André Teófilo Beck Member, ABCM University of São Paulo USP Structural Engineering Department São Carlos, SP, Brazil

Transcription

1 A First Attempt Towards Reliability-based Calibratio of Brazilia Structural esig Codes Adré Teófilo Beck Member, ABCM Uiversity of São Paulo USP Structural Egieerig epartmet São Carlos, SP, Brazil Atôio C. de Souza Jr. Uiversity of São Paulo USP Structural Egieerig epartmet São Carlos, SP, Brazil A First Attempt Towards Reliabilitybased Calibratio of Brazilia Structural esig Codes This paper addresses the reliability-based calibratio of partial safety factors for Brazilia desig codes NBR8681:2003 (Actios ad Safety of Structures) ad NBR8800:2008 (esig of Steel ad Steel-Cocrete Composite Structures). To the author s kowledge, these codes have ever bee subject to reliability-based calibratio of partial ad load combiatio factors. This paper represets a first effort i reachig this goal. The preset calibratio effort is based o actual data for wid loads i south east of Brazil, but uses maily iteratioal data for other problem parameters. So far, the ivestigatio is limited to structural steel members. The ivestigatio leads to a set of optimized partial safety factors, which are compared to partial factors curretly used i Brazilia desig codes. Results show that the optimized set of partial factors leads to more uiform reliability for differet desig situatios ad load combiatios. The ivestigatio icludes a aalysis of the ecoomical impact of replacig the curret set of partial factors by the calibrated factors foud i this paper. It is show that, with a optimized set of partial safety factors, it is possible to maitai the curret level of reliability ad still produce a 5% average reductio i expediture with structural materials, atiowide. The paper also poits to geeral similarities betwee Brazilia desig codes, the ew geeratio of EUROCOES ad America ANSI/AISC codes. These codes are compared i terms of their ability to produce uiform reliability for differet desig situatios. Keywords: structural reliability, desig codes, code calibratio Itroductio 1 I the late 70 s ad early 80 s, ANSI/AISC structural desig codes have udergoe sigificat improvemets, with a migratio from allowable stress to limit state (partial factor) desig. This migratio process was guided by reliability-based calibratio of the partial safety factors of the ew codes. The calibratio was coducted esurig that the safety level of the ew codes reflected the geeral safety level of old codes, which was regarded as result of collective kowledge ad experiece-based optimizatio over the years. This process was well documeted i the literature (Elligwood et al., 1980; Elligwood ad Galambos, 1982). More recetly, Europea desig codes also migrated to a limit state partial factor format. Europea Uio member states are ow idividually defiig the partial safety factors to be used withi each coutry. Idividual efforts i performig the reliability based calibratio of these partial factors have bee reported (Gayto et al., 2004; Gulvaessia ad Holicky, 2005). The Brazilia Actios ad Safety of Structures: Procedures (NBR8681:2003) ad esig of Steel ad Steel-cocrete Composite Buildigs (NBR8800:2008) were also recetly coverted to a limit state partial factor format. Both codes are largely based o the EUROCOES, but their coceptio was ot accompaied by a cosistet (e.g., reliability-based) tropicalizatio of partial safety factors. Adaptatios of partial safety factors were based o subjective judgmet, derived from the experiece of committee members. The same partial ad load combiatio factors are curretly beig revised i applicatio to the EUROCOES. The preset paper is a first effort towards verifyig the adequacy of partial factors curretly used i Brazilia desig codes. Statistical descriptio of resistace ad load parameters for the Brazilia reality is curretly icomplete. There have bee o comprehesive studies of ucertaities i dead ad life loads reported i the literature. For these parameters, iteratioal data is used i the preset ivestigatio. Some iformatio o material resistace for steel, cocrete ad wood is available, but i scattered form. This Paper accepted ecember, Techical Editor: Nestor A. Zouai Pereira iformatio still has to be collected ad compiled for future calibratio work. Statistics for the occurrece of wids i Brazil have bee reported (Satos, 1989; Riera ad Rocha, 1998) ad are used i the preset study. The article is laid out as follows. Sectio 1 itroduces the subject ad locates it i a historical ad geographical perspective. Sectio 2 presets the basic formulatios of codified limit state desig, with particular attetio to similarities betwee the ANSI/AISC, EUROCOE ad Brazilia code formats. Sectio 3 presets de statistical data used i the preset calibratio effort. I sectio 4, the