Computers and Structures 98-99 (2012) 7–16

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Computers and Structures

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

Optimal performance-based design of wind sensitive tall buildingsconsidering uncertainties

M.F. Huang a,⇑, C.M. Chan b, W.J. Lou a

a Institute of Structural Engineering, Zhejiang University, Hangzhou 310058, PR Chinab Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Hong Kong

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 August 2011Accepted 24 January 2012Available online 24 February 2012

Keywords:Performance-based designServiceability designOccupant comfortReliability-based design optimization

0045-7949/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.compstruc.2012.01.012

⇑ Corresponding author. Tel.: +86 571 88208728; faE-mail address: mfhuang@zju.edu.cn (M.F. Huang)

Two major wind-induced performance indexes of tall buildings excited by dynamic and random windcould be the lateral drift and acceleration. The wind-induced performance-based design optimizationframework has been developed to take into account uncertainties in the vibration related occupant com-fort problems of tall buildings. An innovative decoupling strategy is adopted to transform the originalcoupled reliability-based optimization problem into two separated sub problems, which are then solvedusing the inverse reliability approach and Optimality Criteria (OC) algorithm respectively. A 60-storybuilding example is employed to demonstrate the effectiveness and practicality of the proposed reliabil-ity performance-based design optimization method.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Performance-based design is an attempt to design buildingswith predictable loading-induced performance, rather than beingbased on prescriptive mostly empirical code specifications. Theearthquakes and strong winds are the two major loading condi-tions imposed on buildings. While the performance-based seismicdesign (PBSD) is becoming well accepted in professional practicefor the design of buildings under seismic loading, the wind-in-duced performance-based design (WPBD) is also emerging as apromising design methodology to improve the current practice inthe tall building design against wind.

Modern tall buildings are wind sensitive structures. In additionto the strength-based safety design considerations, the major designperformance objectives of a tall building are related to the checkingof the wind-induced serviceability design requirements in terms oflateral deflection and motion perception criteria. With the emer-gence of the performance-based engineering, there is an urgentneed to develop innovative design optimization methodology to en-sure satisfactory performance of tall buildings at various risk levelsof wind hazards (e.g., monsoon, tropical storm and typhoon).

The important step for developing a wind-induced perfor-mance-based design methodology is to establish a compatible setof wind design criteria which could be ‘‘risk-consistent’’ with theframework for performance-based seismic engineering [1,2]. Gen-erally, design wind speed, defined as annual maximum wind speedcorresponding to 50-year (or any other interval) return period of

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wind, could be estimated by statistical analysis to best fit observedor simulated wind speed data into the cumulative distributionfunction of Type I asymptotic extreme value distribution [3,4].After establishing the design wind speed corresponding to variousrisk levels (defined by specific probabilities of exceedance duringthe building lifetime), the wind-induced response could be pre-dicted corresponding to the given design wind speed.

The two major predictable wind-induced performances forstructural design of tall buildings are the lateral deflection andthe excessive vibration. The extreme wind loading conditions of50-year return period are generally used for checking the service-ability lateral drift criteria and the ultimate strength limit state de-sign for safety requirements. However, modern high-rise buildingsdesigned to satisfy static lateral drift requirements may still oscil-late excessively during wind storms [5]. The checking procedurefor motion perception can be expressed in terms of the dynamic re-sponse and acceptability threshold of motion. It has been widelyaccepted that the perception of wind-induced motion is closely re-lated to the acceleration response of buildings [6]. Both peak accel-eration and standard deviation acceleration under extreme windconditions of 10-year, 5-year and 1-year return period are com-monly used to represent building motion [7–14].

It is increasingly recognized that optimization methodologiesshould account for the stochastic nature of engineering systemsoperated in a random environment [15–17]. A great extent ofuncertainty on the structural systems as well as wind characteris-tics has been addressed in the context of wind engineering[18–22]. Performance-based design methodology allows a signifi-cantly different approach for formulating optimization problems,leading to the field of performance-based design optimization

8 M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16

(PBDO) incorporating either load uncertainties or structural systemuncertainties in the optimal performance-based design problemformulation [23–26]. Although the previous research achieve-ments of PBDO in both earthquake and wind engineering representsignificant advances in the use of structural optimization tech-niques for performance-based tall building designs, there is asyet no effective performance-based design optimization methodtaking into account the wind-induced drift as well as accelerationperformances of tall buildings together. Moreover, the uncertainnature of wind loads and building systems has not been rigorouslyconsidered in these previous developments of design optimizationframework for tall buildings [24–26].

In order to optimize structures to account for uncertainties suchas wind loads, structural reliability theory should be incorporatedinto the structural optimization process. Reliability-based struc-tural optimization can provide a good balance between the struc-tural reliability (e.g., the safety needs of the structure) andoptimal design objectives (e.g., reducing its cost), while specifiedperformance requirements are satisfied. In recent years, extensiveresearch has been carried out on RBDO problems [15,27–30]. Ingeneral, reliability-based optimization is a two-level optimizationprocedure, in which the outer loop is an optimization for the de-sign variables and the inner loop for reliability analysis. For mostengineering problems, the performance function is always a highlynonlinear and implicit function with respect to the basic randomvariables, whose reliability evaluation needs high computationalcost. The direct solving of the loop-nested optimization problemis computationally expensive [29,31,32]. In order to improvecomputational efficiency in the RBDO methodology, efficient ap-proaches in which reliability analysis and optimization performedsequentially have been developed [31,32]. Royset et al. [31] pro-posed a new approach for solving the reliability-based optimalstructural design problem by representing the reliability terms intraditional RBDO into deterministic functions, which define theminimum of the corresponding limit state function within a ballof specified radius. This reformulation has a limitation and cannotbe applied to the common RBDO problems, in which the distribu-tion parameters of some random variables are design variables[33–36]. Recently, various decoupling approaches were developedusing sequential approximate programming concept [37,38], theOptimality Criteria (OC) algorithm [39], and a line search strategy[40].

