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

Topology optimization for buildings in seismic zones within a PBEE framework Bobby, Sarah; Spence, Seymour M. J.; Kareem, Ahsan Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Topology Optimization for Buildings in Seismic Zones within aPBEE FrameworkSarah BobbyGraduate Student, Dept. of Civil and Environmental Engineering and Earth Sciences,Univ. of Notre Dame, Notre Dame, USASeymour M.J. SpenceAssistant Professor, Dept. of Civil Engineering, Univ. of Michigan, Ann Arbor, USAAhsan KareemRobert M. Moran Professor of Engineering, Dept. of Civil and EnvironmentalEngineering and Earth Sciences, Univ. of Notre Dame, Notre Dame, USAABSTRACT: This paper presents a probabilistic performance-based topology optimization frameworkfor the conceptual design of uncertain building systems in seismic zones. The stochastic nature of theground motions is rigorously considered in a simulation-based probabilistic performance assessmentframework that allows for the definition of a novel decoupling technique that efficiently separates theprobabilistic analysis from the optimization loop. In particular, the methodology is based on the construc-tion of a series of approximate sub-problems with simplified governing equations, which convenientlyallows their evaluation using established techniques for static, deterministic topology optimization prob-lems. The applicability and efficiency of the method is demonstrated on a case study.1. INTRODUCTIONTopology optimization techniques have recentlybeen recognized as a powerful tool for conceptualbuilding design (Kareem et al., 2013; Bobby et al.,2014). Although originally developed in a deter-ministic setting, it is necessary to rigorously con-sider the effect of the uncertainties affecting theproblem (e.g. stochastic loading conditions, un-certainties in structural parameters, model idealiza-tion) to ensure satisfactory structural performanceas these techniques essentially obtain efficient de-signs by eliminating redundancies and pushing de-signs to their limiting capacities. Furthermore,it is of interest to formulate the problem withina setting that is already acceptable to the build-ing design community for straightforward integra-tion into the structural design process. Probabilis-tic Performance-Based Design (PPBD) techniqueshave recently gained momentum in the structuralengineering community for the design of struc-tures within a rigorous probabilistic framework,and therefore it is of interest to frame the topol-ogy optimization methodology within the setting ofPPBD.Within the context of topology optimization,Reliability-Based Topology Optimization (RBTO)methods are capable of explicitly accounting for theeffect of uncertainties during the optimization pro-cess using constraints written in terms of acceptablefailure probabilities. The majority of these meth-ods have focused on including time-invariant uncer-tainties in the topology optimization problem in anefficient manner, while few methods have thus farbeen developed for the incorporation of stochasticloading. Chun et al. (2013) accounts for stochas-tic seismic excitation using the instantaneous fail-ure probability as opposed to the more complex firstexcursion probability, which is typically used to de-scribe the reliability of structures. Although themethod described in Bobby et al. (2014) does con-sider the first excursion probability, the structuralperformance is described using fragility constraints112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015(conditional failure probabilities), which are lesscomplete than a full probabilistic analysis. Fur-thermore, only stationary stochastic excitation, i.e.wind loading, was considered. Thus there is a needfor a framework able to consider non-stationary ex-citation developed in a full PPBD setting.This research will present a framework forthe topology optimization of buildings in seismi-cally