International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Environmental contours for determination of seismic design response spectra Loth, Christophe; Baker, Jack W. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Environmental Contours for Determination of Seismic DesignResponse SpectraChristophe LothModeler, Risk Management Solutions, Newark, USAJack W. BakerAssociate Professor, Dept. of Civil and Env. Eng., Stanford University, Stanford, USAABSTRACT: This paper presents a procedure for using environmental contours and structural reliabilitydesign points for the purpose of deriving seismic design response spectra for use in structural engineeringdesign checks. The proposed approach utilizes a vector-valued probabilistic seismic hazard analysis tocharacterize the multivariate distribution of spectral accelerations at multiple periods that may be seen at agiven location, and a limit state function to predict failure of the considered system under a given level ofshaking. A reliability assessment is then performed to identify the design point - the response spectrumwith the highest probability of causing structural failure. This is proposed as the spectrum for whichengineering design checks can be performed to evaluate performance of a given structure. While thefull structural reliability analysis would not be performed in any practical application, this analysis doesprovide three major insights into appropriate response spectra to use in engineering evaluations. First,when the structure’s limit state function is dependent on a spectral acceleration at a single period, thisapproach produces risk-targeted spectral accelerations consistent with those recently adopted in severalbuilding codes. Second, the design point spectrum can be approximated by a Conditional Mean Spectrum(conditioned at the spectral period most closely related to the structure’s failure). This motivates recentproposals to use the Conditional Mean Spectrum for engineering design checks. Third, the design pointspectrum will vary depending upon the structural limit state of interest, meaning that multiple ConditionalMean Spectra will be needed in practical analysis cases where multiple engineering checks are performed(though this can be avoided, at the expense of conservatism, by using a uniform risk spectrum). With theabove three observations, this work thus adds theoretical support for several recent advances in seismichazard characterization.This paper develops a procedure for using envi-ronmental contours and structural reliability designpoints to formulate improved design spectra for usein assessing the performance of buildings underearthquakes. Most seismic building codes and de-sign guidelines are based on implicit performancegoals that structures should achieve. Despite thesignificant uncertainty in future ground motion oc-currence, building codes commonly check a struc-ture’s behavior under a single level of earthquakeloading, quantified with a design spectrum. How-ever, this explicit design check is often not definedwith respect to the performance goals.The objective of this paper is to provide the linkbetween the explicit design check and the implicitperformance goals. Using structural reliability ap-proaches, with environmental contours of spectralaccelerations at multiple periods, a justification ofthe use of multiple conditional mean spectra (CMS)(Baker, 2011) for design checks is detailed. Morespecifically, it is shown that exceedance levels ofparticular engineering demand parameters (EDP)can be estimated with a small number of structuralanalyses based on the calibrated CMS.1. DESCRIPTION OF THE PROBLEMFor a structure located a given site, we intend to es-timate the value ed p f , the level of EDP exceeded112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015with a given annual rate ν f . This is an importantquantity in the context of structural performanceassessments, where we might want structures toachieve performance goals such as an annual ex-ceedance rate of a given ed pallowable being less thanν f . In this case, we would simply need to verify thated p f is less than ed pallowable, this inequality beingseen as an explicit design check of the performancegoal. The estimation of ed p f may be achieved byconducting Probabilistic Seismic Demand