calibratio (optimizatio) problem is formulated. Results of a first calibratio are preseted i Sectio 5. Sectio 6 presets a compariso of differet code formats (ANSI/AISC versus EUROCOE) i terms of calibratio of partial factors. Sectio 7 presets a study of the potetial ecoomical impact of implemetig the partial safety factors foud i the preset calibratio effort. Sectio 8 closes the paper with some coclusios. Nomeclature = dead load, o-dimesioal L = live (accidetal) load, o-dimesioal W = wid load, o-dimesioal L a.p.t. = average-poit-i-time value of live load L 50years = fifty-year extreme live load W 1year = aual extreme wid load W 50years = fifty-year extreme wid load R[] = geeral resistace fuctio S[] = geeral load effect fuctio Greek Symbols = partial factor applied o omial loads ad resistaces φ R = partial factor applied o omial resistace ψ = load combiatio factors Subscripts k relative to characteristic value relative to omial value relative to desig value J. of the Braz. Soc. of Mech. Sci. & Eg. Copyright 2010 by ABCM April-Jue 2010, Vol. XXXII, No. 2 / 119

2 Adré Teófilo Beck ad Atôio C. de Souza Jr. Codified Limit State esig ANSI/AISC Load ad Resistace Factor esig The desig format used i curret ANSI/AISC desig codes came to be kow as Load ad Resistace Factor esig (LRF). The geeral desig equatio is give by (ASCE, 2006): + i Qi + j Qj ϕr R (1) where idex stads for omial value, are partial factors applied o (omial values of) loads, is the (omial value of) dead load (self-weight), Q i is the pricipal variable load, Q j is the accompayig variable load, R is a omial member resistace ad φ R is a partial safety factor applied o omial member resistace. Reliability-based calibratio of partial factors of ANSI/AISC codes has bee facilitated by the simplicity of these i compariso to Europea ad Brazilia codes. This simplicity arises from use of a sigle resistace factor for member resistace, but also from usig a set of pre-defied load combiatio equatios. A differet set of partial safety factors is used for each load combiatio, ad the combied factored load effects are give by: S L S + (0.5 L or 0.8 W ) = max W L E + (0.5 L or 0.2 S) 0.9 (1.3 W or 1.5 E ) (2) where S is the desig load, L is the live load (called accidetal load i Brazilia codes), W is the wid load, S is the sow load ad E is the earthquake load. etermiatio of load factors for ANSI/AISC codes was also simplified by the rule used i load combiatios. These codes are based o Turkstra s load combiatio rule (Turkstra ad Madse, 1980), where the extreme of oe variable load (called pricipal load i the combiatio) is combied with the average-poit-i-time value of the accompayig loads. This simplifies the reliability problem, which becomes time-ivariat, with radom load processes replaced by (extreme-value) radom variables. NBR8800 ad EUROCOE Formats The geeral format adopted i Brazilia NBR8800:2008 ad i Europea desig codes uses a sigle load combiatio expressio: r k + i i + ψ j j j j i R S Q Q R where S[.] is a load effect fuctio, R[.] is a resistace fuctio, r k is a characteristic material resistace, is the resistace factor (oe for each structural material), i are partial load factors, ψ j are load combiatio factors ad other symbols are equivalet to Eq. (1). Here, oe resistace factor is used for each structural material ivolved. The format is more flexible i terms of resistace modelig, but less flexible is terms of load combiatios, as will be show i Sectio 6: Calibratio i Other Code Formats. I this study, load combiatios ivolvig permaet ad variable actios are cosidered. Accidetal (live) ad wid variable (3) actios are cosidered i the study, leadig to five load combiatio equatios, followig Eq. (3): LL + S = max + WW + LL + WψWW + WW + Lψ LL (4) The first three combiatios are particular cases of the last two, whe variable loads L ad/or W are zero or o-existet. The first three equatios are reproduced here for completeess, to stress the fact that load combiatios ivolvig oly some of the variable loads are also relevat i calibratio of Brazilia desig codes. ead ad Live Loads Actual data o dead ad live load ucertaity is ot available for the Brazilia reality. I this ivestigatio, it is assumed that o sigificat differeces exist i dead ad life loads betwee Brazil ad what is reported i iteratioal refereces. This assumptio should be ivestigated i the future. ead ad live load statistics used i this paper are based o data reported by Elligwood et al. (1980), as preseted i Table 1. NBR8681 Code Format The geeral desig format adopted i Brazilia code NBR8681:2003 differs slightly from the format adopted i NBR8800: r k S + i Qi + ψ j Qj R j i R The symbols are similar to those used i Eq. (3). I this format, the partial factor for the pricipal variable actio is multiplied by the combiatio value of