent damage to the onrete and renforement ([1]-[3]), the maxmum serve load may be sgnfantly redued. Current prate fousses on destrutve and non-destrutve testng to assess the onrete degradaton due to hgh temperatures [4]. However, test results ndate a sgnfant satter of the resdual mehanal propertes for a gven maxmum temperature [1], and even pror to fre large unertantes may exst wth respet to the strength haratersts and geometry (e.g. onrete over) of onrete elements. For the desgn of new strutures aordng to the Euroodes, these unertantes are taken nto aount through a sem-probablst methodology where haraterst values for the mehanal propertes are ombned wth partal safety fators to provde an adequate level of safety [5]. In EN 1990 [6] the target relablty ndex β t for normal strutures s 3.8 (for a 50 year referene perod).

3 Department of Strutural Engneerng, Faulty of Engneerng and Arhteture, Ghent Unversty, Tehnologepark-Zwjnaarde, 904, 905 Zwjnaarde, Belgum. A smlar relablty-based approah should be used when determnng the maxmum allowable serve load after fre exposure n order to ensure that the struture has the same strutural relablty for ontnued use as a new struture. One possble approah would be to perform a probablst evaluaton usng for example the assessment method presented n [7]. However, n prate these fully probablst alulatons are too omplex and tme-onsumng for most projets. In ths ontrbuton these dffultes are overome by ntrodung a smplfed relabltybased assessment method for determnng the maxmum serve load for onrete beams after fre exposure. The methodology presented here s an extenson and mprovement of a onept ntally presented by the authors n [8]. THE ASSESSMENT INTERACTION DIAGRAM (AID) The smplfed methodology s based on the applaton of what the authors all an assessment nteraton dagrams (AID). The AID gves a vsual representaton of the maxmum allowable load rato χ whh orresponds wth a spef target relablty ndex β t, where χ s defned by equaton (1) wth Q k the haraterst value of the mposed load effet and G k the haraterst value of the permanent load effet. In most stuatons the permanent load G k an easly be determned and an be onsdered unaffeted by the fre exposure. Consequently, assessng the maxmum allowable load after fre exposure omes down to alulatng the maxmum allowable haraterst value of the mposed load effet Q k,max. The AID orrespondng wth β t = 3.8 (50 year referene perod) s gven n Fgure 1. Qk Q G k k (1) Fgure 1. Assessment nteraton dagram for β t = 3.8 (50 year referene).

4 The AID gven by Fgure 1 s based on equaton (), onsderng a Gumbel dstrbuton for the mposed load Q, a normal dstrbuton for the permanent load G, lognormal dstrbutons for the model unertantes K R and K E, and a lognormal dstrbuton for the resstane effet R. All dstrbutons have been hosen n aordane wth [9]. R E Z K R K G Q () For gven ombnatons of µ R / µ G and V R the AID provdes the maxmum allowable load rato χ max, wth V R the oeffent of varaton of the resstane effet, µ R the mean resstane effet, and µ G the mean value of the permanent load (whh an be assumed equal to G k n aordane wth [8] and an easly be determned). As the permanent load effet G k s assumed to be known, the maxmum allowable mposed load Q k s gven by equaton (3): Q max k,max Gk (3) 1 max ANALYTICAL FORMULAS FOR µ R AND V R In order to apply the AID for the post-fre assessment of onrete beams, the mean value µ R and oeffent of varaton V R of the resstane effet R have to be determned. A ommon method to evaluate the response of onrete strutures exposed to fre s to neglet the strength loss of the onrete below 500 C and to assume omplete loss of strength of onrete above 500 C. Ths smplfed method s allowed by EN [10] for the desgn of onrete strutures exposed to fre (.e. durng fre) and has been appled by Kodur et al. [7] for the post-fre assessment of onrete olumns. The onept of ths sotherm method s llustrated by Fgure. Fgure. Coneptual vsualzaton of the lmtng sotherm method for the onrete ompressve strength

