Integrated Wind Load Analysis and Stiffness Optimization

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Engineering Structures 32 (2010) 1252–1261

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Engineering Structures

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

Integrated wind load analysis and stiffness optimization of tall buildings with3D modes

C.M. Chan a, , M.F. Huang b , K.C.S. Kwokc

a Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong b Institute of Structural Engineering, Zhejiang University, Hangzhou 310058, PR Chinac School of Engineering, University of Western Sydney, NSW, Australia

a r t i c l e i n f o

Article history:Received 17 September 2008Received in revised form24 December 2009Accepted 4 January 2010Available online 20 January 2010

Keywords:Tall buildingsWind loadsLateral–torsional motionsThree-dimensional modesStiffness optimization

a b s t r a c t

Recent trends towards constructing taller buildings with irregular geometric shapes imply that thesestructures are potentially more responsive to wind excitation. The wind-induced motion of modern tallbuildings is generally found to involve with significant coupled lateral and torsional effects, which areattributed to the asymmetric three-dimensional (3D) mode shapes of these buildings. The 3D coupledmodes also complicate the use of high frequency force balance (HFFB) technique in wind tunnel testingfor predicting thewind-induced loads andeffects on tall buildings. This paper firstlypresentsthe analysisof equivalent static wind loads (ESWLs) on tall buildings with 3D modes provided that the wind tunnelderived aerodynamic wind load spectra are given. Then an integrated wind load updating analysis andoptimal stiffness design technique is developed for lateral drift design of tall asymmetric buildingsinvolving coupled lateral–torsional motions. The results of a practical 40-storey building example withsignificant swaying and torsional effects arepresented. Notonly is thetechnique able to produce themostcost efficient element stiffness distribution of the structure satisfying multiple serviceability wind driftdesign criteria, but a potential benefit of reducing the wind-induced loads can also be achieved by thestiffness design optimization method.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Recent trends towards constructing taller buildings with irreg-ular geometric shapes imply that these structures are potentiallymore responsive to wind excitation. The wind-induced motion of modern tall buildings is generally found to involve with significantcoupled lateral and torsional effects, which are attributed to theasymmetric three-dimensional (3D) mode shapes of these build-ings. Torsional twisting effects, resulting from an imbalanced dis-tribution of wind loads on a building surface, are further amplifiedin tall asymmetric buildings by the presence of significant eccen-tricities between thecenterof stiffnessof thestructural systemandthe center of wind forces on the building. Making accurate predic-tions of wind loads and their effects on such complex asymmet-ric buildings involving lateral and torsional motion is therefore animportant step in the design process of such structures. For de-sign purposes, marked improvements have been made in manywind codes and standards to provide reasonable estimation of wind-inducedstructural loadson isolated buildings. However, cur-rent wind codes, which are developed for general design purpose,

Corresponding author. Tel.: +852 23587173; fax: +852 23581534.E-mail address: [email protected] (C.M. Chan).

still cannot account for wind-induced effects on specific buildingshapes significantly different from simple geometric shapes, andfor the interference effects caused by the surrounding objects andstructures in a complex terrain.

Inthe past few decades, wind tunnels have emerged asan indis-pensable experimental tool for assessing wind-induced loads andeffects on complex tall buildings. Due to its accuracy and relativeease in testing, the high frequency force balance (HFFB) techniquehas been one of the most widely used methods for determiningaerodynamic wind forces on buildings. Despite its popularity, the

HFFB technique is primarily suitable for buildings with uncoupledmodes. The limitations inherent in the HFFB technique and its ap-plicability to buildings with 3D coupled modes have been firstlyinvestigated by Yip and Flay and then by other researchers [ 1–5 ].Indeed, modern buildings with complex geometric shapes and3D mode shapes often complicate the use of the HFFB techniquewhich was traditionally developed for uncoupled buildings withone-dimensional (1D) modes [ 2,3]. To overcome these limita-tions, much effort has been made on refining the force balancedataanalysis technique to take intoaccount coupling effectson tallasymmetricbuildings with3D modes [ 4,5]. Furthermore, Chen andKareem [ 2] presented a framework for the coupled building re-sponse analysis and the modeling of the associated equivalentstatic wind loads (ESWLs) using HFFB measurements.

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C.M. Chan et al. / Engineering Structures 32 (2010) 1252–1261 1253

In a newer wind tunnel testing method using the synchronousmulti-pressure sensing system (SMPSS), the aerodynamic windforces can be derived from simultaneous multiple point pressuremeasurements on the surface of a building test model. Thistechnique allows direct computation of mode-generalized forcesfor any number of modes of vibration of the building with linearor nonlinear 3D modes. Using the SMPSS or HFFB technique,the aerodynamic wind loads can be estimated experimentallyon a rigid scaled model of the prototype. Once the aerodynamicwind load spectra derived from wind tunnel testing are obtained,the wind-induced response as well as the ESWLs can then becomputed, given the dynamic propertiesof the prototypebuilding.As long as the geometric configuration of the building remainsconstant, the same set of aerodynamic load spectra can be used toestimate thewind-induced responsesand ESWLs duringthe designoptimization process of the structural system of a tall buildingwithout making further wind tunnel tests.

