Safety ad Security Egieerig IV 161 The target reliability ad desig workig life M. Holický Kloker Istitute, CTU i Prague, Czech Republic Abstract Desig workig life ad target reliability levels recommeded i various atioal ad iteratioal documets are icosistet. Idicative values of desig workig life are withi a rage from 10 to 100 years for differet types of structures, recommeded target reliability idexes are usually give for oe or two referece periods (1 year ad 50 years), without a explicit lik to the desig workig life. The cotributio attempts to clarify the relatioship betwee the desig workig life ad the reliability idex ad to provide guidace for specificatio of the target reliability level for give cosequeces, desigs workig life ad discout rate. The theoretical study based o probabilistic optimizatio is supplemeted by practical recommedatios. It appears that the optimum reliability idexes deped primarily o the ratio of cost of structural failure (malfuctioig costs) ad the cost per uit of structural parameter, less sigificatly o the desig workig life ad discout rate. Keywords: desig workig life, target reliability, optimisatio. 1 Itroductio Desig workig life is uderstood as a assumed period of time for which a structure is to be used for its iteded purpose without ay major repair beig ecessary. Idicative values of desig workig life (10 to 100 years for differet types of structures) are give i EN 1990 [1]. Recommeded target reliability idexes are give for two referece periods (1 year ad 50 years), without ay explicit lik to the desig workig life (see Table 1). It should be uderlied that a couple of values (for 1 year ad 50 years) give i Table 1 for each reliability class correspods to the same reliability level. Practical applicatio of these values, however, depeds o the time period cosidered i the verificatio, which may be liked to available probabilistic doi:10.2495/safe110151
162 Safety ad Security Egieerig IV iformatio cocerig time variat basic variables (imposed load, wid, earthquake, etc.). Table 1: Reliability classificatio i accordace with EN 1990 [1]. Reliability Cosequeces Reliability idex Examples of buildigs classes of structural for referece period ad civil egieerig failure 1 year 50 years works RC3 high High 5,2 4,3 Bridges, public buildigs RC2 Medium 4,7 3,8 Resideces ad offices ormal RC1 low Low 4,2 3,3 Agricultural buildigs For example, cosiderig a structure of reliability class 2 ad the desig workig life 50 years, the reliability idex = 3,8 should be used provided that probabilistic models of basic variables are available for this period. The same reliability level is achieved whe the referece period 1 year ad = 4,7 are applied usig the theoretical models for oe year. A more detail recommedatio cocerig is provided by ISO 2394 [2] where the target reliability idexes are idicated for the whole desig workig life (without ay limitatio) ad related ot oly to the cosequeces but also to the relative costs of safety measures (see Table 2). Table 2: Target reliability idexes (life-time, examples) i accordace with ISO 2394 [2]. Relative costs of Cosequeces of failure safety measures small some moderate great High 0 1,5 2,3 3,1 Moderate 1,3 2,3 3,1 3,8 Low 2,3 3,1 3,8 4,3 Similar recommedatio is provided i the JCSS Probabilistic model code [3] (Table 3). Recommeded target reliability idexes are also related to both the cosequeces ad to the relative costs of safety measures, however for the referece period 1 year. The cosequeces classes i [3] (similar to those i EN 1990 [1]) are liked to the ratio ρ defied as the ratio betwee the total costs (cost of costructio plus direct failure costs) ad costructio costs as follows: - Class 1 Mior Cosequeces: ρ is less tha approximately 2; risk to life, give a failure, is small to egligible ad ecoomic cosequeces are small or egligible (e.g. agricultural structures, silos, masts); - Class 2 Moderate Cosequeces: ρ is betwee 2 ad 5; risk to life, give a failure, is medium or ecoomic cosequeces are