the secodary variable actio. I the author s uderstadig, ad from a theoretical poit of view, there is o justificatio for this format; hece the NBR8800 format should be preferred. Reliability idexes obtaied for curret ad for calibrated factors, for these two formats, are compared i Sectio 6. The load combiatio equatios correspodig to Eq. (4) are: + LL S = max + WW + L( L + ψww ) + W ( W + ψ LL ) (6) Compariso of Code Formats The symbols used i this paper to describe the differet code formats differ somewhat from the symbolism used i each of the described codes. This was doe o purpose, sice our objective is to stress the similarities betwee the code formats, ad later compare these formats (see Sectio 6). Oe of the mai differeces betwee the ANSI/AISC ad Brazilia (or Europea) code formats is the way the partial resistace factor is applied. I ANSI/AISC codes, a sigle partial resistace factor is applied to omial member resistace. I (5) 120 / Vol. XXXII, No. 2, April-Jue 2010 ABCM

3 A First Attempt Towards Reliability-based Calibratio of Brazilia Structural esig Codes Brazilia ad Europea desig codes, a differet partial resistace factor is used for each structural material. It is widely accepted (Kogut ad Chou, 2004; Mohamed et al., 2001) that the ANSI/AISC format lacks flexibility, ad results i o-uiform reliability i problems ivolvig more tha oe structural material (for example, reiforced cocrete or steel-cocrete composites). Use of a sigle resistace factor i ANSI/AISC codes makes the reliability-based calibratio easier. I calibratio of EUROCOES, fixed sesitivity factors have bee used to split the load ad resistace parts of the problem (Holicky, 2008), so that calibratio ca be performed separately. iffereces betwee differet code formats, with respect to resistace modelig, are ot explored herei, but will be addressed i future studies. The preset study is limited to structural steel members. Moreover, i the preset study, a liear resistace model R[.] is assumed, hece the partial resistace factors i the ANSI/AISC ad Brazilia formats become equivalet: 1 = ϕ R R The preset study explores differeces related to load combiatios i distict code formats. I this regard, it is oted that the ANSI/AISC code format is more flexible tha the Europea or Brazilia couterparts, sice it uses oe set of partial factors for each load combiatio equatio, whereas Brazilia codes use oe sigle set of partial (ad combiatio) factors for all load combiatio expressios. The geeral format of the ANSI/AISC code, for load combiatios ivolvig live ad wid loads, would lead to the followig desig equatios: 1 2 6L + S = max 3 + 7W 4 + 8L + 10W + + (7) (8) These are ot the desig equatios of the ASCE:2006 code, Eq. (2). Equatio (8) is the ANSI/AISC equivalet to Eqs. (4) ad (6). It is oted that this format, which uses differet load factors i each load combiatio, has 11 degrees of freedom, whereas combiatios (4) ad (6) have 5 degrees of freedom each. This differece betwee desig code formats is explored i Sectio 6. Statistics of Load ad Resistace Parameters Structural Steel Members Statistics o structural steel member resistace is take from Elligwood et al. (1980). Statistics for differet steel members (tesio members, compact beams, cocetrically loaded colums ad beam colums) do ot vary too much. Hece, as approximatio, a sigle set of structural steel statistics is used i the preset calibratio effort: mea of 1.18R, coefficiet of variatio (c.o.v.) of This c.o.v. accouts for material ucertaity ad model error. The mea of 1.18 R is already adjusted to take ito accout the rate of loadig, for combiatios ivolvig wid. These values are i agreemet with results reported by Pimeta et al. (2008) regardig bedig of steel beams i Brazil. Table 1 summarizes statistics of load ad resistace variables cosidered i this study. I Table 1, L 50years ad W 50years are the 50 year extreme live ad wid loads, respectively, which are geerally combied with the aual extreme wid (W 1year ) ad with the average-poit-i-time live load (L a.p.t ), followig Eq. (13). Table 1. Statistics of resistace ad load variables. Variable mea c.o.v. distributio Res. of steel members 1.18 R 0.15 logormal ead load ormal L a.p.t L 0.55 Gamma Life load: L 50years 1.00 L 0.25 Gumbel W 1year 0.33 W 0.47 Gumbel Wid load W 50years 0.90 W 0.34 Gumbel Wid Loads i Cetral-South of Brazil Statistics of aual extreme storm wids for the cetral-south of Brazil have bee reported by Satos (1989) ad Riera ad Rocha (1998). Wid speed series of 15 to 29 years, recorded o 11 meteorological statios, were used by Satos (1989) to divide the cetral-south of Brazil i 5 meteorological provices. The author costructed regressio curves, which allow oe to evaluate mea ad stadard deviatio of maximum aual wid speeds at ay locatio withi the 5 provices, accordig to