Analy-sis (PSDA) (Shome and Cornell, 1999), which pro-vides the exceedance rates of all EDP levels basedon a large number of structural analyses at varyingground motion intensity levels. However, this ap-proach may require too much computational effortfor the current purpose. Alternatively, we want todetermine one target response spectrum that can beused in a structural analysis to more directly deter-mine the value ed p f .We quantify the ground motion at a given sitewith a set of spectral accelerations Sa at vari-ous periods T1,...,Tn, and denote the vector X =[Sa(T1), ...,Sa(Tn)]. The EDP of interest corre-sponds here to an implicit function of X, with ad-ditional variability corresponding to the structuralresponse uncertainty given X. Section 2 will firstexamine how to quantify the joint distribution of Xand how to use this information to estimate EDPexceedance levels. While this first approach willinitially assume EDP to be an explicit function ofX, two key simplifications will be introduced insection 3 to consider the general case of EDP asan unknown function of X. A brief explanation ofthe incorporation of structural response uncertaintyis shown in section 4, and a performance assess-ment of a tall structure using the calibrated CMS isdescribed in section 5.2. ENVIRONMENTAL CONTOURS WITHVECTOR-VALUED SEISMIC HAZARD2.1. Quantification of the seismic hazardWhen considering a single spectral accelerationSa(T ) at period T , accounting for the aggregationof various possible earthquake scenarios is clas-sically achieved by conducting Probabilistic Seis-mic Hazard Analysis (PSHA) (McGuire, 2004).The main result of such an approach is a hazardcurve, which provides MRESa(T )(x), the mean rateof Sa(T ) exceeding the value x. Using the MRE, wemay also define a mean rate density (MRD) by dif-ferentiating the MRE and taking the absolute value.Either the MRD or the MRE can be used to quan-tify the rate of occurrence of spectral accelerationvalues within a specified interval.In this paper, we are interested in using thejoint hazard associated with a vector of spectralaccelerations at multiple periods. The joint dis-tribution of a vector of spectral accelerations X =[Sa(T1), ...,Sa(Tn)] can be obtained using Vector-valued PSHA (VPSHA) (Bazzurro and Cornell,2002). Similar to scalar PSHA, we can character-ize this joint distribution with a multivariate MRDof the vector X. For instance, in the case of n =2 periods, the mean occurrence rate ν(Sa(T1) ∈[b11,b12],Sa(T2) ∈ [b21,b22]) of events where bothb11 ≤ Sa(T1) ≤ b12 and b21 ≤ Sa(T2) ≤ b22 can bedetermined as:ν(Sa(T1) ∈ [b11,b12],Sa(T2) ∈ [b21,b22])=∫ b22b21∫ b12b11MRDSa(T1),Sa(T2)(x1,x2)dx1dx2 (1)An exact computation of a multivariate MRDmay not be tractable when considering many spec-tral acceleration periods (typically, for n ≥ 4), orwhen there are many possible earthquake sources.Therefore, a simpler method has been proposed andreferred to as an indirect approach to VPSHA (Baz-zurro et al., 2010), where the results from a singleSa deaggregation (Bazzurro and Cornell, 1999) arejointly used with a marginal hazard curve in orderto obtain the desired MRD with a reduced computa-tional effort. For example, a two-period MRD willbe given as:MRDSa(T1),Sa(T2)(x1,x2)≈MRDSa(T1)(x1)∫∫fSa(T2)|Sa(T1),M,R(x2|x1,m,r)×fM,R|Sa(T1)(m,r|x1)dmdr (2)where fM,R|Sa(T1)(m,r|x1) is the joint probabilitydensity function (pdf) of magnitude and distancegiven occurrence of Sa(T1) = x1, obtained fromdeaggregation, and fSa(T2)|Sa(T1),M,R(x2|x1,m,r) is212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015the conditional pdf of Sa(T2) given Sa(T1), m andr. This pdf can be evaluated using the joint log-normality of [Sa(T1),Sa(T2)] for a given magnitudeand distance.It should be noted that this formula is not sym-metric in Sa(T1) and Sa(T2), unlike the exact com-putation from the direct approach. In Equation 2,Sa(T1) is the conditioning variable, while Sa(T2)is the conditioned variable. This indirect approachformulation introduces small errors in the marginaldistribution of the conditioned variable, but themarginal distribution of the conditioning variable ispreserved.Example of 2-IM MRD calculationIn this section, we show an example of VPSHAresults carried out at a site in Berkeley (latitude:37.87o, longitude: −122.29o, average