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

The SL-AVV approach to system level reliability-based design optimization of large uncertain and stochastic… Spence, Seymour M. J. 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The SL-AVV Approach to System Level Reliability-Based DesignOptimization of Large Uncertain and Stochastic Dynamic SystemsSeymour M.J. SpenceAssistant Professor, Department of Civil Engineering and Environmental Engineering,University of Michigan, Ann Arbor, MI 48109, USAABSTRACT: Recently a number of efficient reliability-based design optimization methodologies havebeen proposed for optimizing uncertain dynamic systems subject to stochastic excitation. While thesemethods are capable of handling a large number of uncertain parameters, they are generally applicable toproblems characterized by small design variable vectors. This paper focuses on the development of a newreliability-based design optimization methodology for uncertain dynamic systems subject to stationarystochastic wind excitation that is capable of handling large design variable vectors, a characteristic ofmany practical design problems, while considering system level performance constraints.1. INTRODUCTIONReliability based design optimization (RBDO) isa powerful tool for obtaining structural systemsthat satisfy a number of probabilistic constraintsposed with the aim of ensuring a satisfactory per-formance of the system. The advantages of suchan approach over more classic deterministic op-timization strategies are well known. However,this approach is far more computationally involvedcompared to its deterministic counterpart. Thishas hindered the widespread adoption of RBDO,especially for large and dynamic structural sys-tems. This computational burden led to many ofthe early approaches to RBDO being based onclassic analytical approximations of the reliabil-ity integrals of the probabilistic constraints. Re-cent advances in computational capabilities as wellas achievements in the field of simulation-basedreliability assessment have spawned a new gen-eration of simulation-based RBDO strategies thatare capable of handling problems characterizedby large numbers of random variables, a typicalproperty of uncertain structural systems subject tostochastic excitation (Schuëller and Jensen, 2008;Valdebenito and Schuëller, 2010). This paper fo-cuses on the development of a novel simulation-based RBDO strategy aimed at solving problemsthat are not only characterized by a high number ofrandom variables, but also by a high-dimensionaldesign variable vector which, significantly com-plicates the situation (Valdebenito and Schuëller,2010). This class of problems is often encoun-tered in structural design optimization where alarge number of members are to be designed(Spence and Kareem, 2014), or in applications oftopology optimization where a high number ofdesign variables are required in order to providean adequate discretization of the design domain(Bobby et al., 2014). In particular, the method pro-posed in this work is based on the generalizationof a recently proposed component-level simulation-based RBDO strategy (Spence and Kareem, 2014;Spence and Gioffrè, 2012) to wind excited systemscharacterized by system level constraints.2. PROBLEM DEFINITIONThe RBDO problems of interest to this work my becast in the following form:Find x = {x1, ...,xm}T (1)to minimize W = f (x) (2)s. t. Pf (x) ≤ P0 (3)xk ∈ Xk k = 1, ...,m (4)where x is a high-dimensional design variable vec-tor containing the parameters that fully define the112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015state of the system, e.g. section sizes, W is the ma-terial weight of the structural system, Pf is the sys-tem level failure probability, P0 is the target systemlevel failure probability while Xi is the discrete setto which the kth design variable must belong.The RBDO problem outlined above is character-ized, for the problems that are of interest to thisstudy, by a high-dimensional design variable vectorx, a high-dimensional uncertain vectorU describingthe model uncertainty as well as a multi-variate sta-tionary stochastic process, F, describing the windexcitation. In order to describe damage, the follow-ing system level demand to capacity ratio will beconsidered:d(u, rˆ,x) = maxi=1,...,Nr{maxt∈[0,T ]|ri(t;u,x)|ci}(5)where Nr is the total number of components defin-ing the system