While wind tunnel techniques can provideaccurate predictionsof structural loads fora building, current practice normally regardswind tunnel derived ESWLs as being similar to the code specifiedwind loads that are kept constant throughout the structuraldesign synthesis process. It has not been commonly recognizedby structural engineers that the ESWLs are indeed directly relatedto the dynamic properties (such as the stiffness, mass, damping)of a building. Today’s design practice does not normally takeaccount of the frequency dependent characteristics of ESWLs inthe structural design process. Chan et al. [6] highlighted theimportance of updating the ESWLs on a building during the lateralstiffness design optimization process of the building. Evidence hasindicated that increasing the natural frequency of wind sensitivetall buildings by stiffness optimization can generally result in thebenefits of reducing the equivalent static wind loads [ 6]. However,the integrated analysis and stiffness designoptimization techniquedeveloped by Chan et al. [6] is only applicable to symmetric tallsteel buildings with uncoupled mode shapes. It is necessary thatthe integrated technique be further extended to optimize the

lateral–torsional stiffness of general tall buildings with complex3D modes while accurately accounting for the potential benefitof reducing the wind-induced forces and moments during thestiffness design optimization process.

In this paper, an equivalent static wind load analysis methodbased on the HFFB or SMPSS technique using measured aerody-namic force data is firstly presented. While the importance of instantaneously updating the wind-induced load during the struc-tural design synthesis is emphasized, attention is also paid tothe proper evaluation of the dynamic wind loads on asymmet-ric tall buildings with due consideration of the lateral–torsionalmechanical coupling effects as well as the intermodal coupling of modal responses. The developed optimization technique applica-ble to general tall buildings is based on the Optimality Criteria (OC)

approach, which has beenwidelyused and shown particular effec-tive for element sizing design optimization of large building struc-tures [7,8]. The instantaneous change in theESWLs can be updatedwhenever there exist considerable structural modifications duringthe structural design process causing a substantial change in themodal frequencies of thebuilding.Finally,a practical40-storeyres-idential building with 3D mode shapes involving significant sway-ing and torsional effects is presented to illustrate the applicabilityand effectiveness of the computer-based technique for integratedwind-induced load analysis and stiffness design optimization of tall building structures.

2. Determination of equivalent static wind loads

Using the aerodynamic loading measured in the wind tunnel,the response of a building with 3D coupled mode shapes can be

calculated by modal analysis for any combination of the building’smass, stiffness, damping and the oncoming wind speed. As theaerodynamic wind force is a random process, the jth modal dis-placement response of the building in the generalized coordinatesystem can be computed based on random vibration theory in thefrequency domain [ 2] as

σ 2q j =

∞0 |H ( f )|

2

S Q jj ( f )d f (1)

where σ q j =thestandard deviationvalueof the jth modal displace-ment. S Q jj( f ) is the input mode-generalizedwind load spectrum; H is the mechanical admittance function which can be expressed as

H j ( f )2

= 1

m2 j 2π f j

4

1

1 − f / f j2 2

+ 4ζ 2 j f / f j2

(2)

where f j is the jth modal frequency of the building; ζ j is the jthmodal damping ratio; and m j is the jth modal mass.

In the HFFB technique, the jth mode-generalized wind loadspectrum can be given in terms of measured base moment spectraas

S Q jj ( f ) =s= x, y,θ l= x, y,θ η jsη jlS M sl ( f ) (3)

where S M sl ( f ) denotes the auto or cross power spectral densitybetween the respective base moments M s( t ) and M l( t ) ; η s (s = x, y, θ ) represents the mode shape correction that depends onthe manner of mode shapes as well as the local wind pressuredistribution and its correlation with height over a buildingsurface [2]. In the event that the aerodynamic wind loads are to bemeasured by the SMPSS technique, the jth modal force can thenbe more accurately evaluated without the need for mode shapecorrections [ 3].

In general, the mean square modal response in Eq. (1) can beapproximated as the sum of the background component and theresonant component as follows

σ 2q j = σ 2q jb + σ 2q jr (4)

in which σ q jb = the standard deviation value of the backgroundcomponent of the jth modal displacement; σ q jr = the standarddeviation value of the resonant component of the jth modaldisplacement,which can be simplified into the following algebraicequation using the white noise assumption as

σ 2q jr ≈ S Q jj ( f j) ∞0H j( f )

2d f =

1

m2 j 2π f j

4

π4ξ j

f jS Q jj( f j). (5)

Similarly, the mean square base moment or torque responsecan be written as the sum of the background and the resonantcomponent as follows [ 10 ]

σ 2M s = σ 2M bs + σ 2M rs (6)

in which the first part of the right-hand side is referred to as thebackground component, which can be considered as quasi-staticand can be directly quantified by integrating the area underneaththe corresponding base moment auto spectrum curve of S M ss ( f ) .The second part of the right-hand side of Eq. (6) represents theresonant dynamic amplification. Fora lightly dampedsystem,suchas a tall building, the resonant component of the mean squarebase moment response can be obtained approximately using thecomplete quadratic combination (CQC) method as follows