Safety ad Security Egieerig IV 163 cosiderable (e.g. office buildigs, idustrial buildigs, apartmet buildigs); - Class 3 Large Cosequeces: ρ is betwee 5 ad 10; risk to life, give a failure, is high, or ecoomic cosequeces are sigificat (e.g. mai bridges, theaters, hospitals, high rise buildigs). Table 3: Tetative target reliability idexes (ad associated target failure rates) related to oe year referece period ad ultimate limit states i accordace with JCSS [3]. Relative costs of safety measures Mior cosequeces of failure Moderate cosequeces of failure Large cosequeces of failure Large = 3,1 (p 10 3 ) = 3,3 (p 510 4 ) = 3,7 (p 10 4 ) Normal = 3,7 (p 10 4 ) = 4,2 (p 10 5 ) = 4,4 (p 510 6 ) Small = 4,2 (p 10 5 ) =4,4 (p 510 6 ) = 4,7 (p 10 6 ) Both documets [2] ad [3] seem to recommed the reliability idexes lower tha those give i EN 1990 [1] eve for the small relative costs of safety measures. It should be oted that EN 1990 [1] gives the reliability idexes for two referece periods 1 ad 50 years that may be accepted as the desig workig life for commo structures (see also discussio provided i [4]). ISO 2394 [2] recommeds idexes for life-time, examples, thus related to the desig workig life ad Probabilistic Model Code [3] provides reliability idexes for the referece period of 1 year. However, a clear lik betwee the desig workig life ad the target reliability level is ot apparet from ay of the above metioed documets. Thus, it is ot clear what the target reliability idex should be used for a give desig workig life differet from 50 years (say 10 years). The basic aim of this cotributio is to clarify the lik betwee the desig workig life ad the reliability idex ad to provide guidace for specificatio of the target reliability level for a give desig workig life. Submitted theoretical study based o probabilistic optimizatio is supplemeted by practical recommedatios. 2 Geeral priciples of probabilistic optimizatio Probabilistic optimizatio is based o fudametal form of the objective fuctio expressed as the total cost C tot (x,q,) C tot (x,q,) = C f Pf ( x,i) Q( q,i) + C 0 + x C 1 (1) i1 Here x deotes a decisio parameter of the optimizatio (a parameter of structural resistace), q is aual discout rate (e.g. 0,03, a average log ru value of the real aual discout rate i Europea coutries), the umber of years of a cosidered desig workig life (e.g. 50, 100), P f (x,i) failure probability
164 Safety ad Security Egieerig IV at the year i, C f malfuctioig costs (due to loss of structural utility), Q(q,i) discout factor depedet o the aual discout rate q ad the umber of years i, C 0 iitial cost idepedet of decisio parameter x, ad C 1 cost per uit of the decisio parameter x. Note that the desig workig life is cosidered here as a give determiistic quatity characterized by the umber of years. I reality the workig life for a give desig may be a radom quatity depedig o social ad physical factors. The desig itself may aim at some optimum. This optio of radom desig workig life is, however, eglected i this study. Assumig almost idepedet failure evets i subsequet years, the aual probability of failure P f (x,i) at the year i is give by the geometric sequece P f (x,i) = p(x) (1 p(x)) i 1 (2) where p(x) deotes the iitial probability of failure that is depedet o the decisive parameter of structural resistace x. Note that aual failure probabilities ca be assumed to be idepedet whe failure probabilities are domiatly iflueced by time-variat loads (climatic actios, traffic loads). The the failure probability P f (x) durig years ca be estimated by the sum of the sequece P f (x,i) give as P f (x) = 1 (1 p(x)) p(x) (3) Note that the approximatio idicated i equatio (3) is acceptable for small probability p(x) < 10 3. The discout factor of the expected future costs at the year i is cosidered i a usual form as Q(q,i) = 1 / (1+q) i (4) Thus, the cost of malfuctioig C f is discouted by the factor Q(q,i) depedig o the discout rate q ad the poit i time (umber of year i) whe the loss of structural utility occurs. Cosiderig equatios (2) ad (4) the total costs C tot (x,q,) described by equatio (1) may be writte as C tot (x,q,) = C f p(x) PQ (x,q,) + C 0 + x C 1 (5) Here the total sum of expected malfuctio costs durig the period of years is depedet o the product of the preset value of malfuctio cost C f, aual probability p(x) ad a sum of the geometric sequece havig the quotiet (1 p(x))/(1+ q), deoted as time factor PQ(x,q,): (1 p( x)) 1 (1 q) PQ(x,q,) (6) (1 p( x)) 1 (1 q) I geeral the total cost C tot (x,q,) depeds o the costs C 0, C 1, C f, aual probability of failure p(x), discout rate q ad o the umber of years. Note