response, T is the event duration,ri(t) are the component response processes, rˆ isthe vector collecting the largest values of ri to oc-cur during a given event of duration T while ci isthe capacity of the system. Obviously ci is charac-terized by a significant amount of uncertainty andwill therefore be modeled as a random variable inthis work. Under these circumstances, a predefineddamage state will occur if d is larger than 1. There-fore the limit state function of interest is the follow-ing:g(u, rˆ,x) = 1−d(u, rˆ,x) (6)while the failure probability of interest is:Pf (x) =P(g(u, rˆ,x) ≤ 0)=∫ ∫g(u,rˆ,x)60p(rˆ|u)p(u)drˆdu (7)where p indicates the conditional and non-conditional joint probability density functions ofthe uncertain vectors U and Rˆ. As indicated inEq. (7), the random Nr-dimensional vector Rˆ willin general depend on U.3. MECHANICAL MODELING3.1. ExcitationIt is common to describe the intensity of windstorms through the maximum wind speed, vH tooccur during the event at a height of interest H(e.g. building height) averaged over a fixed intervalT (e.g. an hour) while considering a site specificroughness length z0. Generally, wind speed data, v,is only available averaged over a period τ (often 3 s)and collected at a meteorological height Hmet at re-gional airports characterized by a roughness lengthz01. A probabilistic model for transforming this in-formation into site specific data is the following:vH(T,z0) = e7e3(τ,T )(e5z0e6z01)e4δln[H/(e5z0)]ln[Hmet/(e6z01)]e2e1v(τ,Hmet ,z01)(8)where e1 and e2 are random variables modeling ob-servational and sampling errors in v; e4, e5, ande6 are random variables modeling the uncertaintieswith respect to the actual values of the empiricalconstant δ = 0.0706 and of the roughness lengthsz0 and z01, respectively; e3(τ,T ) is the conversionfactor that accounts for the uncertainty in convert-ing between wind speed averaging times; while e7is a model uncertainty to be used in the case of hur-ricanes and tornadoes.The stochastic wind loads can be estimated di-rectly from wind tunnel tests carried out on rigidscale models. In particular, each realization of themulti-variate stationary stochastic process definingthe wind loads may be related to a scaled (lengthand time scales) realization of its wind tunnel coun-terpart, fw(t), through the expression:f(t;u) = w1w2w3(vHvHm)2fw(t) (9)where vHm is the simulated hazard intensity used inthe wind tunnel tests while w1, w2 and w3 are com-ponents of U and model the uncertainties associ-ated with the estimation of building aerodynamicsthrough the use of wind tunnels.3.2. Component ResponseIn this work it is assumed that the response process,ri(t), associated with the ith failure mode of the sys-tem may be written in the following form:r(t;u) = s1ΓTr[KΦnqrn(t)+ f(t)](10)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where Γr is a vector of influence functions givingthe response in r due to a unit load acting at eachdegree of freedom of the system, Φn = [φ1, ...,φn]is the mass normalized mode shape matrix of ordern, qrn(t) = {qr1(t), ...,qrn(t)}T is the vector of reso-nant modal displacement responses, K is the nom-inal (mean) stiffness matrix while s1 is a randomvariable modeling the epistemic uncertainty in theload effect model of Eq. (10).In Eq. (10) each component of qrn(t) is given bythe solution of the following uncertain modal equa-tion:q¨ j(t)+2s3 jζ js2 jω jq˙ j(t)+(s2 jω j)2 q j(t) = φTj f(t)(11)where q j, q˙ j and q¨ j are the jth generalized displace-ment, velocity and acceleration response, ω j is thenominal value of the jth circular frequency, s2 j isan uncertain parameter modeling the variability inthe estimate of ω j, while ζ j is the nominal valueof the generalized damping ratio with s3 j an uncer-tain parameter modeling the uncertainty that existsin the nominal value of ζ j.The jth component of qrn(t) is simply givenby qr j(t) = q j(t)− qb j(t), where the backgroundmodal displacement qb j is given by:qb j(t) =1(s2 jω j)2φTj f(t) (12)4. RELIABILITY PROBLEMAs outlined in Eq. (7), in order to calculate the fail-ure probability of the system, the conditional distri-butions of the largest values of the response func-tions ri(t) are needed. In particular, to this end, itis convenient to consider the following random re-duced variate:Ψˆi(u) =Rˆi(u)−µri(u)σri(u)(13)where µri and σri are the mean and standard devia-tion of the stationary response