shear wavevelocity in the top 30 meters Vs30 = 300 m/s), us-ing the USGS model for seismic sources (Petersenet al., 2008) and the Chiou and Youngs (2008)ground motion prediction equation. Events withmoment magnitudes ranging from 4.5 to 8.5 andsource-to-site distances from 0 to 200 km were con-sidered. VPSHA code from Barbosa (2011) hasbeen used to compute the joint mean rate densityof spectral accelerations at periods T1 = 1s andT2 = 0.3s. Figure 1 shows the contours of the ob-tained joint MRD with Sa(1s) as the conditioningvariable.2.2. Incorporation of VPSHA distributions in astructural reliability frameworkGiven this hazard information, we now considera structural response function of spectral acceler-ations at periods T1 = 1s and T2 = 0.3s:EDP =√α1Sa(T1)2 +α2Sa(T2)2 (3)Note that this EDP functional form was previouslyused as an example in Loth and Baker (2014). Here,we retain the values α1 = 0.75 and α2 = 0.25.For a given ν f , we seek the value of ed p f(the EDP level exceeded with rate ν f ), using acalibrated response spectrum, which will here becharacterized by a particular vector value for X,[Sa∗(T1),Sa∗(T2)], also referred to as the “designpoint”. In particular, from a structural reliabilityperspective, this design point should correspond tothe most likely realization of X yielding the de-sired EDP value. Using the joint mean rate densityof [Sa(T1),Sa(T2)] from VPSHA, we can directlyquantify EDP exceedence rates with numerical in-tegration. More precisely, we define the associatedfailure function as:g(X) = ed p f −√α1Sa(T1)2 +α2Sa(T2)2 (4)with X = [Sa(T1),Sa(T2)]. Note that by definition,the function g(X) must verify:ν f =∫∫g(X)≤0MRDSa(T1),Sa(T2)(x1,x2)dx1dx2 (5)This equation is used to find the value for ed p f ,with a trial and error approach.Using the obtained value for ed p f , we can alsocompute the failure contour defined as the set of[Sa(T1),Sa(T2)] values producing g(X) = 0. Theassociated design point [Sa∗(T1),Sa∗(T2)], by def-inition the most likely set of spectral accelerationvalues causing “failure”, is the point having thehighest mean rate density on the failure contour:[Sa∗(T1),Sa∗(T2)]= argmaxg(X)=0MRDSa(T1),Sa(T2)(x1,x2)(6)This solution can be determined using a simple trialand error approach, by searching all g(X) = 0 val-ues. This type of optimization problem is a particu-lar example of environmental contours (Haver andWinterstein, 2009), which provide reliability-baseddemand exceedance levels.Figure 1 shows the failure function and designpoint for the example problem for a ν f value of0.0004 yr−1 (equivalent to an occurrence probabil-ity of 2% in 50 years). The shaded region repre-sents the area where√α1Sa(T1)2 +α2Sa(T2)2 >ed p f . The obtained value for ed p f is 1.691, and thecorresponding design point is [Sa∗(T1),Sa∗(T2)] =[1.760g,1.464g].While this approach can be used for any type ofEDP functional form, an analytical equation willgenerally not be known in practice (i.e., it will resultfrom structural analysis software). The next section312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 1: Two-period failure function g(X) and associ-ated design point [Sa∗(T1), Sa∗(T2)]will show how to remediate to this issue by a sim-plification of the failure function. Another simpli-fication of the joint distribution of X will also bedetailed.3. PROPOSED SIMPLIFICATIONSIn the previous section, we illustrated how to useenvironmental contours with a joint distribution ofspectral accelerations to estimate a seismic demandlevel exceeded with a particular annual rate. Thisapproach is limited to cases where EDP is an ex-plicit function of X. The first simplification we pro-pose will allow us to address cases where the EDPfunctional form is unknown (only spectral acceler-ation periods of interest are known). The secondsimplification will discuss a simplified model forthe joint spectral acceleration hazard, correspond-ing to a simplified version of VPSHA that iden-tifies multiple single event scenarios. The use ofthese single event scenarios should provide compa-rable seismic demands as the ones obtained usingthe VPSHA joint spectral acceleration hazard.3.1. Simplification 1: Single period failure func-tionThe first simplification we propose is to replaceg with single period failure failure functions gi,i ∈ {1, ...,n}, which depend