5 The beam n Fgure s assumed to be exposed to fre from three sdes. Applyng the onept of a lmtng sotherm for the onrete ompressve strength f, the resdual bendng apaty after fre exposure s gven by equaton (4), wth F s,res the sum of the resdual yeld fore of the tensle renforement bars, h the beam heght, the onrete over, Ø the renforement bar dameter, and θ the depth of the lmtng sotherm. Fs, res M R, res Fs, res h 0.5 b f,0 (4) If all renforement bars have the same dameter and an be onsdered to have attaned the same maxmum temperature θ max, or f an averaged resdual yeld stress s appled, F s,res s gven by equaton (5), wth A s the total renforement area, k fy,res the reduton fator for the resdual renforement yeld stress, and f y,0 the ntal 0 C renforement yeld stress. F A k f (5) s, res s fy, res y,0 Applyng Taylor approxmatons, the mean value µ R an be approxmated by (6), whle the standard devaton σ R s approxmated by (7), and V R s gven by σ R / µ R. The onsttuents S 1 to S 9 ontrbutng to σ R are gven by equatons (8) to (16). In the equatons below µ and σ are the mean value and standard devaton of the total model unertanty K T, defnng the resstane R of the lmt state equaton as K T M R,res. µ µ µ µ R R µ µ µ fy,0 µ h µ 0.5 µ As kfy, res fy,0 b f,0 (6) R µ 9 R X X X 1 µ µ fy,0 1 As kfy, res fy,0 h 0.5 b µ f,0 S µ µ µ µ µ µ µ fy,0 kfy, res fy,0 h As b µ f,0 S µ µ µ µ µ µ S (7) (8) (9) As kfy res fy h S µ µ µ µ (10) 3,,0 As kfy res fy S µ µ µ µ (11) 4,,0 µ µ fy,0 5 As fy,0 h kfy, res b µ f,0 S µ µ µ µ µ µ (1)

6 µ µ fy,0 6 As kfy, res h fy,0 b µ f,0 S µ µ µ µ µ µ µ fy, f,0 b µ f,0 S µ (13) (14) µ fy, b b µ f,0 S µ (15) µ fy, µ f,0 S µ (16) EVALUATION OF THE BASIC VARIABLES The mean value µ and standard devaton σ of the bas varables an be determned usng data from nspetons or an be based on lterature data. An overvew of standard values for the mean µ and standard devaton σ, or oeffent of varaton V are lsted n Table 1, based on [9], wth the subsrpt nom ndatng the nomnal desgn value. The model unertanty K T has been albrated omparng the results of the smplfed formula (4) wth numeral alulatons based on [11] and also norporates K R and K E. In ase tests are performed to estmate for example the ntal (0 C) onrete ompressve strength, the sample mean and standard devaton are an estmaton of µ f,0 and σ f,0 respetvely, and these values should be used nstead of the default lterature data gven n Table I. The depth θ of the lmtng sotherm an be estmated dretly usng test results (for example usng the methodology desrbed n [4]), or an be based on an estmaton of the fre severty by a fre expert. The latter method has the advantage that no tests are requred whh an be very valuable for an early prelmnary evaluaton of the safety of the struture mmedately after the fre. When the fre expert assgns dfferent probabltes p to dfferent fre severtes t E, the depth of the lmtng sotherm θ, an be evaluated for eah of these fre severtes usng a smple thermal alulaton tool. If ths thermal alulaton s onsdered too omplex or tmeonsumng, the fre expert an assgn probabltes to equvalent ISO 834 fre duratons for whh temperature dagrams are lsted n EN [10], allowng for an easy evaluaton of the assoated depth θ,. One both the probabltes p and the depths θ, are evaluated, µ θ and σ θ are gven by equatons (17) and (18). Note that t s possble to assgn a probablty of 1 to a sngle onservatvely assessed fre severty t E. Ths results n µ θ = θ, and σ θ = 0. p (17),, µ p (18)

7 TABLE I. PROBABILISTIC MODELS FOR BASIC VARIABLES, BASED ON [9]. Symbol Dm. µ σ V f,0 MPa fk 1 V f y,0 MPa f yk 1 V f fy A s mm² A s,nom mm nom 5 - h mm h nom 5 - b mm b nom 5 - K T The mean value µ kfy,res of the reduton fator k fy,res for the renforement yeld stress s gven by equatons (19) to (1). Frst k fy,res s evaluated for eah of the renforement bars for dfferent fre severtes t E and a dsrete set of possble postons x, y. The onsdered postons x, y are gven n Table II for orner renforement bars and entral renforement bars together wth ther assoated ourrene probabltes based on a Beta dstrbuton of the onrete over. Subsequently, these values for k fy,res are ombned by equaton (19) aross the dfferent fre severtes t E, after whh equaton (0) ntegrates aross the dfferent postons x, y. Fnally, equaton (1) takes the average of the dfferent rebars to obtan an average k fy,res as n equaton (5). Note that for many pratal stuatons t suffes to evaluate k fy,res,j for a sngle orner rebar and a sngle entral rebar and apply these values for other rebars. For the standard devaton σ kfy,res a onservatve assessment s made by onsderng only the orner rebar as ths rebar experenes the hghest varablty of the reduton fator k fy,res. The fnal value for σ kfy,res s alulated through equatons () and (3). The model for k fy,res as a funton of the maxmum attaned renforement temperature θ s based on [1] and [3] and s llustrated by Fgure 3, as ntrodued n detal n [8]. A more straghtforward but more onservatve alternatve method s to evaluate both µ kfy,res and σ kfy,res for the orner rebar onsderng a sngle onservatve axs poston {µ - σ + Ø/; µ - σ + Ø/}. Ths further onservatve smplfaton an be partally ompensated by a hange of the mean value of the model unertanty K T. kfy, res, j x, y kfy, res, x, y p (19) kfy, res, j µ kfy, res, j x, y pxy (0) xy kfy, res kfy, res, j j (1) n x, y kfy, res,,, t, x, y p k t x y x y p fy, res E, kfy, res, kfy, res E,, ()