type of storm ad wid directio. Maximum aual wid speeds, idepedet of storm type ad directio, ca also be evaluated. The preset study is based o evaluatio of the model of Satos at 16 locatios (11 meteorological statios ad geometrical ceters of 5 provices). Mea ad stadard deviatio of maximum aual wid speeds were evaluated at these locatios ad fitted to Gumbel distributios. These were the coverted to 50 year extremes usig: W50years ( W ) 1year F ( x) = F ( x) 50 By comparig the mea of extreme wid speeds at the 16 locatios with omial (desig) wid speeds (NBR6123:1988), the wid statistics show i Table 2 were obtaied. Table 2. Statistics of wid speeds. Variable mea* c.o.v. distributio Wid V 1year 0.57 V 0.21 Gumbel speed: V 50years 0.95 V 0.13 Gumbel *where V is the omial (desig) wid speed, followig NBR6123:1988. Wid speed statistics were coverted ito wid pressure, usig: 1 W = ρ c V 2 2 where ρ is air desity ad c is a aerodyamic coefficiet, accoutig for shape, gust (turbulece) ad expositio. The quadratic relatio betwee wid speeds ad wid pressures applies to bias factors. Bias factors for wid pressure, preseted i Table 1, were obtaied from: 2 µ W µ V = W V (10) ue to the quadratic relatioship, the coefficiet of variatio of wid pressure (V W ) is obtaied as V W 2 V V, for a determiistic coefficiet c. The ucertaity of shape (V a ), gust (V g ) ad expositio (V r ) factors is take ito accout by meas of a secod momet mea value approximatio (JCSS, 2001): (9) J. of the Braz. Soc. of Mech. Sci. & Eg. Copyright 2010 by ABCM April-Jue 2010, Vol. XXXII, No. 2 / 121

4 Adré Teófilo Beck ad Atôio C. de Souza Jr. V = V + V + V + (2 V ) W a g r V (11) This leads to the wid pressure statistics reported i Table 1 (Souza Jr., 2009). As poited out by Riera ad Rocha (1998), modelig both wid speeds ad pressures by Gumbel distributios is formally icosistet. However, Elligwood et al. (1980) foud that icorporatio of ucertaities i aerodyamic coefficiets leads to best fit of wid pressure data by Gumbel distributios. Reliability-Based Code Calibratio Reliability-based code calibratio is the determiatio of the partial safety factors i Eqs. (1), (3) or (5), i order for the rage of structural desigs resultig from these equatios to preset miimal variatios with respect to a pre-selected target reliability level (β T ). ifferet desig situatios, covered by the code beig calibrated, have to be cosidered. This icludes structural elemets uder tesile, compressive or bedig loadig, istability of colums, ad so o. This also icludes desigs resultig i distict ratios betwee the differet loads beig combied. Hece, it is customary i load calibratio work to cosider differet ratios betwee live ad dead loads (L / ) ad betwee wid ad dead loads (W / ). The calibratio of partial factors to be used i a give desig code is accomplished by solvig a ucostraied optimizatio problem: m i= 1 j= 1 ( ) 2 miimize: Ψ = β mi[ β ] w T ijk ij k (12) where ad m are the umber of load ratios (desig situatios) cosidered, β T is the target reliability idex, β ijk is the reliability idex for desig ij ad for the k th load combiatio. Expressio (12) does ot pealize reliability idexes that are below the target, as proposed by Sorese et al. (1994). For each load ratio cosidered, a differet weight is used i the combiatio (w ij i Eq. (12)), accordig to the relative sigificace of that load ratio i actual code use. Load ratios ad weights used i this paper are show i Table 3. These values were adapted from Elligwood et al. (1980). Table 3. Weights of the differet load ratios cosidered i the calibratio. w ij L / W / The reliability problem correspodig to each desig situatio is costructed from the omial loads. For example: cosiderig a uitary (o-dimesioal) omial dead load = 1, load ratios W / = 1 ad L / = 1, usig Eq. (4) (load combiatios for NBR8800) ad partial factors recommeded i this code (reproduced i Table 3), oe obtais the desig load as 3.8 uits. Usig the resistace factor, the omial resistace is obtaied as R = = From the omial values of loads ad resistaces, ad usig the data preseted i Table 1, statistics of the radom variable loads ad resistaces are re-costructed. These are used i the reliability aalyses. Two liear limit state fuctios are used i the reliability aalyses: g ( X) = R ( + L + W ) = years 1year g ( X) = R ( + L + W ) = 0 2 a.p.t. 50years (13) where X is a vector of radom variables. These two equatios accout for the five load combiatio cases, Eqs. (4) ad (6), sice some load ratios are zero (followig Table 3). It is iterestig to ote the equivalece betwee limit state fuctios used i reliability aalyses (Eq. (13)) ad the correspodig code desig equatios (Eqs. (1), (3) ad (5)). This equivalece explais why moder codes are said to be based o limit state desig. The limit state fuctios, Eq. (13), defie boudaries