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J. Wind Eng. Ind. Aerodyn. ] (]]]]) ]]]–]]]

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Journal of Wind Engineeringand Industrial Aerodynamics

0167-61

doi:10.1

� Corr

E-m

mami@

PleasInd.

journal homepage: www.elsevier.com/locate/jweia

Efficient algorithms for the reliability optimization of tall buildings

S.M.J. Spence �, M. Gioffr�e

CRIACIV/Department of Civil and Environmental Engineering (DICA), University of Perugia, Italy

a r t i c l e i n f o

Keywords:

Reliability-based design optimization

Tall building design

Wind loads

Directional wind climates

05/$ - see front matter & 2011 Published by

016/j.jweia.2011.01.017

esponding author. Tel.: +39 0755853956; fax

ail addresses: [email protected] (S.M.

unipg.it (M. Gioffr�e).

e cite this article as: Spence, S.M.J., GAerodyn. (2011), doi:10.1016/j.jweia

a b s t r a c t

Modern tall buildings are often characterized by their slenderness and sensitivity to extreme wind

events. For these buildings traditional least weight optimization procedures based on a few idealized

equivalent static wind loads derived from directionless wind models may be inadequate. This is

especially true considering traditional models used for combining aerodynamics and site specific

climatological information. Indeed these methods were developed for buildings with statistically and

mechanically uncoupled systems exhibiting strong preferential behavior for certain wind directions.

Using these models during a traditional deterministic optimization may lead to unsafe designs. In this

paper a recently developed component-wise reliability model is used to rigorously combine the

directional building aerodynamics and climatological information. An efficient reliability-based design

optimization scheme is then proposed based on decoupling the traditionally nested optimization loop

from the reliability analysis. The decoupling is achieved by assuming the level cut sets containing the

mean wind speeds generating a response with specified exceedance probability independent of

changes in the design variable vector. The decoupled optimization problem is solved by defining a

series of approximate explicit sub-problems in terms of the second order response statistics of the

constrained functions.

& 2011 Published by Elsevier Ltd.

1. Introduction

The member size optimization of tall buildings is a well-established field of application for large scale optimization algo-rithms. This is easily explained through the obvious economicadvantages that can be had through this type of application due tothe high initial costs of such constructions. Research into this areahas been vigorously explored since the early Nineties. While sizeand scope of problems examined have grown over the years, thebasic approaches to solve these problems have remained unaltered.In particular they are based on the resolution of a static responseoptimization problem and therefore the definition of an adequatenumber of idealized Equivalent Static Wind Loads (ESWLs) (Baker,1990; Chan et al., 1995; Park and Adeli, 1997; Kim et al., 1998,2008; Chan et al., 2009, 2010; Li et al., in press). However theprocess of transforming an inherently dynamic phenomenon,such as the response of wind sensitive tall buildings, into a staticresponse problem is not an easy task especially consideringmodern tall buildings which tend to be characterized by a coupledresponse (Holmes, 2002; Chen and Kareem, 2005a,b). Traditionalmethods for treating the wind hazard through the definition of anappropriate number of ESWLs are based on several simplifications

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J. Spence),

ioffr�e, M., Efficient algorith.2011.01.017

including: the combination of the directional aerodynamics andwind climate information through approximate, often directionless,models; the definition of the ESWLs that reproduce the maximumvalue of a limited number of critical load effects, often the basebending moments or top floor displacements (Zhou and Kareem,2001); the assignment of the occurrence probability of a genericload effect equal to the occurrence probability of the wind speedthat produces the maximum response of one of the critical loadeffects. These simplifications introduce a number of shortcomings inthe response estimation among which are the choice of the criticalload effects and the inaccurate estimation of the occurrence prob-abilities of both the critical and non-critical load effects. This last isdue to the fact that the mean hourly wind speed with a certainoccurrence probability will not in general produce a load effect withthe same occurrence probability due to the inherently directionalnature of both the wind climate and building aerodynamics. Con-sidering the computational power now available, the investigationof more thorough approaches based on reliability models thatrigorously account for problems such as that concerning theaccurate combination of directional aerodynamic and climatologicalinformation is overdue. From an optimization viewpoint this impliesthe definition of appropriate Reliability-Based Design Optimization(RBDO) algorithms (Nikolaidis and Burdisso, 1988; Enevoldsen andSorensen, 1994) that must be capable of handling the sheer size ofmodern tall buildings. Indeed the inherent computational effortnecessary for solving such large scale (thousands of probabilisticconstraints) RBDO problems implies the need for defining

ms for the reliability optimization of tall buildings. J. Wind Eng.