8 kfy, res xy xy x, y kfy, res kfy, res, j xy x, y p kfy, res xy p (3) TABLE II. POSITIONS (x, y ) AND ASSOCIATED PROBABILITY p xy FOR CORNER AND CENTRAL REINFORCEMENT x [mm] y [mm] p xy CORNER REINFORCEMENT µ σ + Ø / µ σ + Ø / 0.03 µ σ + Ø / µ + σ + Ø / 0.03 µ σ + Ø / µ + Ø / 0.11 µ + Ø / µ σ + Ø / 0.11 µ + σ + Ø / µ σ + Ø / 0.03 µ + Ø / µ + Ø / 0.69 CENTRAL REINFORCEMENT µ + Ø / µ σ + Ø / 0.17 µ + Ø / µ + Ø / 0.83 EXAMPLE APPLICATION After a severe offe fre, a onservatve assessment by a fre expert ndates an ISO 834 standard fre duraton of 90 mnutes. Smply supported beams wth a heght of 800 mm, wdth of 500 mm, and span of 8 m support the elng (.e. the floor of the story above). Table IV gves an overvew of the alulated and assessed values for the bas varables. Applyng equatons (6)-(16), µ R = 1065 knm and V R = 0.0. Consderng the layout of the struture the bendng moment ndued by the permanent load (nludng self-weght of the beam) s 408 knm, and therefore µ R / µ G =.61. Applyng the AID of Fgure 1, χ max s 0.17, and onsequently M Qk,max = 69.4 knm. For the spef buldng 8 m of elng wdth s transferred to the beam, and therefore the maxmum allowable haraterst value of the mposed load on the floor above s 1.08 kn/m². If the requred value of the mposed load on the floor above the fre ompartment s larger than 1.08 kn/m², the beams should be strengthened. TABLE IV. PARAMETERS FOR THE INVESTIGATED BEAM (t E = 90 mn) Symbol Dm. µ σ f,0 MPa f y,0 MPa A s mm² v mm 0 5 h mm 30 5 h mm b mm mm 9 0 k fy,res K T

9 Fgure 3. Resdual renforement yeld stress rato k fy,res = f y,res / f y,0 as a funton of θ. CONCLUSIONS A relablty-based methodology for determnng the maxmum allowable mposed load on a onrete beam after fre exposure has been presented. The methodology s easy-to-use for prattoners as only smple analytal formulas have to be evaluated. Subsequently, the maxmum allowable load s determned by applyng pre-alulated graphs, alled assessment nteraton dagrams (AID). Whle the method an easly be mplemented n normal spreadsheet software ts relablty-based bakground ensures a ratonal answer to the queston f the onrete beam s safe enough for ontnued use, or whether strengthenng s neessary. REFERENCES 1. fb, fb bulletn 46: Fre Desgn of Conrete Strutures Strutural Behavour and Assessment, State-of-the-art Report, Internatonal Federaton for Strutural Conrete, Lausanne, Bngöl, A.F., Gül, R., Effet of elevated temperatures and oolng regmes on normal strength onrete, Fre and Materals, 33, p.79-88, Elghazoul, A.Y., Cashell, K.A., Izzudn, B.A., Expermental evaluaton of the mehanal propertes of steel renforement at elevated temperature, Fre Safety Journal, 44, , Annerel, E., Assessment of the Resdual Strength of Conrete Strutures after Fre Exposure, dotoral thess, Ghent Unversty, Gulvanessan, H., Calgaro, J.-A., Holký, M., Desgner s gude to EN 1990: Euroode 0: Bass for strutural desgn, Thomas Telford, London, CEN. (00a). EN 1990: Euroode 0: Bass of strutural desgn. European Standard. 7. Kodur, V.K.R., Raut, N.K., Mao, X.Y., Khalq, W. (013). Smplfed approah for evaluatng resdual strength of fre-exposed renfored onrete olumns. Mater. Strut.: 46, Van Cole, R., Caspeele, R., Taerwe, L. (014b). Towards a relablty-based post-fre assessment method for onrete slabs norporatng nformaton from nspeton. Strutural Conrete: JCSS. (007). Probablst Model Code. The Jont Commttee on Strutural Safety. 10. CEN. (004b). EN : Euroode : Desgn of onrete strutures Part 1-: General rules Strutural fre desgn. European Standard. 11. Van Cole, R., Caspeele, R., Taerwe, L. (014). Relablty-based evaluaton of the nherent safety presumptons n ommon fre safety desgn. Engneerng Strutures: 77,