In this paper, a reliability performance-based design optimiza-tion method for the wind-induced performance design of tallbuildings is developed. Specifically, wind-induced accelerationhas been probabilistically assessed and rigorously considered inthe newly developed method. Performance-based design windspeed is carried out to estimate the design wind speed and toquantify uncertainties in the design wind speed estimation. Themajor sources of building system uncertainty are also identifiedand described in terms of the probability density function (PDF)and the coefficient of variance (COV). An innovative decouplingstrategy is adopted to transform the original coupling reliability-based design optimization problem into two separated sub prob-lems, which are then solved using the inverse reliability algorithmand OC algorithm, respectively. The effectiveness and practicalapplication of the reliability performance-based design techniqueare illustrated by a full-scale 60-story building example subjectto dynamic wind loads derived from the wind tunnel test.

2. Occupant comfort criteria and acceleration performancefunctions

As there are currently no universally accepted acceleration cri-teria which govern acceptable levels of wind-induced vibration in

tall buildings, the occupant comfort criteria used in the developeddesign optimization framework are carefully selected based on theavailable literatures and guidelines [6–8,41]. The ISO-6897 [8] pro-vide a guideline for evaluating the acceptability of low-frequency,in the frequency range of 0.063 Hz to 1.0 Hz, horizontal motion ofbuildings subjected to wind forces. The frequency dependentguideline specifies the acceptable standard deviation accelerationcriterion for occupancy comfort for 10 min duration in 5-year re-turn period wind as

rU€qjj¼ expð�3:65� 0:41 ln fjÞ ð1Þ

where superscript U denotes the upper limits in the standard devi-ation of wind-induced acceleration; fj = the jth modal frequency of atall building considered. The latest development on occupant com-fort and motion perception leads into the adoption of frequency-based peak acceleration criteria in both AIJ and ISO [41]. For sim-plicity, the latest occupant comfort criteria for office buildingsmay be approximated by a formula derived from ISO-6897 as [6]

aUj ¼ 3:5 � 0:72 � expð�3:65� 0:41 ln fjÞ ð2Þ

where 3.5 is a typical value of the peak factor; 0.72 is a multiplica-tion factor as recommended by ISO-6897 [8], which converts thelimiting acceleration from a recurrence interval of five years toone year. Kwok et al. [6] demonstrated that Eq. (2) shows fairlygood agreements compared with the curve of acceptable peakacceleration of one year recurrence for office buildings as adoptedin AIJ-GEH [13].

Upon to the determination of acceleration criterion, the perfor-mance function of occupant comfort could be written by combingthe wind-induced modal acceleration formula [26] and the peakacceleration criterion in Eq. (2) as

gjðV1; fj;njÞ¼3:5�0:72�expð�3:65�0:41ln fjÞ�gf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipfj

4njm2j

SQjjðfj;V1Þ

sð3Þ

where V1 denotes the design wind speed with a recurrence intervalof one year; nj = the jth modal damping ratio of the tall building;SQjj

= the power spectral density of the jth modal wind force [26];gf = the peak factor. For a Gaussian process, the peak factor gf canbe calculated as follows [42]

gf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln fjs

qþ 0:5772=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln fjs

qð4Þ

where s representing the observation time duration for assessingacceleration response may be normally taken as 600 s.

The probability of failure due to the violation of acceleration cri-teria could be defined by the limit state gj(V1, fj,nj) = 0 with gj 6 0defining the failure state. The function g is called the limit stateperformance function, and it separates the multidimensional vari-able space into the failure domain (gj 6 0) and the safe domain(gj > 0). Theoretically, the probability of failure could be evaluatedby a multidimensional integral of the form

Pf ¼ Pðgðx;dÞ 6 0Þ ¼Z

g60. . .

Zfxðx;dÞdx1 dx2 � � �dxn ð5Þ

where x denotes the random variable vector, which is used to mod-el uncertainty in the occupant comfort design problem; d repre-sents the deterministic design parameter vector, and the optimumvalue of d defines an optimized design solution.

3. Uncertainties in occupant comfort problems

For wind-excited buildings, major uncertainties rise from eitheraerodynamic wind loading characteristics or building system prop-erties. The uncertainty of wind loading characteristics is directlyaffected by statistical properties of design wind speeds, which

M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16 9

could be quantified based on the probabilistic wind speed dataanalysis as presented in the following section. Building systemproperties, including mass, stiffness and damping, are physicallydetermined by the construction material, the structural membersand their configuration. These structural properties further influ-ence the dynamic characteristics of structures, i.e., the natural fre-quency and vibration mode shapes. Various levels of practicalrandomness due to fabrication and construction process arisingfrom the material, member or form may cause the structural sys-tem exhibiting uncertain behavior. In addition, the use of computermodels to predict system behavior inevitably introduces the modelerror due to the limits on resolution of the model or lack of suffi-cient knowledge and data.