only on Sa(Ti). In thecontext of the two-period EDP example from Equa-tion 3, we define:{g1(X) = x1 f −Sa(T1)g2(X) = x2 f −Sa(T2)(7)where x1 f and x2 f are constants obtained by target-ing the same rate ν(gi ≤ 0) = ν f for i ∈ {1,2}. Thecorresponding design points are denoted:[Sa∗i (T1),Sa∗i (T2)]= argmaxgi(X)=0MRDSa(T1),Sa(T2)(x1,x2)(8)As detailed in Loth and Baker (2014), we may useg1 and g2 to approximate the initial failure func-tion g. We then evaluate seismic demands based onEquation 3 with the obtained single period designpoints:ed pi =√α1Sa∗i (T1)2 +α2Sa∗i (T2)2 (9)and obtain the following approximation:ed p f ≈max{ed p1,ed p2} (10)Figure 2 depicts the single period examples.Figure 2: Simplification 1: single period failurefunctions and associated design points; (a): g1,[Sa∗1(T1),Sa∗1(T2)]; (b):g2, [Sa∗2(T1),Sa∗2(T2)]This approximation is critical as the problem cannow be solved with no knowledge of the actualEDP functional form, by using scalar PSHA. Forinstance, g1(X) = 0 corresponds to Sa(T1) = x1 fwhere x1 f is the spectral acceleration value ex-ceeded with rate ν f , obtained with the hazard curvefor Sa(T1) (i.e., x1 f will satisfy MRESa(T1)(x1 f ) =ν f ). Similarly, using MRESa(T2), g2(X) = 0 cantheoretically be determined as Sa(T2) = x2 f whereMRESa(T2)(x2 f ) = ν f . The point [x1 f ,x2 f ], which412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015can be seen as a UHS point associated with the ex-ceedence rate ν f , is shown in Figure 2. We ob-serve that while the UHS point lies precisely onthe g1(X) = 0 contour (i.e., Sa∗1(T1) = x1 f , Figure2a), it is slightly above the g2(X) = 0 contour (i.e.,Sa∗2(T2) < x2 f , Figure 2b). This error originatesfrom the fact that in the indirect approach (Equation2) used here, Sa(T1) is the conditioning variable,while Sa(T2) is the conditioned variable. Therefore,the marginal distribution of Sa(T2) is approximatewhereas the marginal distribution of Sa(T1) is pre-served. It should be noted that these discrepanciesare also due to the binning of the various randomvariables (Sa, M, R). The quantification of thesevarious sources of error has been studied by Baz-zurro et al. (2010).3.2. Simplification 2: Single event scenariosThe second simplification complements the first byconsidering an approximation of the MRD fromVPSHA with multiple joint lognormal distribu-tions, corresponding to MRD’s from single eventscenarios.3.2.1. ObjectiveWhile using numerical integration with environ-mental contours provides fairly accurate results, itis rather complicated to carry out. Ideally, wewould like to be able to conduct reliability calcu-lations with an analytical representation of the jointMRD from VPSHA, such as the approach detailedin Loth and Baker (2014). A first potential solutionwould be to transform the full joint MRD from VP-SHA to the standard normal space (see AppendixB.2. in Melchers, 1999). This approach is rathercomplex and may generate significant approxima-tion in the computation of the transformed joint dis-tribution. However, for the present applications, thejoint distribution is only needed at high amplitudeSa values associated with structural failures. Forsuch Sa values, we might be able to fit a joint log-normal distribution to the VPSHA contours, whichwill further simplify the estimation of seismic de-mands. In this section, we show how to approxi-mate the joint MRD from VPHSA by multiple sin-gle event MRD’s.3.2.2. Single event approximations based ondeaggregationThe proposed fitting approach is based on deaggre-gation results. First, we obtain the Sa(T1) valuex1 f exceeded with a given rate ν f , and conductdeaggregation at this exceedance level to obtain themean magnitude M1 and distance R1. Let us denote“event 1” the earthquake scenario with magnitudeM1 and distance R1, associated with a rate of occur-rence ν1. The value for ν1 is determined such thatthe single event distribution associated with event 1provides the same exceedance rate ν f for x1 f . Thiscan be achieved by using the integration method de-scribed in the previous section to the single eventMRD of event 1. This results in fixing ν1 as the ratioof ν f divided by the probability of Sa(T1) exceed-ing x1 f given the occurrence of event 1, which canbe computed from the chosen ground motion pre-diction equation. We