σ 2M rs =n

j

=1

n

k

=1

σ M jrs σ M krs r jk =n

j

=1

n

k

=1

Γ jsΓ ksσ q jr σ qkr r jk (7)

where σ M jrs = the standard deviation value of the jth modal



1254 C.M. Chan et al. / Engineering Structures 32 (2010) 1252–1261

component of the resonant base bending moment or torque;r jk = the intermodal correlation coefficient for the jth and kthmodal response, which can be referred to the recent work of Chenand Kareem [2], Xie et al. [11 ] and Huang et al. [ 12 ]; Γ js, Γ ks = themodal participation coefficients of the sth component of the basemoments or base torque, which can be defined as

Γ js =2π f j

2

H

0 m ( z ) φ js ( z ) z d z , s = x, y

2π f j2

H

0I m ( z ) φ jθ ( z )d z , s = θ

(8)

where m( z ) is the mass per unit height; I m ( z ) is the rotationalmass moment of inertia about the vertical axis per unit height;φ js( z ) (s = x, y, θ ) = the component of the jth 3D mode shape.By substituting Eq. (5) into Eq. (7) , the standard deviation resonantcomponent σ M rs , of the base moment response M s can be explicitlyexpressed in terms of generalized force spectra as

σ 2M rs =n

j=1

n

k=1

Γ jsΓ ksr jk

64 π 3 m jmk f j f k2 f j f kξ jξ k

S Q jj( f j)S Q kk ( f k). (9)

Using the gust response factorapproach, the peak base momentor torque response can be rewritten as [13 ]

M s = M s + M 2bs + M 2rs (10)

where the background component M bs and the resonant compo-nent M rs can be calculated respectively as

M bs = g bσ M bs = g b ∞0S M ss ( f )d f (11)

M rs = g r σ M rs

= g r n

j=1

n

k=1

Γ jsΓ ksr jk

64 π3

M jM k f j f k2

f j f kξ jξ k

S Q jj ( f j)S Q kk ( f k)

1/ 2

(12)

in which the background peak factor, g b , can be approximated bythe gust factor of the oncoming wind velocity, the value of whichis usually taken to be about 3 to 4 [ 14 ]. For a Gaussian process, theresonant peak factor, g r , can be given as [ 15 ]

g r =√ 2 ln v · T + 0.577√ 2 ln v · T

(13)

where T is the observation time (usually 3600 s) for the givenwind conditions under consideration; v indicates the mean zero-crossing rate for the base moment component process, and couldbe fairly approximated by the first modal frequency value.

Based on the calculated peak base moments or torque, thewind-induced structural loads (or the so-called ESWLs) on thebuilding can then be determined by distributing the peak basemoments or torque to the floor levels over the building height[16 ,17 ]. Similar to the base moments, the equivalent static windloads expressed in terms of peak load F s at each floor level, canalso be written into a linear combination of the mean (F s) , thebackground (W b F b) and the resonant ( n

j=1 W jrs F jrs) componentsas

F s = F s + W b F bs +n

j=1

W jrsF jrs , ( s = x, y, θ ) (14)

where W b = σ M b /(σ 2M b + σ 2M r )

1/ 2 , W jrs =nk=1 σ M jrs r jk /(σ 2

M b +σ 2M r )

1/ 2 ; F bs is the sth component peak background wind loads and

ˆF jrs is the sth component of the jth modal peak resonant windloads. Using Eq. (14) , the ESWLs can be expressed respectively in

the two translational and one torsional directions of a building inassociation with the default global coordinate system. In theory,the mean component of both the crosswind and torsional windloads should be equal to zero. The alongwind mean force per unitheight can be related to the approaching wind velocity profile andwritten as follows

¯F ( z )

= 1

2ρ

U 2

H

z

H

2αBC

D (15)

where ρ = the air density; U H = the wind speed at the top of thebuilding; α = the power law exponent of wind profile; B = thewidth of the building; C D = the drag force coefficient of the build-ing. Due to the quasi-static nature of the background componentof the wind loads, the distribution of the background componentESWLs to the floor levels over the building height can sometimesbe assumed to follow the distribution of the multiplication of theturbulence intensity profile and the mean alongwind loading pro-file given in Eq. (15) as

F bx, by ( z ) =F ( z )I ( z )

H

0 F ( z )I ( z ) z d z M bx,by (16)

F bθ ( z ) = ¯F x( z )I ( z ) z

H

0 F x( z )I ( z ) z d z M bθ (17)

where F bs( z ) ( s = x, y, θ ) is the distributed peak background windload; I ( z ) denotes the turbulence intensity profile.

