Robustness theoretical framework

Transcription

1 Jont Workshop of COST Actons TU0601 and E55 September , Lublana, Slovena Robustness theoretcal framework John D. Sørensen Aalborg Unversty, Denmark Enrco Rzzuto Unversty of Genoa, Italy Mchael H. Faber ETH Zurch, Swtzerland Summary More frequent use of advanced types of structures wth lmted redundancy and serous consequences n case of falure combned wth ncreased requrements to effcency n desgn and executon followed by ncreased rsk of human errors has made the need of requrements to robustness of new structures essental. Further, the collapse of the World Trade Centre towers and a number of collapses of structural systems durng the last 10 years has ncreased the nterest n robustness. Typcally modern structural desgn codes requre that the consequence of damages to structures should not be dsproportonal to the causes of the damages. However, despte the mportance of robustness for structural desgn such requrements are not substantated n more detal, nor have the engneerng professon been able to agree on an nterpretaton of robustness whch facltates for ts quantfcaton. The an of ths fact sheet s to descrbe a theoretcal and rsk based framework to form the bass for quantfcaton of robustness and for pre-normatve gudelnes. Keywords Robustness of structures, robustness ndcators. Background / Introducton Robustness of structures has been recognzed as a desrable property because of a several hgh system falures, such as the Ronan Pont Buldng n 1968, where the consequences were deemed unacceptable relatve to the ntatng damage. After the collapse of the World Trade Centre, robustness has obtaned a renewed nterest, prmarly because of the serous consequences related to falure of advanced types of structures and that consequences due to structural collapse may exceed the mere rebuldng costs by orders of magntudes. Further, t was confrmed that robustness s strongly related to nternal structural characterstcs such as redundancy, ductlty and ont behavor characterstcs, but also that the consequences of structural collapse strongly depend on the specfc scenaro of events startng wth some trggerng event over a complex seres of ntermedate events nvolvng 27

2 Robustness theoretcal framework more localzed damages whch fnally lead to the collapse. In ths scenaro the extent to whch consequences are generated depend not only on nternal structural characterstcs but may even more pronounced depend on passve and actve measures for damage reducton as well as possble non-conformtes wth desgn assumptons due to the qualty of executon and or mantenance. In order to mnmze the lkelhood of falures as those mentoned above many modern buldng codes consder the need for robustness n structures and provde strateges and methods to obtan robustness. In fact, n all modern buldng codes, one can fnd a statement (n ths or a slghtly dfferent form): total damage resultng from an acton should not be dsproportonal to the ntal damage caused by ths acton. Durng the last decades there have been sgnfcant efforts to quantfy aspects of robustness. When modellng robustness, system effects are very mportant. However, the prmary crtera n buldng code are related to desgn and verfcaton of suffcent relablty of components. It should also be noted that redundancy n systems s closely related to robustness. In prncple redundant system are beleved to be more robust than non-redundant systems but ths s not always the case as llustrated by the falures of the Ballerup Super Arena and the Bad Rechenhall Icehall, see (Früwald et al. 2007) and (Wnter & Kreuznger 2008). In the Ballerup Super Arena collapse two out of 12 man trusses faled. The trusses n the prmary load bearng system were made of glued lamnated tmber by a new, nnovatve desgn. A human error n desgn of the onts of the trusses was the man reason for the collapse. The transverse purlns (secondary system) were desgned n such a way that progressve collapse of the whole roof should not occur n case of falure of a sngle man truss. The roof system can therefore be consdered as a robust system n the sense that the whole roof dd not collapse. Ths seems to be a good strategy n case of desgn/human errors occurrng n many places / onts (hgh correlaton) for new, unconventonal structures. In the Bad Rechenhall Icehall, the total roof collapsed progressvely startng n one end of the arena. The prmary structural system conssts of very hgh box-grder beams wth no prevous experence. The secondary system was relatvely stff mplyng that the roof could be consdered as a parallel system. Desgn, executon and operatonal errors n all man beams mpled that the load bearng capacty was sgnfcantly lower than requred and the roof collapsed wth a snow load about ½ of the desgn snow load. The roof system can therefore not be consdered as a robust system n the sense that the whole roof collapsed. It seems not to be a good strategy to use a parallel system when desgn/human errors occur n many places / beams (hgh correlaton). Methodology Robustness s related to scenaros where exposures ncludng unntentonal and unforeseen loads and defects result n local damage to the structural system, and where ths damage may lead to further collapse of the structure. 28