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dx.doi.org/10.1016/j.jweia.2011.01.017

mailto:[email protected]



dx.doi.org/10.1016/j.jweia.2011.01.017

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S.M.J. Spence, M. Gioffr�e / J. Wind Eng. Ind. Aerodyn. ] (]]]]) ]]]–]]]2

efficient algorithms that overcome the coupled or nested natureof traditional RBDO algorithms.

In this paper an efficient RBDO procedure is proposed for themember size optimization of tall buildings subject to multipleloading conditions and time variant experimentally determinedwind loads. In particular the procedure is based on the concept ofdecoupling the reliability analysis from the optimization loop(Der Kiureghian and Polak, 2001; Royset et al., 2001; Du andChen, 2004; Zou and Mahadevan, 2006; Jensen et al., 2008b;Spence, 2009; Spence and Gioffr�e, 2010a,b). This allows thereliability analysis to be performed separately from the successivedeterministic optimization loop, guaranteeing far greater effi-ciency than traditional approaches. The reliability of the systemis guaranteed through a component-wise strategy that rigorouslycombines experimentally determined directional aerodynamicswith site specific directional climatological information whilefully accounting for the non-linear nature of the limit statefunctions (Gioffr�e and Spence, 2010). The results of the reliabilityanalysis are then used to define a deterministic optimizationproblem in terms of the second order response statistics. Thisproblem is characterized by the replacement of the reliabilityconstraints of the original optimization problem with responseconstraints evaluated in the level cut sets of the componentresponse surfaces with prescribed vulnerability. The applicabilityof the proposed procedure is then demonstrated on a full scaleplanar frame set in a 3D wind environment.

2. The RBDO procedure

2.1. Problem formulation

Consider a tall building with j¼1,y,N floors and i¼1,y,Mmembers making up the structural system with k¼1,y,K columnlines for which the peak inter-story drift, djks, is to be controlled intwo orthogonal directions (s¼X,Y). With the objective of mini-mizing the weight of the structure, W, in terms of a vector ofdeterministic design variables, x¼{x1,y,xn}T, while ensuring thestructural integrity of the system through component reliabilityconstraints on the peak inter-story drift and member capacityratios, bil, under l¼1,2,y,L static loading conditions, the followingtime invariant RBDO problem may be posed:

Find x¼ fx1, . . . ,xngT ð1Þ

to minimize W ¼ f ðxÞ ð2Þ

subject to:

Pf ,djksðxÞ ¼

ZGdjksðx,v,z,TÞr0

pvðvÞ dvrPacceptf ,djks

ðj¼ 1, . . . ,NÞ ðk¼ 1, . . . ,KÞðs¼ X,YÞ ð3Þ

Pf ,bilðxÞ ¼

ZGbilðx,v,z,TÞr0

pvðvÞ dvrPacceptf ,bil

ði¼ 1, . . . ,MÞ ðl¼ 1, . . . ,LÞ ð4Þ

where Gdjksðx,v,z,TÞ and Gbil

ðx,v,z,TÞ are the limit state functionswith specified first excursion probabilities; z is the responsevector calculated from the dynamic equilibrium problem govern-ing the response of the structural system; T is the observationperiod for which the first excursion probabilities are calculated; vis a vector of random variables defining the intensity of the windhazard taken as v ¼ V H ,y

� �where V H is the maximum mean

wind speed at the top of the building of height H blowingfrom direction y (Gioffr�e and Spence, 2010); f is the objectivefunction; Pf ,djks

ðxÞ and Pf ,bilðxÞ are the failure probabilities asso-

ciated with the drift and capacity ratios while Pacceptf ,djks

and Pacceptf ,bil

are

Please cite this article as: Spence, S.M.J., Gioffr�e, M., Efficient algorithInd. Aerodyn. (2011), doi:10.1016/j.jweia.2011.01.017

their acceptable values respectively; pv(v) is the joint probabilitydensity function of V H and y.

The optimization problem outlined in Eqs. (1)–(4) is charac-terized by the limit state functions Gdjks

ðx,v,z,TÞ and Gbilðx,v,z,TÞ

that assume the following form:

Gdjksðx,v,z,TÞ ¼ LSdjks

�ðmdðzðx,vÞÞþgdðzðx,vÞ,TÞsdðzðx,vÞÞjks ð5Þ

Gbilðx,v,z,TÞ ¼ LSbil

�ðmbðxi,zðx,vÞÞþgbðzðx,vÞ,TÞsbðxi,zðx,vÞÞil ð6Þ

where LSdjksand LSbil

are the limit states, md, mb, sd, sb, gd and gb

are the means, standard deviations and peak factors, withspecified first excursion probabilities, of the response processesdjks(z(t,x)) and bil(xi,z(t,x)) calculated from the following dynamicequilibrium problem:

M €zðtÞþC _zðtÞþKzðtÞ ¼ fðt,vÞ ð7Þ

where M, C and K are the mass, damping and stiffness matricesrespectively while f(t,v) is the vector of the time varying forcingfunctions evaluated for the hazard intensity v¼ V H ,y

� �and

considered stationary. It should be observed that the limit statefunctions of Eqs. (5) and (6) are implicit functions of the designvariables x because of their dependency on the response vector zwhich can be shown to be an implicit function of the designvariables (Arora, 2004).