For a building having 3D coupled mode shapes, the distributionof the resonant component of the wind loads follows basically themass-related modal inertial force distribution as shown in Eq. (18)

F jrs ( z ) = g r σ M jrs µ s ( z ) = g r Γ jsσ qr µ s ( z ) (18)

where F jrs( z ) (s = x, y, θ ) is the peak resonant wind loadcorresponding to the jth mode per unit height; µ s( z ) ( s = x, y, θ )is the inertial distribution factor, and is given as

µ s( z ) =

m ( z ) φ jx, jy ( z )

H

0 m ( z ) φ jx, jy ( z ) z d z , s = x, y

I m ( z ) φ jθ ( z )

H

0 I m ( z ) φ jθ ( z )d z , s = θ .

(19)

3. Dependence of wind-induced loads on natural frequency of the structure

For multistorey buildings, the ESWLs corresponding to thespecific incident wind angle are generally expressed in terms of the alongwind, crosswind and torsional directions; and each di-rectional ESWL consists of the mean, background and resonant

components. Generally speaking, the mean and background com-ponents are the major parts of wind loads in the alongwind direc-tion, since the mean wind loads as given by Eq. (15) theoreticallyonly exist in the alongwind direction. For wind sensitive tall build-ings, significant contributions of the resonant component to thetotal wind effects on buildings can be found in the crosswind andtorsional directions. The crosswind loading effects is mainly dueto wake excitation by vortex shedding as well as turbulence andbuffeting by wind flow re-attachment on the faces of the building.Aerodynamically, the presence of turbulent flows of various scalesand the existence of neighboring building obstacles may cause theuneven distribution of fluctuating wind pressures. The imbalancein aerodynamic forces and uneven distribution of fluctuating windpressures are the major aerodynamic sources of torsional wind

loads on buildings. Even for a symmetric building, instantaneousfluctuating torsional aerodynamic forces may still exist.




Since the mean and background wind loading components aremainly related to the approaching wind conditions and the aero-dynamic shape of the building, they are very much independentof the building natural frequency [10 ]. As a result, varying thebuilding’s natural frequency by modifying the structural stiffnessof the building normally has only a negligible effect on the meanand background values of the wind-induced structural forces. Onthe other hand, as given in Eq. (12) , the resonant component of the wind loads is directly related to the natural frequency, f j, of the building such that any changes in the natural frequencies of the structure will subsequently lead to a direct impact on thewind-induced resonant effects on the building, particularly in thecrosswind and torsional directions. For slender and flexible tallbuildings, the wind-induced responses are mostly dominated bydynamic amplification effects dueto turbulence andbuffetingbothin the alongwind and crosswind directions, or sometimes by vor-tex shedding resonant effects particularly in the crosswind direc-tions [ 18 ]. For buildings with elongated plan forms, the resonantcomponent of torsional wind loading also has a significant contri-bution to the total wind-induced response of such buildings [ 19 ].Torsional loading effects can be further accentuated in the pres-ence of eccentricities betweenthe centerof stiffness andthe centerof wind loads. In general, modern tall buildings are wind sensitiveand the ESWLs of these structures are more influenced by the res-onant effects in the crosswind and torsional directions.

Once the wind tunnel derived aerodynamic wind load spectraare available, they can be explicitly expressed in the form of an al-gebraic function of the modal frequency using regression analy-sis [20 ,21 ]. Such algebraic functions can then be used to directlyupdate the prediction of wind-induced structural loads on thebuilding for any instantaneous change in the dynamic propertiesof the building during the design optimization synthesis. With theaid of piece-wise regression analysis, the power spectral densityfunction of modal wind forces (as shown in Fig. 1) for a typicaltall building can be inversely related to the modal frequency of the building and is explicitly expressed as an algebraic function in

terms of the modal frequencywithinthe typical range of frequencyfor serviceability check, characterizing the descending part of thepower spectra, as follows

S Q jj ( f j) ≈ β j f −α j j (20)

where α j and β j are regression constants and normally α j > 0 andβ j > 0. It is worth noting that some tall buildings with well orga-nized aerodynamic shapes may result in flat modal force spectrawith α j ≈ 0. In such instances, the modal force spectra and in turnthe dynamic responses of these buildings become less sensitive tothe modification of building stiffness and some other measures,such as installing mechanical dampers, may become necessary tomitigate wind-induced vibration of buildings. Substituting Eq. (20)into Eq. (5) , and subsequentlyinto (18) gives thedirectdependence

of the resonant components of ESWLs to the modal frequency as

F jrs ( z ) ≈ f −(α j+3) / 2 j

g r Γ js8m jπ 3/ 2 β j

ζ jµ s ( z ) . (21)

For wind sensitive tall buildings where the value of the expo-nent α j is normally greater than 0, the resonant ESWLs can be re-duced by increasing modal frequency according to Eq. (21) .

The resonant components of ESWLs would be combinedtogether with the mean components as well as background com-ponents using Eq. (14) to obtain peak load at each floor level fordesign purposes. For normal low-rise buildings, wind-inducedstructural loads are dominated by static mean components andquasi-static background components. The wind-induced resonant

effects on low-rise buildings are small and negligible such thatwind loads can be considered as constant static design loads.

However, for dynamically sensitive tall buildings, wind-inducedresonant effects may become critical. In order to make an accu-rate prediction of the wind-induced structural loads on the build-ing, it is necessary that the ESWLs be always updated wheneverthere exists a significant change in the structural properties of thebuilding.In general, the resonant component of theESWLs fora tallbuilding can be reduced by increasing the modal frequencies of thebuilding through structural optimization by efficientlydistributingstructural materials to improve the lateral and tensional stiffnessof the building.