3 Robustness theoretcal framework Fgure 1: Illustraton of the basc concepts n robustness (Eurocode EN ). An llustraton s presented n Fgure 1 (from Eurocode EN ). Due to an exposure (a) of any knd, local damage (b) may occur. Ths local damage s defned as the drect consequence of the exposure. Gven ths local damage the structure may survve or (a substantal part) may collapse. Robustness requrements are especally related to step from b) to c),.e. to avod that a local damage develops to total collapse. Durng the last decades there has been a sgnfcant effort to develop methods to assess robustness and to quantfy aspects of robustness. The basc and most general approach s to use a rsk analyss where both probabltes and consequences are taken nto account. Approaches to defne a robustness ndex can be dvded n the followng levels wth decreasng complexty: A rsk-based robustness ndex based on a complete rsk analyss where the consequences are dvded n drect and ndrect rsks A probablstc robustness ndex based on probabltes of falure of the structural system for an undamaged structure and a damaged structure A determnstc robustness ndex based on structural measures, e.g. pushover load bearng capacty of an undamaged structure and a damaged structure Fgure 2: An event tree for robustness quantfcaton, (Baker et al. 2008). Fgure 2 presents the same dea as n fgure 1 n a more general way n the form of an event tree. The assessment starts wth the consderaton and modellng of exposures (EX) that can cause damage to the components of the structural system. The term exposures refers to extreme values of desgn loads, accdental loads and deteroraton processes but also ncludes human errors n the desgn, executon and use of the structure. The term damage refers to reduced performance or falure of ndvdual components of the structural system. 29

4 Robustness theoretcal framework After the exposure event occurs, the components of the structural system ether reman n an undamaged state ( D ) as before or change to a damage state (D). Each damage state can then ether lead to the falure of the structure (F) or no falure ( F ). Consequences are assocated wth each of the possble damage and falure scenaros, and are classfed as ether drect (C dr ) or ndrect (C nd ). Drect consequences are consdered to result from damage states of ndvdual component(s). Indrect consequences are ncurred due to loss of system functonalty or falure and can be attrbuted to lack of robustness (Baker et al. 2008) and (JCSS 2008). The basc framework for rsk analyss s based on the followng equaton wth rsk contrbutons from local damages (drect consequences) and comprehensve damages (follow-up / ndrect consequences), are added, see (Baker et al. 2008) and (JCSS 2008): R = C where P ( D E ) P( EX ) + C P( S D EX ) P( D EX ) P( EX ) dr, nd, k k (1) k C dr, C nd, consequence (cost) of damage (local falure) D due to exposure EX consequence (cost) of comprehensve damages (follow-up / ndrect) S k gven local damage D due to exposure EX P(EX) probablty of exposure EX P(D EX ) probablty of damage D gven exposure EX P(Sk...) probablty of comprehensve damages S k gven local damage D due to exposure EX The optmal desgn (decson) s the one mnmzng the sum of costs of mtgatng measures and the total rsk R. A detaled descrpton of the theoretcal bass for rsk analyss can be found n (JCSS 2008). It s noted that an mportant step n the rsk analyss s to defne the system and the system boundares. The total probablty of comprehensve damages / collapse assocated to (1) s: ( ) = P( collapse D EX ) P( D EX ) P( EX ) P collapse (2) where P( ) local damage collapse D EX s the probablty of collapse (comprehensve damage) gven D due to exposure damage (collapse) s ncluded n (2). EX. Note that compared to (1) only one comprehensve For damages related to key elements the probablty of collapse s P ( collapse D EX ) 1. From equaton (2) t s obvous that the probablty of collapse can be reduced by: Reducng one or more of the probabltes of exposures P(EX )- preventon of exposure or event control 30

5 Robustness theoretcal framework Reducng one or more of the probabltes of damages P(D EX ) - related to element/component behavour Reducng one or more of the probabltes P( collapse D EX ) If the consequences are ncluded n a rsk analyss then also reducton of drect (local) consequences, C dr, and comprehensve (ndrect) consequences, C nd, are mportant. Accordng to the descrpton above and the robustness defnton n (EN1990:2002), robustness s manly related to the reducton of the probablty P(collapse D EX ). Increasng the robustness at the desgn stage wll n many cases only ncrease the cost of the structural system margnally the key pont s often to use a reasonable combnaton of a sutable structural system and materals wth a ductle behavour. In other cases ncreased robustness wll nfluence the cost of the structural system. Rsk-based robustness ndex (Baker et al. 2008) proposed a defnton of a robustness ndex. The approach dvdes consequences nto drect consequences assocated wth local component damage (that mght be consdered proportonal to the ntatng damage) and ndrect consequences assocated wth subsequent system falure (that mght be consdered dsproportonal to the ntatng damage). An ndex s formulated by comparng the rsk assocated wth drect and ndrect consequences. The ndex of robustness ( I rob ) s defned as I rob Dr = (3) R Dr R + R Ind where R Dr and R Ind are the drect and ndrect rsks assocated wth the frst and the second term n equaton (1). The ndex takes values between zero and one, wth larger values ndcatng larger robustness. As mentoned above the optmal decson s the one whch mnmzes the total rsk obtaned by equaton (1). Ths could equally well be by reducng the frst or the second term n equaton (1). Ths mples that the defnton of a robustness ndex by equaton (3) s not always fully consstent wth a full rsk analyss, but should be consdered as a helpful ndcator based on rsk analyss prncples. It s noted that snce the drect rsks typcally are related to code based lmt states they can generally be estmated wth hgher accuracy than the ndrect rsks. Relablty-based robustness ndex (Frangopol & Curley 1987) and (Fu & Frangopol 1990) proposed some probablstc measures related to structural redundancy whch also ndcates the level of robustness. A redundancy ndex (RI) s defned by: RI Pf (damaged) Pf (ntact) = (4) P f (ntact) where P f (damaged) s the probablty of falure for a damaged structural system and P f (ntact) s the probablty of falure of an ntact structural system. The redundancy ndex provdes a 31