active zones that rigorously accounts for thestochastic nature of the ground accelerations andthe dynamic nature of the structural response withina PPBD setting. An efficient decoupling methodis presented in which the topology optimizationand probabilistic performance-based analysis aredecoupled through the definition of a number of ap-proximate sub-problems. Examples are presentedin order to illustrate the proposed methodology.2. PERFORMANCE-BASED DESIGNThe framework presented in this paper performstopology optimization within the PPBD setting pro-posed by the PEER Center (Porter, 2003). The par-ticular focus of this research is the estimate andcontrol of non-structural damage. It is therefore ofinterest to use the PEER methodology to estimatethe mean annual rate of exceedance of a damagemeasure as follows:λ (dm) =∫edp∫imG(dm|edp) · |dG(edp|im)|· |dλ (im)|(1)where λ (a) is the mean annual rate of exceedanceof event A = a, where capital letters indicate vari-ables and lower case letters indicate their realiza-tions, G(a|b) denotes the complementary cumu-lative distribution function of random variable Agiven B = b, dm denotes a damage measure indi-cating damage to non-structural components, edpis an engineering demand parameter characterizinga structural response, im is the measure of the inten-sity of the event, and |dλ (im)| can be interpreted asν |dG(im)| = ν · f (im)dim where f (a) is the proba-bility density function of a and ν is the mean annualrate of arrival of significant events (Der Kiureghian,2005). The various terms that appear in Eq. (1) canbe obtained through separate analyses: hazard anal-ysis for λ (im), structural analysis for G(edp|im),and damage analysis for G(dm|edp).3. THE PERFORMANCE-BASED TOPOL-OGY OPTIMIZATION PROBLEMThe probabilistic performance-based topology opti-mization problem of interest considers probabilisticperformance constraints written in terms of λ (dm).Assuming a design domain that is discretized usingfinite elements, the topology optimization problemof interest may be written as:minρV (ρ) =n∑e=1∫ΩeρedΩs.t. λ (dm j)−λ0 j ≤ 0, j = 1, . . . ,NM(ρ,U)z¨(t)+C(ρ ,U)z˙(t)+K(ρ ,U)z(t) = f(t;U,W)0 ≤ ρe ≤ 1, e = 1, ...,n(2)where ρ = {ρ1, . . . ,ρn}T is the element-wise nor-malized material density design variable vector, nis the total number of elements in the discretizedstructure, Ωe denotes the domain of element e, Vis the volume of material in the design domain,dm j and λ0 j are the damage threshold and targetmean annual exceedance rate for the jth probabilis-tic constraint, respectively, N is the total number ofperformance constraints, U is a vector containingthe uncertain model parameters, z denotes the dis-placement vector, M, C, and K denote the mass,damping, and stiffness matrices, respectively, and fis the seismic loading vector that depends on U aswell as a white-noise stochastic process, W.4. PROBABILISTIC PERFORMANCE AS-SESSMENT FRAMEWORKThis section describes the uncertainty models usedduring the assessment of the performance con-straints for the subtasks of hazard analysis, struc-tural analysis, and damage analysis and addition-ally defines the associated random variables, con-tained in U= {UTh ,UTs ,UTd }T .4.1. Hazard analysisA stochastic ground motion model, specifically thepoint-source model described in Boore (2003), will212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015be used to obtain a complete and detailed proba-bilistic description of the seismic hazard. In partic-ular, the ground acceleration is defined using a ra-diation spectrum, A( f ,Uh), describing the groundmotion’s frequency content, f , and an envelopefunction, et(t;Uh), describing the ground motion’svariation in time t. To generate a ground motionacceleration time history, z¨g(t), first a white noisesequence, w (lower case indicating a realization),is generated for the time duration of interest andmultiplied by et(t;uh). The resulting sequence istransformed