process ri(t) condi-tioned on u. For Gaussian systems (a typical char-acteristic of the stochastic response of multistorybuildings), the conditional distribution of ψˆi can beestimated from classic results of time-variant relia-bility:P(Ψˆi ≤ ψˆi|u) = [1−G0(ψˆi)]exp[−υ+(ψˆi,u)T1−G0(ψˆi)](14)where υ+(ψˆi,u) is the up-crossing rate of thenormalized threshold ψˆi conditional on u whileG0(ψˆi,u) is the probability that the system responseis above the threshold ψˆi at time equal to zero andis given by:G0(ψˆi) = exp[−ψˆi2](15)The conditional crossing rate may be estimated as:υ+(ψˆi,u) = κ(ψˆi,u)r+(ψˆi,u) (16)where r+ is given by the conditioned Rice formulaas:r+(ψˆi,u) =σr˙i(u)2piσri(u)G0(ψˆi) (17)with σr˙i the standard deviation of the derivative ofthe response process ri(t) while κ is the correc-tion factor accounting for any dependency betweensuccessive crossings of the normalized threshold ψˆiand can be modeled as:κ(ψˆi,u) = 1− exp[−(k(u))1.2(2pi)0.1ψˆi](18)where k is the spectral shape factor given by:k(u) =√2pi(1−γ21 (u)γ0(u)γ2(u))(19)where γp for p = 0,1,2 are the spectral moments ofri given by:γp(u) =∫∞0ωpSri(ω;u)dω (20)where Sri is the one-sided spectrum of ri while ω isthe circular frequency.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20154.1. Conditional Response StatisticsIn defining the distributions of the reduced vari-ates ψˆi, the conditional (on u) response statisticsare needed. To this end, the mean conditioned re-sponse, µri , is simply given by the expected valueof Eq. (10) given u and therefore by:µri(u) = s1ΓTri f¯ (21)where f¯ is the expected value of f. The second or-der conditioned response statistic, σ2ri , may be esti-mated in the frequency domain as:σ2ri(u) = σ2rib(u)+∫∞0Srrn (ω ;u)dω (22)where σ2rib is the background response variancewhile Srrn is the one-sided resonant response spec-trum estimated considering the participation of nvibration modes. The estimate of the backgroundcontribution to σ2r is given by:σ2rib(u) = s21ΓTriCf(u)Γri (23)where Cf(u) is the covariance matrix of the excita-tion f conditioned on u. The resonant response con-tribution to σ2ri can be efficiently estimated througha double modal spectral proper orthogonal de-composition (POD) of f(t) (Carassale et al., 2001;Spence and Kareem, 2013). Following this frame-work, Srrn is estimated as:Srrn (ω;u) =l∑i=1F˜i(ω;u)F˜∗i (ω;u) (24)where the symbol ∗ indicates the transposed com-plex conjugate, l is the number of spectral load-ing modes used in the estimation of Srrn while F˜iis given by:F˜i(ω ;u) = s1ΓTriKΦnHrn(ω ;u)ΦTn χ i(ω)√Λi(ω)(25)where χ i is the ith frequency dependent load-ing eigenvector, Λi is the corresponding ith fre-quency dependent loading eigenvalue while Hrn isthe diagonal resonant mechanical transfer function(Spence and Kareem, 2013).The other spectral moments, i.e. γ1 and γ2 =σ2r˙i , may be efficiently estimated through Eq. (20)where Sri is assessed in a strictly modal setting bysimply substituting into F˜i (Eq. 25) the classic me-chanical transfer function instead of Hrn .4.2. Solution StrategyIn defining an efficient solution strategy for the reli-ability integral Eq. (7), it is convenient to write theintegral in terms of the vector of reduced variates,Ψˆ = {ψˆ1, ..., ψˆNr}T :Pf (x) =∫ ∫g(u,ψˆ,x)60p(ψˆ|u)p(u)dψˆdu (26)For the problems of interest to this work, the aboveintegral will be of high dimensions therefore rul-ing out the use of classical reliability methods(Schuëller et al., 2003). In particular, in this workMonte Carlo simulation is adopted. In generatingconditional samples of the random vector Ψˆ, thecomponents are considered independent and there-fore fully described by their marginal distributionsgiven in Eq. (14). The grounds for making thisassumption lies in the fact that the dependency be-tween the components of Ψˆmodels the dependencybetween the peaks of the normalized response pro-cesses, which are only weakly dependent on uthrough the crossing rate. Therefore Ψˆ will be onlyweakly dependent on parameters such as the meanwind speed vH defining the intensity of the windevent, which significantly contributes to the depen-dency of the peaks of the various response functionsri(t).5. THE SYSTEM LEVEL RBDO APPROACHThis section presents a new system level RBDOalgorithm, SL-AVV, that leverages the recentlyintroduced concept of Auxiliary Variable Vector(Spence and Kareem, 2014).5.1. Problem DefinitionThe failure probability of Eq. (7) is a system fail-ure probability defined as the union of the followingcomponent level failure events:Fi ={|rˆi|ci> 1}i = 1, ...,Nr (27)412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Here the failure probability of Eq. (7) is first writtenin terms of the component failure modes as:Pf = P{Nr⋃i=1Fi}=Nr∑i=1P{Fi}−α(Nr∑i=1P{Fi})(28)where α is simply given by:α =(Nr−1∑i=1Nr∑j>iP{Fi∩Fj}−Nr−2∑i=1Nr−1∑j>iNr∑k> jP{Fi∩Fj ∩Fk}+ . . .