6 Robustness theoretcal framework measure on the robustness / redundancy of the structural system. The ndex takes values between zero and nfnty, wth smaller values ndcatng larger robustness. They also consdered the followng related redundancy factor: β = β ntact R (5) βntact βdamaged where β ntact s the relablty ndex of the ntact structural system and β damaged s the relablty ndex of the damaged structural system. The ndex takes values between zero and nfnty, wth larger values ndcatng larger robustness. Determnstc robustness ndex A smple and practcal measure of structural redundancy (and robustness) used n the offshore ndustry s based on the so-called RIF value (Resdual Influence Factor), (ISO ). A Reserve Strength Rato (RSR) s defned as: R c RSR = (6) Sc where R c denotes characterstc value of the base shear capacty of an offshore platform (typcally a steel acket) and S c s the desgn load correspondng to ultmate collapse. In order to measure the effect of full damage (or loss of functonalty) of structural member no on the structural capacty the so-called RIF value (sometmes referred to as the Damaged Strength Rato) s defned by: RSRfal, RIF = (7) RSR ntact where RSR ntact s the RIF-value of the ntact structure and RSR fal, s the RIF-value of the structure where member no s faled/removed. The RIF takes values between zero and one, wth larger values ndcatng larger robustness. Other smple measures of robustness have been proposed based on e.g. the determnant of the stffness matrx of structure wth and wthout removal of elements. 32

7 Robustness theoretcal framework Robustness n codes of practce Fgure 3: Code based desgn. In many codes of practce as e.g. the Eurocodes the prmary desgn requrements are related to checkng that each component / element / connecton has suffcent relablty. A suffcent relablty level s secured by usng characterstc values and partal safety factors calbrated to a relablty level whch typcally correspond to an annual probablty of falure of the order However, addtonal requrements / measures are needed to secure that the structure also as a system has suffcent relablty. Further, provsons are needed to reduce / elmnate the effect of desgn errors, executon errors, unexpected deteroraton of components, etc. Robustness requrements n codes of practce should cover these aspects together wth qualty control systems and applcaton of best practces n desgn, executon and operaton & mantenance as llustrated n fgure 3. It s noted that many codes of practce contan some robustness rules, e.g. requrements to te together concrete elements, but the rules / provsons are not formulated n a consstent way on a ratonal bass. In countres where structures are desgned for sesmc loads the requrements to obtan earthquake resstant structures nclude many of the same aspects as those consdered good for robustness, e.g. redundancy and ductlty. References Baker, J.W., Schubert M., Faber, M.H On the assessment of robustness. Journal of Structural Safety, vol. 30, pp EN 1990:2002. Bass of structural desgn. EN :2006. Actons on structures - Part 1-7: General actons - Accdental actons. ISO 19902:2007. Petroleum and Natural Gas Industres Fxed Steel Offshore Structures. 33

8 Robustness theoretcal framework Frangopol D.M., Curley J.P Effects of damage and redundancy on structural relablty. ASCE Journal of Structural Engneerng, 113(7), Früwald, E., Serrano, E., Toratt, T., Emlsson, A., Thelandersson, S Desgn of safe tmber structures - How can we learn from structural falures n concrete, steel and tmber?. Report TVBK Lund Unversty. Fu G., Frangopol D.M., Balancng weght, system relablty and redundancy n a multobectve optmzaton framework. Structural Safety, 7(2 4), Jont Commttee on Structural Safety (JCSS) Rsk Assessment n Engneerng Prncples, System Representaton & Rsk Crtera. JCSS Publcaton, Schubert, M., Faber, M.H On the modelng and analyss of robustness of systems. Proceedngs EM08, Inaugural Internatonal Conference of the Engneerng Mechancs Insttute, Mnneapols, USA, May Wnter, S., Kreuznger, H., The Bad Rechenhall ce-arena collapse and the necessary consequences for wde span tmber structures. Proceedngs WCTE 2008 Conference 2008, Myazak, Japan. 34