The peak factors gd and gb are of fundamental importance forthe present formulation. Indeed the solutions of the followingtime variant reliability problems:

Pð ~Gdjks¼ djks�

~djksðzðt,xÞÞjvo0Þ

Pð ~Gbil¼ bil�

~b ilðxi,zðt,xÞÞjvo0Þ

8<: 8tA ½0,T� ð8Þ

may be estimated as the first excursion probabilities passed thelevels djks and bil of the response processes ~djks(z(t,x)) and~bil(xi,z(t,x)). Because the wind hazard may be considered station-ary, these probabilities obviously coincide with the exceedanceprobabilities associated with the peak factors that give theresponse levels djks and bil estimated as:

djks ¼ ðmdjksðzðxÞÞþgdjks

ðzðxÞ,TÞsdjksðzðxÞÞÞjv

bil ¼ ðmbilðxi,zðxÞÞþgbil

ðzðxÞ,TÞsbilðxi,zðxÞÞÞjv

(ð9Þ

In particular by defining the component vulnerability as theconditional probability of exceedance Pðdpeak

jks 4djksjvÞ orPðbpeak

il 4biljvÞ (Gioffr�e and Spence, 2010), the definition ofresponse levels with assigned excursion probabilities, djks andbil, is equivalent to the definition of acceptable vulnerability levelsfor the various response components that are to be constrained inthe optimization problem. Therefore the accurate estimation ofpeak factors with specified excursion probabilities is central tothis formulation as they define the limit state functions withspecified first excursion probabilities. It should also be observedthat any non-Gaussian response feature seen in ~d jksq(z(t,x)) and~bil(xi,z(t,x)) may be accounted for through appropriate peak factormodels such as those proposed in Gioffr�e and Gusella (2007).

2.2. Proposed resolution setting

The main difficulty in solving the RBDO problem outlined inEqs. (1)–(4) is posed by the non-linear nature of the limit states(Gioffr�e and Spence, 2010) and by the large scale (thousands ofprobabilistic constraints) of the problems that are of practicalinterest. This is particularly true for traditional general purposeRBDO algorithms which are in general coupled, or nested,implying the fact that the reliability analysis is carried out simulta-neously with the optimization causing the procedures to be extre-mely computationally cumbersome (Harish, 2004). To overcomethis problem, various methods have been developed (Schueller and


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S.M.J. Spence, M. Gioffr�e / J. Wind Eng. Ind. Aerodyn. ] (]]]]) ]]]–]]] 3

Jensen, 2008). In particular methods based on decoupling theoptimization loops for the reliability analysis have received con-siderable attention (Der Kiureghian and Polak, 2001; Royset et al.,2001; Du and Chen, 2004; Zou and Mahadevan, 2006; Jensen et al.,2008b; Spence, 2009; Spence and Gioffr�e, 2010a,b), as they allow theadoption of efficient deterministic optimization algorithms to beapplied to the results of the reliability analysis.

In the following a method is proposed that decouples theoptimization problem of Eqs. (1)–(4) from the reliability analysiswhich is carried out using the model proposed in Gioffr�e and Spence(2010) and therefore rigorously accounting for the non-linear natureof the limit state functions. The method is based on firstly expandingthe constraints in Eqs. (3) and (4) in the following form:

Pf ,djksðxÞ ¼ Pðdjks4LSdjks

Þ ¼ 1�

Z VLSdjksH

0

Z 2p

0pvðV H ,yÞ dV H dy

264

375l

rPacceptf ,djks

ð10Þ

Pf ,bilðxÞ ¼ Pðbil4LSbil

Þ ¼ 1�

Z VLSbilH

0

Z 2p

0pvðV H ,yÞ dV H dy

24

35lrPaccept

f ,bil

ð11Þ

where l is the intensity of the directional extreme wind climate

(Grigoriu, 2006, 2009), while VLSdjks

H and VLSbil

H are the level cut sets of

the response surfaces with prescribed vulnerability containing allthose wind speeds associated with the limit states LSdjks

and LSbil.