3.1. Estimation of design wind speeds

For structural design purpose, extreme wind speed rather thandaily wind speed is needed to consider. Utilizing statistics of ex-tremes, it seems to be reasonable to fit observed extreme windspeed data to the Type I Gumbel extreme value distribution [3,4].Denote the annual largest wind speed as a random variable V,which could be modeled by the Type I Gumbel extreme value dis-tribution as

FV ðvÞ ¼ PðV 6 vÞ ¼ 1� p ¼ exp � exp �v � ub

� �� �ð6Þ

where u is modal wind speed or the location parameter; b is the dis-persion or a scale wind speed; p is the probability of the designwind speed V being exceeded by a chosen value of v. Given a sampleof V with size N, the modal wind speed u is defined as the particularvalue of V such that the expected number of sample values largerthan u is one. The distribution parameters u and b could be conve-niently estimated by the probability paper method. Upon takinglogarithms twice, Eq. (6) becomes

v ¼ uþ b � ln½� lnð1� pÞ�f g ð7Þ

Hence the reciprocal of the slope of a graph of �ln[�ln(1 � p)]against v is the estimation of b. The estimation of modal windspeed can also be obtained from a zero intercept on this graph.For a design wind speed of VR corresponding to a return periodof R years, the probability p of the design wind speed VR being ex-ceeded in one year is p = 1/R, then the design wind speed VR couldbe estimated by

bV R ¼ uþ b � ln � ln 1� 1R

� �� �� �ð8Þ

3.2. Uncertainties in estimation of design wind speeds

The uncertainty in estimation of design wind speeds could bequantified by studying the mean and variance of the estimatorfor design wind speeds. Denote the mean and standard deviationof the sample of extreme wind speeds by E(X) and D(X), respec-tively. From the knowledge of Type I extreme value distribution,the mean value and the variance of extreme wind speeds couldbe calculated form the distribution parameters, u and b

EðXÞ ¼ uþ cb ð9Þ

DðXÞ ¼ pffiffiffi6p b ð10Þ

where c = 0.5772 (the Euler constant).The estimator in Eq. (8) could be rewritten in term of E(X) and

D(X) asbV R ¼ bEðxÞ þ aðRÞbDðxÞ ð11Þ

where aðRÞ ¼ffiffiffi6p� ln½� lnð1� 1=RÞ�f g=p�

ffiffiffi6p

c=p; the samplemean and sample standard deviation must be estimated from a re-cord {x1,x2, . . . ,xn} of the extreme wind speeds as

bEðxÞ ¼ 1n

Xi

xi ð12Þ

bDðxÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Xi

½xi � bEðxÞ�2sð13Þ

The value of bV R is uncertain because the sample mean and sam-ple standard deviation are random variables whose variance de-pends on the sample size N. Based on the first and secondmoment characterization of the sample mean and sample standarddeviation, the mean and variance of bV R can be approximated by [3]

EðbV RÞ � EðXÞ þ aðRÞDðXÞ ð14Þ

DðbV RÞ �ffiffiffiffiffiffiffiffiffiffiffiDðXÞ

n

r1þ a2ðRÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2D2ðXÞ � 1

4N

s24 35 ð15Þ

where c2 denotes the kurtosis, a measure of the degree of flatteningof the curve of a probability density function near its center. Thekurtosis for the Type I extreme value distribution is c2 = (3p4b4)/[20D4(X)]. Then the uncertainty in the estimation of design windspeeds could be quantified by the coefficient of variation (COV),i.e., COVðbV ÞR ¼ DðbV RÞ=EðbV RÞ.

In this way, not only the design wind speed corresponding toany time interval but also the uncertainties involved in the windspeed estimation could be readily quantified. To facilitate the reli-ability-based structural design optimization against strong wind.While 50-year return period wind speed is used to predict wind-in-duced response, 1-year return period wind speed is adopted toestimate peak acceleration response to check for the occupantcomfort of wind-excited buildings. It is worth to note that in thispaper wind-induced drift response is calculated by the equivalentstatic wind loading (ESWL) approach [26,43] in a deterministicmanner. Correspondingly, the design constraints of the wind-in-duced drift will be treated traditionally as deterministic con-straints to limit static response in the performance-based designframework. On the other hand, a probabilistic performance func-tion involving both the peak acceleration response and the occu-pant comfort threshold as expressed in Eq. (3) is employed tocapture the major uncertainties of loading characteristics and sys-tem properties and to define a comprehensive probabilisticconstraint.

3.3. Building system properties

Since the design optimization problem studied here is mainlyabout the occupant comfort of tall buildings, natural vibration fre-quencies (modal frequencies) become one of the critical system-le-vel parameters as reflected in the frequency dependent occupantcomfort performance function of Eq. (3). The modal frequency valuein one way is indicating the stiffness of a tall building and in anotherway determines the magnitude of wind load spectra representingthe aerodynamic loading on the tall building. In an approximationsense, the wind-induced modal force spectra SQjj

ðfj;V1Þ in Eq. (3)could be explicitly expressed as a function of modal frequency aswell as design wind speed [26]. This frequency dependent featureof the wind load spectra could be used to approximate the modalacceleration responses and then facilitate to explicitly express theoccupant comfort performance function in terms of modal fre-quency and design wind speed. The lognormal distribution with aCOV of 0.10 was assumed for the modal frequencies of the building

Table 1Random variables in wind-induced occupant comfort problem.

Random variables Distribution type Mean Standard deviation COV Location Scale

Wind speed (m/s) Gumbel 22.50 1.125 0.05 21.994 0.8772Natural frequency (Hz) Lognormal 0.185 0.0185 0.10 �1.692 0.0998Damping ratio Lognormal 0.010 0.0015 0.15 �4.616 0.1492

Note: the wind speed given is for one-year return period and referred at the height of 90 m in Hong Kong area.

Table 2Performance-based wind hazard design level.