4. Integrated design optimization

4.1. Formulation of the drift design problem

Consider a general tall building consisting of various numbersof steel frame elements, concrete frame elements and concreteshear wall elements. The design variables are taken as the cross-section area of steel elements, the breadth and depth dimensionsof the rectangular concrete frame elements, and the thicknessof the concrete shear wall elements. For simplicity, note that all

element sizing design variables are represented by a collective setof i = 1, 2, . . . , N i generic variables X i. The minimum materialcost design of the building structure subject to j = 1 , 2 , . . . , N g multiple lateral drift design constraints can be stated as:Minimize

W ( X i) =N i

i=1

w i X i (22)

subject to

d j ≤ dU j ( j = 1 . . . N g ) (23)

X Li ≤ X i ≤ X U i (i = 1 . . . N i). (24)

Eq. (22) defines the design objective of minimizing the materialcost, in which w i represents the corresponding unit material costforelement i.Eq. (23) defines theset of j = 1 , 2 , . . . , N g interstoreydrift or top deflection constraints under the equivalent static windload conditions where dU

j represents the predefined allowableinterstorey drift or overall top deflection limit. In general, theallowable drift ratio for buildings appears to be within the rangeof 1 / 750 to 1 / 250, with 1 / 500 being typical [ 22 ]. Eq. (24) definesthe element sizing constraints in which superscript L denotes thelower size bound and superscript U denotes the upper size boundof member i.

To facilitatenumerical solution of the designoptimization prob-lem, the implicit drift constraints (Eq. (23) ) must be formulated ex-plicitly in terms of various design variables. Using the principle of virtual work, the collective set of lateral drift constraints can be ex-

pressed explicitly asd j( X i) = d j( Ais , Bic , Dic , t iw )

=N s

is=1

eis j

Ais + eis j +N c

ic =1

e0ic j

Bic Dic + e1ic j

Bic Dic 3 +

e2ic j

Bic 3Dic

+N w

iw =1

e0iw j

t iw + e1iw j

t 3iw ≤ dU j

( j = 1 , . . . , N g ) (25)

in which Ais is the cross-section area of is = 1 , 2, . . . , N s steelframe elements, Bic , Dic are the breadth and depth dimensions of ic = 1, 2 , . . . , N c rectangular concrete frame elements, and t iw isthe thickness of iw = 1 , 2, . . . , N w concrete shear wall elements,

respectively; eij and eij are the virtual strain energy coefficient andits correction factor of steel members respectively; e0ij, e1ij and e2ij




(a) 0 ◦ wind. (b) 90 ◦ wind.

Fig. 1. Wind-induced modal force spectra for the building.

are the virtual strain energy coefficients of concrete members [ 7].Once the finiteelement analysis is carried out fora given structuraldesignunder theESWLs andvirtual loading conditions, theinternalelement forces and moments are obtained and the element’svirtual strain energy coefficients can then be readily calculated.

4.2. Optimality Criteria method

Upon establishing the explicit formulation of the drift designconstraints, the next task is to apply a suitable numerical tech-nique for solving the optimal design problem. A rigorously derivedOptimality Criteria (OC) method [7–9 ], which has been shown tobe computationallyefficient for large-scale structures, is employedin this paper. In this OC approach, a set of optimality criteria cor-responding to multiple design constraints for the optimal designis first derived and a recursive algorithm is then applied to indi-rectly solve the optimal solution by satisfying the derived optimal-ity criteria. To seek for a numerical solution using the OC method,the constrained optimal design problem must be transformed intoan unconstrained Lagrangian function which involves both the objective function (Eq. (22) ) and the set of explicit drift constraints(Eq. (25) ) associatedwith corresponding Lagrangianmultipliers.Bydifferentiating the Lagrangian function with respect to each sizingdesign variable and setting the derivatives to zero, the necessarystationary optimality conditions can be obtained and then utilizedin the following recursive relation to resize the active sizing vari-ables Ais , Bic , Dic , t iw , which are represented by the generic vari-ables X i for simplicity:

X v

+1

i = X vi · 1 −

1

η

N g

j=1λ j

∂d j/∂ X i∂W /∂ X i + 1

v

( i = 1 , 2 , . . . , N ) (26)

where λ j denotes the associated Lagrangian multiplier for the jthdrift constraint, v represents the current iteration number; and ηis a relaxation parameter. During the resizing iterations, anymem-ber reaching its size bounds is deemed as an inactive member, anditssize is set at its corresponding size limit. The two partial deriva-tives with respect to each element sizing design variable involvedin Eq. (26) can be analytically determined, since the structure ma-terialcost function W (Eq. (22) ) andthe drift constraint d j (Eq. (25) )are explicit expressions given the element sizing design variables.