The geometric meaning of the level cut sets is shown in Fig. 1 for ageneric response R. If in Eqs. (10) and (11) the probability of

exceedance is fixed coincident to Pacceptf ,djks

and Pacceptf ,bil

respectively, then

the corresponding level cut sets Vdjks

H and Vbil

H give the wind speeds

over all wind directions that will give the response levels, djks and bil

corresponding with Pacceptf ,djks

and Pacceptf ,bil

. The knowledge of the level cut

sets Vdjks

H and Vbil

H allow for the definition of the following alternative

optimization problem:

Find x¼ fx1, . . . ,xngT ð12Þ

0

90

180

270

360

Fig. 1. Level cut set, VR

H , through the response


to minimize W ¼ f ðxÞ ð13Þ

subject to:

ðmdðzðx,vÞÞþgdðzðx,vÞ,TÞsdðzðx,vÞÞÞjks|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}V

djksH

�LSdjksr0 ð14Þ

ðmbðxi,zðx,vÞÞþgbðzðx,vÞ,TÞsbðxi,zðx,vÞÞÞil|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}V

bilH

�LSbilr0 ð15Þ

The solution of this optimization problem formulated in the levelcut sets will at convergence give the same solution as the originalproblem. In particular if a constraint is active at the optimum thisimplies that the corresponding limit state function is in thelimit state.

The problem outlined in Eqs. (12)–(15) is still coupled, as avariation in the design variable vector x will cause a change in thelevel cut set causing the responses djks and bil. Therefore duringthe optimization the reliability model has to be updated. Todecouple the process it is proposed to consider the level cut sets,V

djks

H and Vbil

H , independent of the design vector during theoptimization loop. This makes the process iterative as at theend of an optimization loop the reliability model must be updatedtherefore yielding the new level cut sets.

By decoupling the optimization, the problem of Eqs. (12)–(15)may be solved by any appropriate deterministic algorithm.

2.3. Optimization procedure

For practical applications the size of the problem outlined inEqs. (12)–(15) implies the need for optimization algorithms withhigh convergence rates. These will in general be achieved byadopting gradient-based algorithms (Arora, 2004; Haftka andGurdal, 1992). However, these methods require design sensitivityanalysis which in light of the high number of implicit non-linearprobabilistic constraints, second order nature of the governingequations and onerous reanalysis can require excessive computa-tional effort (Kang et al., 2006; Arora and Wang, 2005). Variousmethods have been developed for increasing the overall efficiencyof the optimization process. Among these are those based on the

VH

surface of R with prescribed vulnerability.


dx.doi.org/10.1016/j.jweia.2011.01.017


concept of defining a sequence of approximate explicit sub-problems in terms of the design variables around the currentdesign points (Schmit and Miura, 1976; Jensen, 2000; Bruyneelet al., 2002; Jensen et al., 2008a,b). The approximate problems aresolved by conventional optimization algorithms without the needto perform any additional structural analyses due to the explicitnature of sub-problem. Convergence is reached when the magni-tude of change in the objective function or optimality conditionsis below a fixed level. The next section will present a case studyon which the proposed RBDO procedure is implemented.

3. Case study

3.1. Building description

The planar frame considered in this application is taken from afull 3D model of a regular 74 story tall building with a typicalstory height of 3.98 m. The original structure is a steel outriggerframework. The planar frame considered is one of the innerframes, part of the steel core and connected to the external tubethrough three outriggers placed at 1/3 and 2/3 of the height aswell as at the top of the building (Fig. 2). The central frame isstiffened through a series of diagonals. All beams are rigidlyconnected to the columns while the diagonals and K-bracing aresimply connected. The outriggers are also simply connected. Thebeams and diagonals are American Institute of Steel Construction(AISC, 2001) standard wide-flange W sections. The initial beamsizes are W24�176 while the diagonals are W14�370. Thecolumns and outriggers are steel box sections of flange thickness1/20 of the mid-line diameter. The frame is analyzed in thevertical plane parallel to the global X-direction. The nodes at eachfloor level of the frame are rigidly connected concerning transla-tions in the global X-direction simulating the presence of the rigid

Fig. 2. Planar frame as set in the 3D wind environment.


floor diaphragms. The rigid connections do not apply toZ-direction translations or rotations around the Y-axis.

The goal of this case study is to minimize the weight of theframe while ensuring the lateral drift (in the global X-direction)and member capacity performance of the frame while consideringthe extreme 3D wind environment reported in Spence and Gioffr�e(2010b). The aerodynamic loads considered in this study arethose described in Spence (2009).