which divide the failure ad survival domais: Ω = { x g( x) 0} is the failure domai f Ω = { x g( x) > 0} is the safety domai s For each limit state, the failure probability is evaluated as: P P[ g( x) 0] f ( x) dx f = = X Ω f (14) (15) Equatio (15) is solved via de First Order Reliability Method (FORM), usig the StRA (Structural Reliability Aalysis ad esig) program (Beck, 2008). This solutio ivolves a trasformatio of the radom variables to the stadard ormal space, ad a search for the most probable failure poit, or desig poit. The reliability idex β is the (miimal) distace betwee the desig poit ad the origi of the stadard ormal space. The reliability idex is related to the failure probability by the expressio: P = Φ[ β ] f (16) For the two limit states cosidered herei (Eq. (13)), ad followig Turkstra s rule, the reliability idex for desig situatio ij is give by the smallest value betwee the two load combiatios: β = mi[ β, β ] ij ij1 ij2 (17) The optimizatio problem (Eq. (12)) is solved usig a particle swarm optimizatio (PSO) algorithm, implemeted i the StRA program. Calibratio Results for NBR8800 ad β T = 3.0 Partial Factors to Obtai β T = 3.0 Exactly Recet studies by the authors (Beck ad ória, 2008; Oliveira et al., 2008; Beck et al., 2009) reveal that the reliability idex of structures desiged accordig to Brazilia code NBR8800:2008 lie i the rage from β = 2.3 to β = 4.5. Therefore, i this study, the target reliability idex was first selected as β T = 3.0. A prelimiary aalysis was made to fid the partial safety factors that would lead to the target reliability idex exactly, i the NBR8800 code format (Eq. (4)). Results of this aalysis are show i Figs. 1 ad 2. Partial factors show i these figures serve as a guide to the fixed set of partial factors to be foud i the calibratio 122 / Vol. XXXII, No. 2, April-Jue 2010 ABCM

5 A First Attempt Towards Reliability-based Calibratio of Brazilia Structural esig Codes process. The figures show early costat partial factors for resistace ad dead loads ( = 1.1, = 1.1), ad some variatio o the ideal partial safety factors for the variable actios. Partial Safety Factors 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0 L / for W / = 0,5 for W / = 1,0 for W / = for W / = 5,0 for W / = 0,5 for W / = 1,0 for W / = for W / = 5,0 for W / = 0,5 for W / = 1,0 for W / = for W / = 5,0 for W / = 0,5 for W / = 1,0 for W / = for W / = 5,0 Figure 1. Partial safety factors to obtai β T = 3.0 exactly i +L+W load combiatio, L mai load. Partial Safety Factors 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0 W / for L / = 0,5 for L / = 1,0 for L / = for L / = 5,0 for L / = 0,5 for L / = 1,0 for L / = for L / = 5,0 for L / = 0,5 for L / = 1,0 for L / = for L / = 5,0 for L / = 0,5 for L / = 1,0 for L / = for L / = 5,0 Figure 2. Partial safety factors to obtai β T = 3.0 exactly i +L+W load combiatio, W mai load. Calibratio Results for NBR8800 Reliability-based calibratio of the partial factors used i desig code NBR8800 amouts for fidig the set of five partial ad combiatio factors i Eq. (4), plus the resistace factor, i order to produce the smallest variatios of reliability idexes with respect to the target reliability. The problem is ot easy to solve whe the six factors are sought simultaeously. Some covergece problems were ecoutered sice the optimum dead load factor, for combiatios ivolvig oly dead load (L / = W / = 0), is quite differet from the optimum for combiatios ivolvig variable actios. Hece, the author s experiece i prelimiary calibratio rus was used to fix the values = 1.10 ad = The optimizatio process was the ru agai to fid the remaiig partial factors. Results are preseted i Table 4. I Table 4, the partial factors foud i the preset calibratio effort are compared with the factors curretly used i Brazilia desig code NBR8800:2008. Table 4. Partial safety factors for Brazilia codes ad calibrated i this paper. Partial or comb. factor NBR8800 NBR8681 Calibratio for β T = ψ L ψ W ψ L ψ W Calibratio for β T = 2.8 Table 4 shows sigificat differeces betwee the partial factors curretly i use ad the factors foud through the calibratio process. It is observed that the calibrated factors are larger for the pricipal loads ( ad ), but the combiatio factors are sigificatly smaller, resultig i sigificatly smaller values of the accompayig variable actio (product ψ). Reliability idexes resultig from use of these partial factors are compared graphically i Figs. 3 ad 4. Figure 3 shows reliability idexes obtaied for all desigs (load ratios) cosidered, with β give as the miimum betwee the two (formally, five) limit state (or load combiatio) equatios. The desig load is the largest amogst the combiatios cosidered (followig Eq. (4)), ad the reliability idex is the smallest betwee the two limit state equatios (Eq. (13)). I Figure 3, reliability idex results are show for all desig situatios. Figure 4 shows oly the lower ad upper bouds of these idexes, i terms of load ratios W / (left) ad L / (right). The bouds show the uiformess of (or lack of) reliability idexes obtaied usig the differet sets of partial safety factors. I Figures 3 ad 4, it ca be clearly see that the calibrated set of partial ad combiatio factors leads to more uiform reliability idexes. Moreover, the average reliability idex obtaied with the calibrated set is larger tha the average reliability obtaied with the curret set of factors. The set of calibrated partial factors is a optimum (or, at least, a better) set, i compariso to what is curretly used i Brazilia desig codes. Of course, this observatio must be subject to further cofirmatio, give the limitatios of the preset calibratio effort. Sice there is o evidece of previous calibratio or of proper tropicalizatio of partial safety factors used i Brazilia desig codes, the results obtaied i this paper are very sigificat, ad should be regarded as a stimulus for further ivestigatio. 3,5 3,3 3,1 2,9 2,7 2,5 2,3 2,1 W / (NBR 8800 format) ad: L / = 0,0 L / = 0,5 L / = 1,0 L / = 1,5 L / = L / = 5,0 (NBR8800) ad: L / = 0,0 L / = 0,5 L / = 1,0 L / = 1,5 L / = L / = 5,0 Figure 3. Reliability idexes for Brazilia NBR8800 code ad for calibrated partial factors (β T = 3.0). J. of the Braz. Soc. of Mech. Sci. & Eg. Copyright 2010 by ABCM April-Jue 2010, Vol. XXXII, No. 2 / 123

6 Adré Teófilo Beck ad Atôio C. de Souza Jr. Calibrate factors (NBR8800 format): (NBR8800): miimum betas miimum betas (NBR8800 format): (NBR8681 format) miimum betas miimum betas W / W / (NBR8800 format): (NBR8800): miimum betas miimum betas (NBR8800 format): (NBR8681 format) miimum betas miimum betas L / Figure 4. Reliability idex bouds for Brazilia NBR8800 code ad for calibrated partial factors (β T = 3.0). L / Figure 5. Reliability idex bouds for calibrated partial factors (β T = 3.0), NBR8800 ad NBR8681 code formats. Calibratio i Other Code Formats I this sectio, the calibratio procedure is used to fid the optimum set of partial factors i the NBR8681 ad ANSI/AISC code formats. Calibratio i the NBR8681 format (Eq. (6)) actually resulted i the same set of partial factors preseted i Table 4 (secod colum for β T = 3.0). Clearly, this is due to the fact that partial factors ad foud i the calibratio for NBR8800 are very similar. Figure 5 shows that both formats, usig the calibrated factors, lead to virtually the same rage of variatio of reliability idexes. I Figure 6, the two desig formats are compared whe usig the set of partial factors curretly i use i the Brazilia codes. The differeces betwee the two formats are more oticeable i this case, although still acceptably small. It turs out that, i practice, the two desig formats (NBR8800 ad NBR8681) are equivalet. The authors, however, still cosider that the format of NBR8800 is more logical, sice it maitais the theoretical idepedecy betwee the partial factors for pricipal ad accompayig variable actios. For the purpose of illustratio, the calibratio was also doe i the ANSI/AISC code format, that is: usig a differet set of partial factors for each load combiatio equatio (Eq. (8)). The resultig calibrated partial factors are = 1.10 (or φ R = 0.91) ad: L S = max W L W L W (18) These load combiatios are equivalet to those used i calibratio of Brazilia codes, but they ca be grossly compared with the partial factors curretly i use i ANSI/AISC codes, Eq. (2). I their calibratio effort, Elligwood et al. (1980) also foud the optimum partial factor for dead loads to be equal to 1.10, as i the preset study. They, however, preferred to set the dead load factor to 1.2, followig Eq. (2). The rage of reliability idexes obtaied usig the partial factors calibrated i the NBR8800 ad ANSI/AISC code formats are compared i Fig. 7. It ca be see that the ANSI/AISC format leads to greater uiformity of reliability idexes. This happes because the ANSI/AISC format is more flexible i terms of represetig the differet load combiatios. As argued above, the ANSI/AISC format has 11 degrees of freedom to deal with the same load combiatios that, i the Brazilia format, takes 5 partial safety ad combiatio factors. This differece i flexibility leads to the results observed i Fig / Vol. XXXII, No. 2, April-Jue 2010 ABCM