Safety ad Security Egieerig IV 165 that for small probabilities of failure p(x) (for appropriate structural parameter x) ad small discout rate q, the time factor PQ(x,q,). The ecessary coditio for the miimum of the total cost follows from (1) as thus C tot ( x, q, ) C x f i1 Pf ( x, i) Q( q, i) x xxopt C 1 0 Pf ( x, i) C1 Q( q, i) x (8) C i 1 xx f opt Equatio (6) represets a geeral form of the ecessary coditio for the miimum of total cost C tot (x,q,) ad the optimum value x opt of the parameter x ad the optimum aual probability of failure p opt = p(x opt ). The optimum probability for the total desig workig life T d = years follows from equatio (2) as P f,opt = 1 (1 p opt ) p opt (9) The correspodig optimum reliability idex opt = -1 (P f,opt ). These quatities are i geeral depedet o the cost ratio C f /C 1, discout rate q ad umber of years. 3 Failure probability of a geeric structural member Cosider a geeric structural member described by the limit state fuctio Z(x) = x f (G+Q) (10) Here x deotes a determiistic structural parameter (say cross-sectio area), f stregth of the material, G appropriate load effect due to permaet load ad Q load effect due to variable load. Theoretical models of the radom quatities f, G ad Q cosidered i the followig example are give i Table 4. (7) Table 4: Theoretical models of the radom variables f, G ad Q (aual extremes). Variables Distributio The mea Stadard deviatio Coef. of variatio f Logormal 100 10 0,10 G Normal 40 4 0,10 Q Gumbel 10 5 0,50 Cosiderig the theoretical models give i Table 4, the reliability margi Z(x) may be (coservatively) approximated by a ormal distributio (the coefficiet of skewess is aroud 0,1 oly). The aual failure probability p(x) is the give as p(x) = Z(x) (Z(x) = 0) (11)
166 Safety ad Security Egieerig IV where Z (Z(x) = 0) deotes the ormal distributio of the reliability margi Z(x) for Z(x) = 0. 4 A example The followig example illustrates the geeral priciples ad a special case of probabilistic optimizatio. To simplify the aalysis the total costs C tot (x,q,) give by equatio (1) are trasformed to the stadardized form tot (x,q,) as C tot (x,q,) = tot ( x, q, ) C0 p( x) PQ( x, q, ) + x C 1 / C f (12) Cf Obviously, both the costs C tot (x,q,) ad tot (x,q,) achieve the miimum for the same parameter x opt. It is assumed that the discout rate is q = 0,03 ad the desig workig life is = 50 years. Uder these assumptios Figure 1 shows variatio of the total stadardized costs tot (x,q,) (give by equatio (12)), ad the optimum reliability idex opt correspodig to the probability P f,opt (give by equatio (9)), with structural parameter x for selected costs ratio C f /C 1. The optimum values x opt (q,) of the structural parameter x are idicated by the dotted vertical lies. x opt 1.4 1.3 C f / C 1 = 100000 1.2 1.1 1 10000 1000 0.9 0 20 40 60 80 100 Figure 1: Variatio of the total stadardized costs tot (x,q,) ad the reliability idex with structural parameter x for q =0,03, = 50 ad selected costs ratios C f /C 1. Figure 2 shows variatio of the optimum structural parameter x opt (q,) with the costs ratio C f /C 1, agai for q =0,03, =50.
Safety ad Security Egieerig IV 167 0.012 tot (x,q,) 7 0.01 C f / C 1 = 10 6 0.008 5 0.006 0.004 C f / C 1 =200 4 3 0.002 C f / C 1 =1000 C 1 / C f = 100000 0 0.8 0.9 1 1.1 1.2 1.3 x 2 1 Figure 2: Variatio of the optimum structural parameter x opt with the life time for selected costs ratios C f /C 1, ad the discout rate q =0,03. 5 The optimum reliability idex The optimum reliability idex opt (q,,c f /C 1 ) depeds o the discout rate q, desig workig life ad the cost ratio C f /C 1. However, the idex opt is primarily depedet o the cost ratio C f /C 1, ad its depedece o the discout rate q ad the desig workig life seems to be isigificat. This is well illustrated by Figure 3 that shows variatio of the optimum reliability idex opt with the cost ratio C f /C 1 for selected desig workig life = 1, 50, 100, ad the discout rate q = 0,03. Figure 4 shows variatio of the optimum reliability idex opt with the cost ratio C f /C 1 for discout rates q = 0,01, 0,03 ad 0,05 ad for the desig workig life = 50. It follows from Figures 3 ad 4 that the optimum reliability idex opt slightly decreases with icreasig workig life ad icreasig discout rate q. Figure 5 shows cotour lies of the optimum reliability idex opt as a fuctio of the desig workig life ad logarithm of the cost ratio log(c f /C 1 ) for q = 0.03. It ca be operatioally used to assess the optimum reliability idex opt for give ad the cost ratio C f /C 1. For example, for = 50 ad the cost ratio C f /C 1 = 10000 (log(c f /C 1 ) = 4), the optimum reliability idex opt 4,3. The optimum opt is primarily depedet o the cost ratio C 1 /C f, less sigificatly o the desig workig life ad the discout rate q.