3.2. Explicit approximate constraints

3.2.1. Inter-story drift ratios

For the planar case study under consideration the inter-storydrift ratio constraints become djks¼djX where the subscript k isdropped as all column lines obviously have the same inter-storydrift ratio response. Numerically the constraints of Eq. (14) areimposed in a discrete number of points, q¼1,y,Q, belonging tothe level cut sets V

djX

H evaluated in the current design point x0. Theproblem is therefore to find an explicit approximate expression interms of the design variables around the current design point, x0,of the following peak response function of the response process~djXq(z(t,x)):

djXq ¼ ðmðzðx0ÞÞþgðzðx0Þ,TÞsðzðx0ÞÞjdjXqðj¼ 1, . . . ,NÞ ðq¼ 1, . . . ,Q Þ

ð16Þ

By taking advantage of the fact that the structure under con-sideration is composed of monodimensional elements it can beshown that the peak response function djXq may be approximatelyexpressed explicitly in terms of the design variable vector as(Spence, 2009):

djXq �XMi ¼ 1

X3

p ¼ 1

wpiðxiÞmGpiðzðx0ÞÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}jXq

þgdðzðx0Þ,TÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}jXq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXMi ¼ 1

XMw ¼ 1

X3

p ¼ 1

X3

m ¼ 1

wpiðxiÞwmwðxiÞCGpiGmwðzðx0ÞÞ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

jXq

vuuutð17Þ

where mG1iðzðx0ÞÞ is the mean of the function G1iðzðx0ÞÞ,

CGpiGmwðzðx0ÞÞ are the zero-lag covariance coefficients between

Gpiðzðx0ÞÞ and Gmwðzðx0ÞÞ while wpiðxiÞ are the following explicitfunctions linking the design variables xi to the mechanical proper-ties of the cross section of the ith member:

w1iðxiÞ ¼1Ai

w2iðxiÞ ¼1

AYi

w3iðxiÞ ¼1

IX i

8>>><>>>: ð18Þ

where X and Y represent the local reference system of the ithmember while Ai, AYi and IX i are the axial area, shear areaand moment of inertia respectively of the ith member. Thefunctions Gpiðzðx0ÞÞ for p¼1,2,3 are functions of the time varyinginternal forces calculated from the dynamic equilibrium of Eq. (7)and of the internal forces due to a virtual loading system (Spence,2009). Eq. (17) is an approximate expression for djXq as anychanges in the internal forces due to a change in the designvariable vector x are ignored, that is the internal forces areconsidered invariant for small changes in the design variables.

3.2.2. Capacity ratios

Concerning the local member level response the followingcapacity ratios, suggested in AISC (2001), were considered:

~bilðxi,zðt,x,vÞÞ ¼jNilðxi,zðt,x,vÞÞj

afNniðxiÞþb

jMX ilðxi,zðt,x,vÞÞj

fbMXniðxiÞ

� �ð19Þ


dx.doi.org/10.1016/j.jweia.2011.01.017


subject to the conditions:

jNilðxi,zðt,x,vÞÞj

fNniðxiÞZ ~N )

a¼ 1

b¼8

9

8<:

jNilðxi,zðt,x,vÞÞj

fNniðxiÞo ~N )

a¼ 2

b¼ 1

(

Nilðxi,zðt,x,vÞÞZ0) f¼ft

Nilðxi,zðt,x,vÞÞo0) f¼fc

8>>>>>>>>>>><>>>>>>>>>>>:

ð20Þ

where Nni and MXni are the nominal axial and flexural strengthsaround the local X- axis of member i, f and fb are axial andflexural resistance factors while Nil(xi,z(t,x,v)) and MX ilðxi,zðt,x,vÞÞare the total internal forces due to the specified factored combi-nation l¼1,y,L.

As for the inter-story drift constraints the capacity constraintsmust be imposed in a discrete number of points, q¼1,y,Q,belonging to the level cut sets V

bil

H evaluated in the current designpoint x0. In particular what is needed is an approximate explicitexpression of the following peak response function:

bilq ¼ ðmðxi,zðx0ÞÞþgðzðx0Þ,TÞsðxi,zðx0ÞÞÞjbilq

ði¼ 1, . . . ,MÞ ðl¼ 1, . . . ,LÞ ðq¼ 1, . . . ,Q Þ ð21Þ

The main difference between the peak response function ofEq. (21) and that of the peak inter-story drift ratio response ofEq. (16) is the non-continuous nature of the response processes~b il(xi,z(t,x,v)) compared to the continuous nature of the processes~djXq(z(t,x,v)). This makes the evaluation of the crossing rate of~b il(xi,z(t,x,v)) far more involved compared to that of the processes~djXq(z(t,x,v)). The method adopted in this paper for overcoming thisdifficulty is that proposed in Spence (2009) and is based on defining,from the conditions in Eq. (20), the following four, id¼1,y,4, peakresponse functions derived from continuous processes:

bðidÞilq ¼X2

j¼1

IðidÞj ðxiÞmjIF j jðxi; zðxÞÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

lq

þgbðzðxÞ; TÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}lq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2

j¼1

X2

k¼1

IðidÞj ðxiÞIðidÞk ðxiÞCjIFj jjIFk jðxi ;zðxÞÞ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

lq

vuuut ð22Þ

where jIF1j¼jNilq(xi, z(t, x))j and jIF2j¼jMXilq(xi, z(t, x))j whileCjIF j jjIFk j