into the frequency domain and normal-ized by the square root of the mean of the amplitudespectrum. The normalized spectrum is then mul-tiplied by A( f ,uh) and transformed back into thetime domain, resulting in a realization of a groundacceleration time history.The radiation spectrum is described as a prod-uct of the source spectrum E( f ;Uh), path effectP( f ;Uh), and site effect G( f ;Uh) as follows:A( f ;Uh) = (2pi f )2E( f ;Uh)P( f ;Uh)G( f ;Uh) (3)The source spectrum is given by:E( f ;Uh) = cMwS( f ;Uh) (4)where c is a constant, Mw is the seismic momentand is related to the moment magnitude M by therelationship log10Mw = 1.5(M+10.7) and S( f ;Uh)is the displacement source spectrum. The two-corner point-source displacement source spectrumdeveloped by Atkinson and Silva (2000) for the re-gion of California is used for this research:S( f ;Uh) =[1− e1+( f/ fa)2+ e1+( f/ fb)2](5)where the lower corner frequency fa, higher cor-ner frequency fb, and weighting parameter e are re-lated to M by the relationships log10 fa = 2.181−0.496M, log10 fb = 2.41− 0.408M, and log10 e =0.605−0.255M respectively.The path effect is defined as (Boore, 2003):P( f ;Uh) = Z(Rr)exp [−pi f Rr/(Q( f )cQ)] (6)where Z(Rr) and Q( f ) are the geometrical spread-ing and regional attenuation functions, respectively,cQ is the seismic velocity, and Rr =√h2 + r2 is theradial distance from the source to the site, wherer is the closest distance to the fault plane and h isa moment-dependent equivalent point-source depthand is related to M by the relationship log10 h =−0.05+0.15M (Atkinson and Silva, 2000).The site effect may be given as follows:G( f ;Uh) = D( f ;Uh)Am( f ) (7)where Am( f ) and D( f ;Uh) are the amplifica-tion and diminution functions, respectively. Thediminution may be accounted for using the κ0 fil-ter:D1( f ;Uh) = exp[−piκ0 f ] (8)or the fmax filter:D2( f ;Uh) =[1+( f/ fmax)8]−1/2(9)or a combination of both filters (Boore, 2003).The envelope function is given as follows:et(t;Uh) = at (t/tn)bt exp [−ctt/tn] (10)where the parameters at , bt , and ct are given by:bt = −λt ln(ηt)/(1+λt [ln(λt)−1]) (11)ct = bt/λt (12)at = (exp[1]/λt)bt (13)where tn = 2Tw is the time duration parame-ter, Tw = 12 fa + 0.05Rr is the duration of strongground motion (Boore, 2003; Atkinson and Silva,2000) while λt and ηt are parameters definingthe temporal envelope. The parameters Uh ={M,r, fa, fb,e,h,κ0, fmax,λt ,ηt ,Tw}T are modeledas uncertain using the distributions in Table 1.4.2. Structural analysisThe stochastic nature of the structural responsemust be considered during the structural analysis.This may be achieved by solving the first excursionof the engineering demand parameter response pro-cess, REDP, over the threshold edp during time pe-riod T , which is taken as the earthquake duration.In other words, the EDP of interest is the largestvalue of REDP occurring during T . It is, therefore,312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 1: Suggested distributions for parameters in U.Variable P1∗ P2∗∗ Distribution Ref.M 5lb 8ub Trunc. Exp. [1]r 30 km 0.40 Lognormal [1]fa ? 0.25 Lognormal [2]fb ? 0.25 Lognormal [2]e ? 0.25 Lognormal [2]h ? 0.25 Lognormal [2]κ0 0.02lb 0.04ub Uniform [1]fmax 25 Hz 0.25 Lognormal [2]λt 0.2 0.40 Lognormal [2]ηt 0.05 0.40 Lognormal [2]Tw ? 0.40 Lognormal [2]s 1 0.025 Normal [3]ωi § 0.01 Lognormal [4]ξi § 0.30 Lognormal [4]Cn 0† 1‡ Lognormal -∗ Median unless otherwise stated∗∗ C.o.V. unless otherwise statedlb Lower boundub Upper bound? Given by respective equation in text§ Mean value, given by structural model† Mean value‡ Standard deviation[1] Vetter and Taflanidis (2012)[2] Vetter and Taflanidis (2014)[3] Minciarelli et al. (2001)[4] Bashor et al. (2005)the estimate of this distribution that is of interestand although there is no closed-form solution forthis quantity it may be estimated using simulationtechniques.Assuming linear structural behavior during theearthquake