− (−1)Nr−1P{Fi∩ ...∩FNr})/(Nr∑i=1P{Fi})(29)It is then assumed that the component failure prob-abilities can be approximated as exponential dis-tributions. This allows the failure probability tobe written as (where the dependency on the designvariables is now indicated):Pf (x) = (1−α(x))(Nr∑i=1exp[− 1µDi(x)])+∆(x)(30)where µDi is the expected value of the random com-ponent demand to capacity ratio given by Rˆi/Ciwhile ∆ is the error term introduced to account forthe assumption made on the component failure dis-tributions. If it is assumed that α and ∆ of Eq.(30) are independent of the design variable vectorx, then the dependency of the system level fail-ure probability on x is exclusively in terms of themean values of the random component demand tocapacity ratios Di. The problem therefore becomesthe description of the dependency of µDi on x. Tothis end the concept of AVV (Spence and Kareem,2014) can be leveraged.Before continuing, it should be observed that theassumption of independence of α from x is equiv-alent to assuming that the change in the probabilityof the joint occurrence of the various failure modesdue to a change in x, will in general follow the trendof the sum of the probabilities of the individual fail-ure events.5.2. The AVVsIn order to derive the aforementioned AVVs, it isfirst convenient to consider the following variablesdefined for each realization of U and Ψˆ and for thecurrent design variable vector x0:ϒi(u, ψˆi,x0) =ψˆi(u,x0)CL(u,x0)Γri(x0)σri(u,x0)(31)where CL is the covariance matrix of the followingvector:L(t;u,x0) = s1[KΦnqrn(t)+ f(t)](32)Dividing by the value of the capacity contained inu, the following static relationship is determined forthe damage ratio di associated with the realizationsof u and ψˆ :di(u,x0) =1ciΓTri(x0)ϒi(u,x0) = ΓTri(x0)ϒdi(u,x0)(33)where ϒdi is simply given by the ratio between ϒiand ci. The realizations of the vector ϒd j , generatedduring the simulation process used to estimate thereliability of the system, may be used to define thefollowing AVV:ϒ˜i(x0) = ϒ¯di(x0) (34)where ϒ¯di is the expected value of ϒdi .The significance of the AVV, and so of ϒ˜i, is thatif it is statically applied to the nominal structure itwill cause a response in ri that is equal to the ex-pected value of the damage ratio Di. In other wordsthe following relationship holds:µDi(x0) = ΓTri(x0)ϒ˜i(x0) (35)This relationship is particularly useful as it allowsthe failure probability of Eq. (30) to be written interms of what may be considered, for all intents andpurposes, a series of static load distributions.5.3. The Approximate SubproblemThe definition ϒ˜i together with the formulation ofthe failure probability given in Eq. (30) can be usedto define a approximate optimization subproblem,the sequential definition and solution of which will512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015lead to a final optimal solution. To this end, if theassumption of independence of α and ∆ from thedesign variable vector is extended to the AVV ϒ˜i,then the following optimization problem can be castthat is completely defined from the results of a sin-gle reliability analysis carried out in the current de-sign point x0:Find x = {x1, ...,xm}T (36)to minimize W = f (x) (37)s. t.(1−α)(Nr∑i=1exp[− 1ΓTri(x)ϒ˜i])+∆6 P0(38)xk ∈ Xk k = 1, ...,m (39)Not only does the subproblem outlined above de-couple the reliability problem from the optimiza-tion loop, but it also takes on an extremely conve-nient form as it can be easily made explicit in termsof x by simply defining an explicit expression of Γriin terms of x. In defining this explicit relationship, anumber of classic results can be used that were de-veloped for the optimization of deterministic struc-tures under static loads. In particular, here the solu-tion proposed by Chan et al. (1995) was adopted.Once the subproblem has been made explicit inx, any gradient-based optimization strategy can beused to find solutions. Here the