ðxi,zðxÞÞ is the zero-lag covariance coefficient between theinternal forces jIFjj and jIFkj with Ij for j¼1,2 given by:

IðidÞ1 ¼1

aðidÞfðidÞNniðw1iðxiÞÞ

IðidÞ2 ¼bðidÞ

fbMXniðw2iðxiÞÞ

8>>>><>>>>:

ð23Þ

Which of the peak response functions is governing may then bedetermined from the maximum and minimum axial force occurringin the member given by:

Nþilq ðxi,zðxÞÞ ¼ mNðxi,zðxÞÞþgþN sNðxi,zðxÞÞ

N�ilqðxi,zðxÞÞ ¼ mNðxi,zðxÞÞþg�NsNðxi,zðxÞÞ

(ð24Þ

where g+N and g�N are the peak factors giving the maximum and

minimum axial force calculated for an observation period T and firstexcursion probability equal to that specified for gbil

.If the internal forces are once again considered invariant for

small changes in the design vector x then Eq. (22) can beconsidered as explicit approximate expressions for the peakresponse functions id¼1,y,4 around the current design pointx0. By defining similar expressions for Eq. (24) the governing peakresponse function may be approximately estimated for the designvector x0þDx. From the explicit approximate expressions it is


then possible to define probabilistic lower limits (Spence, 2009)on the design variables, x¼ fx1,x2, . . . ,xMg by solving member bymember the problem outlined above in terms of xi while max-imizing the utilization of the member.

3.2.3. Explicit sub-problem

Given an initial design point x0 and the level cut sets Vdjks

H andV

bil

H derived from the reliability model, the approximate expres-sions so far formulated allow for the definition of the followingstatic sub-problem:

Find x¼ fx1,x2, . . . ,xMg ð25Þ

to minimize WðxÞ ¼XMi ¼ 1

g Li

w1iðxiÞð26Þ

subject to:

djXq �XMi ¼ 1

X3

p ¼ 1

wpiðxiÞmGpiðzðx0ÞÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}jXq

þgdðzðx0,TÞÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}jXq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXMi ¼ 1

XMw ¼ 1

X3

p ¼ 1

X3

m ¼ 1

wpiðxiÞwmwðxiÞCGpiGmwðzðx0ÞÞ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

jXq

vuuut�LSdjX

r0 ðj¼ 1, . . . ,NÞ ðq¼ 1, . . . ,Q Þ ð27Þ

max ½xLi , �xL

i �rxirxUi

ði¼ 1, . . . ,MÞ ðq¼ 1, . . . ,Q Þ ð28Þ

where g is the material specific weight while the lower limits onthe design variables are taken as the maximum between thelower limit due to construction issues, �xL

i , and those evaluatedfrom the results of the reliability model xL

i : xUi is the upper limit on

the design variable. Once the sub-problem has converged it isreformulated in the new design point xDC Z1

0 and the optimizationrepeated. Each reformulation is termed a Design Cycle (DC). Aftera number of redesigns the problem will converge and theoptimization process will terminate. Being a static responseproblem with explicit representation, any gradient-based optimi-zation algorithm can be used to find its solution. In this paper anOptimality Criteria algorithm (OC) (Haftka and Gurdal, 1992;Chan et al., 1995; Spence, 2009) was adopted.

3.3. Initial performance

The initial performance of the inter-story drift ratios andfactored member capacity ratios is estimated from the reliability

model proposed in Gioffr�e and Spence (2010) while assigning an

acceptable failure probability of Pacceptf ,djX

¼ Pacceptf ,bil

¼ 0:02 for an epoch

of one year. These failure probabilities obviously coincide withMean Recurrence Intervals (MRIs) of 50 years. In particular theacceptable vulnerability/first excursion probability of the framecomponents is taken as the probability associated with theexpected value of the conditional probabilities of exceedance

PðdpeakjX 4djX jvÞ or Pðbpeak

il 4biljvÞ. The response processes are

considered Gaussian in nature. Therefore response surfaces withassigned vulnerability may be obtained by considering the fol-lowing peak factor (Davenport, 1964):

g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lnðn0TÞ

pþ

0:5772ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lnðn0TÞ

p ð29Þ

where n0 is the mean 0-level up-crossing rate which for tallbuildings may be taken equal to the first natural frequency while


dx.doi.org/10.1016/j.jweia.2011.01.017

0 1 2 3 4 5

x 10−3

0

10

20

30

40

50

60

70

Fig. 3. Initial drift performance for Pacceptf ,djX

¼ 0:02.