7 A First Attempt Towards Reliability-based Calibratio of Brazilia Structural esig Codes (NBR8800): (NBR8681): miimum betas miimum betas (ANSI/AISC format): (NBR8800 format): miimum betas miimum betas W / W / (NBR8800): (NBR8681): miimum betas miimum betas L / Figure 6. Reliability idex bouds for partial factors curretly i use i Brazilia codes NBR8800 ad NBR8681. Ecoomical Impact I this paper, a set of optimal partial safety factors for Brazilia desig codes NBR8800 ad NBR8681 is obtaied. This set is optimum i the sese that it produces more uiform reliability, i compariso to the partial factors curretly i use i these codes. A recommedatio of chage i the partial safety factors used i these codes must be accompaied by a aalysis of the ecoomical impact of such chages. The weights used to describe the relative importace of each desig situatio (Table 3) ca be used for this purpose. Cosiderig a uitary omial dead load ( = 1) as referece, the distict load ratios show i Table 3 ad the partial factors curretly i use i the codes, a weighted sum of desig loads is obtaied as: S L W + L + W ψw m wij i j = max i= 1 j= 1 wij L W + L ψ L + W i j (19) (ANSI/AISC format): (NBR8800 format): miimum betas miimum betas L / Figure 7. Reliability idex bouds for calibrated partial factors (β T =3.0), NBR8800 ad ANSI/AISC code formats. This weighted sum of desig loads ca the be compared with the same sum obtaied with the calibrated partial safety factors. The differece i the weighted sums is a measure of the impact of chagig the partial safety factors. The ecoomical impact ca be assumed to be proportioal to this weighted sum of desig loads. Applyig this procedure to the partial safety factors show i Table 4 (ad calibrated for β T = 3.0), it is foud that the calibrated set produces a icrease of 1% i the weighed sum of desig loads, i compariso to the partial factors curretly i use (NBR8800:2008). Figure 4 shows that this icremet i desig loads is associated with a icrease i the lower rage of reliability idexes. The calibratio ot oly produces more uiform reliability, but also a slight icrease i reliability levels. The calibratio procedure was repeated for a target reliability idex of β T = 2.8, which is compatible with the lower rage of reliability idexes obtaied with the curret partial factors of NBR8800. The rage of reliability idexes obtaied with this ew set of partial factors is compared with NBR8800 i Fig. 8. The figure shows that the set of partial factors calibrated for β T = 2.8 produces more uiform reliability, which is greater tha or equal to what is obtaied with the coefficiets of NBR8800. The set of partial safety ad combiatio factors obtaied i the calibratio with β T = 2.8 is show i the third colum of Table 4. This ew calibrated set maitais curret reliability levels, but produces a reductio of the order of 5% i the weighted sum of desig loads. To uderstad the impact of this result, we must recall that it is assumed to be equivalet to a 5% reductio i expediture with structural materials atiowide. J. of the Braz. Soc. of Mech. Sci. & Eg. Copyright 2010 by ABCM April-Jue 2010, Vol. XXXII, No. 2 / 125

8 Adré Teófilo Beck ad Atôio C. de Souza Jr. (NBR8800 format): (NBR8800): miimum betas miimum betas W / (NBR8800 format): (NBR8800): miimum betas miimum betas L / Figure 8. Reliability idex bouds for Brazilia NBR8800 code ad for calibrated partial factors (β T = 2.8). Coclusios This paper preseted a ivestigatio of partial safety factors used i Brazilia desig codes NBR8681:2003 ad NBR8800:2008. A reliability-based calibratio of partial safety ad combiatio factors used i these codes was performed. The calibratio effort resulted i a optimized set of partial factors, which was show to produce more uiform reliability for differet desig situatios, i compariso to the partial factors curretly i use i these codes. The calibratio performed for a target reliability idex of β T = 3.0 resulted i more uiform reliability, ad i a slight icrease i mea reliability levels, at a "cost" of a 1% icrease i average desig loads. For a target reliability of β T = 2.8, the calibratio produced more uiform reliability, which is equal to or greater tha the curret miimum reliability levels, ad with a reductio of early 5% i the weighted sum of desig loads. The ecoomical impact of this reductio is quite sigificat. I the set of partial ad combiatio factors calibrated for β T = 2.8, the wid load factor ( ) is icreased from 1.4 to 1.6, but the combiatio values of secodary loads are reduced. The combiatio value for live load as secodary actio is reduced from 1.05 to 0.45, ad the combiatio value for wid is reduced from 0.84 to For this ew, optimized balace of partial ad combiatio factors, more uiform reliability is obtaied, with a 5% reductio i average desig loads. The study has also show that the ANSI/AISC code format, which uses oe set of partial factors for each load combiatio expressio, is more flexible tha the Brazilia or EUROCOE formats, where a sigle set of partial ad combiatio factors is used. The larger flexibility of ANSI/AISC codes i represetig the differet load combiatios results i more uiform reliability. Results obtaied i this paper have to be cofirmed by further ivestigatio, due to