pseudo-discrete op-timality criteria (Chan et al., 1995) is used, there-fore allowing the discrete nature of the design spaceto be fully considered.Once a solution is found to a subproblem definedin x0, a new sub-problem must be formulated in thedesign point found from the solution of the previoussub-problem. This updating procedure defines a de-sign cycle and will continue until the design pointof two successive design cycles are identical.6. CASE STUDY6.1. DescriptionThe case study consists of a 45-story rectangu-lar building with an offset core (Fig. 1a). Thecolumns consist of steel box sections and aregrouped in plan as indicated in Figure 1b (C1 toC18). In particular, it is required that the mid-line diameter of the box sections, Di, belong to theHV YX(a)180 mXY(b)BracingB1B2B2B3B3 B4B5B6C1C1C2C2C3C4C5C6C7C8C7C6C5C4C3C8C9 C10C11C12C13C12C11C10C9C17C18C17C15C14C14C15C16C1672 m 36 mDesignColumnFigure 1: (a) 3D view of the 45-story building; (b)structural layout of the building showing beam andcolumn assignments and the critical column.discrete set {0.2 m,0.25 m,0.3 m,0.35 m, ...,2 m}while the flange thickness is fixed at Di/20. Thecolumns are grouped over three consecutive floors.The beams, B1 to B6, are grouped in plan as in-dicated in Figure 1b and are required to belong tothe family of AISC W24 steel profiles. The beamsare also grouped three floors at a time (for a total of6× 15 groups). The diagonal bracings are also re-quired to belong to the AISC W24 steel profiles andare grouped as pairs over the height of the building.The aforementioned grouping results in 375 inde-pendent design variables. Initially the structure isdesigned with all columns having a mid-line diame-ter equal to 0.6 m while all beams and diagonals areset to W24×176 profiles. The first three structuralmodes, with mean circular frequencies ω1 = 1.023rad/s, ω2 = 1.118 rad/s and ω3 = 1.847 rad/s, areconsidered sufficient for describing the resonant re-sponse. The mean modal damping ratios were takenas 1.5 %. The uncertain parameters S1, S2 j and S3 jwith j = 1, ...,3 were modeled as independent log-normal random variables with coefficients of varia-tion 0.025, 0.015, and 0.3 respectively.Non-structural damage is to be controlled forwind blowing down the X direction (Fig. 1). In par-ticular, it is assumed that damage is associated withthe X and Y interstory drift response of the criti-cal column line illustrated in Fig. 1b. The capaci-ties,Ci, are taken as independent lognormal random612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015variables with mean 1/400 of the story height andcoefficient of variation 0.25.The building is considered to be located inthe Miami area of Florida, USA. The extremewind hazard is given by the risk of hurricanes. Inparticular, the site of the building is characterizedby a roughness length z0 = 2 m while an averagingtime of T = 3600 s was considered. The set of windspeeds used to defined the distribution of v (Eq.(8)) was obtained from the simulated hurricanedatabase of the National Institute of Standardsand Technology while considering milepost 1450where a roughness length of z0 = 0.05 m wasassumed together with a meteorological height ofHmet = 10 m. The averaging time for this datawas τ = 60 s. The distributions and parametervalues assumed for E1 to E7 can be found in Table2 of Spence and Kareem (2014) while consideringa mean for E3 of 0.8065. The stochastic loadsacting on the building were derived from the TokyoPolytechnic University Wind Pressure Database.In particular, the mean wind speed during thetests was vHm = 11 m/s. Through integrationand appropriate scaling, realizations of fw wereobtained. The distributions and parameter valuesassumed for W1 to W3 can be found in Table 2 ofSpence and Kareem (2014) while considering acoefficient of variation of 0.05 for W1. For the casestudy under investigation the reliability integralneeds to be calculated over the space of U ={C1, ...,C90,S1,ST2n,ST3n ,W1, ...,W3,E1, ....,E7,V}T ,with n = 3, and Ψˆ, resulting in a total dimensionof 198 which constitutes a large scale reliabilityproblem. Considering the size of x, m = 375,the problem truly represents a large scale RBDOoptimization problem.6.2. ResultsTwo cases are presented in this section with thefirst considering the components of random vec-tor Ψˆ independent and the second considering thecomponents of Ψˆ fully correlated with the aim ofquantifying the assumption of Sec. 4.2. P0 wasset at 2× 10−2. Figure 2 shows the convergencehistory of the objective function which here coin-cides with the weight of the material composing thestructural system while Fig. 3 reports the history ofDesign Cycle0 2 4 6 8 10W[kN]×10511. 