event, REDP may be described by a loadeffect model using a vector of influence functionsand quasi-static loads. For a displacement-basedresponse this may be written as:REDP(t;ρ,Us,Uh,W) = sΓTR(ρ)F(t;ρ ,Us,Uh,W)(14)where s models the uncertainty in the transforma-tion of seismic effects into structural effects, ΓR isa vector of influence functions that, when multi-plied by the quasi-static loading vector, gives theresponse of interest, while F is the following vectorof quasi-static loads:F(t;ρ,Us,Us,W) =K(ρ)Φk(ρ)qk(t;ρ,Us,Uh,w)(15)where Φk = [φ1, . . . ,φk] is a matrix containing thestructure’s first k mode shape vectors and qk(t) ={q1(t), . . . ,qk(t)}T is a vector containing the struc-ture’s first k modal responses at time t. The dis-placement of the ith mode may be determined bysolving the following equation:q¨i(t)+2ξiωiq˙i(t)+ω2i qi(t) = φTi f(t;ρ ,Us,Uh,W)(16)where the damping ratio, ξi, and natural frequency,ωi, of the ith mode, respectively, are modeled asuncertain, and the external loading vector for theseismic hazard is given by:f(t;ρ ,Us,Uh,W) =−M(ρ)iz¨g(t;Us,Uh,W)mi(17)where z¨g is the ground motion acceleration time his-tory, i is the vector of earthquake influence coef-ficients, and mi is the ith modal mass. Suggesteddistributions for the components of the uncertainvector Us = {s,ω1, ...,ωk,ξ1, ...,ξk}T are given inTable 1. The distribution of EDP may then be esti-mated through simulation by taking, for each real-ization ofUs, Uh andW fed through the system, thelargest value of REDP to occur during the durationof the simulated event.4.3. Damage analysisFor the damage model, an intuitive damage mea-sure DM defined by the following demand-to-capacity ratio is considered (Jalayer et al., 2007):DM = EDP/C(Ud) (18)whereC is the capacity corresponding to the partic-ular EDP under consideration. A simple model forthe uncertain capacity is C = C¯(1+δCCn) where C¯is the nominal (mean) capacity associated with theresponse of interest, δC is the coefficient of varia-tion, andCn is the normalized capacity distribution.A suggested distribution for Ud = {Cn} is given inTable 1.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20154.4. Solution strategyIn order to estimate the integral of Eq. (1) whileconsidering the hazard, structural and damage mod-els defined in the previous sections, Monte Carlosimulation is used while writing the integral in thefollowing form (Spence and Kareem, 2014):λ (dm) = ν∫w∫uh∫us∫edpG(dm|edp)× p(edp|us,uh,w)p(us)p(uh)× p(w)dedp dus duh dw(19)where the components of the vectors Us, Uh andW are independent therefore allowing their jointprobability functions, p, to be simply written as theproducts of their marginal distributions.5. THE AVV SOLUTION STRATEGYThe solution of Eq. (2) is not trivial due to the pres-ence of the probabilistic performance constraints.As the design variable vector is typically of high di-mension it is advantageous to use efficient gradient-based algorithms in the solution strategy. However,the implicit dependence of the probabilistic con-straints on ρ and the large size of vector W makesthe solution of the problem, as written in Eq. (2),computationally impractical. Thus a simulation-based decoupling technique has been developed toefficiently solve the performance-based topologyoptimization problem of interest. To define the ap-proximate sub-problem it is convenient to re-writethe probabilistic constraints in Eq. (2) in their in-verse form:dm(λ0 j )j (ρ)−dm j ≤ 0 (20)where dm(λ0 j )j is the damage measure with targetmean annual exceedance rate λ0 j .5.1. Model descriptionA fine finite element discretization, though ineffi-cient during the reliability analysis, is necessary fora detailed topology design. In order to increasecomputational efficiency of the reliability analysis,a “reduced system” comprised of a limited num-ber of “master” degrees of freedom (DOFs) is usedto evaluate the dynamic response of the structure.The master DOFs