Wind speed level Average return period Performance objective

Frequent 1 year Occupant comfortRare 50 years DriftVery rare 475 years Repairable damage

10 M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16

[18,44]. The mean value of the natural frequencies in the first threemodes may be determined from a computer-based finite elementmodel (FEM) by eigenvalue analysis.

Structural damping, recognized as the most uncertain parame-ter, is another important factor in the serviceability design of tallbuildings. Available data from full-scale measurements has beengathered and analyzed for the estimation of the means and COVsof damping values for a wide class of tall buildings [45]. The log-normal distribution was shown to provide the best fit to thosedamping data. In this paper, the mean value and COV of the struc-tural damping ratio in the first three fundamental modes were as-sumed to be 1% and 15%, respectively. In Table 1, a summary of thedistribution type, statistical properties and distribution parameters(i.e., location and scale) for random variables are reported.

It is worth to note that in Table 1 three random variables repre-senting major uncertainties in the occupant comfort problem havebeen listed. The way to model uncertainty is rather than a unique,but a simplified rational approach. With accumulating more andmore basic data about building mass and stiffness, these factorsmay be directly expressed as random variables in the generalframework of reliability performance-based design optimization.From practical point of view, building mass and stiffness, or under-line construction material, structural element dimension andstructural form, are much less uncertain than damping and, there-fore, can be treated as deterministic variable. For the sake of sim-plicity and practicality, structural element dimension would betreated as deterministic design variable in the following formula-tion of optimization [39].

Fig. 1. Structural Plan of t

4. Reliability performance-based design optimization

Based on a wind and hurricane design framework recom-mended in [1,22], multiple design wind hazard levels as shownin Table 2 could be explicitly defined by specific probabilities ofexceedance to cover a greater spectrum of possible extreme windevents that threaten to wind-sensitive building structures. Whenapplied to wind-resistant design, performance-based design notonly can address issues of occupant comfort and drift serviceabilityfor relative frequent wind environmental conditions, but also treatthe rare extreme events for minimizing any possible future dam-age. In this paper, the frequent and rare levels of wind will be for-merly treated by the performance-based optimization techniquewith the assumption that under these wind events structural re-sponses are in the linear-elastic stage. It is noted that the very rarewind with a recurrence interval of 475 years may cause nonlinear-inelastic responses of building structures. Therefore, nonlinear dy-namic analysis or an inelastic ‘‘pushover’’ analysis has to be used inthe future development to cover the design against the very rarewind.

The reliability performance-based design optimization problemof tall buildings would then be formulated with various constraintsin order to satisfy different design targets at specified performanceobjective levels. While the deterministic threshold target is usedfor drift constraints, the probabilistic target in terms of target fail-ure probability is set for the failure events of occupant comfort,defining as an exceedance of peak acceleration response over thelimiting acceleration criterion value. Consider a mixed steel andconcrete building having i = 1,2, . . . ,N structural elements, includ-ing steel frame, concrete frame as well as shear wall elements.For simplicity, all element sizing design variables (i.e., crossing sec-tion area of steel section, dimension size of concrete section) canbe represented by a collective set of generic sizing variables di,which are assembled as the deterministic design parameter vectord. Then the minimum structural material cost design of a buildingstructure subject to the deterministic and probabilistic perfor-mance-based constraints can be stated as:

he 60-story building.

Fig. 2. 3D views of the 60-story hybrid building.

VR = 22.53+5.391[-ln(-ln(1-1/R)]

-2

-1

0

1

2

3

4

5

6

7

10 20 30 40 50 60 70

Mean speed (m/s)

-ln[

-ln(

1-p)

]

34.7

2.250 (R=10)

43.6

3.902 (R=50)

6.162 (R=475)

55.7

u=22.53

Fig. 4. Design hourly-mean wind speed in Hong Kong (1953–2006).

M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16 11

Minimize

WðdÞ ¼XN

i¼1

widi ð16Þ

Subject to Eqs. (17)–(19).Deterministic serviceability drift performance constraints under

rare wind of 50-year return period:

gs;l ¼ds;l � ds�1;l

hs6 gU

s;l ðs ¼ 1;2; . . . ; SÞ ðl ¼ 1;2; . . . ;NlÞ ð17Þ

in which s denotes the story number; hs indicates the story height ofthe sth story; l denotes the different incident wind angle conditionsunder consideration.

Probabilistic occupant comfort acceleration performance con-straints utilizing 1-year recurrence interval wind speed:

020406080

100120140160180200220240

-1.0 -0.5 0.0 0.5 1.0 Mode shape

Ele

vati

on (

m)

ΦxΦyr*Φz

020406080

100120140160180200220240

-1.0 -0.5 0 Mod

ΦxΦyr*Φz

(a) Mode 1 (b) M

0.185Hz

Fig. 3. Mode shapes of th

Pðgjðx;dÞ 6 0Þ 6 PUj ðj ¼ 1;2; . . . ;nÞ ð18Þ

Element sizing constraints:

dLi 6 di 6 dU

i ði ¼ 1;2; . . . ;NÞ ð19Þ

Eq. (16) defines the objective function of the minimum materialcost, in which wi = the respective unit cost of the steel sections,concrete sections and shear walls. Eq. (17) expresses the lateraldrift performance constraints under 50-year return period wind,where dU

s;l denotes the design target of the drift performance. Ingeneral, the allowable wind-induced drift ratio for tall buildingsappears to be within the range of 1/750 to 1/250, with 1/400 beingtypical. Eq. (18) represents the set of j = 1,2, . . . ,n probabilistic con-straints for occupant comfort performances of a tall building underthe most critical incident wind angle conditions with 1-year recur-rence interval wind speed, where x denotes the random variablevector, including design wind speed, modal frequency and damp-ing ratio; PU

j denotes allowable failure probability of occupantcomfort performance of a tall building vibrating in a jth mode.Eq. (19) defines the element sizing constraints in which superscriptL denotes lower size bound and superscript U denotes upper sizebound of member i.