2: Design history of the objective function.Design Cycle0 2 4 6 8 10P f10-210-1100UncorrelatedCorrelatedFigure 3: Failure probability history.the system level failure probability. From Figs.2 and 3 it is immediately evident the strong con-vergence properties of the proposed SL-AVV algo-rithm. Indeed, the problem practically convergesafter only 6 design cycles which means that thesimulation-based reliability analysis only had to beinvoked 6 times before solutions are found. It isalso evident that the assumption that the randomvector Ψˆ has independent components would, atleast for this example, seem to be perfectly reason-able with little difference between the two extremecases. Figure 4 reports the variation of α (Eq. (29))during the optimization process. While α is seento change, its variation quickly drops off after de-sign cycle 4 therefore allowing for the quick over-all convergence seen in Figs 2 and 3. Fig. 5 showsthe initial and final failure distributions for the twocases. Once again the effectiveness of the proposed712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Design Cycle0 2 4 6 8 10α00. 4: History of the parameter α (Eq. (29)).D10-1 100 101P f10-210-1100Initial uncorr.Initial corr.Final uncorr.Final corr.Figure 5: Initial and final failure probability distribu-tions.method is evident. Finally, it should be observedthat the failure probability of the converged designis exact as for any design cycle in which x does notchange (e.g. at convergence), all approximationsused in the proposed approach are exact.7. CONCLUSIONSThis paper presented a new simulation-based sys-tem level reliability based design optimization al-gorithm, SL-AVV, for large scale uncertain anddynamic systems excited by stationary stochasticwind excitation. The method was based on theconcept of decoupling the reliability analysis fromthe optimization loop through the definition of asequence of high quality subproblems defined interms of a number of Auxiliary Variable Vectors(AVVs) that allow the system failure probability tobe written in terms of a sum of exponential func-tions modeling the failure modes of the system. Thestructure of the AVVs allows the subproblem to beeasily made explicit in the design variable vector,and therefore any gradient-based optimization al-gorithm may be used for its resolution. The effec-tiveness of the proposed approach lies in how eachsubproblem can be defined exclusively in terms of asingle simulation-based reliability analysis carriedout in the current design point. The practicalityand strong convergence properties of the proposedapproach were illustrated on a full scale buildingexample characterized by a high-dimensional de-sign variable vector as well as a high-dimensionalsystem-level reliability integral.8. REFERENCESBobby, 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.Carassale, L., Piccardo, G., and Solari, G. (2001). “Dou-ble modal transformation and wind engineering appli-cations.” J. Eng. Mech., 127(5), 432–438.Chan, C. M., Grierson, D. E., and Sherbourne, A. N.(1995). “Automatic optimal design of tall steel build-ing frameworks.” J. Struct. Eng., 121(5), 838–847.Schuëller, G. I. and Jensen, H. A. (2008). “Computa-tional methods in optimization considering uncertain-ties - an overview.”Comput. Methods Appl. Mech. En-grg., 198(1), 2–13.Schuëller, G. I., Pradlwarter, H. J., and Koutsoure-lakis, P. S. (2003). “A comparative study of reliabil-ity estimation procedures for high dimensions.” 16thASCE Engineering Mechanics Conference, July 16-18, Seattle, Washington, USA.Spence, S. M. J. and Gioffrè, M. (2012). “Large scalereliability-based design optimization of wind excitedtall buildings.” Prob. Eng. Mech., 28, 206–215.Spence, S. M. J. and Kareem, A. (2013). “Data-enableddesign and optimization (DEDOpt): Tall steel build-ings frameworks.” Comput. Struct., 134(12), 134–147.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.Valdebenito, M. A. and Schuëller, G. I. (2010). “A sur-vey on approaches for reliability-based optimization.”Struct. Multidisc. Optim., 42, 645–663.8


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