0 136 272 408 544 680 816 952 10880

0.5

1

1.5

Fig. 4. Initial factored member capacity performance for Pacceptf ,bil

¼ 0:02.

1.25 2.5 3.75 5 6.25 7.5

x 10−3

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Non feasible design region

Fig. 5. Failure distributions, Pf ,djX, of the inter-story drift ratios.

0.7 1 1.3 1.6 1.90

0.02

0.04

0.06

0.08

0.1


Fig. 6. Failure distributions, Pf ,bil, of the capacity ratios.


T is the observation period equal to 3600 s. The local capacityrequirements are those suggested in AISC (2001) while the loadcombination used in this study was the following (ASCE, 2005):

1:2ðDLÞþ1:0ðLLÞþ1:6ðWind LoadÞ ð30Þ

where DL is the dead load and LL is the static live load. Fig. 3shows the initial drift performance of the frame against the limit


state LSdjXset at 1/400 while Fig. 4 shows the initial factored

member capacity ratios against the limit state LSbilset at unity.

In Fig. 5 are shown the failure distributions, Pf ,djX, for the inter-

story drift ratios highlighting the non-feasible design regionimposed by the limit state LSdjX

¼ 1=400 while in Fig. 6 are shown

the analogous distributions, Pf ,bil, for the factored capacity ratios

and feasible design region for LSbil¼ 1.

4. Optimization

The calibration of the optimization algorithm is achievedthrough the definition of the design variable vector, x, andtherefore the functions w1iðxiÞ, w2iðxiÞ and w3iðxiÞ. By choosing forthe members with standard AISC (AISC, 2001) sections their crosssectional areas as design variables, xi¼Ai, it can be shown that thefunctions w1i, w2i and w3i are given by:

w1iðAiÞ ¼ 1=Ai

w2iðAiÞ ¼ cAY=Aiþ �cAY

w3iðAiÞ ¼ cIX =Aiþ �cIX

8><>: ð31Þ

where c and �c are the regressional constants derived under theassumption that the cross section maintains within a constantshape group as it changes size (Chan et al., 1995). For thisexample the beams and diagonals are allowed to vary withinthe groups W24 and W14 respectively.

If for the box sections of the columns and outriggers the flangeto diameter ratio is kept constant, the functions w1iðxiÞ, w2iðxiÞ andw3iðxiÞ may be written in terms of the mid-line diameter, Di, as

w1iðDiÞ ¼ 5=D2i

w2iðDiÞ ¼ 19=2D2i

w3iðDiÞ ¼ 30=D4i

8>><>>: ð32Þ

The lower limit on these design variables was fixed at Di¼0.3 mwhile the upper limit was fixed at Di¼1.8 m.

The design variables were then gathered into a number ofdesign groups. In particular the symmetry of the frame wasguaranteed by grouping symmetrically with respect to the centralvertical axis. Also beams and columns were grouped two levels ata time. The outriggers were designed member by memberensuring however symmetry. The final number of independentdesign variables is 287, therefore n¼287.


dx.doi.org/10.1016/j.jweia.2011.01.017


4.1. Results

As mentioned in Sections 3.2.1 and 3.2.2 the constraints ofEqs. (14) and (15) will be imposed in a discrete number of points Q.The exact number of points to consider, and therefore additionalconstraints, should take into account the fact that a change in thestructural behavior due to a change in the design vector x for aspecific wind direction and response is likely to positively affectthe same response for another wind direction. Therefore it isunlikely a large number for Q is necessary. It should be appreciatedthat the choice of Q can only affect eventual convergence of theoptimization loop, and not the accuracy of the reliability

0 1 2 3 4 5

x 10−3

0

10

20

30

40

50

60

70

Fig. 7. Optimized drift ratio performance for Pacceptf ,djX

¼ 0:02.

0 1 2 3 4 5 6 7 8 9 10 11 12 133.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4x 10

4

Fig. 8. Objective function design history.


estimation as this is carried out for all wind directions and passedto the optimization loop through the level cut sets V

djX

H and Vbil

H . Inparticular, due to the simplicity of the frame in this example, Q¼1is considered. To investigate the effect on the results due to thechoice of Q, the optimization was performed for two separate winddirections. That is, two separate optimization problems weresolved considering the constraints constructed for wind directionsof 01 and 901.