obtain M1 = 6.98, R1 = 4 km,ν1 = 0.0103 yr−1. Similarly, we define “event 2”,based on the deaggregation results for Sa(T2) at theamplitude x2 f . Event 2’s characteristics are M2 =6.64, R2 = 6 km, and ν2 = 0.0389 yr−1. Theseevents represent an approximation of the full earth-quake hazard (accounting for all sources) at the siteof interest. Figure 3 shows contours correspondingto the demand of Equation 3 with the rate of ex-ceedance ν f = 0.0004 yr−1.Figure 3: Simplification 2: single event distributionfitting; (a): event 1; (b): event 2.3.3. Practical implications of the two simplifica-tions for seismic demand estimationsUsing both simplifications (i.e., the single periodfailure functions and the joint hazard from single512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015event scenarios), the computation of the seismic de-mand ed p f is further simplified by choosing thesingle period failure function corresponding to thespectral acceleration at which the single event MRDis fitted. In practice, this amounts to evaluatinged p f as the maximum of the seismic demands ob-tained with multiple CMS conditioned on the UHSamplitudes xi f . Using multiple CMS as described isthus implicitly accounting for the underlying jointspectral acceleration hazard.In the developed example, ed p1 is thus evaluatedusing the MRD from event 1 with the failure func-tion g1. Equivalently, ed p1 can be obtained usingthe CMS conditioned at 1s on x1 f based on event1’s magnitude and distance. Similarly, ed p2, eval-uated using the MRD from event 2 with the failurefunction g2, can be obtained using the CMS con-ditioned at 0.3s on x2 f based on event 2’s magni-tude and distance. Here, we obtain ed p1 = 1.617,and ed p2 = 1.459. An estimate of ed p f is finallycomputed by taking the maximum of the two seis-mic demands: we get ed p f ≈ 1.617, which is al-most equal to the value of 1.691 found in section2.2 using the numerical integration of the full VP-SHA distribution.4. INCORPORATION OF STRUCTURALRESPONSE UNCERTAINTYWe first discuss a theoretical approach to accountfor structural response uncertainty in the previouslydescribed environmental contours. We then showan adaptation of the calibration of the CMS, condi-tioned on risk-targeted spectral acceleration ampli-tudes.4.1. Theoretical considerationsIn order to incorporate the presence of structural re-sponse uncertainty, the considered EDP will gen-erally be expressed as a function of a level of re-sponse spectrum and a structural response uncer-tainty given that spectrum. EDP|X is now con-sidered as a lognormal random variable, with fixedlogarithmic standard deviation σ∆. This assumptionis consistent with commonly observed responsehistory analyses results (for instance, with drift de-mands (Shome and Cornell, 1999)). Due to thisadditional variability, we define a new type of per-formance goal to estimate ed p f , where we target aspecific level of performance given a fixed responsespectrum, characterized by X= xd:P(EDP > ed p f |X= xd) = pd (11)with pd a given probability. Target spectra arethen calibrated by solving Equation 11 for xd , us-ing the structural reliability techniques presented inthis section. More details of this approach can befound in Loth (2014).4.2. Adapted CMS calibration accounting forstructural response uncertaintyGiven ν f , pd , and σ∆, the calibration of the CMSconditioned at Ti will be based on the correspondingrisk-targeted spectral accelerations SaRT(Ti), whichcan be obtained by solving Equation 11 assumingEDP to be a function of only Sa(Ti). Note thatthis approach to compute a risk-targeted spectralacceleration is also suggested in section ofASCE 7-10 (ASCE/SEI, 2010).The next section will show an application of theuse of these calibrated CMS to the estimation ofEDP exceedance levels of a high rise structure.5. EXAMPLE & COMPARISONS5.1. Considered case studyIn this section, we summarize results from realis-tic performance assessments of a MDOF structurebased on the use of multiple CMS. The obtained re-sults are validated against PSDA results developedfor the same case study by Lin (2012).The considered structure is a 20-story reinforcedconcrete moment frame designed for the FEMAP695 project (2009), in which it has the ID 1020.The structural model accounts for cyclic and in-cycle strength