are located on the floor systems,which are part of the underlying secondary sys-tem of the structure, which is classified as a “non-designable” domain and thus is not altered duringthe optimization. The mass of the system is mod-eled as lumped at the master DOFs and thus is lo-cated at the floor levels, which is a typical assump-tion for buildings. The loading vector f, definedin Eq. (17), thus has nonzero loads located onlyat these master DOFs. The “complete” system isobtained as the superposition of the design domainand secondary system. Using this classification, thedynamic response may be estimated in the masterDOFs as:M˜(ρ ,U) ¨˜z(t)+ C˜(ρ,U) ˙˜z(t)+ K˜(ρ,U)z˜(t) = f˜(t;U,W)(21)where M˜ is the reduced mass matrix and is obtainedby eliminating the rows and columns of M corre-sponding to the DOFs with zero mass, C˜ is the re-duced order damping matrix, z˜ is the displacementvector describing the displacement of the completesystem in the master DOFs, f˜ is the reduced exter-nal load vector obtained by eliminating the com-ponents of f corresponding to the DOFs with zeromass (which are, by definition, zero-valued), andK˜ is the reduced order stiffness matrix and can beestimated using static condensation.5.1.1. Approximate sub-problemTo formulate the approximate sub-problem, firstconsider the system defined in the current designpoint ρ0 and conditioned on a particular realizationof the random vectors U= u andW=w. The EDPfor one event may be represented as follows:edp j(ρ0,u,w) = REDP(tˆ;ρ0,u,w)= sΓTR(ρ0)F˜(tˆ;ρ0,u,w)= Γ˜TR j(ρ0)ϒ˜ j(ρ0,u,w)(22)where tˆ is the time instant at which the largest re-sponse of interest occurs and ϒ˜ j = sF˜(tˆ;ρ0,u,w).The value of the damage measure associated withthis edp may be obtained as follows:dm j(ρ0,u,w) = Γ˜TR j(ρ0)ψ˜ j(ρ0,u,w)/C¯ j (23)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where ψ˜ j = ϒ˜ j(ρ0,u,w)/(1+ δC jCn, j) is a vectorthat, when statically applied to the reduced system,will produce a response level that if divided by C¯ jyields the damage measure dm j.Considering the system unconditioned, the dam-age parameter with exceedance rate λ0 j may be rep-resented as:dm(λ0 j )j (ρ0) = µdm j(ρ0)+ηdm j(λ0 j ,ρ0)σdm j(ρ0)(24)where ηdm j is the reduced variate giving the num-ber of standard deviations, σdm j , by which the dam-age parameter with mean annual exceedance rateλ0 j exceeds the mean value, µdm j , of the damagemeasure. By observing that the mean value and thevariance, σ2dm j , of the damage measure may be ex-pressed as:µdm j(ρ0) = Γ˜TR j(ρ0)ψ˜ j(ρ0)/C¯ j (25)σ2dm j(ρ0) = Γ˜TR j(ρ0)Cψ˜ j(ρ0)Γ˜R j(ρ0)/C¯2j (26)where ψ˜ j andCψ˜ j are the mean and covariance ma-trix of the vectors ψ˜ j generated during the simu-lation, the damage measure with mean annual ex-ceedance rate λ0 j may be written as:dm(λ0 j )j (ρ0) = Γ˜TR j(ρ0)Ψ˜ j(λ0 j ,ρ0)/C¯ j (27)where Ψ˜ j is the AVV for the jth constraint and isgiven by:Ψ˜ j(λ0 j ,ρ0) = ψ˜ j(ρ0)+ηdm j(λ0 j ,ρ0)Cψ˜ j(ρ0)Γ˜R j(ρ0)σdm j(ρ0)C¯ j(28)The significance of Eq. (27) is that it allows dm(λ0 j )jto be calculated through the static application ofthe AVV to the master degrees of freedom of thenominal/mean system. In particular, the effectsof the uncertainties considered in the problem arecontained in the AVV, therefore the AVV providesa static and deterministic relationship between thenominal system and the damage measure with tar-get mean annual exceedance rate.The definition of the AVV for the reduced systemmay be used to define the following AVV for thecomplete system as follows:Ψij(λ0 j ,ρ0) ={Ψ˜kj(λ0 j ,ρ0), i = k ∈ Ξ0, i 6= k ∈ Ξ(29)where Ξ is the set of degrees of freedom definingthe master nodes while i = 1, . . . ,Ddo f where Ddo findicates the total number of degrees of freedom ofthe complete system. The inverse reliability con-straints given in Eq. (20) may be written using theAVV for the complete system as follows:ΓTR j(ρ0)Ψ j(λ0 j ,ρ0)/C¯ j −dm j ≤ 0 (30)or using the following