In Fig. 7 the optimized drift performance for Pacceptf ,djX

¼ 0:02 isshown. The high quality of the approximate sub-problem isclearly visible from the number of active or near active con-straints at the optimum design. This is again seen in Fig. 8 fromthe rapid and steady convergence of proposed RBDO algorithm.The limited number of design cycles is very encouraging as it isthe updating of the reliability model which represents the mosttime-consuming part of the proposed algorithm. The differencebetween the two optima seen in Fig. 8 is mainly due to thedifference in the resolution of the response surfaces with pre-scribed vulnerability. Indeed the two optima seem to coincide ascan be seen from Fig. 7 for the optimized drift ratios andfrom Fig. 9 for the optimized capacity ratios. From Fig. 9 thesmall number of active capacity ratios is evident. Figs. 10 and 11show the failure distributions of the inter-story drift ratios, Pf ,djX

,and of the capacity ratios, Pf ,bil

, after the proposed optimizationalgorithm has been applied. It should be appreciated that the

0 41 82 123 164 205 246 2870

0.5

1

1.5

Fig. 9. Optimized factored member capacity performance for Pacceptf ,bil

¼ 0:02.

1.25 2.5 3.75 5 6.25 7.5

x 10−3

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Fig. 10. Failure distributions, Pf ,djX, of the optimized inter-story drift ratios.


dx.doi.org/10.1016/j.jweia.2011.01.017

0.7 1 1.3 1.6 1.90

0.02

0.04

0.06

0.08

0.1


Fig. 11. Failure distributions, Pf ,bil, of the optimized member capacity ratios.

0 45 90 135 180 225 270 315 36045

50

55

60

65

70

75

80

Fig. 12. Level cut set, VdjX

H , for the 60th floor drift ratio response with Pacceptf ,djX

¼ 0:02.

0 45 90 135 180 225 270 315 36045

50

55

60

65

70

75

80

Fig. 13. Level cut set, Vbil

H , for the factored capacity ratio response for member 177

with Pacceptf ,bil

¼ 0:02.

0 45 90 135 180 225 270 315 36040

50

60

70

80

90

100

110

Fig. 14. Level cut set, Vbil

H , for the factored capacity ratio response for member 295

with Pacceptf ,bil

¼ 0:02.


failure distributions rigorously account for the site specific direc-tional aerodynamics and climatological information. Finally, inFigs. 12–14 are shown some example level cut sets V

djX

H and Vbil

H

for the drift and capacity ratios. In particular, from Fig. 12 for the60th floor drift ratio, the level cut set seems to be quite insensitiveto the optimization process. This result will obviously depend on


the structure. For the 2D frame, this result was to be expected. It isinteresting to observe, however, that this is not always true even forthis simple example as can be seen for member 295, Fig. 14.

5. Conclusions

In this paper a procedure for the efficient reliability-baseddesign optimization of wind excited tall buildings has beenproposed based on the concept of decoupling the traditionallynested reliability and optimization procedures. The methodrigorously accounts for the directional site specific aerodynamicand climatological characteristics through a recently proposedcomponent-wise reliability model. In particular this model yieldsrigorous reliability estimates even for highly non-linear limitstates as no underlying approximations of the these last arenecessary. The results of the reliability analysis are then used todefine an implicit deterministic optimization problem in terms ofthe second order response statistics. In particular the optimiza-tion problem is characterized by replacing the traditional integralform component reliability constraints with equivalent con-straints on the peak response functions that are to be constrained.These last are evaluated in a discrete number of points belongingto the level cut sets of the component response surfaces withprescribed vulnerability/first excursion probabilities. The optimi-zation loop is then decoupled from the reliability analysis byassuming the mean wind speeds and directions of the level cutsets, derived from the reliability analysis, independent of thedesign variable vector. The decoupled optimization problem isthen solved by defining a sequence of approximate explicit sub-problems in terms of the second order response statistics eval-uated in the wind speeds of the level cut sets. At convergence ofeach sub-problem the reliability model is updated thereforeensuring consistent reliability level at the final optimum design.Finally the proposed algorithm is tested on a full scale planarframe subject to thousands reliability constraints and analyzed inan extreme directional 3D wind environment. The efficiency ofthe proposed algorithm is clearly seen from the rapid and steadyconvergence history. In particular an impressive number ofconstraints are seen to be active or near active at the optimum.

Acknowledgement

The research presented in this paper was partially financed bythe Italian Ministry for Education, University and Research


dx.doi.org/10.1016/j.jweia.2011.01.017


(MIUR), under the WI-POD project (PRIN 2007) which is greatlyacknowledged by the authors.

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