deterioration as well as stiffness de-terioration. The building is assumed to be locatedat a real site in Palo Alto (latitude: 122.143o W,longitude: 37.442o N, shear wave velocity: Vs30 =400 m/s). It should be noted that this structure wasdesigned for a different site, located in NorthernLos Angeles (Haselton et al., 2010).We will consider two different EDP’s: the maxi-mum story drift ratio over all stories (SDR), and thepeak floor acceleration at the 15th story (PFA(15)).Given ν f = 0.001 yr−1, we estimate sdr f and612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015pfa(15)f as theoretically defined by the demand lev-els exceeded with rate ν f .5.2. ApproachThe procedure used in this example was proposedin Loth (2014). For a calibration at pd = 0.5, whichis retained here, it consists in the following steps:(1) Identify a set of periods T1, ...,Tn relevant tothe EDP of interest; (2) Obtain the hazard curvesfor Sa(Ti) at the site, and compute the risk-targetedSaRT(Ti) at each period Ti, for a chosen value forpd; (3) Compute the corresponding CMSi condi-tioned on SaRT(Ti); (4) Select and scale ground mo-tion records for each CMSi; (5) Conduct responsehistory analyses, and compute the geometric meansof responses associated with each CMS; (6) Esti-mate ed p f as the maximum of the geometric meansobtained with each CMS.5.3. ResultsFor both SDR and PFA(15), the same set of periodsof interest will be retained: the first three modalperiods T1 = 2.63s, T2 = 0.85s, T3 = 0.46s, andan elongated period of TL = 6.31s obtained from apushover analysis (see Chapter 5 in Loth (2014)).Three modal periods were selected here becausewe expect PFA(15) to have significant high modeparticipation. Enough modal periods should gen-erally be considered to account for sufficient massparticipation (for instance, ASCE/SEI (2010) rec-ommends 90% for response spectrum method), es-pecially when considering higher mode dominatedEDP’s. An elongated period was also included toaccount for the inelastic effect lengthening the pe-riod of the structure. The first four rows of Table1 show the obtained SDR and PFA(15) values forthe four considered CMS. We observe that the storydrift generally appears to be first mode dominated,as CMS conditioned at T1 leads to the highest de-mand (sdr f ≈ 1.8%), while the other CMS providelower values. In the case of PFA(15), the CMS lead-ing to the highest responses is the one conditionedon the third modal period T3 (pfa(15)f ≈ 0.40g). Thisis an expected result, as floor accelerations of highrise buildings are often driven by higher mode ef-fects.Table 1: Summary of the response history analyses re-sults (SDR and PFA(15)) for the EDP-based assessmentwith pd = 0.5.T [s] SDR[%] PFA(15)[g]0.45 0.9 0.40geomean 0.85 1.4 0.34from CMS at 2.63 1.8 0.266.31 1.5 0.25ed p f 1.8 0.405.4. Comparison with PSDALin’s PSDA data provides values for sdr f rangingfrom 1.6% to 1.9% 1, which is in good agreementwith our results (1.8%). Similarly, for PFA(15), ourestimate for pfa(15)f is 0.40g, whereas Lin’s esti-mates lie between 0.40g and 0.43g. This externalanalysis further confirms the accuracy of obtainedvalues for sdr f and pfa(15)f , and the adequacy of theproposed use of multiple CMS.6. CONCLUSIONSWe have first described the definition and computa-tion of joint mean rate density of spectral accelera-tions at multiple periods using vector-valued prob-abilistic seismic hazard analysis (VPSHA). Whilestandard PSHA provides occurrence rates of a spec-tral acceleration at a single period, VPSHA allowsthe quantification of seismic demands influenced byspectral accelerations at multiple periods. A re-liability assessment, based on environmental con-tours, of an example structure characterized witha known EDP function of spectral accelerations attwo distinct periods was shown, using a numericalintegration of the corresponding VPSHA distribu-tion. We then proposed two key simplifications: (1)an approximation of the multivariate failure func-tion with multiple univariate failure functions; (2)an approximation of the VPSHA distribution withthe use of multiple single event scenarios. The jointapplication of these two simplifications is shownto be equivalent to the use of multiple CMS con-ditioned at periods of interest for the considered1Lin obtained slightly different values when integratingnonlinear response history analysis results using ground mo-tion records selected from various target spectra.