It is noted that the expression of performance function gj(x,d) isin a general form and is an implicit function of the deterministicdesign parameter vector d, which would influence the dynamicproperties of the building and successively affect the occupantcomfort performance function approximated by Eq. (3).

.0 0.5 1.0e shape

020406080

100120140160180200220240

-1.0 -0.5 0.0 0.5 1.0Mode shape

ΦxΦyr*Φz

ode 2 (c) Mode 3

0.327Hz 0.410Hz

e 60-story building.

0.0%

5.0%

10.0%

15.0%

20.0%

20 40 60 80 100 120 140 160 180 200Sample size N

Coe

ffic

ient

of

vari

atio

n

3.5%

Fig. 5. The uncertainties in estimation of design wind speeds for 50-year returnperiod.

Table 4Iteration history results using inverse reliability method.

Iteration number Random vector: uk l⁄(f1) ||uk|| Gk(u,p)

1 0.0746 0.1773 0.0499 0.185 0.1987 0.02812 -0.3385 0.9171 -0.3904 0.1975 1.0526 0.01123 -0.4821 1.3308 -0.5711 0.2134 1.5263 0.00524 -0.5231 1.5598 -0.6326 0.2243 1.7626 0.00295 -0.5281 1.6851 -0.6472 0.2302 1.8808 0.00176 -0.5227 1.7531 -0.6462 0.2332 1.9401 0.0017 -0.5158 1.7898 -0.6413 0.2346 1.9699 0.00068 -0.5101 1.8095 -0.6366 0.2352 1.9849 0.00039 -0.5021 1.8307 -0.6296 0.2358 2 0

10 -0.4996 1.8323 -0.6271 0.2357 2 0

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1 2 3 4 5 6 7 8 9Design cycle

Nor

mal

ized

cos

t

Drift optimization only

Drift and frequency optimization

0.96

1.07

Fig. 6. Design history of structure cost for the tall building.

0.150.200.250.300.350.400.450.500.550.600.650.70

1 2 3 4 5 6 7 8 9

Design cycle

Mod

al f

requ

ency

(H

z)

Mode 1 frequency

Mode 2 frequency

Mode 3 frequency

Target frequency for first mode

Fig. 7. Design history of modal frequencies for the tall building.

12 M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16

4.1. Decoupling of sizing optimization and probabilistic constraints

It is noted that the original problem defined through Eqs. (16)–(19)is a nested optimization problem due to the close relationship be-tween the objective function of Eq. (16) and the probabilistic con-straints of Eq. (18) connected by the design parameter vector d.Therefore each move in the design parameter space as one iterativestep in a design optimization algorithm to search for the optimumsolution would cause the need to reevaluation of reliability of theoccupant comfort performance, in which reliability analysis itselfwould involve a computationally intensive numerical procedure.The numerical procedure aiming to conduct the reliability analysiscould be implemented by the first-order reliability method (FORM)or Monte Carlo Simulation method. A FORM approximation to Pf isobtained by locating the most probable failure point (MPFP) onthe failure surface (gj(x,d) = 0) with minimum distance to the originin a standard normal space. Based on reliability theory, the requiredminimum distance as a measure of reliability may be determined bysolving an equality constrained minimization problem. The inneroptimization loop for performing reliability analysis and the outerdesign optimization loop for searching optimum deterministic siz-ing design variables constitute the bi-level nested optimizationproblem.

The decoupling strategy was employed in this paper by intro-ducing intermediate design parameter, the mean value of modalfrequency lfj

; in lightning of the fact that the generic implicit formof performance function could be approximated by an explicitfunction of modal frequency as shown in Eq. (3). By varying themean value of modal frequency, the probabilistic constraints onoccupant comfort performance may be satisfied with the aid ofthe inverse reliability algorithm [33,34,46]. Since the mean valueof modal frequency could be computed using a FEM model, the de-sired value of the intermediate design parameter lfj

is achievableby modifying the FEM model through the deterministic designparameter vector d. Once the optimum mean value of modal fre-quency is obtained, the searching for the optimum value of the de-sign parameter vector d could be implemented by solving adeterministic design optimization problem with multiple fre-quency and drift constraints. Hence the initial reliability perfor-mance-based design optimization problem could be decoupledinto two sub problems as:

Sub problem 1: Probabilistic design optimization.

Table 3The reliability index and probability of failure for the initial 60-story building using both

Mean frequency (Hz) FORM

Reliability index

Mode 1 0.185 1.163Mode 2 0.327 6.483Mode 3 0.41 2.641

Given PUj .

Find: the optimum mean value of modal frequency l�fj; and the

MPFP point u⁄.Subject to

Pðgjðx;pÞ 6 0Þ 6 PUj ð20Þ

where p indicates the deterministic intermediate design parametervector (including distribution parameters of random variables). Inthis paper, the vector p is reduced to one single intermediate designparameter as lfj

.

FORM and Monte Carlo simulations.

Monte Carlo simulation

Probability of failure Pf Sample number

1.224E�01 1.158E�01 10004.504E�11 0.000E+00 500,0004.128E�03 4.006E�03 50,000

Table 5The reliability index and probability of failure for the optimized 60-story building using both FORM and Monte Carlo simulations.