equivalent formulation:ΛTj z j(ρ0)/C¯ j −dm j ≤ 0K(ρ0)z(ρ0) = Ψ j(λ0 j ,ρ0)(31)where Λ j =KΓR j is a vector of constants extractingthe displacement-based response to be constrained.The relationship given by Eq. (31) is exact for thecurrent design point. If, however, it is assumed thatthe vectors Ψ j are weakly dependent on the designvariables, the following approximate topology sub-problem may be defined:minρV (ρ) =n∑e=1∫ΩeρedΩs.t. ΛTj z(ρ)/C¯ j −dm j ≤ 0, j = 1, . . . ,NK(ρ)z(ρ) = Ψ j, j = 1, . . . ,N0 ≤ ρe ≤ 1(32)The definition of the approximate sub-problemnot only decouples the probabilistic analysis fromthe optimization but also allows the use of opti-mization algorithms typically used to solve static,deterministic topology optimization problems dueto the static nature of the governing equations. Asthis problem is approximate for changes in thedesign point, upon the convergence of the sub-problem the AVV must be updated for all con-straints and the sub-problem resolved. This se-quential solution procedure until convergence of612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015the AVVs and structural design guarantees an op-timum structure conforming to the reliability con-straints defined in the original optimization prob-lem of Eq. (2). The efficiency of the proceduremay be estimated by the number of design cycles,or updates and solutions of the sub-problem, thatare required for convergence.6. CASE STUDYThe case study presented in this paper considers theoptimum topology design of the bracing for a pla-nar lateral load resisting frame of a 3-story build-ing subject to earthquake excitation. The columnsand floor beams of the secondary system are mod-eled using W8x10 steel members and are indicatedby vertical and horizontal lines, respectively, in Fig.1(a). Q4 elements (dimensions 0.1m x 0.1m x 0.1mwith material properties of concrete) are used tomodel the design domain as indicated by the shadedgray area in Fig. 1(a). The initial structure had auniform design variable vector with ρ = 0.8. A lin-ear density filter with a radius of 0.3 m was usedand the Solid Isotropic Material with Penalization(SIMP) method was implemented with a penaltyp = 5 (Bendsøe, 1989). The Method of MovingAsymptotes (Svanberg, 1987) was used as the opti-mization algorithm for this research.Ground accelerations were generated using themethod described in Sec. 4.1 where the annualrate of arrival of significant events was taken asν = 2.927 for events of magnitude 5 ≤ M ≤ 8, asdetermined using data from Caltech (2014). Time-invariant uncertain parameters were modeled us-ing the distributions given in Table 1. The firstthree modes were used to calculate the structuralresponse, where the mean value of the ith modaldamping ratio was taken as ξi = 0.01 for i =1, . . . ,3. The response processes of interest REDPwere the inter-story drifts for each floor of the struc-ture. The nominal capacity indicating damage wasgiven as C¯ = h f loor/400 and δC = 0.01. The targetmean annual exceedance rates for all probabilisticconstraints were taken as λ0 j = 0.01 for a damagemeasure dm j = 1.The final bracing scheme for the case study isshown in Fig. 1(b). The volume history, given inFig. 2, shows rapid and steady convergence andattests to the efficiency of the method. The originalconstraints on the target mean annual failure ratesfor all floors were achieved for the final design, asindicated in Fig. 3. Finally, the AVV shows rapidand steady convergence, as shown in Fig. 4 for asample AVV on the 3rd floor of the structure. OtherAVVs showed similar convergence properties.