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015structural demand. The incorporation of structuralresponse uncertainty can be achieved by condition-ing the CMS on risk-targeted spectral amplitudes.Results from a performance assessment of a 20-story building using these CMS are shown.Future work may consist of analyzing jointMRD’s from VPSHA corresponding to differentsites with various seismic regimes, along with morecomplex structures and EDP’s. This might result inthe need to use a higher number of CMS. However,recent work (e.g., Carlton and Abrahamson, 2014;Loth, 2014) seems to indicate that a reduced num-ber of "broadened" CMS (i.e., CMS with increasedamplitudes over some period range) may be consid-ered instead, which will reduce the effort involvedwith this approach.7. ACKNOWLEDGEMENTSThis work was supported in part by the Na-tional Science Foundation under NSF grant num-ber CMMI 0952402. Any opinions, findings andconclusions or recommendations expressed in thismaterial are those of the authors and do not nec-essarily reflect the views of the National ScienceFoundation.8. REFERENCESASCE/SEI (2010). Minimum design loads for buildingsand other structures. ASCE 7-10; American Societyof Civil Engineers.Baker, J. W. (2011). “Conditional mean spectrum: Toolfor ground-motion selection.” Journal of StructuralEngineering, 137(3), 322–331.Barbosa, A. R. (2011). “Simplified vector-valued prob-abilistic seismic hazard and seismic demand anal-ysis of a 13-story reinforced concrete frame-wallbuilding.” Ph.D. thesis, University of California, SanDiego, California.Bazzurro, P. and Cornell, C. A. (1999). “Disaggrega-tion of seismic hazard.” Bulletin of the SeismologicalSociety of America, 89(2), 501–520.Bazzurro, P. and Cornell, C. A. (2002). “Vector-valuedprobabilistic seismic hazard analysis (VPSHA).” Pro-ceedings of the 7th US National Conference on Earth-quake Engineering, 21–25.Bazzurro, P., Park, J., and Tothong, P. (2010). “Vector-valued probabilistic seismic hazard analysis of corre-lated ground motion parameters.” Report No. USGSaward G09AP00135.Carlton, B. and Abrahamson, N. (2014). “Issues and ap-proaches for implementing conditional mean spectrain practice.” Bulletin of the Seismological Society ofAmerica.Chiou, B. S. J. and Youngs, R. R. (2008). “An NGAmodel for the average horizontal component of peakground motion and response spectra.” EarthquakeSpectra, 24(1), 173–215.Federal Emergency Management Administration, U. S.(2009). Quantification of Building Seismic Perfor-mance Factors, FEMA P695.Haselton, C. B., Liel, A. B., Deierlein, G. G., Dean,B. S., and Chou, J. H. (2010). “Seismic collapsesafety of reinforced concrete buildings. I: Assessmentof ductile moment frames.” Journal of Structural En-gineering, 137(4), 481–491.Haver, S. and Winterstein, S. R. (2009). “Environmentalcontour lines: A method for estimating long term ex-tremes by a short term analysis.” Transactions of theSociety of Naval Architects and Marine Engineers,116, 116–127.Lin, T. (2012). “Advancement of hazard-consistentground motion selection methodology.” Ph.D. thesis,Stanford University, California.Loth, C. (2014). “Multivariate ground motion intensitymeasure models, and implications for structural relia-bility assessment.” Ph.D. thesis, Stanford University,California.Loth, C. and Baker, J. W. (2014). “Rational designspectra for structural reliability assessment using theresponse spectrum method.” Earthquake Spectra (inpress).McGuire, R. K. (2004). Seismic hazard and risk analy-sis. Earthquake Engineering Research Institute.Melchers, R. E. (1999). Structural reliability analysisand prediction. John Wiley New York.Petersen, M. D., Frankel, A. D., Harmsen, S. C.,Mueller, C. S., Haller, K. M., Wheeler, R. L., Wesson,R. L., Zeng, Y., Boyd, O. S., Perkins, D. M., et al.(2008). “Documentation for the 2008 update of theUnited States national seismic hazard maps.” ReportNo. 2008-1128, U.S. Geological Survey Open-File.Shome, N. and Cornell, C. A. (1999). Probabilistic seis-mic demand analysis of nonlinear structures. Dept. ofCivil and Environmental Engineering, Stanford Univ.,Stanford, Calif.8


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