Mean frequency (Hz) FORM Monte Carlo simulation

Reliability index Probability of failure Pf Sample number

Mode 1 0.2357 1.999 2.276E�02 2.410E�02 1000Mode 2 0.4959 7.028 1.050E�12 0 500,000Mode 3 0.5893 3.801 7.206E�05 8.000E�05 200,000

M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16 13

Sub problem 2: Deterministic design optimization.Find d = d⁄, which minimizes W(d).Subject to: lfj

ðdÞP l�fj ; gs;lðdÞP gUs;l.

4.2. Reliability index approach and inverse reliability algorithm

Define the function Gj(u,p) represents the Rosenblatt transfor-mation of the limit-state function gj(x,p) from the original param-eter space into the new space of the uncorrelated standard normalvariates, i.e., Gj(u,p) = gj[x(u),p]. It is noted that the comprehensivedefinition of the probabilistic constraints in Eqs. (18) and (20) in-clude two inequality relations. One is representing the definitionof occupant comfort failure event, the other is to imposing the lim-its on the probability of failure being within the allowable failureprobability value. With the aid of reliability index approach[34,35], the two-inequality probabilistic constraint has been con-verted into one inequality relationship on the reliability index. Inaccordance to the first-order reliability method (FORM), the pre-scribed failure probability limit PU

j could be approximated in termsof the target reliability index bL as

PUj � Uð�bLÞ ð21Þ

Similarly, for the current design with a mean value of modal fre-quency, the probability of failure could also be related to the corre-sponding design reliability index as

PfGjðu; pÞ < 0g � Uð�bjÞ ð22Þ

Therefore, the original probabilistic constraints could be trans-formed into the form of one inequality

Uð�bjÞ 6 Uð�bLÞ j ¼ 1;2; . . . ;n ð23Þ

Since the cumulative distribution function (CDF) of the standardnormal distribution is a non-decreasing monotonic function aboutits variable, the probabilistic constraints finally could be repre-sented in terms of the reliability index

bj P bL; j ¼ 1;2; . . . ;n ð24Þ

Utilizing the reliability index approach, the sub problem 1 could berewritten as

Given bL.Find: the optimum mean value of modal frequency p� ¼ l�fj

; andthe MPFP point u⁄

Subject to

minðkuk ¼ffiffiffiffiffiffiffiffiffiuTup

Þ ¼ bL and Gjðu; pÞ ¼ 0 ð25Þ

where superscript T denotes transpose operation.The method of Lagrange’s multiplier has been used to obtain the

Kaush–Kuhn–Tucker optimality condition for the sub problem 1 as[33,46]

uþ bLruGj

kruGjk¼ 0 ð26Þ

Gjðu;pÞ ¼ 0 ð27Þ

where ruGj = gradient operator with respect to u, which is readilyavailable since the approximated performance function of Eq. (3)

is an explicit function of random variables. The modified Hasofer–Lind–Rackwitz–Fiessler (HLRF) algorithm was employed to solvingthe inverse reliability problem of the sub problem 1 based on thefollowing recursive formulas [47]

u�kþ1 ¼ruGTðu�k;pkÞu�k � Gðu�k;pkÞ

kruGðu�k; pkÞk2 ruGðu�k;pkÞ ð28Þ

pkþ1 ¼ pk

þruGTðuk;pkÞuk � Gðuk; pkÞ þ bLkruGTðukþ1; pkþ1Þk@G@fj

f¼pk

ð29Þ

where k indicates the iteration number; the partial derivative of theperformance function with respect to modal frequency is also ana-lytically available based on the performance function of Eq. (3). Theiterative apply of Eq. (28) and (29) to search for the optimum inter-mediate design parameter p and the MPFP point satisfying the giventarget reliability index for the specified performance function con-stitutes the inverse reliability algorithm.

4.3. Optimality Criteria algorithm

To facilitate a numerical solution of the deterministic designoptimization problem of the sub problem 2, it is necessary thatthe implicit drift and drift constraints be formulated explicitly interms of design variables dk. Using the principal of virtual work,the elastic drift response of a building under the actions of ESWLscan be formulated explicitly by the mechanical properties of mate-rials, and the sizing design variables [48].

Using the Rayleigh quotient method, the modal frequency fj ornatural period Tj of a building system can be related to the totalinternal strain energy Uj of the system due to the jth modal inertiaforce applied statically to the system as follows [26]:

Uj ¼ cj=f 2j ¼ cjT

2j ð30Þ

where cj denotes a proportionality constant relating the internalstrain energy Uj to the square of the modal frequency fj or thesquare of the nature period Tj of the system. For a tall building ofmixed steel frame and concrete core construction, the total internalstrain energy of the building structure due to an externally appliedmodal inertia force can be obtained by summing up the internalwork done of each member as:

UjðdiÞ¼XNs

is¼1

eis j

disþe0is j

� �þXNw

iw¼1

e0iwj

diwþe1iwj

d3iw

!ðj¼1;2; . . . ;nÞ ð31Þ

where eisj; e0isj = the internal strain energy coefficients and its cor-rection factor of steel member is; e0iwj; e1iwj = the internal strain en-ergy coefficients of concrete wall section iw. All these energycoefficients can be calculated from the internal forces of each mem-ber after the static analysis of the building structure subjected tothe applied jth modal inertia force [48]. Substituting Eq. (31) intoEq. (30), the modal frequency can then be expressed as a functionin terms of the element sizing design variables.