Before Eq. (26) can be used to resize X i, the Lagrangian

multipliers λ j must first be determined. Considering the sensitivityof the kth drift constraint in response to changes in the design

variables, one can derive a set of N g simultaneous linear equationsto solve for the N g unknown λ j [7]:

N g

j=1

λ v j

N i

i=1

X i∂dk∂ X i

∂ d j∂ X i

∂W ∂ X i

= −N i

i=1

X i∂dk

∂ X i v − η dU k − dv

k

(k = 1 , 2, . . . , N g ). (27)

Having the current design variables X vi the corresponding λ v j

values are readily determined by solving Eq. (27) . Having thecurrent values of λ v

j , the new set of design variables X v+1i can, in

turn, be obtained from Eq. (26) . Therefore,the recursiveapplicationof the simultaneous equations of Eq. (27) to find the λ v

j and theresizing formula of Eq. (26) to find the design variables constitutethe OC algorithm. By successively applying the OC algorithm untilconvergence occurs, the optimal design solution is then found.More details of the OC algorithm can be referred to the work of Chan [ 7,8].

4.3. Procedure of integrated design optimization

The proposed integration of the wind-induced dynamic loadanalysis and optimal structural design procedure can be outlinedstep by step as follows:

1. Given the geometric shape of a tall building, determine theaerodynamic wind load spectra by wind tunnel tests.

2. Develop the finite element modelfor thebuilding and carry outan eigenvalue analysis to obtain the natural frequencies andmode shapes of the building.

3. Based on the current set of dynamic properties of the building,determine the floor-by-floor ESWLs Eq. (14) for the buildingusing the wind tunnel derived aerodynamic wind load spectra.

4. Apply the derived ESWLs to the building and carry out a staticstructural analysis to estimate the peak drift responses of thebuilding.

5. Establish the explicit expression of the drift constraints Eq. (25)and formulate explicitly the optimal drift design problem.

6. For thecurrent set of elementsizingdesignvariable X i, solve thesystem of simultaneous linear equations Eq. (27) for the set of Lagrange multiplier λ j.

7. For the current value of λv j , find the new set of design variables

X v+1i from Eq. (26) .

8. Repeat steps6 and 7 and check the convergence of the recursiveprocess: if all X v+1i = X vi and λ v

j = λ v−1 j , then proceed to step 9.




9. Check the convergence of the material cost objective function:if the cost of the structure for three consecutive reanalysis-and-redesign cycles is within certain prescribed convergencecriteria, for example within 0.1% difference in the current totalmaterial cost, then terminate the design process with the min-imum material cost design for the structure; otherwise, returnto step 2, update the finite element model using the current setof design variables and repeat the eigenvalue analysis and de-sign optimization process.

Fig. 2 shows a schematic flow chart for the proposed integratedwind load analysis and stiffness optimization technique for thelateral drift design of a tall building structure. The overallprocedure can be programmed and integrated with commerciallyavailable finite element method (FEM) software such as SAP2000.Eigenvalueanalysis and structural analysis can therefore be carriedout by the FEMsoftware within the integrated design optimizationprocedure.

5. Illustrative example: 40-storey shear wall residentialbuilding

5.1. The 40-storey building and the wind tunnel test

An optimization study of a 40-storey residential building com-missioned by the Hong Kong Housing Authority was carried out atthe Hong Kong University of Science and technology. A 3D com-puter model of the building and a structural layout plan are givenin Figs. 3 and 4, respectively. With an elongated width of 73m, anarrow depth of 12 m and a total height of 122 m, the buildinghas a critical aspect ratio (height/depth) of over 10.4. In view of itselongated and slender configuration, the building is anticipated tobe wind sensitive and to exhibit significant swaying and twistingresponses. The building is of reinforced concrete construction withcoupled shear walls. As shown in Figs. 3 and 4, multiple structuralshear walls are coupled by lintel beams whenever possible to pro-

vide the total lateral and torsional resistance of the building. Theeffectiveness of the structural resisting system depends on manyfactors, such as the configuration of the structural form, the vari-able thickness of theshear walls andthe variable dimensionsof thelintel beams.

A wind tunneltest wascarried outat theCLPPowerWind/WaveTunnel Facility (WWTF) of the Hong Kong University of Scienceand Technology. One 1:2000 scale topographical model incorpo-rating relevant parts of the Hong Kong territory was firstly usedto determine the representative approaching wind profiles for thebuilding. Wind loads on the building were then measured by theHFFB technique using a 1:400 scale rigid model subjected to thespecific wind profiles obtained from the topographical study.Windtunnel measurements were taken for totally 36 incident wind an-

gles at 10 ◦ intervals for the full 360 ◦ azimuth. Two critical inci-dent wind directions corresponding to two perpendicular incidentwind angles were identified by the wind tunnel test. One was the0-degree wind perpendicular to the wide face acting in the shortdirection (i.e. along the Y -axis) of the building; another one wasthe 90-degree wind perpendicular to the narrow face acting in thelong direction (i.e. along the X -axis). It was found that while theglobal maximum overturning moment about X -axis occurs at inci-dent wind angle of 0 ◦, the global maximum overturning momentabout Y -axis and torsional moment about the vertical Z -axis occurat incident wind angle of 90 ◦ during the wind tunnel study. Thepower spectral densityfunctions of modal forces forthe building attwo critical incident wind angles are shown in Fig. 1. Based on thetopographical study and theHong Kong wind code [ 23 ], thedesign

wind speeds with a 50-year return period were adopted as 54.0m/s and 47.7 m/s at the gradient height of300 m for the 0 ◦ and the

Fig. 2. Flow chart of integrated analysis and design optimization process.