168 Safety ad Security Egieerig IV 6 opt 5 4 = 1 5 0 100 3 2 1 1 10 100 1 10 3 1 10 4 1 10 5 1 10 6 C f / C 1 Figure 3: Variatio of the optimum reliability idex opt with the cost ratio C f /C 1 for selected desig workig life = 1, 50, 100, ad the discout rate q = 0,03. 6 opt 5 4 q = 0.01 0.03 0.05 3 2 1 1 10 100 1 10 3 1 10 4 1 10 5 1 10 6 C f / C 1 Figure 4: Variatio of the optimum reliability idex opt with the cost ratio C f /C 1 for selected discout rates q = 0.01, 0.03, 0.05, ad the desig workig life = 50.
Safety ad Security Egieerig IV 169 -log(c f /C 1 ) opt Figure 5: Cotour lies of the optimum reliability idex opt as a fuctio of the desig workig life ad the cost ratio C 1 /C f for the discout rate q = 0,03. It should be metioed that the target reliability idex t ca ot be chose as the optimum reliability idex opt whe the cost ratio C f /C 1 is ukow or difficult to assess. The a coservative value assessed for reasoable lower bouds of the desig workig life (say 50 years) ad the discout rate (say 0,02) ca be used. 6 Coclusios ad recommedatios Preset documets icludig codes for structural desig provide o clear lik betwee the desig workig life ad the target reliability level ad o recommedatios are offered to specify the target reliability idex for a give desig workig life differet from 50 years (say 10 years). Probabilistic optimizatio may provide valuable backgroud iformatio for specificatio of the target probability of failure or the reliability idexes. It appears that the optimum reliability idexes deped o: - the ratio of cost of structural failure (malfuctioig costs) ad cost per uit of structural parameter, - the desig workig life, - discout rate.
170 Safety ad Security Egieerig IV Results obtaied from the aalyzed example idicate more specific coclusios, validity of which should be coditioed by the accepted assumptios cocerig the objective fuctio ad aual failure probability. It appears that with icreasig malfuctioig cost, the optimum reliability idex ad the optimum structural parameter icrease (Figures 1 ad 2). The desig workig life seems to have a very limited ifluece o the optimum reliability (Figure 3). Eve less sigificat seems to be effect of the discout rate (Figure 4). For practical purposes the optimum target reliability idex ad the correspodig structural parameter ca be well assessed usig Figure 5 cosiderig reasoable lower bouds for the desig workig life (say 50 years) ad the discout rate (say 0,02). Available experiece idicates that applicatios of the optimizatio approach i practice should be primarily based o properly formulated objective fuctios, ad o credible estimates for the cost per uit of structural parameter ad the cost of structural failure (malfuctioig costs). The results of this study ca be implemeted i practical codified desig based o the Eurocodes as follows: the characteristic values of the basic variables icludig time varyig loads (wid, sow etc) may remai idepedet of the desig workig life; the desig values are specified o the bases of appropriate reliability idex assessed for give cost ratio (of malfuctioig costs ad cost per uit of the structural parameter), desig workig life ad discout rate; the partial factors are determied cosiderig specified desig values ad uchaged characteristic values of basic variables. Alteratively the partial factors may be kept uchaged ad the characteristic values may be adjusted to achieve appropriate desig value of the basic variables. However, this approach seems to be less suitable for time ivariat variables. It should be stated that further ivestigatios are plaed to aalyze the importat aspects of reliability differetiatio takig ito accout cosequeces, desig workig life ad discout rate ad to illustrate implemetatio of achieved results i practical desig. Ackowledgemet This study is a outcome of the research project GAČR 103//09/0693. Refereces [1] EN 1990: 2002. Eurocode - Basis of structural desig. CEN/TC 250, 2002. [2] IS 2394: 1998. Geeral priciples o reliability for structures. ISO, 1998. [3] JCSS: 2001. Probabilistic Model Code. http://www.jcss.ethz.ch/, 2001. [4] Dimitris Diamatidis: Reliability differetiatio. I.: Holicky et al.: Guidebook 1, Load effects o Buildigs, CTU i Prague 2009.