Upon establishing the explicit formulation of the drift and fre-quency design constraints, the next task is to apply a suitablenumerical technique for solving the deterministic optimal design

14 M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16

problem of the sub problem 2. A rigorously derived Optimality Cri-teria (OC) algorithm, which has been shown to be computationallyefficient for large-scale structures is herein employed [25,26,48].To seek for numerical solution using the OC algorithm, the con-strained optimal design problem must be transformed into anunconstrained Lagrangian function which involves both the objec-tive function and the set of explicit drift and frequency constraintsof associated with corresponding Lagrangian multipliers. By differ-entiating the Lagrangian function with respect to each sizing de-sign variable and setting the derivatives to zero, the necessarystationary optimality conditions can be obtained and then utilizedin a recursive relation to resize the active sizing variables until con-vergence occurs.

4.4. Procedure of reliability performance-based design optimization

The proposed reliability performance-based design optimiza-tion procedure can be outlined step by step as follows:

1. Carry out statistical analysis to estimate design wind speed cor-responding to various levels of wind hazard defined in Table 2.

2. Given the geometric shape of a building, determine the aerody-namic wind load spectra and modal force spectra by wind tun-nel tests.

3. Develop the finite element model (FEM) for the building andcarry out an eigenvalue analysis to obtain the modal frequen-cies and mode shapes of the vibrations of the building.

4. Based on the current set of dynamic properties of the building,calculate the dynamic drift and acceleration response of thebuilding; determine the equivalent static wind loads and applythem to the building.

5. Establish the explicit occupant comfort performance function inEq. (3), and solve the probabilistic sub problem 1 using inversereliability algorithm to obtain the optimum mean value ofmodal frequency.

6. Establish the explicit expression of the drift and frequency con-straints and formulate explicitly the deterministic design opti-mization problem of the sub problem 2, and solve it based onthe OC algorithm.

7. Check the convergence of the design objective function: if thecost of the structure for three consecutive reanalysis-and-rede-sign cycles is within certain prescribed convergence criteria, forexample within 0.1% difference in the structural material cost,then terminate the deterministic design optimization processto retrieve the optimized building structure with the minimummaterial cost and the target modal frequency, which ensuresthe occupant comfort performance having a required reliability.

5. Illustrative example

A 60-story building having a height of 240 m and a rectangularfloor plan dimension of 24 m by 72 m is used to illustrate the effec-tiveness and practicality of the reliability performance-based de-sign optimization technique. The hybrid building as shown inFigs. 1 and 2 has adopted basically an outrigger braced system,which consists of a perimeter steel frame connected to a reinforcedconcrete core by two levels of 4-story steel outriggers and belttrusses. Two double-bay X-braced frames are also established attwo end faces of the building to enhance the torsion rigidity ofthe building. Since the concrete lift core was located eccentrically,significant wind-induced lateral-torsional coupling effects on thebuilding could be found.

The exterior steel frames of the building were to be designedusing AISC standard steel sections as follows: W30 shapes forbeams; W14 shapes for columns, braces, belt and outrigger trusses.The initial member sizes for the buildings were established on the

basis of a preliminary strength check. Once the finite element mod-els were developed for the building, their initial dynamic proper-ties (i.e., the natural frequencies and mode shapes) were thendetermined by eigenvalue analysis and are given in Fig. 3. Themode shapes were found to be coupled and three-dimensional.The first three fundamental vibration modes of the building havecorresponding natural frequencies of 0.185 Hz (swaying primarilyin the short direction of the building), 0.327 Hz (swaying primarilyin the long direction) and 0.410 Hz (mainly torsional vibration). Awind tunnel test was carried out at the CLP Power Wind/WaveTunnel Facility (WWTF) of the Hong Kong University of Scienceand Technology. Wind forces acting on the building were mea-sured by the synchronous multi-pressure sensing system (SMPSS)technique using a 1:400 scale rigid model. The PSD functions of themodal forces were obtained from the wind tunnel test.

One-year recurrence interval wind speed in a typhoon-proneurban environment like Hong Kong was considered for calculatingpeak acceleration response of the building, while a 50-year returnperiod of wind was used for predicting drift performance of thebuilding. The modal damping ratio of 1% and 1.5% for the first threefundamental vibration modes of the building were assumed foracceleration and drift response prediction, respectively. Two spe-cific wind conditions corresponding to two perpendicular incidentwind directions were simultaneously considered in the perfor-mance-based design synthesis process. One was the 0-degree windperpendicular to the wide face acting in the short direction (i.e.along the Y-axis) of the building; another one was the 90-degreewind perpendicular to the narrow face acting in the long direction(i.e. along the X-axis).

5.1. Site-specific design wind speed estimation

The most valuable source of information in the estimation of fu-ture wind speeds at the building site is the historical wind speeddata that has been recorded at the building site or at a nearbyobservatory site. The wind speed data used in this paper has beenpublished by Hong Kong Observatory. The basic reference windspeeds adopted by the Hong Kong Wind Code 2004 for the con-struction of the design wind profile are derived from extreme windanalysis of the measured wind data at Waglan Island, Hong Kong.All available typhoon data measured at Waglan Island since 1953to 2006 form the basis for wind speed analysis in this paper. Totalnumber of typhoon wind speeds of observation used in the analy-sis is 110.

Fig. 4 presents the Gumbel plot of typhoon hourly-mean windspeeds. As shown in Fig. 4, the design hourly-mean wind speedcorresponding to 10-year, 50-year and 475-year return periodsare 34.7 m/s, 43.6 m/s and 55.7 m/s, respectively. It is worth notingthat theoretically Eq. (8) does not apply to the case of 1-year-recur-rence wind speed, which should be established based on the daily-maximum wind speed data collected from an observatory. In thispaper, for simplicity, the mode value of yearly extreme wind speed22.53 m/s was conservatively taken as the design wind speed of 1-year return period. Since the mode value of yearly extreme windspeed corresponds approximately to a return period of 1.584 years,the use of mode value as 1-year-recurrence wind speed isconservative.