90◦ wind, respectively. Provided the wind load spectra, the ESWLson the building corresponding to these two most critical incidentwind directions (0 ◦ and 90 ◦ wind) have been considered and ana-lyzed during the design optimization process.

Table 1 presents a breakdown of the results of the maximumbase shear forces and base torque for theinitial building. It is noted

that in Table 1 the maximum base shear force (F x) and torsionalmoment (M zz ) are found under the 90 ◦ wind, and the maximumbase shear force F y is calculated under the 0 ◦ wind. Both maxi-mum base shear forces ( F x and F y) occur in the alongwind direc-tion corresponding to the two respective incident wind angles. Asshown in Table 1 the mean and background components togetherhave summed up to be slightly over 70% of the total alongwindloads for both base shears F X and F Y respectively, while the reso-nant component contributes the remaining 30% of each base shear.Since the base shear forces acting on the building are dominatedby the quasi-static mean and background components, which areweakly dependent on the dynamic property of the building, onlyslightly reduction in their values can be achieved by the means of stiffness design optimization. However, in the torsional direction

about the vertical axis of the building, the dynamic resonant load-ing is found to be the dominating component that contributes to




Zone 3(22/F-Roof)

Zone 2(12/F-21/F)

Zone 1(G/F-11/F)

Fig. 3. The 3D view of the residential 40-storey building.

Fig. 4. Typical floor layout plan with variable shear wall elements of the residential 40-storey building.

Table 1Breakdown of maximum wind loads for the building before optimization.

Wind loads Mean component Background component Resonant component Sum

Base shear F x (kN) 1075 (33%) 1221 (37%) 962 (30%) 3258 (100%)Base shear F y (kN) 9179 (40%) 7220 (31%) 6644 (29%) 23,040 (100%)

Base torque M zz (kN m) 30,680 (15%) 26,660 (13%) 143,990 (72%) 201,340 (100%)

Table 2Wind-induced base shears and base moments for the building before and after optimization.

Wind loads Hong Kong wind code Wind tunnel results before optimization Wind tunnel results after optimization Wind load reduction (%)

Base shear F x (kN) 6,570 3,258 3,223 −1.1Base shear F y (kN) 24,170 23,040 22,690 −1.5Base moment M xx (kN m) 1,530,500 1,723,700 1,702,800 −1.2Base moment M yy (kN m) 441,070 241,130 239,390 −0.7Base torque M zz (kN m) 0 201,340 183,740 −8.7

72% of the total base torque. As the resonant component of windloads is inverselyproportional to themodal frequencyof the build-ing according to Eq. (21) , any considerable modification in the dy-

namic stiffness of the building may lead to a significant change inthe torsional wind loading on the building.

In the preliminary design phase of the building before the windtunnel test, wind loads on the building were calculated accordingto the Hong Kong Wind Code [23 ]. The alongwind loads in terms

of base shear and base moment derived from the Hong Kong WindCode are given in Table 2 . The code-specified base shear F y is found




Fig. 5. 3D mode shapes of the 40-storey residential building.

to be slightly larger than the value predicted by wind tunnel test,while the base shear F x is substantially overestimated by the windcode. Unlikethe Hong Kong Wind Code in which there is nospecificguideline for predicting the wind-induced torsional load, the windtunnel study gives a significant base torque of201,340kN m for theinitial building, indicating an eccentricity of 8.7 m equal to 11.8%of the width of the building.

The initial member sizes were established on the basis of apreliminary strength design. Once the finite element model having2479 frame elements and 8523 shell panels was set up for thebuilding, an eigenvalue analysis was then carried out to determinethedynamic propertiesof the building (i.e., thenatural frequenciesand mode shapes). The 3D mode shapes of the building are givenin Fig. 5. The three fundamental coupled vibration modes have thefirst natural frequencies of 0.307 Hz (mainly torsional vibration),the second of 0.323 Hz (swaying primarily in the short direction of the building) and the third of 0.464 Hz (swaying primarily in thelong direction). The first vibration mode of the building is indeed atorsional mode, indicating significant dynamic torsional effects onthe building. In the dynamic analysis, the values of the dampingratio for first three vibration modes were taken as 1.5%, 1.5%and 2%.

In this stiffness optimization study, the major design variablesare the thicknesses of variable shear walls. All variable shear wallelements for the design optimization task are highlighted as WallGroup 1–6 as shown in Fig. 3. Due to some practical planning andconstructability design considerations, the shear walls that are nothighlighted in Fig. 3 are kept unchanged during the optimizationprocess. For this 40-storey building, all shear walls are maintained

to have uniform thickness, except Wall Group 3 in which threevariations of wall thickness are allowed along the height of thebuilding. Three vertical zones of wall thickness variations areallowed with zone one from ground floor to the 11th floor, zonetwo from the 12th floor to the 21st floor, and zone three fromthe 22nd floor to the main roof of the building as illustrated inFig. 2. Grade 45 concrete is used for the first 20 stories of thebuilding and Grade 35 concrete for the upper 20 stories. In thedesign optimization process, the top deflection limit of H / 500,where H = 114 .1 m is theheight of themainroof from the ground,was imposed at the four top corners of the building.