(a) (b)h=15 mhfloor hfloor hfloor w=5 mFigure 1: (a) Design domain; (b) Final bracing scheme0 5 10 15 20 25 3001234567Design CycleV[m3 ]Figure 2: Volume History0 5 10 15 20 25 3000.0050.010.015Design Cyclemaxj[λj(dm=1)]Figure 3: Maximum annual exceedance rate for dm = 1over all constraints for each design cycle7. CONCLUSIONSThis paper presented a framework for the topol-ogy optimization of uncertain and stochastic build-712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20 25 300    1Design CycleΨjFigure 4: Convergence history of sample AVV: 3rd floorings within a probabilistic performance-based en-gineering setting. The stochastic nature of theseismic ground accelerations as well as additionaltime-invariant uncertainties describing the knowl-edge/state of the system were rigorously includedusing probabilistic performance constraints writtenin terms of the mean annual rate of exceedance ofa damage measure. A novel decoupling approachwas presented for the efficient solution of the prob-abilistic performance-based topology optimizationproblem in which the optimization loop and proba-bilistic analysis were decoupled using a series of se-quential approximate sub-problems defined using anumber of AVVs. The applicability and efficiencyof the proposed framework was illustrated for thebracing design of a 3-story building.8. ACKNOWLEDGMENTSSupport for this work was in part provided by theNSF Grant No. CMMI-1301008. The authors alsothank Krister Svanberg for providing the MMA al-gorithm, which was used as the optimization algo-rithm in this research.9. REFERENCESAtkinson, G. M. and Silva, W. (2000). “Stochastic mod-eling of California ground motions.” Bulletin of theSeismological Society of America, 90, 255–274.Bashor, R., Kijewski-Correa, T., and Kareem, A. (2005).“On the wind-induced response of tall buildings: theeffects of uncertainties in dynamic properties and hu-man comfort thresholds.” Proc. of the 10th AmericasConference on Wind Engineering.Bendsøe, M. (1989). “Optimal shape design as a mate-rial distribution problem.” Structural Optimization, 1,193–202.Bobby, S., Spence, S. M. J., Bernardini, E., and Kareem,A. (2014). “Performance-based topology optimiza-tion for wind-excited tall buildings: A framework.”Engineering Structures, 74, 242–255.Boore, D. M. (2003). “Simulation of ground motion us-ing the stochastic method.” Pure and Applied Geo-physics, 160, 635–676.Caltech (2014). Earthquake Catalog, URL: SouthernCalifornia Earthquake Data Center.Chun, J., Song, J., and Paulino, G. H. (2013). “Systemreliability based topology optimization of structuresunder stochastic excitations.” Proc. of 11th Interna-tional Conference on Structural Safety & Reliability.Der Kiureghian, A. (2005). “Non-ergodicity andPEER’s framework formula.” Earthquake Engng.Struct. Dyn., 34, 1643–1652.Jalayer, F., Franchin, P., and Pinto, P. E. (2007). “Ascalar damage measure for seismic reliability analy-sis of RC frames.” Earthquake Engng. Struct. Dyn.,36, 2059–2079.Kareem, A., Spence, S., Bernardini, E., Bobby, S., andWei, D. (2013). “Using computational fluid dynamicsto optimize tall building design.”CTBUH Journal, III,38–43.Minciarelli, F., Gioffrè, M., Grigoriu, M., and Simiu, E.(2001). “Estimates of extreme wind effects and windload factors: Influence of knowledge uncertainties.”Prob. Eng. Mech., 16, 331–340.Porter, K. A. (2003). “An overview of PEER’sperformance-based earthquake engineering method-ology.” Proc. of the Ninth International Conferenceon Applications of Statistics and Probability in CivilEngineering (ICASP9).Spence, S. M. J. and Kareem, A. (2014). “Performance-based design and optimization of uncertain wind-excited dynamic building systems.” Eng. Struct., 78,133–144.Svanberg, K. (1987). “The method of moving asymp-totes - a new method for structural optimization.” Int.J. Numer. Meth. Engng., 24, 359–373.Vetter, C. and Taflanidis, A. A. (2012). “Global sensitiv-ity analysis for stochastic ground motion modeling inseismic-risk assessment.” Soil Dynamics and Earth-quake Engineering, 38, 128–143.Vetter, C. and Taflanidis, A. A. (2014). “Comparison ofalternative stochastic ground motion models for seis-mic risk characterization.” Soil Dynamics and Earth-quake Engineering, 58, 48–65.8


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