The uncertainties in estimation of design wind speeds for 50-year return period were quantified in terms of coefficients of vari-ation based on the parameters of Type I extreme value distribution.The coefficient of variation is represented as a function of the sam-ple size N. As shown in Fig. 5, the value of COV is monotonicallyreduced as the number of sample data increasing. The sampling er-ror due to using N = 110 was calculated as 3.5%, which means thatthe design wind speed for 50-year return period would be withinthe range of (43.6 ± 3.5%) m/s. For occupant comfort reliability

M.F. Huang et al. / Computers and Structures 98-99 (2012) 7–16 15

assessment, one-year recurrence interval wind speed was treatedas a random variable with a mean value of 22.5 m/s and a COV va-lue of 5%.

5.2. Results and discussion

Due to vortex shedding effects, significant crosswind vibrationsof the building in the Y-direction (short direction) induced by the90-degree wind was found. The 90-degree wind was then identi-fied as a critical wind condition for the occupant comfort reliabilityoptimization. The results of reliability assessment for the occupantcomfort performance of the initial building design were obtainedusing both FORM and Monte Carlo simulations, and presented inTable 3. It was found only in case of the first mode the reliabilityindex is less than the target reliability index of 2, which is normallyadopted for the serviceability design [49]. The small value of reli-ability index means that the occupant comfort performance func-tion of first mode is violating the corresponding probabilisticconstraint. Hence, the inverse reliability algorithm is employed tosearch for the optimum mean value of the first modal frequency,by which the target reliability index of 2 will be achieved as de-fined in the sub problem 1 of Eq. (25). The iteration history resultsfor the optimum mean value of the first modal frequency were re-ported in Table 4. After 10 iterations, the intermediate designparameter was obtained as 0.2357 Hz, which satisfies the peakacceleration criteria with a target reliability of 2. The optimizedmean value of first modal frequency was increased by 27% com-pared with the initial modal frequency value of 0.185 Hz. Thenthe deterministic optimization process for the sub problem 2 wascarried out to obtain the optimized sizing solution.

Fig. 6 presents the material cost design history of the building inthe deterministic drift and frequency optimization process. Thenormalized cost with respect to the initial cost of the building is gi-ven for each design cycle, which includes the process of one formalstructural analysis and one deterministic resizing optimization.Two history curves are presented in Fig. 7. One cost history was ob-tained using the developed drift and frequency optimization meth-od, while another was obtained using the conventional static driftoptimization technique without considering the frequency con-straint. Although the structural costs of the building are foundsomewhat fluctuating at the first few design cycles, steady conver-gence to the final optimum solution is achieved at the seventh de-sign cycle for the drift optimization only and the eighth designcycle for the drift and frequency optimization. If only the drift per-formance is considered, the optimization procedure was able toachieve about 4% saving in the material cost. When taking into ac-count the frequency constraint in the design optimization process,an increase of about 11% in the structural cost was needed to fulfillthe first modal frequency constraint in addition to the multipledrift constraints. Fig. 7 presents graphically the design history ofthe first there modal frequencies. As evidently shown in Fig. 7,steady and rapid convergence of the first modal frequency to thetarget frequency value of 0.2357 Hz has been successfully achievedfor the large-scale 60-story building structure. As demonstrated,the developed performance-based design optimization method iscapable of searching for the optimal distribution of element stiff-ness of practical building structures to satisfy simultaneously var-ious drift and frequency design constraints.

For comparison, the reliability assessment of occupant comfortperformance for the optimized building design was also performedusing both FORM and Monte Carlo simulations. It was found fromthe final reliability results as shown in Table 5 that while the reli-ability index for the first vibration mode is very close to the targetvalue of 2, the reliability indexes for both the second and thirdmodes are still well above the target reliability index. Actually,due to the improvement of the overall building lateral-torsional

stiffness, the reliability indexes for both the second and thirdmodes were increased by 8.4% and 43.9% respectively, comparedto the initial reliability indexes given in Table 3. The larger increaseof the reliability index for mode 3 experiencing mainly torsionalvibration indicates that the torsional stiffness has been more en-hanced by the deterministic sizing optimization process than thelong-direction lateral stiffness.

6. Conclusions

This paper presents a reliability performance-based optimiza-tion method for the wind-induced drift and occupant comfort per-formance design of tall buildings. The occupant comfortperformance function of a wind-excited tall building is expressedexplicitly in terms of random variables, which represent the majoruncertainties of wind characteristics and building system proper-ties. The original coupling reliability-based design optimizationproblem was reformulated into two separated sub problems,which are then effectively solved using the inverse reliability algo-rithm and OC algorithm, respectively. Results of a full-scale 60-story hybrid steel and concrete building have shown that thedeveloped reliability performance-based design optimizationmethod provides a powerful design tool for drift and occupantcomfort serviceability design of tall buildings subjected to variousintensity levels of wind hazard attacking. The proposed perfor-mance-based optimal design method is capable of achieving themost cost efficient distribution of element stiffness of practical tallbuilding structures while satisfying deterministic drift and proba-bilistic occupant comfort performance design constraints.

Acknowledgement

The work described in this paper was partially supported by theNational Natural Science Foundation of China (Project Nos.51008275, 90815023).

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