5.2. Results and discussion

Fig. 6 presents the design history of the total normalized costof the structure. Although relatively stringent restrictions due to

Fig. 6. History of the normalized structure cost for the 40-storey residentialbuilding.

planning and constructability requirements have been imposed,thestiffness designoptimizationtechniquehas achieved a 9.9%de-crease of the total initial material cost of the building. A carefulscrutiny on the optimized structure indicates that such a cost sav-ing has been attained by more effectively stiffening the torsionalresistance of the building through deepening the lintel beams atthe twoendwalls (see Fig.3) andthickening the topparapet,whichprovides a ring beam wrapping along the edge of the main roof level of the building (see Fig. 2). As a result of the torsional stiff-ness enhancement,the thicknessof five outof thesix variableshear

walls, except wall group 5, can be reduced or kept constant asshown in Table 3 . Consequently, the reduction in the wall thick-ness has also resulted in a net increase of 193m 2 usable floor area.

As shown in Fig. 6, a somewhat zigzag design history can beobserved at the first few design cycles. At the beginning stage of the design history, as the structural stiffness of the building is im-proved by the OC optimization technique, there exists an increasein the frequencies of the structure and a corresponding reductionof the ESWLs, leading to a subsequent reduction in the lateral driftresponse. As a result of the reduced drift response, the structuretends to be weakened in the immediate following design cycle,thus causing a drop in the frequencies and subsequently an in-crease of the ESWLs, which, in turn, leads to an increased struc-tural cost of the building. The end result gives a fluctuating zigzag

design process, which converges to the final optimum design after12 successive design cycles, producing a net 9.9% cost saving.




Table 3Original and optimized thickness of variable shear walls.

Wall group 1 2 3 4 5 6

Original thickness (mm) 400 525 525 300 300 525Optimized thickness (mm) 400 300 350 / 250/200 a 200 425 300a The three values of optimized thickness correspond to Zones 1 to 3, respectively.

(a) Before optimization. (b) After optimization.

Fig. 7. Lateral deflection profiles at the center and the corner of the building before and after optimization.

The results of the wind-induced base forces and moments fortheoptimizedstructureare compared to that of theinitial structureas shown in Table 2 . It is evident that only some minor reductionsare found in the base shear forces ( F x and F y) and base overturningmoments ( M xx and M yy), since their values are more dominatedby the mean and background components of the wind loads.However, a more significant reduction in the base torque (M zz ) hasbeen achieved for the optimized structure because of the fact thatthe value of the base torque is more dominated by the dynamicresonant component of the wind load. An increase in the torsionalstiffness, by structural optimization, leading to an increase in thefirst modal frequency (torsional) of the building from 0.307 Hz to0.318 Hz,has consequentlyresultedin thebenefit of8.7%reductionin the base torque.

With a significant wind-induced torsion applied on the residen-tial building, the building is found to sway and twist with substan-tiallylarger deflection at the most distant cornerof the building.Asshown in Fig. 7(a), the maximum topdeflection of theinitial struc-tureat themost criticalcornerpositionis found tobe 0.24 m,givinga slight 5% violation in the top drift limit. It is worth noting that amuch smaller deflection of 0.10 m is found at the center of the topfloor of the structure. An increase of 140% in the lateral deflectionat the top corner of the initial building is due to torsional twistinginduced by wind. After optimization, the maximum top drift ratioof the optimized structure is kept within the allowable drift ratioof 1/ 500 as shown in Fig. 7(b). It is evident that the stiffness op-timization technique is capable of achieving a more cost efficientdesign by redistributing the structural material to maximum thelateral and torsional stiffness of the building while satisfying thespecified drift constraints.

6. Conclusions

This paper presents an integration of an aerodynamic wind load

analysis and a lateral–torsional stiffness optimization techniquefor wind-induced drift design of tall buildings with 3D modes.

Encouraging results have been found in the serviceability drift de-sign optimization of a 40-storey practical residential building. Thecomputer-based integrated design optimization method is ableto produce the most cost efficient structural stiffness distributionof the building satisfying multiple lateral drift design constraints

incorporating with torsional effects under multiple wind loadingconditions. The integrated design optimization technique is alsocapable of achieving an additional benefit of wind load reductionby instantaneously updating wind-inducedstructural loads duringthe design synthesis process. The results of the 40-storey build-ing indicate that the wind-induced torsional loads on the buildingwith the asymmetric elongated plan form have been substantiallyreduced by the stiffness optimization method.

Acknowledgements

Thework described in this paper was partiallysupportedby theResearch Grants Council of the Hong Kong Special AdministrativeRegion, China (Project Nos. CA04/05.EG01 and 611006) andthe National Natural Science Foundation of China (Project No.90815023), andwas based upon research conducted by M.F. Huangunder the supervision of C.M. Chan for the degree of Doctorof Philosophy in Civil Engineering at the Hong Kong Universityof Science and Technology. Special thanks are due to the HongKong Housing Authority for taking the lead to support the use of the structural optimization technique and providing consent topresent the results of the example building.

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Integrated Wind Load Analysis and Stiffness Optimization

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