limitatios of the preset calibratio effort. Further ivestigatio is already uderway, ad icludes other structural materials as reiforced cocrete, steel-cocrete composites, masory ad timber structures. Sice partial factors o structural materials are expected to accout for differet ucertaities i structural materials, the calibrated partial factors for loads obtaied i this paper are expected to chage oly margially. Ackowledgemets The authors would like to express their gratitude to Brazilia agecies CNPq ad CAPES for fiacial support of this research project. Refereces ANSI/AISC 360:2005, 2005, Specificatio for Structural Steel Buildigs. America Istitute of Steel Costructio, Chicago, Illiois. ASCE, 2006, Miimum esig Loads for Buildigs ad Other Structures. America Society of Civil Egieerig. Beck, A.T., 2008, Object-orieted time variat reliability aalysis. I: 8th World Cogress o Computatioal Mechaics, Veice, Italy. Beck, A.T. ad oria, A.S., 2008, Reliability aalysis of I-sectio steel colums desiged accordig to ew Brazilia buildig codes, J. of the Braz. Soc. of Mech. Sci. ad Eg., Vol. 30, No. 2, pp Beck, A.T., Oliveira, W.L.A., enardim, S., Elebs, A.L.H.C., 2009, Reliability-based evaluatio of desig code provisios for circular cocretefilled steel colums, Egieerig Structures, Vol. 31, p Elligwood, B. ad Galambos, T.V., 1982, Probability-based criteria for structural desig, Structural Safety, Vol. 1, pp Elligwood, B., Galambos, T.V., MacGregor, J.G. ad Corell, C.A., 1980, evelopmet of a Probability Based Load Criterio for America Natioal Stadard A58. EUROCOE 2001, pren 1990: Basis of Structural esig Aex C: Basis for Partial Factor esig ad Reliability Aalysis. Europea Committee for Stadardizatio, Brussels, Fial raft. Gayto, N., Mohamed, A., Sorese, J.., Pedola, M. ad Lemaire, M., 2004, Calibratio methods for reliability-based desig codes, Structural Safety, Vol. 26, pp Gulvaessia, H. ad Holicky, M., 2005, Eurocodes: usig reliability aalysis to combie actio effects, Proceedigs of the Istitutio of Civil Egieers, Structures ad Buildigs, Thomas Telford. Holicky, M., 2008, Reliability-based aalysis of codified desig allowig for productio quality, Proceedigs of 4 th Iteratioal ASRANet Colloquium, Athes. JCSS, 2001, Probabilistic Model Code, Joit Committee o Structural Safety. Kogut, G.F., Chou, K.C., 2004, Partial resistace factor desig o steelcocrete beam-colums, Egieerig Structures, Vol. 26, pp NBR8681:2003, 2003, Actios ad Safety of Structures: Procedures. ABNT - Brazilia Associatio of Techical Codes, Rio de Jaeiro (i Portuguese). NBR6123:1988, 1988, Wid Loads i Buildigs. ABNT Brazilia Associatio of Techical Codes, Rio de Jaeiro (i Portuguese). NBR8800:2008, 2008, esig of Steel ad Steel-Cocrete Composite Structures: Procedures. ABNT Brazilia Associatio of Techical Codes, Rio de Jaeiro (i Portuguese). Mohamed, A., Soares, R., Veturii, W.S., 2001, Partial safety factors for homogeeous reliability of oliear reiforced cocrete colums, Structural Safety, Vol. 23, pp Oliveira, W.L., Beck, A.T., Elebs, A.L.H.C., 2008, Safety evaluatio of circular cocrete-filled steel colums desiged accordig to Brazilia buildig code NBR 8800:2008, IBRACON Structures ad Materials Joural, Vol. 1, pp Pimeta, R.J., Gozaga, L.G.M., Queiroz, G. ad iiz, S.M.C., 2008, Basic requisites for reliability aalysis of siusoidal web beams subject to lateral-torsioal bucklig, Revista Escola de Mias (to appear). 126 / Vol. XXXII, No. 2, April-Jue 2010 ABCM

9 A First Attempt Towards Reliability-based Calibratio of Brazilia Structural esig Codes Rackwitz, R., 2000, Optimizatio the basis for code-makig ad reliability verificatio, Structural Safety, Vol. 22, pp Riera, J.. ad Rocha, M.M., 1998, Load defiitio for wid desig ad reliability assessmets: extreme wid climate. I: Rieira ad aveport (eds.), Wid effects o Buildigs ad Structures, Balkema, Rotterdam. Satos, M. dos, 1989, Regioalizatio of Extreme Wid Velocities ad Temperatures i Ceter-South of Brazil, M.Sc. Thesis, UFRGS (i Portuguese). Sorese, J.., Kroo, I.B. ad Faber, M.H., 1994, Optimal reliabilitybased code calibratio, Structural Safety, Vol. 15, pp Souza Jr., A.C. de, 2009, Applicatio of Reliability i Calibratio of Partial Safety Factors for Brazilia Structural esig Codes, Masters Thesis, Escola de Egeharia de São Carlos, USP (i Portuguese). Stucchi, F.R. ad Satos, S.H.C., 2007, Reliability based compariso betwee ACI ad NBR6118. Revista IBRACON de Estruturas, Vol. 3, No.2, pp Turkstra, C.J. ad Madse, H.O., 1980, Load combiatios icodified structural desig, Joural of the Structural ivisio, ASCE, Vol. 116, No. ST12, pp Vrouwevelder, A.C.W.M., 2002, evelopmets towards full probabilistic desig codes, Structural Safety, Vol. 24, pp J. of the Braz. Soc. of Mech. Sci. & Eg. Copyright 2010 by ABCM April-Jue 2010, Vol. XXXII, No. 2 / 127